![LIST OF FIGURES vii
5.3 Example with Convex Objective Function, µ > µ0 194
5.4 Long-Run Demand and Supply Curves 203
5.5 Uniform [d, a] Value Distribution Long-Run Demand and
Supply Curves, Case 1 205
5.6 Uniform [d, a] Value Distribution Long-Run Demand and
Supply Curves, Case 2 205
5.7 Long-Run Demand and Supply Curves; Uniform [0, a] Value
Distribution 207
5.8 Convergence of Iterative Algorithm for Case of Uniform [0, a]
Demand 208
6.1 Comparison of S.O. and F.O. Supply-Demand Curves for
Variable λ 239
6.2 Nash Equilibrium for Two Competitive M/M/1 Facilities 246
6.3 Waiting-Cost Function for M/M/1 Queue 251
6.4 Illustration of Sequential Discrete-Time Algorithm 254
6.5 Facility Dominance as a Function of λ 266
6.6 Graphs of U0
(λ) and C0
(λ) for Parallel-Facility Example 269
7.1 First Example Network for Braess’s Paradox 286
7.2 Second Example Network for Braess’s Paradox 288
7.3 Example Network with α(λ) < π(λ) 293
7.4 Illustration of Theorem 7.2 300
7.5 Illustration of Derivation of Upper Bound for Affine Waiting-
Cost Function 302
7.6 Graph of φ(ρ) 304
7.7 Table: Values of σ = φ(ρe
) and (1 − σ)−1
304
A.1 Graph of V (t): Work in System 345](https://image.slidesharecdn.com/2207852-250605111415-ac5fac2e/85/Optimal-Design-Of-Queueing-Systems-1st-Edition-Shaler-Stidham-Jr-12-320.jpg)
![4 INTRODUCTION TO DESIGN MODELS
Figure 1.1 Total Cost as a Function of Service Rate
Let C(µ) denote the expected steady-state total cost per unit time, when
service rate µ is chosen. Then
C(µ) = c · µ + h · L(µ) ,
where L(µ) is the expected steady-state number in system. For a FIFO M/M/1
queue, it is well known (see, e.g., Gross and Harris [79]) that
L(µ) = λW(µ) =
λ
µ − λ
, (1.1)
where W(µ) is the expected steady-state waiting time in system.∗
Thus our
optimization problem takes the form:
min
{µ:µ>λ}
C(µ) = c · µ + h ·
λ
µ − λ
. (1.2)
Note that
C00
(µ) =
2hλ
(µ − λ)3
0 , for all µ λ ,
so that C(µ) is convex in µ ∈ (λ, ∞). Moreover, C(µ) → ∞ as µ ↓ λ and as
µ ↑ ∞. (See Figure 1.1.) Hence we can solve this problem by differentiating
C(µ) and setting the derivative equal to zero:
C0
(µ) = c −
hλ
(µ − λ)2
= 0 . (1.3)
∗ The expression (1.1) holds more generally for any work-conserving queue discipline that
does not use information about customer service times. See, e.g., El-Taha and Stid-
ham [60].](https://image.slidesharecdn.com/2207852-250605111415-ac5fac2e/85/Optimal-Design-Of-Queueing-Systems-1st-Edition-Shaler-Stidham-Jr-21-320.jpg)
![OPTIMAL SERVICE RATE 5
This yields the following expression for the unique optimal value of the
service rate, denoted by µ∗
:
µ∗
= λ +
r
λh
c
. (1.4)
The optimal value of the objective function is thus given by
C(µ∗
) = c
λ +
p
λh/c
+ λh/
p
λh/c = cλ +
√
λhc +
√
λhc .
This expression has the following interpretation. The term c · λ represents
the fixed cost of providing the minimum possible level of service, namely,
µ = λ. The next two terms – both equal to
√
λhc – represent, respectively,
the service cost and the waiting cost associated with the optimal “surplus”
service level, µ∗
− λ. Note that an optimal solution divides the variable cost
equally between service cost and waiting cost.
More explicitly, if one reformulates the problem in equivalent form with the
surplus service rate, µ̃ := µ−λ, as the decision variable and removes the fixed-
cost term, cλ, from the objective function, then the new objective function,
denoted by C̃(µ̃), takes the form
C̃(µ̃) = cµ̃ + hλ/µ̃ . (1.5)
The optimal value of µ̃ is given by
µ̃∗
=
r
λh
c
,
and the optimal value of the objective function by
C̃(µ̃∗
) = c
p
λh/c) + λh/
p
λh/c =
√
λhc +
√
λhc .
It is the particular structure of the objective function (1.5) – the sum of a term
proportional to the decision variable and a term proportional to its reciprocal
– that leads to the property that an optimal solution equates the two terms,
a property that of course does not hold in general when one is minimizing
the sum of two cost terms. The general condition for optimality (cf. equation
(1.3)) is that the marginal increase in the first term should equal the marginal
decrease in the second term, not that the terms themselves should be equal.
It just happens in this case that the latter property holds when the former
does.
Readers familiar with inventory theory will note the structural equiva-
lence of the objective function (1.5) to the objective function in the classical
economic-lot-size problem and the resulting similarity between the formula for
µ̃∗
and the economic-lot-size formula.
1.1.2 Extensions and Exercises
1. Constraints on the Service Rate. Suppose the service rate is constrained
to lie in an interval, µ ∈ [µ, µ̄]. Characterize the optimal service rate, µ∗
,](https://image.slidesharecdn.com/2207852-250605111415-ac5fac2e/85/Optimal-Design-Of-Queueing-Systems-1st-Edition-Shaler-Stidham-Jr-22-320.jpg)
![6 INTRODUCTION TO DESIGN MODELS
in this case. Do the same for the case where the feasible values of µ are
discrete: µ ∈ {µ1, µ2, . . . , µm}.
2. Nonlinear Waiting Costs. Suppose in the above model that the cus-
tomer’s waiting cost is a nonlinear function of the time spent by that
customer in the system: h · ta
, if the time in system equals t, where
a 0. (Note that for a 1 the waiting cost h·ta
is concave in t, whereas
for a 1 it is convex in t.) Set up and solve the problem of choosing
µ to minimize the expected steady-state total cost per unit time, C(µ).
For what values of a is C(µ) convex in µ?
3. General Service-Time Distribution. Consider an M/GI/1 model, in which
the generic service time S has mean E[S] = 1/µ and second moment
E[S2
] = 2β/µ2
, where β ≥ 1/2 is a given constant and µ is the decision
variable. (Thus the coefficient of variation of service time is given by
p
var(S)/E[S] =
√
2β − 1, which is fixed.) In this case the Pollaczek-
Khintchine formula yields
W(µ) =
1
µ
+
λβ
µ(µ − λ)
.
Set up the problem of determining the optimal service rate µ∗
, with linear
waiting cost rates. For what values of β is C(µ) convex? If possible, find
a closed-form expression for µ∗
in terms of the parameters, λ, c, h, and β.
(The easy cases are when β = 1 (e.g., exponentially distributed service
time) and β = 1/2 (constant service time, S ≡ 1/µ).)
1.2 Optimal Arrival Rate
Now consider a FIFO M/M/1 queue in which the service rate µ is fixed and
the arrival rate λ is a decision variable.
Examples
1. A machine center: at what rate λ should incoming parts (or subassem-
blies) be admitted into the work-in-process buffer?
2. A communication system: at what rate λ should messages (or packets)
be admitted into the buffer before a communication channel?
Performance Measures and Trade-offs
As λ increases, the throughput (number of jobs served per unit time) in-
creases. (For λ µ, the throughput equals λ; for λ ≥ µ, the throughput
equals µ.) This is clearly a “good thing.” On the other hand, the congestion
also increases as λ increases, and this is just as clearly a “bad thing.” Again a
simple linear model offers one way of capturing the trade-off between the two
performance measures.](https://image.slidesharecdn.com/2207852-250605111415-ac5fac2e/85/Optimal-Design-Of-Queueing-Systems-1st-Edition-Shaler-Stidham-Jr-23-320.jpg)
![OPTIMAL ARRIVAL RATE 9
where x+
:= max{x, 0}. Note that in Case 1 we have h/µ ≥ r; that is, the
expected waiting cost is at least as great as the reward even for a customer
who enters service immediately. Hence it is intuitively clear that λ∗
= 0: there
is no economic incentive to admit any customer. If r h/µ, then it is optimal
to allocate λ so that the surplus capacity, µ − λ, equals the square root of
µh/r.
1.2.2 Extensions and Exercises
1. Constraints on the Arrival Rate. Suppose the feasible set of values for λ
is the interval, [λ, λ̄], where 0 ≤ λ λ̄ ≤ ∞. The problem now takes the
form:
max
{λ∈[λ,λ̄]}
{λ · r − hL(λ)} . (1.10)
Since B(λ) = −∞ for λ ≥ µ, we can rewrite the problem in equivalent
form as
max
{λ∈[λ,min{λ̄,µ}]}
λ · r − h
λ
µ − λ
. (1.11)
(Note that the feasible region reduces to [λ, µ) when λ̄ ≥ µ.) Characterize
the optimal arrival rate, λ∗
, for this problem.
2. General Service-Time Distribution. Consider an M/GI/1 model, in which
the generic service time S has mean E[S] = 1/µ and second moment
E[S2
] = 2β/µ2
, where β ≥ 1/2 is given. The Pollaczek-Khintchine for-
mula yields
W(λ) =
1
µ
+
λβ
µ(µ − λ)
.
Set up the problem of determining the optimal arrival rate, λ∗
, with
deterministic reward and linear waiting cost. Show that λ∗
is again char-
acterized by (1.8) and (1.9), and use this result to derive an explicit
expression for λ∗
, in terms of the parameters, µ, β, r, and h.
1.2.3 An Upper Bound on the Optimal Arrival Rate
Note that
B(λ) = λr − hλW(λ) = λ(r − hW(λ)) , (1.12)
so that B(λ) 0 for positive values of λ such that r hW(λ) and B(λ) ≤ 0
for values of λ such that r ≤ hW(λ). If r ≤ hW(0) then r ≤ hW(λ) for all
λ ∈ [0, µ), since W(·) is an increasing function. In this case λ∗
= 0. Otherwise,
we can restrict attention, without loss of optimality, to values of λ such that
r hW(λ). In the M/M/1 case, W(λ) = 1/(µ − λ), so that r ≤ hW(0) if
and only if r ≤ h/µ. Moreover, r = hW(λ) if and only if λ = µ − h/r. These
observations motivate the following definition.](https://image.slidesharecdn.com/2207852-250605111415-ac5fac2e/85/Optimal-Design-Of-Queueing-Systems-1st-Edition-Shaler-Stidham-Jr-26-320.jpg)
![12 INTRODUCTION TO DESIGN MODELS
positive joining probability, a 0, is a(r−hW(0)), which is less than or equal
to zero, the expected net benefit from the joining probability ae
= λe
/Λ = 0.
Hence, in Case 1 there is no incentive for a customer to deviate unilaterally
from ae
= 0. In Case 2, since r − hW(λe
) = 0, the expected net benefit is
a(r − hW(λe
)) + (1 − a) · 0 = 0 ,
and hence does not depend on the joining probability a. Thus, customers are
indifferent among all joining probabilities, 0 ≤ a ≤ 1, so that once again there
is no incentive to deviate from ae
= λe
/Λ.
Since W(λ) = 1/(µ − λ) in the M/M/1 case, we see that the individually
optimal arrival rate λe
coincides with λ̄ as defined by (1.13) and (1.14). But
we have shown that λ∗
= λs
≤ λ̄ = λe
. In other words, the socially optimal
arrival rate, λs
, is less than or equal to the individually optimal arrival rate,
λe
.
The following theorem summarizes these results:
Theorem 1.1 The socially optimal arrival rate is no larger than the individ-
ually optimal arrival rate: λs
≤ λe
. Moreover, λs
= λe
= 0 , if r ≤ h/µ , and
0 λs
λe
, if r h/µ .
A review of our arguments above will show that this property is not re-
stricted to M/M/1 systems and is in fact quite general. In fact, this theorem
is valid for any system (for example, a GI/GI/1 queue) in which the following
conditions hold:
1. W(λ) is strictly increasing in 0 ≤ λ µ ;
2. W(λ) ↑ ∞ as λ ↑ µ ;
3. W(0) = 1/µ .
1.2.5 Internal and External Effects
Suppose r h/µ. It follows from (1.12) that
B0
(λ) = r − [h · W(λ) + h · λW0
(λ)] ,
and that λs
is found by equating h·W(λ)+h·λW0
(λ) to r, whereas (cf. (1.16))
λe
is found by equating h·W(λ) to r. We can interpret h·W(λ) as the internal
effect and h·λW0
(λ) as the external effect of a marginal increase in the arrival
rate. The quantity h · W(λ) is the waiting cost of the marginal customer who
joins when the arrival rate is λ. It is “internal” in that it is a cost borne only by
the customer itself. On the other hand, the quantity h·λW0
(λ) is the marginal
increase in waiting cost incurred by all the customers as a result of a marginal
increase in the arrival rate. It is “external” to the marginal joining customer,
since it is a cost which that customer does not incur. The fact that λs
≤ λe
(that is, customers acting in their own interest join the system more frequently
than is socially optimal) is due to an individually optimizing customer’s failure
to take into account the external effect of its decision to enter. The formula
for λe
only takes into account the internal effect of the decision to enter, that](https://image.slidesharecdn.com/2207852-250605111415-ac5fac2e/85/Optimal-Design-Of-Queueing-Systems-1st-Edition-Shaler-Stidham-Jr-29-320.jpg)
![OPTIMAL ARRIVAL RATES FOR A TWO-CLASS SYSTEM 17
al. [16]):
ri = hiW(λ) + δ and λi 0 (1.19)
or ri ≤ hiW(λ) + δ and λi = 0 (1.20)
for i = 1, 2, and
λ = λ1 + λ2 , (1.21)
δ = (λ1h1 + λ2h2)W0
(λ) . (1.22)
Now consider this system from the perspective of individual optimization.
Suppose a fixed, arbitrary toll, δ, is charged to each entering customer. Each
customer of class i takes W(λ) as given and chooses the probability ai of
joining to maximize
ai · (ri − hiW(λ) − δ) + (1 − ai) · 0 , ai ∈ [0, 1] .
In other words, a class-i customer who joins receives the net benefit, ri −
hiW(λ), minus the toll, δ, paid for the use of the facility. A customer who balks
receives (pays) nothing. Then it is easy to see that arrival rates, λi = ai · Λi,
that satisfy equations (1.19) and (1.20) will be individually optimal for the
customers of both classes. Moreover, for the given toll δ, a solution to (1.19),
(1.20), and (1.21) is a Nash equilibrium.
As expected, equation (1.22) reveals that the socially optimal toll is just the
external effect, defined (as usual) as the marginal increase in the total delay
cost incurred as a result of a marginal increase in the flow, λ. By charging this
socially optimal toll, the system operator can induce individually optimizing
customers to behave in a socially optimal way, thereby making the Nash-
equilibrium allocation coincide with the socially optimal allocation (λs
1, λs
2, λs
)
(cf. Section 1.2).
1.4.1 Solutions to the Optimality Conditions: the M/M/1 Case
Let us now examine the properties of the solution(s) to the KKT conditions,
using the explicit expression (1.18) for W(λ) for an M/M/1 system. The prob-
lem of finding a socially optimal allocation of flows takes the form
max
{λ1,λ2}
r1λ1 −
h1λ1
µ − λ1 − λ2
+ r2λ2 −
h2λ2
µ − λ1 − λ2
s.t. λ1 + λ2 µ
λ1 ≥ 0 , λ2 ≥ 0
Without loss of generality, we may assume that µ = 1. (Equivalently, measure
flows in units of fraction of the service rate µ.) Let a := r1/h1, b := r2/h2,
c := h1/h2. Then an equivalent form for the above problem is
max
{λ1,λ2}
c
aλ1 −
λ1
1 − λ1 − λ2
+ bλ2 −
λ2
1 − λ1 − λ2
(1.23)
s.t. λ1 + λ2 1](https://image.slidesharecdn.com/2207852-250605111415-ac5fac2e/85/Optimal-Design-Of-Queueing-Systems-1st-Edition-Shaler-Stidham-Jr-34-320.jpg)
![26 INTRODUCTION TO DESIGN MODELS
Figure 1.9 Arrival Control to Parallel Queues: Explicit Individually Optimal Solu-
tion
follows: for k = 1, . . . , n, if sk ≤ λ ≤ sk+1, then
λe
j(λ) = µj − [
k
X
i=1
µi − λ]/k , j = 1, . . . , k ,
= 0 , j = k + 1, . . . , n .
Figure 1.9 illustrates the behavior of the individually optimal facility arrival
rates as a function of the total arrival rate. Note that the positive λe
j(λ) are
piecewise linear in λ, with nonincreasing slope. The slopes of all positive λe
j(λ)
are equal in this case.
In Figure 1.10, the individually optimal allocation is superimposed on the
socially optimal allocation, for purposes of comparison. As a general observa-
tion, we can say that the individually optimal allocation assigns more (fewer)
customers to faster (slower) servers than the socially optimal allocation. More
specifically, for the example in Figure 1.10, the individually optimal allocation
always assigns more arrivals to facility 1, the fastest one, and fewer arrivals
to facility 3, the slowest one, than the socially optimal allocation does. As λ
increases, facility 2 first receives fewer, then more, arrivals in the individually
optimal than in the socially optimal allocation. Thus, facility 2 plays the role
of a “slower” server in light traffic and a “faster” server in heavy traffic.
1.6 Endnotes
Over the past forty years, there have been a number of survey papers and
books that discuss optimal control of queues, including Sobel [181], Stid-](https://image.slidesharecdn.com/2207852-250605111415-ac5fac2e/85/Optimal-Design-Of-Queueing-Systems-1st-Edition-Shaler-Stidham-Jr-43-320.jpg)
![ENDNOTES 27
Figure 1.10 Arrival Control to Parallel Queues: Comparison of Socially and Indi-
vidually Optimal Solutions
ham and Prabhu [191], Crabill, Gross, and Magazine [46], Serfozo [174], Stid-
ham [184], [185], [186], Kitaev and Rykov [111], and Hassin and Haviv [86].
Optimal design is touched on in some of these references but, to the best of my
knowledge, the present book is the first to provide a comprehensive treatment
of optimal design of queues.
Section 1.1
The model and results in this section were introduced in a pioneering paper
by Hillier [93]. Indeed, the emergence of optimization of queueing systems
(both design and control) as a legitimate subject for research owes a great deal
to Hillier and his PhD students in operations research at Stanford University,
beginning in the mid 1960s.
Section 1.2
Edelson and Hildebrand [59] introduced the basic model of this section.
They compared the socially optimal toll with the facility optimal toll, that is,
the toll that maximizes revenue to the toll collector (e.g., the facility oper-
ator). They showed that the two are equal when all customers received the
same reward, r, from joining and receiving service. When customers are het-
erogeneous – that is, the reward is a random variable, R – the facility optimal
toll (arrival rate) is in general larger (smaller) than the socially optimal toll
(arrival rate) (see Chapter 2).
Section 1.3
Surprisingly, I could find no published reference in which exactly this model](https://image.slidesharecdn.com/2207852-250605111415-ac5fac2e/85/Optimal-Design-Of-Queueing-Systems-1st-Edition-Shaler-Stidham-Jr-44-320.jpg)
![28 INTRODUCTION TO DESIGN MODELS
is considered. The material in this section is taken largely from my class notes
for a course on Optimization of Queueing Systems which I have taught, in
various versions, since the early 1970s. Dewan and Mendelson [54] consid-
ered a model for combined choice of the arrival and service rate, but with
heterogeneous rewards only. They examined only the solution to the neces-
sary first-order optimality conditions, without considering the possibility that
these conditions might not be sufficient. (In the examples they presented, the
conditions were, fortuitously, always sufficient.) Stidham [187] considered es-
sentially the same model as Dewan and Mendelson [54] and pointed out the
possible failure of the objective function to be jointly concave and the resulting
insufficiency of the first-order conditions.
Section 1.4
The model and results of this section come primarily from Stidham [189],
which considered a more general model for a multiclass network of queues.
Chapter 4 expands on the material in this section and Chapter 8 considers
the extension to networks.
Section 1.5
This model was introduced in an unpublished paper (Stidham [182]) and
then elaborated and extended in Bell and Stidham [18]. We return to the topic
of parallel queues in Chapter 6.](https://image.slidesharecdn.com/2207852-250605111415-ac5fac2e/85/Optimal-Design-Of-Queueing-Systems-1st-Edition-Shaler-Stidham-Jr-45-320.jpg)
![30 OPTIMAL ARRIVAL RATES IN A SINGLE-CLASS QUEUE
infinite time period) the expectation of a steady-state random variable. These
ingredients are now discussed in more detail.
The arrival rate λ measures the average number of customers arriving and
joining the system per unit time during the period of interest. As indicated, λ
is a decision variable. The set of feasible values for λ is denoted A. Our default
assumption will be that A = [0, ∞). Alterations to our model and results to
allow for more general feasible sets are usually straightforward and will be left
to the reader.
To capture the benefit of having a higher throughput, there is a utility
function, U(λ), which measures the average gross value received per unit time
as a function of the arrival rate λ. For example, in the model considered in
Section 1.2 with a deterministic reward, r, per entering customer, U(λ) = r·λ,
so that the utility function is linear. Our default assumption is that U(λ) is
nondecreasing, differentiable, and concave in λ ≥ 0. We allow U0
(0) = ∞. In
Section 2.3 we show how a value function of this form can arise when there
is a renewal process of potential arriving customers (with rate Λ ∞) and
a probabilistic joining rule is followed. We can accommodate a finite upper
bound, Λ, on λ by defining U(λ) = U(Λ) for λ Λ. But this definition may
not be compatible with the differentiability assumption, unless U0
(Λ) = 0.
Later (in Section 2.2) we shall examine the effects of relaxing some of the
regularity conditions satisfied by U(λ), including differentiability.
Balanced against the benefit of throughput is the cost to customers caused
by the time they spend in the system. For a given λ, G(λ) denotes the average
waiting cost of a job, averaged over all customers who arrive during the period
in question. Our default assumption is that G(λ) takes values in [0, ∞] and is
strictly increasing and differentiable in λ ≥ 0. We allow G(λ) to equal ∞ in
order to accommodate, for example, a single-server system with service rate
µ, in which we typically have G(λ) = ∞ for λ ≥ µ. This convention makes it
unnecessary to include the constraint λ µ explicitly in our formulation.
For example, in the case of an infinite time period, it might be that
G(λ) = E[h(W (λ))] , (2.1)
where h(t) is the waiting cost incurred by a job that spends a length of time
t in the system and, for each λ ≥ 0, W (λ) is the steady-state random waiting
time in the system for the queueing system induced by λ. In the case of a
linear waiting cost, h(t) = h · t,
G(λ) = h · W(λ) ,
where W(λ) := E[W (λ)]. For the example of an M/M/1 queue operating in
steady state considered in Section 1.2 of Chapter 1, we have
G(λ) =
h
µ − λ
.
Let H(λ) = λG(λ). Then H(λ) is a measure of the average waiting cost
incurred by the system per unit time, inasmuch as it equals the product of the
average number of customers arriving per unit time and the average waiting](https://image.slidesharecdn.com/2207852-250605111415-ac5fac2e/85/Optimal-Design-Of-Queueing-Systems-1st-Edition-Shaler-Stidham-Jr-47-320.jpg)
![A MODEL WITH GENERAL UTILITY AND COST FUNCTIONS 31
cost per customer.∗
We shall assume that H(λ) is a convex function of λ ≥ 0.
(Note that the assumption that G(λ) is strictly increasing and differentiable
implies that H(λ) is also strictly increasing and differentiable.)
All these properties are weak and common in the queueing literature. A
sufficient condition for H(λ) to be convex is that G(λ) is convex, which is
simply an assumption that each customer’s marginal cost of waiting does not
decrease as the arrival rate increases. As an illustration this property holds in
our canonical example: an M/M/1 queue with linear waiting costs and FIFO
queue discipline.
Remark 1 Customers might be sensitive to losses rather than (or in addition
to) delays. (This situation can arise in a system with a finite buffer, in which
an arriving customer who finds the buffer full is lost.) In this case, G(λ) might
measure the cost incurred if a customer is lost because of buffer overflow. In
the special case in which G(λ) = h · P(λ), P(λ) might measure the steady-
state probability that a customer is lost (or the fraction of customers lost)
and h the sensitivity of customers to such a loss. Although we shall continue
to refer to “delay sensitivity” or “waiting costs” throughout the discussion of
this model, the reader should keep in mind that the results also apply to other
measures of congestion, such as losses.
In addition to incurring the waiting cost G(λ), an entering customer may
have to pay an admission fee (or toll) δ. In the present model, the sum of
the toll and the waiting cost constitutes the full price of admission, which we
denote in general by π, or π(λ), when we want to emphasize its dependence
on λ for a fixed δ. Thus we have
π(λ) = δ + G(λ) .
Remark 2 The concept of the full price of admission is common to many
models for arrival-rate selection, as we shall see. In more complicated systems,
such as a set of parallel facilities (Chapter 6) or a network of queues (Chap-
ters 7 and 8), the derivation of the full price is more complicated, as it may
involve choices among alternate facilities or routes. But the analysis of the
arrival-rate selection problem is basically the same as in the single-facility,
single-class model considered in this chapter. Consequently we shall develop
much of the theory for the present model in a general framework that will
allow our results to be carried over to the more complicated models in subse-
quent chapters without unnecessary repetition. When we are operating in this
general framework, we shall simply assume that π = π(λ) is a given strictly
increasing and differentiable function of λ.
As noted, the arrival rate λ is a decision variable. The solution to the
∗ In queueing terms, the relation – waiting cost per unit time = (arrival rate) × (waiting
cost per customer) – is a special case of H = λG, the generalization of L = λW, which
holds under weak assumptions. (See El-Taha and Stidham [60], Chapter 6. In the case of
linear waiting cost, it just follows from L = λW itself.)](https://image.slidesharecdn.com/2207852-250605111415-ac5fac2e/85/Optimal-Design-Of-Queueing-Systems-1st-Edition-Shaler-Stidham-Jr-48-320.jpg)
Optimal Design Of Queueing Systems 1st Edition Shaler Stidham Jr
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- 6. Optimal Design of Queueing Systems Shaler Stidham, Jr. University of North Carolina Chapel Hill, North Carolina, U. S. A.
- 7. Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487‑2742 © 2009 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid‑free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number‑13: 978‑1‑58488‑076‑9 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher can‑ not assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copy‑ right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978‑750‑8400. CCC is a not‑for‑profit organization that pro‑ vides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data Stidham, Shaler. Optimal design for queueing systems / Shaler Stidham Jr. p. cm. “A CRC title.” Includes bibliographical references and index. ISBN 978‑1‑58488‑076‑9 (alk. paper) 1. Queueing theory. 2. Combinatorial optimization. I. Title. T57.9.S75 2009 519.8’2‑‑dc22 2009003648 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com
- 8. Contents List of Figures v Preface ix 1 Introduction to Design Models 1 1.1 Optimal Service Rate 3 1.2 Optimal Arrival Rate 6 1.3 Optimal Arrival Rate and Service Rate 13 1.4 Optimal Arrival Rates for a Two-Class System 16 1.5 Optimal Arrival Rates for Parallel Queues 21 1.6 Endnotes 26 2 Optimal Arrival Rates in a Single-Class Queue 29 2.1 A Model with General Utility and Cost Functions 29 2.2 Generalizations of Basic Model 42 2.3 GI/GI/1 Queue with Probabilistic Joining Rule 45 2.4 Uniform Value Distribution: Stability 68 2.5 Power Criterion 72 2.6 Bidding for Priorities 77 2.7 Endnotes 80 3 Dynamic Adaptive Algorithms: Stability and Chaos 83 3.1 Basic Model 84 3.2 Discrete-Time Dynamic Adaptive Model 85 3.3 Discrete-Time Dynamic Algorithms: Variants 98 3.4 Continuous-Time Dynamic Adaptive Algorithms 101 3.5 Continuous-Time Dynamic Algorithm: Variants 106 3.6 Endnotes 107 4 Optimal Arrival Rates in a Multiclass Queue 109 4.1 General Multiclass Model: Formulation 109 4.2 General Multiclass Model: Optimal Solutions 113 4.3 General Multiclass Model: Dynamic Algorithms 124 4.4 Waiting Costs Dependent on Total Arrival Rate 129 4.5 Linear Utility Functions: Class Dominance 134 4.6 Examples with Different Utility Functions 153 iii
- 9. iv CONTENTS 4.7 Multiclass Queue with Priorities 158 4.8 Endnotes 170 4.9 Figures for FIFO Examples 172 5 Optimal Service Rates in a Single-Class Queue 177 5.1 The Basic Model 178 5.2 Models with Fixed Toll and Fixed Arrival Rate 182 5.3 Models with Variable Toll and Fixed Arrival Rate 184 5.4 Models with Fixed Toll and Variable Arrival Rate 185 5.5 Models with Variable Toll and Variable Arrival Rate 199 5.6 Endnotes 215 6 Multi-Facility Queueing Systems: Parallel Queues 217 6.1 Optimal Arrival Rates 217 6.2 Optimal Service Rates 255 6.3 Optimal Arrival Rates and Service Rates 258 6.4 Endnotes 277 7 Single-Class Networks of Queues 279 7.1 Basic Model 279 7.2 Individually Optimal Arrival Rates and Routes 280 7.3 Socially Optimal Arrival Rates and Routes 282 7.4 Comparison of S.O. and Toll-Free I.O. Solutions 284 7.5 Facility Optimal Arrival Rates and Routes 307 7.6 Endnotes 314 8 Multiclass Networks of Queues 317 8.1 General Model 317 8.2 Fixed Routes: Optimal Solutions 330 8.3 Fixed Routes: Dynamic Adaptive Algorithms 334 8.4 Fixed Routes: Homogeneous Waiting Costs 338 8.5 Variable Routes: Homogeneous Waiting Costs 339 8.6 Endnotes 342 A Scheduling a Single-Server Queue 343 A.1 Strong Conservation Laws 343 A.2 Work-Conserving Scheduling Systems 344 A.3 GI/GI/1 WCSS with Nonpreemptive Scheduling Rules 351 A.4 GI/GI/1 Queue: Preemptive-Resume Scheduling Rules 355 A.5 Endnotes 357 References 359 Index 369
- 10. List of Figures 1.1 Total Cost as a Function of Service Rate 4 1.2 Optimal Arrival Rate, Case 1: r ≤ h/µ 8 1.3 Optimal Arrival Rate, Case 2: r > h/µ 8 1.4 Net Benefit: Contour Plot 20 1.5 Net Benefit: Response Surface 21 1.6 Arrival Control to Parallel Queues: Parametric Socially Optimal Solution 23 1.7 Arrival Control to Parallel Queues: Explicit Socially Optimal Solution 24 1.8 Arrival Control to Parallel Queues: Parametric Individually Optimal Solution 25 1.9 Arrival Control to Parallel Queues: Explicit Individually Optimal Solution 26 1.10 Arrival Control to Parallel Queues: Comparison of Socially and Individually Optimal Solutions 27 2.1 Characterization of Equilibrium Arrival Rate 33 2.2 Graph of the Function U0 (λ) 40 2.3 Graph of the Function λU0 (λ) 41 2.4 Graph of the Objective Function: λU0 (λ) − λG(λ) 41 2.5 Graph of the Function U0 (λ) 43 2.6 Equilibrium Arrival Rate. Case 1: U0 (λ−) > π(λ) > U0 (λ) 44 2.7 Equilibrium Arrival Rate. Case 2: U0 (λ−) = π(λ) = U0 (λ) 44 2.8 Graphical Interpretation of U(λ) as an Integral: Case 1 50 2.9 Graphical Interpretation of U(λ) as an Integral: Case 2 51 2.10 Graph of λU0 (λ): Pareto Reward Distribution (α < 1) 56 2.11 Graph of Ũ(λ): M/M/1 Queue with Pareto Reward Distribution (α < 1) 56 2.12 Graph of λU0 (λ): Pareto Reward Distribution (α > 1) 57 2.13 Graph of Ũ(λ): M/M/1 Queue with Pareto Reward Distribution (α > 1) 58 2.14 U(λ) for Three-Class Example 60 2.15 U0 (λ) for Three-Class Example 61 2.16 λU0 (λ) for Three-Class Example 63 2.17 Ũ(λ) for Three-Class Example (Case 1) 64 2.18 Ũ(λ) for Three-Class Example (Case 2) 64 v
- 11. vi LIST OF FIGURES 2.19 Ui(λ), i = 1, 2, 3, for Three-Class Example 65 2.20 λU0 (λ) for Example 3 67 2.21 Ũ(λ) for Example 3 68 2.22 Supply and Demand Curves: Uniform Value Distribution 69 2.23 An Unstable Equilibrium 70 2.24 Convergence to a Stable Equilibrium 71 2.25 Graphical Illustration of Power Maximization 74 2.26 Graph of Equilibrium Bid Distribution 81 3.1 Period-Doubling Bifurcations 95 3.2 Chaotic Cobweb 96 3.3 Arrival Rate Distribution 97 4.1 Class Dominance Regions for Individual and Social Optimization 153 4.2 Linear Utility Functions: U(λ1, λ2) = 16λ1 − 4λ1/(1 − λ1 − λ2) + 9λ2 − λ2/(1 − λ1 − λ2) 156 4.3 Linear Utility Functions: U(λ1, λ2) = 16λ1 − 4λ1/(1 − λ1 − λ2) + 9λ2 − λ2/(1 − λ1 − λ2) 156 4.4 Linear Utility Functions: U(λ1, λ2) = 64λ1 − 9λ1/(1 − λ1 − λ2) + 12λ2 157 4.5 Linear Utility Functions: U(λ1, λ2) = 16λ1 − 4λ1/(1 − λ1) + 9λ2 − λ2/((1 − λ1)(1 − λ1 − λ2)) 169 4.6 Linear Utility Functions: U(λ1, λ2) = 4λ1 − .4λ1/(1 − λ1) + 6λ2 − λ2/((1 − λ1)(1 − λ1 − λ2)) 169 4.7 Square-Root Utility Functions: U(λ1, λ2) = 64λ1 + 8 √ λ1 − 9λ1/(1 − λ1 − λ2) + 15λ2 172 4.8 Square-Root Utility Functions: U(λ1, λ2) = 24λ1 + 8 √ λ1 − 9λ1/(1 − λ1 − λ2) + 9λ2 172 4.9 Square-Root Utility Functions: U(λ1, λ2) = 24λ1 + 8 √ λ1 − 9λ1/(1 − λ1 − λ2) + 9λ2 − 0.1λ2/(1 − λ1 − λ2) 173 4.10 Square-Root Utility Functions: U(λ1, λ2) = 16λ1 + 16 √ λ1 − 4λ1/(1 − λ1 − λ2) + 9λ2 + 9 √ λ2 − λ2/(1 − λ1 − λ2) 173 4.11 Logarithmic Utility Functions: U(λ1, λ2) = 16 log(1 + λ1) − 4λ1/(1 − λ1 − λ2) + 3λ2 174 4.12 Logarithmic Utility Functions: U(λ1, λ2) = 16 log(1 + λ1) − 4λ1/(1 − λ1 − λ2) + 4 log(1 + λ2) − 0.1λ2/(1 − λ1 − λ2) 174 4.13 Logarithmic Utility Functions: U(λ1, λ2) = 16 log(1 + λ1) − 4λ1/(1 − λ1 − λ2) + 9 log(1 + λ2) − 0.1λ2/(1 − λ1 − λ2) 175 4.14 Logarithmic Utility Functions: U(λ1, λ2) = 16 log(1 + λ1) − 2λ1/(1 − λ1 − λ2) + 9 log(1 + λ2) − 0.25λ2/(1 − λ1 − λ2) 175 4.15 Quadratic Utility Functions: U(λ1, λ2) = 75λ1 − λ2 1 − 4λ1/(1 − λ1 − λ2) + 14λ2 − 0.05λ2 2 − 0.5λ2/(1 − λ1 − λ2) 176 5.1 M/M/1 Queue: Graph of H(λ, µ) (h = 1) 180 5.2 M/M/1 Queue: Graph of ψ(µ) 190
- 12. LIST OF FIGURES vii 5.3 Example with Convex Objective Function, µ > µ0 194 5.4 Long-Run Demand and Supply Curves 203 5.5 Uniform [d, a] Value Distribution Long-Run Demand and Supply Curves, Case 1 205 5.6 Uniform [d, a] Value Distribution Long-Run Demand and Supply Curves, Case 2 205 5.7 Long-Run Demand and Supply Curves; Uniform [0, a] Value Distribution 207 5.8 Convergence of Iterative Algorithm for Case of Uniform [0, a] Demand 208 6.1 Comparison of S.O. and F.O. Supply-Demand Curves for Variable λ 239 6.2 Nash Equilibrium for Two Competitive M/M/1 Facilities 246 6.3 Waiting-Cost Function for M/M/1 Queue 251 6.4 Illustration of Sequential Discrete-Time Algorithm 254 6.5 Facility Dominance as a Function of λ 266 6.6 Graphs of U0 (λ) and C0 (λ) for Parallel-Facility Example 269 7.1 First Example Network for Braess’s Paradox 286 7.2 Second Example Network for Braess’s Paradox 288 7.3 Example Network with α(λ) < π(λ) 293 7.4 Illustration of Theorem 7.2 300 7.5 Illustration of Derivation of Upper Bound for Affine Waiting- Cost Function 302 7.6 Graph of φ(ρ) 304 7.7 Table: Values of σ = φ(ρe ) and (1 − σ)−1 304 A.1 Graph of V (t): Work in System 345
- 14. Preface What began a long time ago as a comprehensive book on optimization of queueing systems has evolved into two books: this one on optimal design and a subsequent book (still in the works) on optimal control of queueing systems. In this setting, “design” refers to setting the parameters of a queueing sys- tem (such as arrival rates and service rates) before putting it into operation. By contrast, in “control” problems the parameters are control variables in the sense that they can be varied dynamically in response to changes in the state of the system. The distinction between design and control, admittedly, can be somewhat artificial. But the available material had outgrown the confines of a single book and I decided that this was as good a way as any of making a division. Why look at design models? In principle, of course, one can always do better by allowing the values of the decision variables to depend on the state of the system, but in practice this is frequently an unattainable goal. For example, in modern communication networks, real-time information about the buffer contents at the various nodes (routers/switches) of the network would, in principle, help us to make good real-time decisions about the routing of messages or packets. But such information is rarely available to a centralized controller in time to make decisions that are useful for the network as a whole. Even if it were available, the combinatorial complexity of the decision problem makes it impossible to solve even approximately in the time available. (The essential difficulty with such systems is that the time scale on which the system state is evolving is comparable to, or shorter than, the time scale on which information can be obtained and calculations of optimal policies can be made.) For these and other reasons, those in the business of analyzing, designing, and operating communication networks have turned their attention more and more to flow control, in which quantities such as arrival (e.g., packet-generation) rates and service (e.g., transmission) rates are computed as time averages over periods during which they may be reasonably expected to be constant (e.g., peak and off-peak hours) and models are used to suggest how these rates can be controlled to achieve certain objectives. Since this sort of decision process involves making decisions about rates (time averages) and not the behavior of individual messages/packets, it falls under the category of what I call a design problem. Indeed, many of the models, techniques, and results discussed in this book were inspired by research on flow and routing control that has been reported in the literature on communication networks. Of course, flow control is still control in the sense that decision variables can ix
- 15. x PREFACE change their values in response to changes in the state of the system, but the states in question are typically at a higher level, involving congestion averages taken over time scales that are much longer than the time scale on which such congestion measures as queue lengths and waiting times are evolving at individual service facilities. For this reason, I believe that flow control belongs under the broad heading of design of queueing systems. I have chosen to frame the issues in the general setting of a queueing system, rather than specific applications such as communication networks, vehicular traffic flow, supply chains, etc. I believe strongly that this is the most appro- priate and effective way to produce applicable research. It is a belief that is consistent with the philosophy of the founders of operations research, who had the foresight to see that it is the underlying structure of a system, not the physical manifestation of that structure, that is important when it comes to building and applying mathematical models. Unfortunately, recent trends have run counter to this philosophy, as more and more research is done within a particular application discipline and is published in the journals of that discipline, using the jargon of that discipline. The result has been compartmentalization of useful research. Important re- sults are sometimes rediscovered in, say, the communication and computer science communities, which have been well known for decades in, say, the traffic-flow community. I blame the research funding agencies, in part, for this trend. With all the best intentions of directing funding toward “applications” rather than “the- ory,” they have conditioned researchers to write grant proposals and papers which purport to deal with specific applications. These proposals and papers may begin with a detailed description of a particular application in which congestion occurs, in order to establish the credibility of the authors within the appropriate research community. When the mathematical model is intro- duced, however, it often turns out to be the M/M/1 queue or some other old, familiar queueing model, disguised by the use of a notation and terminology specific to the discipline in which the application occurs. Another of my basic philosophies has been to present the various models in a unified notation and terminology and, as much as possible, in a unified ana- lytical framework. In keeping with my belief (expressed above) that queueing theory, rather than any one or several of its applications, provides the appro- priate modeling basis for this field, it is natural that I should have adopted the notation and terminology of queueing theory. Providing a unified ana- lytical framework was a more difficult task. In the literature optimal design problems for queueing systems have been solved by a wide variety of analyt- ical techniques, including classical calculus, nonlinear programming, discrete optimization, and sample-path analysis. My desire for unity, together with space constraints, led me to restrict my attention to problems that can be solved for the most part by classical calculus, with some ventures into elemen- tary nonlinear programming to deal with constraints on the design variables. A side benefit of this self-imposed limitation has been that, although the book
- 16. PREFACE xi is mathematically rigorous (I have not shied away from stating results as the- orems and giving complete proofs), it should be accessible to anyone with a good undergraduate education in mathematics who is also familiar with el- ementary queueing theory. The downside is that I have had to omit several interesting areas of queueing design, such as those involving discrete decision variables (e.g., the number of servers) and several interesting and powerful analytical techniques, such as sample-path analysis. (I plan to include many of these topics in my queueing control book, however, since they are relevant also in that context.) The emphasis in the book is primarily on qualitative rather than quanti- tative insights. A recurring theme is the comparison between optimal designs resulting from different objectives. An example is the (by-now-classical) result that the individually optimal arrival rate is typically larger than the socially optimal arrival rate.∗ This is a result of the fact that individual customers, acting in self-interest, neglect to consider the external effect of their decision to enter a service facility: the cost of increased congestion which their decision imposes on other users (see, e.g., Section 1.2.4 of Chapter 1). As a general principle, this concept is well known in welfare economics. Indeed, a major theme of the research on queueing design has been to bring into the language of queueing theory some of the important issues and qualitative results from economics and game theory (the Nash equilibrium being another example). As a consequence this book may seem to many readers more like an economics treatise than an operations research text. This is intentional. I have always felt that students and practitioners would benefit from an infusion of basic eco- nomic theory in their education in operations research, especially in queueing theory. Much of the research reported in this book originated in vehicular traffic- flow theory and some of it pre-dates the introduction of optimization into queueing theory in the 1960s. Modeling of traffic flow in road networks has been done mainly in the context of what someone in operations research might call a “minimum-cost multi-commodity flow problem on a network with non- linear costs”. As such, it may be construed as a subtopic in nonlinear pro- gramming. An emphasis in this branch of traffic-flow theory has been on com- putational techniques and results. Chapters 7 and 8 of this book, which deal with networks of queues, draw heavily on the research on traffic-flow networks (using the language and specific models from queueing theory for the behavior of individual links/facilities) but with an emphasis on qualitative properties of optimal solutions, rather than quantitative computational methods. Although models for optimal design of queueing systems (using my broad definition) have proliferated in the four decades since the field began, I was surprised at how often I found myself developing new results because I could not find what I wanted in the literature. Perhaps I did not look hard enough. If I missed and/or unintentionally duplicated any relevant research, I ask for- ∗ But see Section 7.4.4 of Chapter 7 for a counterexample.
- 17. xii PREFACE bearance on the part of those who created it. The proliferation of research on queueing design, together with the explosion of different application ar- eas each with its own research community, professional societies, meetings, and journals, have made it very difficult to keep abreast of all the important research. I have tried but I may not have completely succeeded. A word about the organization of the book: I have tried to minimize the use of references in the text, with the exception of references for “classical” results in queueing theory and optimization. References for the models and results on optimal design of queues are usually given in an endnote (the final section of the chapter), along with pointers to material not covered in the book. Acknowledgements I would like to thank my editors at Chapman Hall and CRC Press in London for their support and patience over the years that it took me to write this book. I particularly want to thank Fred Hillier for introducing me to the field of optimization of queueing systems a little over forty years ago. I am grateful to my colleagues at the following institutions where I taught courses or gave seminars covering the material in this book: Cornell University (especially Uma Prabhu), Aarhus University (especially Niels Knudsen and Søren Glud Johansen), N.C. State University (especially Salah Elmaghraby), Technical University of Denmark, University of Cambridge (especially Peter Whittle, Frank Kelly, and Richard Weber), and INRIA Sophia Antipolis (especially François Baccelli and Eitan Altman). My colleagues in the Department of Statistics and Operations Research at UNC-CH (especially Vidyadhar Kulka- rni and George Fishman) have provided helpful input, for which I am grateful. I owe a particular debt of gratitude to the graduate students with whom I have collaborated on optimal design of queueing systems (especially Tuell Green and Christopher Rump) and to Yoram Gilboa, who helped teach me how to use MATLAB R to create the figures in the book. Finally, my wife Carolyn deserves special thanks for finding just the right combination of encourage- ment, patience, and (at appropriate moments) prodding to help me bring this project to a conclusion.
- 18. CHAPTER 1 Introduction to Design Models Like the descriptive models in “classical” queueing theory, optimal design models may be classified according to such parameters as the arrival rate(s), the service rate(s), the interarrival-time and service-time distributions, and the queue discipline(s). In addition, the queueing system under study may be a network with several facilities and/or classes of customers, in which case the nature of the flows of the classes among the various facilities must also be specified. What distinguishes an optimal design model from a traditional descriptive model is the fact that some of the parameters are subject to decision and that this decision is made with explicit attention to economic considerations, with the preferences of the decision maker(s) as a guiding principle. The basic distinctive components of a design model are thus: 1. the decision variables, 2. benefits and costs, and 3. the objective. Decision variables may include, for example, the arrival rates, the service rates, and the queue disciplines at the various service facilities. Typical benefits and costs include rewards to the customers from being served, waiting costs incurred by the customers while waiting for service, and costs to the facilities for providing the service. These benefits and costs may be brought together in an objective function, which quantifies the implicit trade-offs. For example, increasing the service rate will result in less time spent by the customers waiting (and thus a lower waiting cost), but a higher service cost. The nature of the objective function also depends on the horizon (finite or infinite), the presence or absence of discounting, and the identity of the decision maker (e.g., the facility operator, the individual customer, or the collective of all customers). Our goal in this chapter is to provide a quick introduction to these ba- sic components of a design model. We shall illustrate the effects of different reward and cost structures, the trade-offs captured by different objective func- tions, and the effects of combining different decision variables in one model. To keep the focus squarely on these issues, we use only the simplest of descriptive queueing models – primarily the classical M/M/1 model. By further restricting attention to infinite-horizon problems with no discounting, we shall be able to use the well-known steady-state results for these models to derive closed-form 1
- 19. 2 INTRODUCTION TO DESIGN MODELS expressions (in most cases) for the objective function in terms of the decision variables. This will allow us to do the optimization with the simple and famil- iar tools of differential calculus. Later chapters will elaborate on each of the models introduced in this chapter, relaxing distributional assumptions and considering more general cost and reward structures and objective functions. These more general models will require more sophisticated analytical tools, including linear and nonlinear programming and game theory. We begin this chapter (Sections 1.1 and 1.2) with two simple examples of optimal design of queueing systems. Both examples are in the context of an isolated M/M/1 queue with a linear cost/reward structure, in which the objective is to minimize the expected total cost or maximize the expected net benefit per unit time in steady state. In the first example the decision variable is the service rate and in the second, the arrival rate. The simple probabilistic and cost structure makes it possible to use classical calculus to derive analytical expressions for the optimal values of the design variables. The next three sections consider problems in which more than one design parameter is a decision variable. In Section 1.3, we consider the case where both the arrival rate and service rate are decision variables. Here a simple analysis based on calculus breaks down, since the objective function is not jointly concave and therefore the first-order optimality conditions do not identify the optimal solution. (This will be a recurring theme in our study of optimal design models, and we shall explore it at length in later chapters.) Section 1.4 revisits the problem of Section 1.2 – finding optimal arrival rates – but now in the context of a system with two classes of customers, each with its own reward and waiting cost and arrival rate (decision variable). Again the objective function is not jointly concave and the first-order optimality conditions do not identify the optimal arrival rates. Indeed, the only interior solution to the first-order conditions is a saddle-point of the objective function and is strictly dominated by both boundary solutions, in which only one class has a positive arrival rate. Finally, in Section 1.5, we consider the simplest of networks – a system of parallel queues in which each arriving customer must be routed to one of several independent facilities, each with its own queue. A final word before we start. In a design problem, the values of the decision variables, once chosen, cannot vary with time nor in response to changes in the state of the system (e.g., the number of customers present). Design problems have also been called static control problems, in contrast to dynamic control problems in which the decision variables can assume different values at different times, depending on the observed state of the system. In the literature a static control problem is sometimes called an open-loop control problem, whereas a dynamic control problem is called a closed-loop control problem. We shall simply use the term design for the former and control for the latter type of problem.
- 20. OPTIMAL SERVICE RATE 3 1.1 Optimal Service Rate Consider an M/M/1 queue with arrival rate λ and service rate µ. That is, customers arrive according to a Poisson process with parameter λ. There is a single server, who serves customers one at a time according to a FIFO (First- In-First-Out) queue discipline. Service times are independent of the arrival process and i.i.d. with an exponential distribution with mean µ−1 . Suppose that λ is fixed, but µ is a decision variable. Examples 1. A machine center in a factory: how fast a machine should we install? 2. A communication system: what should the transmission rate in a com- munication channel be (e.g., in bits/sec.)? Performance Measures and Trade-offs. Typical performance measures are the number of customers in the system (or in the queue) and the waiting time of a customer in the system (or in the queue). If the system operates for a long time, then we might be interested in the long-run average or the expected steady-state number in the system, waiting time, and so forth. All these are measures of the level of congestion. As µ increases, the congestion (as measured by any of these quantities) decreases. (Of course this property is not unique to M/M/1 systems.) Therefore, to minimize congestion, we should choose as large a value of µ as possible (e.g., µ = ∞, if there is no finite upper bound on µ). But, in all real systems, increasing the service rate costs something. Thus there is a trade-off between decreasing the congestion and increasing the cost of providing service, as µ increases. One way to capture this trade-off is to consider a simple model with linear costs. 1.1.1 A Simple Model with Linear Service and Waiting Costs Suppose there are two types of cost: (i) a service-cost rate, c (cost per unit time per unit of service rate); and (ii) a waiting-cost rate h (cost per unit time per customer in system). In other words, (i) if we choose service rate µ, then we pay a service cost c · µ per unit time; (ii) a customer who spends t time units in the system accounts for h · t monetary units of waiting cost, or equivalently, the system incurs h · i monetary units of waiting cost per unit time while i customers are present. Suppose our objective is to minimize the long-run average cost per unit time. Now it follows from standard results in descriptive queueing theory (or the general theory of continuous-time Markov chains) that the long-run average cost equals the expected steady-state cost, if steady state exists (which is true if and only if µ > λ). Otherwise the long-run average cost equals ∞. Therefore, without loss of generality let us assume µ > λ.
- 21. 4 INTRODUCTION TO DESIGN MODELS Figure 1.1 Total Cost as a Function of Service Rate Let C(µ) denote the expected steady-state total cost per unit time, when service rate µ is chosen. Then C(µ) = c · µ + h · L(µ) , where L(µ) is the expected steady-state number in system. For a FIFO M/M/1 queue, it is well known (see, e.g., Gross and Harris [79]) that L(µ) = λW(µ) = λ µ − λ , (1.1) where W(µ) is the expected steady-state waiting time in system.∗ Thus our optimization problem takes the form: min {µ:µ>λ} C(µ) = c · µ + h · λ µ − λ . (1.2) Note that C00 (µ) = 2hλ (µ − λ)3 0 , for all µ λ , so that C(µ) is convex in µ ∈ (λ, ∞). Moreover, C(µ) → ∞ as µ ↓ λ and as µ ↑ ∞. (See Figure 1.1.) Hence we can solve this problem by differentiating C(µ) and setting the derivative equal to zero: C0 (µ) = c − hλ (µ − λ)2 = 0 . (1.3) ∗ The expression (1.1) holds more generally for any work-conserving queue discipline that does not use information about customer service times. See, e.g., El-Taha and Stid- ham [60].
- 22. OPTIMAL SERVICE RATE 5 This yields the following expression for the unique optimal value of the service rate, denoted by µ∗ : µ∗ = λ + r λh c . (1.4) The optimal value of the objective function is thus given by C(µ∗ ) = c λ + p λh/c + λh/ p λh/c = cλ + √ λhc + √ λhc . This expression has the following interpretation. The term c · λ represents the fixed cost of providing the minimum possible level of service, namely, µ = λ. The next two terms – both equal to √ λhc – represent, respectively, the service cost and the waiting cost associated with the optimal “surplus” service level, µ∗ − λ. Note that an optimal solution divides the variable cost equally between service cost and waiting cost. More explicitly, if one reformulates the problem in equivalent form with the surplus service rate, µ̃ := µ−λ, as the decision variable and removes the fixed- cost term, cλ, from the objective function, then the new objective function, denoted by C̃(µ̃), takes the form C̃(µ̃) = cµ̃ + hλ/µ̃ . (1.5) The optimal value of µ̃ is given by µ̃∗ = r λh c , and the optimal value of the objective function by C̃(µ̃∗ ) = c p λh/c) + λh/ p λh/c = √ λhc + √ λhc . It is the particular structure of the objective function (1.5) – the sum of a term proportional to the decision variable and a term proportional to its reciprocal – that leads to the property that an optimal solution equates the two terms, a property that of course does not hold in general when one is minimizing the sum of two cost terms. The general condition for optimality (cf. equation (1.3)) is that the marginal increase in the first term should equal the marginal decrease in the second term, not that the terms themselves should be equal. It just happens in this case that the latter property holds when the former does. Readers familiar with inventory theory will note the structural equiva- lence of the objective function (1.5) to the objective function in the classical economic-lot-size problem and the resulting similarity between the formula for µ̃∗ and the economic-lot-size formula. 1.1.2 Extensions and Exercises 1. Constraints on the Service Rate. Suppose the service rate is constrained to lie in an interval, µ ∈ [µ, µ̄]. Characterize the optimal service rate, µ∗ ,
- 23. 6 INTRODUCTION TO DESIGN MODELS in this case. Do the same for the case where the feasible values of µ are discrete: µ ∈ {µ1, µ2, . . . , µm}. 2. Nonlinear Waiting Costs. Suppose in the above model that the cus- tomer’s waiting cost is a nonlinear function of the time spent by that customer in the system: h · ta , if the time in system equals t, where a 0. (Note that for a 1 the waiting cost h·ta is concave in t, whereas for a 1 it is convex in t.) Set up and solve the problem of choosing µ to minimize the expected steady-state total cost per unit time, C(µ). For what values of a is C(µ) convex in µ? 3. General Service-Time Distribution. Consider an M/GI/1 model, in which the generic service time S has mean E[S] = 1/µ and second moment E[S2 ] = 2β/µ2 , where β ≥ 1/2 is a given constant and µ is the decision variable. (Thus the coefficient of variation of service time is given by p var(S)/E[S] = √ 2β − 1, which is fixed.) In this case the Pollaczek- Khintchine formula yields W(µ) = 1 µ + λβ µ(µ − λ) . Set up the problem of determining the optimal service rate µ∗ , with linear waiting cost rates. For what values of β is C(µ) convex? If possible, find a closed-form expression for µ∗ in terms of the parameters, λ, c, h, and β. (The easy cases are when β = 1 (e.g., exponentially distributed service time) and β = 1/2 (constant service time, S ≡ 1/µ).) 1.2 Optimal Arrival Rate Now consider a FIFO M/M/1 queue in which the service rate µ is fixed and the arrival rate λ is a decision variable. Examples 1. A machine center: at what rate λ should incoming parts (or subassem- blies) be admitted into the work-in-process buffer? 2. A communication system: at what rate λ should messages (or packets) be admitted into the buffer before a communication channel? Performance Measures and Trade-offs As λ increases, the throughput (number of jobs served per unit time) in- creases. (For λ µ, the throughput equals λ; for λ ≥ µ, the throughput equals µ.) This is clearly a “good thing.” On the other hand, the congestion also increases as λ increases, and this is just as clearly a “bad thing.” Again a simple linear model offers one way of capturing the trade-off between the two performance measures.
- 24. OPTIMAL ARRIVAL RATE 7 1.2.1 A Simple Model with Deterministic Reward and Linear Waiting Costs Suppose there is a deterministic reward r per entering customer and (as in the previous model) a waiting cost per customer which is linear at rate h per unit time in the system. Let B(λ) denote the expected steady-state net benefit per unit time. Then B(λ) = λ · r − h · L(λ) , (1.6) where L(λ) is the steady-state expected number of customers in the system, expressed as a function of the arrival rate λ. As in the previous section, we have L(λ) = λW(λ), where W(λ) is the steady-state expected waiting time in the system, and (assuming a first-in, first-out (FIFO) queue discipline) W(λ) is given by W(λ) = 1 µ − λ , 0 ≤ λ µ , with W(λ) = ∞ for λ ≥ µ. Again it follows from standard results in descriptive queueing theory that the long-run average cost equals the expected steady- state cost, if steady state exists (which is true if and only if λ µ). Otherwise the long-run average cost equals ∞. Therefore, without loss of generality we assume λ µ. For the M/M/1 model, the problem thus takes the form: max {λ∈[0,µ)} r · λ − h · λ µ − λ . (1.7) The presence of the constraint, λ ≥ 0, makes this problem more complicated than the example of the previous section. Since B(λ) → −∞ as λ ↑ µ, we do not need to concern ourselves about the upper limit of the feasible region. But we must take into account the possibility that the maximum occurs at the lower limit, λ = 0. Let λ∗ denote the optimal arrival rate. Note that B00 (λ) = −2hµ (µ − λ)3 0 , for all µ λ , so that B(λ) is strictly concave and differentiable in 0 ≤ λ µ. Therefore its maximum occurs either at λ = 0 (if B0 (0) ≤ 0) or at the unique value of λ 0 at which B0 (λ) = 0 (if B0 (0) 0). It then follows from (1.6) that λ∗ is the unique solution in [0, µ) to the following conditions: (Case 1) λ = 0 , if r ≤ hL0 (0) ; (1.8) (Case 2) r = hL0 (λ) , if r hL0 (0) . (1.9) Now for the M/M/1 queue, L0 (λ) = µ (µ − λ)2 , so that B0 (0) ≤ 0 if r ≤ h/µ and B0 (0) 0 if r h/µ. Therefore (Case 1) λ∗ = 0 , if r ≤ h/µ ;
- 25. 8 INTRODUCTION TO DESIGN MODELS Figure 1.2 Optimal Arrival Rate, Case 1: r ≤ h/µ Figure 1.3 Optimal Arrival Rate, Case 2: r h/µ (Case 2) λ∗ = µ − p µh/r , if r h/µ ; The two cases are illustrated in Figures 1.2 and 1.3, respectively. Since µ − p µh/r 0 if and only if r h/µ, we can combine Cases 1 and 2 as follows: λ∗ = µ − p µh/r + ,
- 26. OPTIMAL ARRIVAL RATE 9 where x+ := max{x, 0}. Note that in Case 1 we have h/µ ≥ r; that is, the expected waiting cost is at least as great as the reward even for a customer who enters service immediately. Hence it is intuitively clear that λ∗ = 0: there is no economic incentive to admit any customer. If r h/µ, then it is optimal to allocate λ so that the surplus capacity, µ − λ, equals the square root of µh/r. 1.2.2 Extensions and Exercises 1. Constraints on the Arrival Rate. Suppose the feasible set of values for λ is the interval, [λ, λ̄], where 0 ≤ λ λ̄ ≤ ∞. The problem now takes the form: max {λ∈[λ,λ̄]} {λ · r − hL(λ)} . (1.10) Since B(λ) = −∞ for λ ≥ µ, we can rewrite the problem in equivalent form as max {λ∈[λ,min{λ̄,µ}]} λ · r − h λ µ − λ . (1.11) (Note that the feasible region reduces to [λ, µ) when λ̄ ≥ µ.) Characterize the optimal arrival rate, λ∗ , for this problem. 2. General Service-Time Distribution. Consider an M/GI/1 model, in which the generic service time S has mean E[S] = 1/µ and second moment E[S2 ] = 2β/µ2 , where β ≥ 1/2 is given. The Pollaczek-Khintchine for- mula yields W(λ) = 1 µ + λβ µ(µ − λ) . Set up the problem of determining the optimal arrival rate, λ∗ , with deterministic reward and linear waiting cost. Show that λ∗ is again char- acterized by (1.8) and (1.9), and use this result to derive an explicit expression for λ∗ , in terms of the parameters, µ, β, r, and h. 1.2.3 An Upper Bound on the Optimal Arrival Rate Note that B(λ) = λr − hλW(λ) = λ(r − hW(λ)) , (1.12) so that B(λ) 0 for positive values of λ such that r hW(λ) and B(λ) ≤ 0 for values of λ such that r ≤ hW(λ). If r ≤ hW(0) then r ≤ hW(λ) for all λ ∈ [0, µ), since W(·) is an increasing function. In this case λ∗ = 0. Otherwise, we can restrict attention, without loss of optimality, to values of λ such that r hW(λ). In the M/M/1 case, W(λ) = 1/(µ − λ), so that r ≤ hW(0) if and only if r ≤ h/µ. Moreover, r = hW(λ) if and only if λ = µ − h/r. These observations motivate the following definition.
- 27. 10 INTRODUCTION TO DESIGN MODELS Define λ̄ by: (Case 1) λ̄ = 0 , if r ≤ h/µ ; (1.13) (Case 2) λ̄ = µ − h/r , if r h/µ ; (1.14) Since B(λ) ≥ 0 for 0 ≤ λ ≤ λ̄, and B(λ) ≤ 0 for λ̄ λ µ, it follows that λ̄ is an upper bound on λ∗ . Moreover, in some contexts λ̄ can be interpreted as the individually optimal (or equilibrium) arrival rate, as we shall see presently. 1.2.4 Social vs. Individual Optimization In our discussion of performance measures and trade-offs, we have been implic- itly assuming that the decision maker is the operator of the queueing facility, who is concerned both with maximizing throughput and minimizing conges- tion. Our reward/cost model assumes that each entering customer generates a benefit r to the facility and that it costs the facility h per unit time per customer in the system. In this section we offer alternative possibilities for who the decision maker(s) might be. But first we must resolve another issue. We have also been implicitly assuming that the decision maker (whoever it is) can freely choose the arrival rate λ from the interval [0, µ). How might such a choice be implemented? Here is one possibility. Suppose that potential customers arrive according to a Poisson process with mean rate Λ (Λ ≥ µ). A potential customer joins (or is accepted) with prob- ability a and balks (or is rejected) with probability 1 − a. The accept/reject decisions for successive customers are mutually independent, as well as inde- pendent of the number of customers in the system. That is, it is not possible to observe the contents of the queue before the accept/reject decision is made. As a result, customers enter the system according to a Poisson arrival process with mean rate λ = aΛ.† Moreover, a customer who enters with probability a when the arrival rate equals λ receives an expected net benefit equal to a(r − hW(λ)) + (1 − a)0 = a(r − hW(λ)) . Now let us consider the possibility that the decision makers are the cus- tomers themselves, rather than the facility operator. We discuss this possibil- ity in the next two subsections. 1.2.4.1 Socially Optimal Arrival Rate Suppose now that benefits and costs accrue to individual customers and the decision maker represents the collective of all customers. In this case, a reason- able objective for the decision maker is to maximize the expected net benefit received per unit time by the collective of all customers: B(λ) = λ(r−hW(λ)). This is precisely the objective function that we have been considering. In this † Note that the assumption that Λ ≥ µ ensures that the feasible region for λ is the interval [0, µ), as in our original formulation.
- 28. OPTIMAL ARRIVAL RATE 11 context, our probabilistic interpretation of the choice of λ still makes sense. That is, the decision maker, acting on behalf of the collective of all customers, admits each potential arrival with probability a = λ/Λ. The optimal arrival rate λ∗ can now be interpreted as socially optimal, since it maximizes social welfare, that is, the expected net benefit received per unit time by the collective of all customers, namely B(λ). To emphasize this interpretation, we shall henceforth write “λs ” instead of “λ∗ ”. In the M/M/1 case, then, the socially optimal arrival rate is given by λs = (µ − p µh/r)+ . (1.15) The system controller can implement λs by admitting each potential arrival with probability as := λs /Λ and rejecting with probability 1 − as . 1.2.4.2 Comparison with Individually Optimal Arrival Rate This interpretation of λs as the socially optimal arrival rate suggests the fol- lowing question: how does the socially optimal arrival rate compare to the individually optimal arrival rate that results if each individual potential ar- rival, acting in its own interest, decides whether or not to join? Suppose (as above) that potential customers arrive according to a Poisson process with arrival rate Λ (Λ ≥ µ) and each joins the system with probability a and balks with probability 1−a. Each customer who enters the system when the arrival rate is λ receives a net benefit r − hW(λ). A customer who balks receives nothing. As is always the case with design (static control) models, we assume that the decision (a = 0, 1) must be made without knowledge of the actual state of the system, e.g., the number of customers present. Now, however, the criterion for choice of a is purely selfish: each customer is concerned only with maximizing its own expected net benefit. Since a sin- gle individual’s action has a negligible effect on the system arrival rate λ, each potential customer can take λ as given. For a given λ, the individually optimizing customer seeks to maximize its expected net benefit, a(r − hW(λ)) + (1 − a) · 0 , by an appropriate choice of a, 0 ≤ a ≤ 1. Thus, the customer will join with probability a = 1, if r hW(λ); join with probability a = 0, if r hW(λ); and be indifferent among all a, 0 ≤ a ≤ 1, if r = hW(λ). Motivated by the concept of a Nash equilibrium, we define an individually optimal (or equilibrium arrival rate, λe (and associated joining probability ae = λe /Λ), by the property that no individual customer trying to maximize its own expected net benefit has any incentive to deviate unilaterally from λe (ae ). From the above observations, it follows that λe = 0 (ae = 0) if r ≤ hW(0) (Case 1), whereas if r hW(0) (Case 2) then λe = ae Λ is the (unique) value of λ ∈ (0, µ) such that r = hW(λ) . (1.16) To see this, first note that in Case 1 the expected net benefit from choosing a
- 29. 12 INTRODUCTION TO DESIGN MODELS positive joining probability, a 0, is a(r−hW(0)), which is less than or equal to zero, the expected net benefit from the joining probability ae = λe /Λ = 0. Hence, in Case 1 there is no incentive for a customer to deviate unilaterally from ae = 0. In Case 2, since r − hW(λe ) = 0, the expected net benefit is a(r − hW(λe )) + (1 − a) · 0 = 0 , and hence does not depend on the joining probability a. Thus, customers are indifferent among all joining probabilities, 0 ≤ a ≤ 1, so that once again there is no incentive to deviate from ae = λe /Λ. Since W(λ) = 1/(µ − λ) in the M/M/1 case, we see that the individually optimal arrival rate λe coincides with λ̄ as defined by (1.13) and (1.14). But we have shown that λ∗ = λs ≤ λ̄ = λe . In other words, the socially optimal arrival rate, λs , is less than or equal to the individually optimal arrival rate, λe . The following theorem summarizes these results: Theorem 1.1 The socially optimal arrival rate is no larger than the individ- ually optimal arrival rate: λs ≤ λe . Moreover, λs = λe = 0 , if r ≤ h/µ , and 0 λs λe , if r h/µ . A review of our arguments above will show that this property is not re- stricted to M/M/1 systems and is in fact quite general. In fact, this theorem is valid for any system (for example, a GI/GI/1 queue) in which the following conditions hold: 1. W(λ) is strictly increasing in 0 ≤ λ µ ; 2. W(λ) ↑ ∞ as λ ↑ µ ; 3. W(0) = 1/µ . 1.2.5 Internal and External Effects Suppose r h/µ. It follows from (1.12) that B0 (λ) = r − [h · W(λ) + h · λW0 (λ)] , and that λs is found by equating h·W(λ)+h·λW0 (λ) to r, whereas (cf. (1.16)) λe is found by equating h·W(λ) to r. We can interpret h·W(λ) as the internal effect and h·λW0 (λ) as the external effect of a marginal increase in the arrival rate. The quantity h · W(λ) is the waiting cost of the marginal customer who joins when the arrival rate is λ. It is “internal” in that it is a cost borne only by the customer itself. On the other hand, the quantity h·λW0 (λ) is the marginal increase in waiting cost incurred by all the customers as a result of a marginal increase in the arrival rate. It is “external” to the marginal joining customer, since it is a cost which that customer does not incur. The fact that λs ≤ λe (that is, customers acting in their own interest join the system more frequently than is socially optimal) is due to an individually optimizing customer’s failure to take into account the external effect of its decision to enter. The formula for λe only takes into account the internal effect of the decision to enter, that
- 30. OPTIMAL ARRIVAL RATE AND SERVICE RATE 13 is the customer’s own waiting cost, hW(λ). By contrast, the formula for λs takes into account both the internal effect, hW(λ), and the external effect, hλW0 (λ). It follows that individually optimizing customers can be induced to behave in a socially optimal way by charging each entering customer a fee or con- gestion toll equal to the external effect, hλW0 (λ). In this way arrival control can be decentralized, in the sense that each individual customer can be left to make its own decision. (Again, note that these results hold for any system in which W(λ) is a well defined function satisfying conditions (1)–(3). See Chapter 2 for further analysis and generalizations.) 1.3 Optimal Arrival Rate and Service Rate Now let us consider an M/M/1 queue in which both the arrival rate λ and the service rate µ are decision variables. We shall use a reward/cost model that combines the features of the models of the last two sections. There is a reward r per entering customer, a waiting cost h per unit time per customer in the system, and a service cost c per unit time per unit of service rate. The objective function (to be maximized) is the steady-state expected net benefit per unit time, B(λ, µ), that is, B(λ, µ) = λ · r − h · L(λ, µ) − c · µ , 0 ≤ λ µ , with B(0, 0) = 0. (Note that B(λ, µ) has a discontinuity at (0, 0).) If c ≥ r, then obviously the optimal solution is λ∗ = µ∗ = 0, with net benefit B(0, 0) = 0, since for all 0 ≤ λ µ we have B(λ, µ) 0. Henceforth we shall assume that c r, in which case we can exclude the point (0, 0) and restrict attention to the region {(λ, µ) : 0 ≤ λ µ}, since it contains pairs (λ, µ) for which B(λ, µ) 0. Note that B(λ, µ) is continuously differentiable over this region. Following the program of the previous two sections, let us use the first-order optimality conditions to try to identify the optimal pair, (λ∗ , µ∗ ). Differenti- ating B(λ, µ) with respect to λ and µ and setting the derivatives equal to zero leads to the equations, ∂ ∂λ B(λ, µ) = r − h · ∂ ∂λ L(λ, µ) = 0 , ∂ ∂µ B(λ, µ) = −h · ∂ ∂µ L(λ, µ) − c = 0 . Since L(λ, µ) = λ/(µ − λ), for 0 ≤ λ µ, we have ∂ ∂λ L(λ, µ) = µ (µ − λ)2 , ∂ ∂µ L(λ, µ) = −λ (µ − λ)2 , from which we obtain the following two simultaneous equations for λ and µ, h · µ (µ − λ)2 = r ,
- 31. 14 INTRODUCTION TO DESIGN MODELS h · λ (µ − λ)2 = c , the unique solution to which is λ = h · c (r − c)2 , µ = h · r (r − c)2 . (1.17) Note that this solution is feasible (that is, λ µ) since c r. To recapitulate, under the assumption that c r, we have identified a unique interior point of the feasible region (0 λ µ) that satisfies the first-order optimality conditions. Surely this must be the optimal solution. After all, we have simply brought together the two models and analyses of the previous sections, in which µ and λ, respectively, were decision variables and in the course of which we verified that our objective function, B(λ, µ), is both concave in λ and concave in µ. What we have not verified, however, is joint concavity in (λ, µ). Without joint concavity, we cannot be sure that a solution to the first-order optimality conditions is a local (let alone a global) maximum. In fact B(λ, µ) is not jointly concave in (λ, µ), because L(λ, µ) = λ/(µ − λ) is not jointly convex . To check for joint convexity, we must evaluate ∆ := ∂2 L ∂λ2 ∂2 L ∂µ2 − ∂2 L ∂λ∂µ 2 and check whether ∆ is nonnegative. Since ∂2 L ∂λ2 = 2µ (µ − λ)3 , ∂2 L ∂µ2 = 2λ (µ − λ)3 , ∂2 L ∂λµ = −(λ + µ) (µ − λ)3 , we have ∆ = 2µ (µ − λ)3 2λ (µ − λ)3 − −(λ + µ) (µ − λ)3 2 = 1 (µ − λ)6 4λµ − (λ2 + 2λµ + µ2 ) = 1 (µ − λ)6 −(λ2 − 2λµ + µ2 ) = 1 (µ − λ)6 −(µ − λ)2 = −1 (µ − λ)4 0 Thus L(λ, µ) is not jointly convex and therefore B(λ, µ) is not jointly concave in (λ, µ).
- 32. OPTIMAL ARRIVAL RATE AND SERVICE RATE 15 It follows that the stationary point (1.17) identified by the first-order con- ditions does not necessarily yield the global maximum net benefit. To gain further insight, let us evaluate B(λ, µ) at this stationary point. Substituting the expressions from (1.17) into the formula for B(λ, µ) and simplifying, we obtain (after simplifying) B(λ, µ) = − h · c r − c 0 = B(0, 0) . So the proposed solution in fact yields a negative net benefit! It is therefore dominated by the point (0, 0) (do nothing) and we know that we can do even better than that when c r. To see how much better, let us examine the problem from a slightly different perspective. Define the traffic intensity ρ (as usual) by ρ := λ/µ and rewrite the net benefit as a function of λ and ρ: B̃(λ, ρ) := r · λ − h · ρ 1 − ρ − c · λ ρ . Now fix a value of ρ such that c r ρ 1 . Then we have B̃(λ, ρ) = λ · (r − c ρ ) − h · ρ 1 − ρ . The second term is constant and the first term is positive and can be made arbitrarily large by choosing λ sufficiently large. Thus B(λ, ρ) → ∞ as λ → ∞ and hence there is no finite optimal solution to the problem. Rather, one can obtain arbitrarily large net benefit by judiciously selecting large values of both λ and µ. Of course these observations raise serious questions about the realism of our model. We shall address these questions later (in Chapter 5). In the meantime, we need to understand what went wrong with our approach based on finding a solution to the first-order optimality conditions. As we saw, the net-benefit function in this model fails to be jointly concave because it contains a congestion-cost term that is proportional to L(λ, µ), the expected steady-state number of customers in the system, which fails to be jointly convex. This congestion-cost term can be written as h · L(λ, µ) = λ(h · W(λ, µ)) , where W(λ, µ) is the expected steady-state waiting of a customer in the sys- tem. In other words, we have a congestion cost per unit time that takes the form (no. customers arriving per unit time) × (congestion cost per customer) . While the congestion cost per customer (in this case, h/(µ − λ)) is jointly convex, the result of multiplying by λ is to destroy this joint convexity.
- 33. 16 INTRODUCTION TO DESIGN MODELS As we shall see in later chapters, this type of congestion cost and its as- sociated non-joint-convexity are not an anomaly but in fact are typical in queueing optimization models. As a result one must be very careful when ap- plying classical economic analysis based on first-order optimality equations. It is not enough to simply assume that the values of the parameters are such that there exists a finite optimal solution in the interior of the feasible region, which then must satisfy the first-order conditions (because they are necessary for an interior maximum). We have seen in the present example that there may be no such interior optimal solution, no matter what the parameter val- ues are. Moreover, there may be an easily identified solution to the first-order conditions which one is tempted to identify as optimal but which may in fact be far from optimal. The literature contains a surprising number of examples in which these kinds of mistakes have been made. 1.4 Optimal Arrival Rates for a Two-Class System Now suppose we have an M/M/1 queue in which there are two classes of customers. The service rate µ is fixed but the arrival rates of the two classes (denoted λ1 and λ2) are decision variables. Customers are served in order of arrival, regardless of class, so that the expected steady-state waiting time in the system is the same for both classes and is a function, W(λ), of the total arrival rate, λ := λ1 + λ2. Recall that in the M/M/1 case W(λ) is given by W(λ) = 1 µ − λ , λ µ ; W(λ) = ∞ , λ ≥ µ . (1.18) We shall assume a reward/cost model like that of Section 1.2, but with class-dependent rewards and waiting cost rates. Specifically, there is a reward ri per entering customer of class i, and a waiting cost hi per unit time per customer of class i in the system. The objective is to maximize the steady-state expected net benefit per unit time: max {λ,λ1,λ2} B(λ1, λ2) = r1λ1 + r2λ2 − (λ1h1 + λ2h2)W(λ) s.t. λ1 + λ2 = λ λ1 ≥ 0 , λ2 ≥ 0 As in the single-class model considered in Section 1.2, if all rewards and costs accrue to the customers, a solution (λs 1, λs 2) to this optimization problem will be socially optimal, in the sense of maximizing the aggregate net benefit ac- cruing to the collective of all customers. Moreover, if potential customers of class i arrive according to a Poisson process with mean rate Λi ≥ µ, then a socially optimal allocation can be implemented by admitting each class-i arrival with probability as i = λs i /Λi. The following Karush-Kuhn-Tucker (KKT) first-order conditions are nec- essary for (λ1, λ2, λ) to be optimal for this problem (see, e.g., Bazaraa et
- 34. OPTIMAL ARRIVAL RATES FOR A TWO-CLASS SYSTEM 17 al. [16]): ri = hiW(λ) + δ and λi 0 (1.19) or ri ≤ hiW(λ) + δ and λi = 0 (1.20) for i = 1, 2, and λ = λ1 + λ2 , (1.21) δ = (λ1h1 + λ2h2)W0 (λ) . (1.22) Now consider this system from the perspective of individual optimization. Suppose a fixed, arbitrary toll, δ, is charged to each entering customer. Each customer of class i takes W(λ) as given and chooses the probability ai of joining to maximize ai · (ri − hiW(λ) − δ) + (1 − ai) · 0 , ai ∈ [0, 1] . In other words, a class-i customer who joins receives the net benefit, ri − hiW(λ), minus the toll, δ, paid for the use of the facility. A customer who balks receives (pays) nothing. Then it is easy to see that arrival rates, λi = ai · Λi, that satisfy equations (1.19) and (1.20) will be individually optimal for the customers of both classes. Moreover, for the given toll δ, a solution to (1.19), (1.20), and (1.21) is a Nash equilibrium. As expected, equation (1.22) reveals that the socially optimal toll is just the external effect, defined (as usual) as the marginal increase in the total delay cost incurred as a result of a marginal increase in the flow, λ. By charging this socially optimal toll, the system operator can induce individually optimizing customers to behave in a socially optimal way, thereby making the Nash- equilibrium allocation coincide with the socially optimal allocation (λs 1, λs 2, λs ) (cf. Section 1.2). 1.4.1 Solutions to the Optimality Conditions: the M/M/1 Case Let us now examine the properties of the solution(s) to the KKT conditions, using the explicit expression (1.18) for W(λ) for an M/M/1 system. The prob- lem of finding a socially optimal allocation of flows takes the form max {λ1,λ2} r1λ1 − h1λ1 µ − λ1 − λ2 + r2λ2 − h2λ2 µ − λ1 − λ2 s.t. λ1 + λ2 µ λ1 ≥ 0 , λ2 ≥ 0 Without loss of generality, we may assume that µ = 1. (Equivalently, measure flows in units of fraction of the service rate µ.) Let a := r1/h1, b := r2/h2, c := h1/h2. Then an equivalent form for the above problem is max {λ1,λ2} c aλ1 − λ1 1 − λ1 − λ2 + bλ2 − λ2 1 − λ1 − λ2 (1.23) s.t. λ1 + λ2 1
- 35. 18 INTRODUCTION TO DESIGN MODELS λ1 ≥ 0 , λ2 ≥ 0 For an interior optimal solution, equation (1.19) must be satisfied for i = 1, 2. The unique solution to these equations is given by λ̃1 = b(c − 1) (ca − b)2 − 1 c − 1 λ̃2 = c c − 1 − ca(c − 1) (ca − b)2 It can be shown that this pair (λ̃1, λ̃2) is an interior point (λ̃1 0, λ̃2 0, λ̃1 + λ̃2 1) if the parameters satisfy the following conditions: b a 1 ; c b − 1 a − 1 ; a (ca − b)2 (c − 1)2 b . So, for an M/M/1 system in which the parameters satisfy these condi- tions, we have established that the first-order optimality conditions have a unique interior-point solution. This result tempts us to conclude that this so- lution is indeed optimal. But the model of Section 1.3, in which the unique interior-point solution to the optimality conditions turned out to be nonop- timal, should serve as a warning to proceed more cautiously. The question remains whether there are other, non-interior-point solutions to the KKT conditions and whether one of these could yield a higher value of the objec- tive function. Put another way: are the KKT conditions sufficient as well as necessary for an optimal solution to our problem? 1.4.2 Are the KKT Conditions Sufficient? To answer this question, let us return to the problem in its original form. The objective function takes the following form (after substituting for λ from the equality constraint), B(λ1, λ2) = r1λ1 + r2λ2 − f(λ1, λ2) , where f(λ1, λ2) := (λ1h1 + λ2h2)W(λ1 + λ2). That is, f(λ1, λ2) is the total delay cost per unit time expressed as a function of λ1 and λ2. The KKT conditions will be sufficient for social optimality if B(λ1, λ2) is jointly concave in (λ1, λ2), which is true if and only if f(λ1, λ2) is jointly convex in (λ1, λ2). It is easily verified that f(λ1, λ2) is convex in λ1 and convex in λ2. To check for joint convexity, we evaluate ∆ := ∂2 f ∂λ2 1 ∂2 f ∂λ2 2 − ∂2 f ∂λ1∂λ2 2
- 36. OPTIMAL ARRIVAL RATES FOR A TWO-CLASS SYSTEM 19 and find that ∆ = −((h1 −h2)W0 (λ1 +λ2))2 , which is strictly negative unless h1 = h2, that is, unless the customer classes are homogeneous with respect to their sensitivity to delay. Thus f(λ1, λ2) is not in general a jointly convex function of λ1 and λ2. Indeed, the conditions for joint convexity fail at every point in the feasible region if the customer classes are heterogeneous, that is, if h1 6= h2. It follows that B(λ1, λ2) fails to be jointly concave unless h1 = h2. Remark 1 Note that we did not use the specific functional form (1.18) of W(λ) in our demonstration of the nonconvexity of f(λ1, λ2). The only properties that we used were that the delay W(λ) for each customer is an increasing, convex, and differentiable function of the sum of the flows, and that the delay cost per unit time for each class i is the product of the flow, λi, and the delay cost per customer, hiW(λ). All these properties are weak and hold for many queueing models, not just for the M/M/1 case. As we shall see in Chapters 4 and 5, nonconvexity is a widely encountered phenomenon in models for the design of queues with more than one decision variable. The nonconcavity of the objective function, B(λ1, λ2), leads one to suspect that the first-order KKT conditions, (1.19)–(1.22), may not be sufficient for an optimal allocation. In particular, an interior-point solution to these conditions – such as the one found in the previous subsection – might not be optimal. Let us now examine that question. First observe that such a solution must lie on the line λ1 + λ2 = λ, where λ satisfies r1 − h1W(λ) = r2 − h2W(λ) . (1.24) Along this line both the total flow λ and the net benefit, B(λ1, λ2), are con- stant: B(λ1, λ2) = B, say. In particular, the two extreme points on this line, namely, (λ, 0), and (0, λ), share this net benefit; that is, B(λ, 0) = B(0, λ) = B . But B(λ, 0) ≤ B(λ∗ 1, 0) , B(0, λ) ≤ B(0, λ∗ 2) , where λ∗ i is the optimal flow allocation to class i when only that class receives positive flow (i = 1, 2). Thus we see that any interior solution to the first-order KKT conditions is dominated by both the optimal single-class allocations. In other words, the system achieves at least as great a net benefit by allocating all flow to a single class, regardless of which class, than by using an interior allocation satisfying the first-order conditions! Our next observation has to do with external effects, congestion tolls, and equilibrium properties. First note that charging each user a toll δ (per unit of flow) equal to the external effect, that is, δ = (λ1h1 + λ2h2)W0 (λ1 + λ2) ,
- 37. 20 INTRODUCTION TO DESIGN MODELS makes (λ̃1, λ̃2) a Nash equilibrium for individually optimizing customers: no customer of either class has an incentive to deviate from this allocation, as- suming that all other customers make no change. Thus, we see that, even by charging the “correct” toll (namely, a toll equal to the external effect), we can- not be certain that the customers will be directed to a socially optimal flow allocation. Rather, the resulting allocation, even though it is a Nash equilib- rium, may be dominated by both of the optimal single-class allocations. Thus we have a dramatic example of the pitfalls of marginal-cost pricing (that is, pricing based on first-order optimality conditions) when the customer classes are heterogeneous in their sensitivities to congestion. As an example, let us return to the M/M/1 example of Section 1.4.1. Let a = 4, b = 9, and c = 4. In this case, the solution to the first-order conditions is λ̃1 = 0.218 ; λ̃2 = 0.354 . The optimal single-user flow allocations are λs 1 = 0.500 and λs 2 = 0.667. The objective function values of these three flow allocations are: B(λ̃1, λ̃2) = 3.81 B(λs 1, 0) = 4.00 B(0, λs 2) = 4.00 Thus we have an illustration of the general result derived above: the interior- point equilibrium flow allocation is dominated by both optimal single-user allocations. For this example, Figure 1.4 and Figure 1.5 show, respectively, a contour plot and graph of the response surface of the net benefit function, B(λ1, λ2). Figure 1.4 Net Benefit: Contour Plot
- 38. OPTIMAL ARRIVAL RATES FOR PARALLEL QUEUES 21 Figure 1.5 Net Benefit: Response Surface 1.5 Optimal Arrival Rates for Parallel Queues Now let us consider n independent M/M/1 queues, with service rates µj and arrival rates λj, j = 1, . . . , n. Suppose that the µj are fixed and that the λj are design variables. Our objective is to minimize the steady-state expected number of customers in the system, subject to a constraint that the total arrival rate should equal a fixed value, λ. Thus the problem takes the form min n X j=1 λj µj − λj s.t. n X j=1 λj = λ (1.25) 0 ≤ λj µj , j = 1, . . . , n . We can interpret this problem as follows. Suppose customers arrive to the system according to a Poisson process with mean arrival rate λ. We must decide how to split this arrival process among n parallel exponential servers, each with its own queue. The splitting is to be done probabilistically, inde- pendently of the state and past history of the system. That is, each arriving customer is sent to queue j with probability aj = λj/λ, so that the arrival process to queue j is Poisson with mean arrival rate λj. We shall use a Lagrange multiplier to eliminate the constraint on the total arrival rate. The Lagrangean problem is: min n X j=1 λj µj − λj − α n X j=1 λj (1.26) s.t. 0 ≤ λj µj , j = 1, . . . , n .
- 39. 22 INTRODUCTION TO DESIGN MODELS The solution is parameterized by α, which can be interpreted as the imputed reward per unit time per unit of arrival rate. Problem (1.26) is separable, so we can minimize the objective function separately for each facility. For facility j, the problem takes the form of the single-facility arrival-rate-optimization problem of Section 1.2, with r = α, h = 1. The solution is: λj = λs j(α) := (µj − q µj/α)+ , j = 1, . . . , n . (1.27) This solution will be optimal for the original problem if α is chosen so that Pn j=1 λs j(α) = λ. Thus an optimal allocation satisfies the following conditions (j = 1, . . . , n): L0 j(λj) = µj (µj − λj)2 = α , if λj 0 , (1.28) L0 j(λj) = 1 µj ≥ α , if λj = 0 , (1.29) for some α such that Pn j=1 λj = λ. These results can be used to solve the original problem (1.25) graphically. First, plot each λs j(α) as a function of α, as shown in Figure 1.6. Define λs (α) := n X j=1 λs j(α) , so that λs (α) is the total arrival rate in an optimal solution of problem (1.26) corresponding to Lagrange multiplier α. We can now find the optimal solution to the original problem for a particular value of λ by drawing a horizontal line from the vertical axis at level λ and finding its intersection with the graph of λs (α), then drawing a vertical line to the α axis. Where this line intersects the graph of λs j(α), we obtain λs j = λs j(λ), the optimal value of λj for the original problem with total arrival rate λ. We can derive an explicit solution for the λs j in terms of the parameter λ (denoted λs j(λ), j = 1, . . . , n) in the following way. First, order the µj so that µ1 ≥ µ2 ≥ · · · ≥ µn. From (1.27) it can be seen that λs (α) is a continuous, strictly increasing function of α, for α ≥ µ−1 1 . In this range, therefore, λs (α) has an inverse, which we denote by α(λ). We solve for α(λ) separately over the intervals induced by µ−1 1 ≤ α ≤ µ−1 2 , µ−1 2 ≤ α ≤ µ−1 3 ,. . . . In particular, for µ−1 1 ≤ α ≤ µ−1 2 , λs 1(α) = µ1 − p µ1/α , λs j(α) = 0 , j = 2, . . . , n . Thus λs 1(α) = λ in this range, so that r 1 α = µ1 − λ √ µ1 , (1.30)
- 40. OPTIMAL ARRIVAL RATES FOR PARALLEL QUEUES 23 Figure 1.6 Arrival Control to Parallel Queues: Parametric Socially Optimal Solution and hence λs 1(λ) = µ1 − √ µ1 √ µ1 (µ1 − λ) = λ . But it follows from (1.30) that µ−1 1 ≤ α ≤ µ−1 2 if and only if 0 ≤ λ ≤ µ1 − √ µ1µ2. Summarizing, for r1 := 0 ≤ λ ≤ r2 := µ1 − √ µ1µ2, we have λs 1(λ) = λ , λs j(λ) = 0 , j = 2, . . . , n . Continuing this argument, we can deduce the general form of the solution for λs j(λ), j = 1, . . . , n. In general, define rk := Pk i=1(µi − √ µiµk), k = 1, . . . , n, rn+1 := Pn i=1 µi. Then, for k = 1, . . . , n, if rk ≤ λ ≤ rk+1, λs j(λ) = µj − √ µj Pk i=1 √ µi ! k X i=1 µi − λ ! , j = 1, . . . , k , = 0 , j = k + 1, . . . , n . Note that each λs j is piecewise linear in λ. Figure 1.7 gives a typical illus- tration. Note that, once λs j(λ) is positive, its rate of increase is nonincreasing in λ (thus λs j(λ) is concave in λ ≥ rj) and that the rates of increase of the λs j(λ) for fixed λ are nondecreasing in j. Individually Optimal Allocation The allocation described above assumes that the allocation of total “de- mand,” λ, to the various facilities is made in accordance with the system-wide
- 41. 24 INTRODUCTION TO DESIGN MODELS Figure 1.7 Arrival Control to Parallel Queues: Explicit Socially Optimal Solution objective of minimizing the total rate of waiting per unit time: Pn j=1 Lj(λj) = Pn j=1 λj/(µj − λj). An equivalent way of viewing this problem is to visual- ize each arriving customer having a probability, aj = λj/λ, of joining facility j, j = 1, . . . , n, where the a0 js are to be chosen (by an omnipotent system designer) to minimize the steady-state expected waiting time of an arbitrary customer: n X j=1 λj λ 1 µj − λj = 1 λ n X j=1 Lj(λj) Now let us consider an allocation (λ1, . . . , λn) (equivalently, a set of joining probabilities (a1, ..., an)) from the point of view of an individual customer who wishes to minimize his expected waiting time. Under the allocation in question, an arriving customer chooses facility j with probability aj = λj/λ; conditional on joining facility j, the expected waiting time is (µj − λj)−1 . (As is always the case in design models, we assume that the fixed mean service rates µj and the arrival rates λj associated with the given allocation are known and the system is in steady state, but the exact number of customers at each facility cannot be observed.) The customer’s unconditional expected waiting time is therefore Pn j=1 aj(µj − λj)−1 . As usual we call an allocation (λ1, . . . , λn) (or a set of joining probabilities (a1, . . . , an)) individually optimal if no customer, acting in its own interest, has an incentive to deviate unilaterally from the allocation. This will be the case if and only if (µj − λj)−1 = (µk − λk)−1 for all j, k such that λj 0 and λk 0, and (µj − λj)−1 ≤ µ−1 k , if λj 0 and λk = 0. Otherwise, e.g., if (µj − λj)−1 (µk − λk)−1 for some j, k such that λj 0, an arriving customer could strictly reduce its expected waiting time
- 42. OPTIMAL ARRIVAL RATES FOR PARALLEL QUEUES 25 Figure 1.8 Arrival Control to Parallel Queues: Parametric Individually Optimal So- lution by joining facility j with probability a0 j := 0 and facility k with probability a0 k := aj + ak, rather than aj = λj/λ and ak = λk/λ, respectively. In other words, an individually optimal allocation satisfies the following conditions, for j = 1, . . . , n: Wj(λj) = 1 µj − λj = α , if λj 0 ; (1.31) Wj(λj) = 1 µj ≥ α , if λj = 0 ; (1.32) for some α 0 such that Pn j=1 λj = λ. We would like to compare such an allocation, denoted λe j(α), or λe j(λ), to the socially optimal allocation, λs j(α), or λs j(λ). First observe from (1.31) and (1.28) that an individually optimal allocation equates average costs, 1/(µj−λj) (internal effects), whereas a socially optimal allocation equates marginal costs, µj/(µj − λj)2 = 1/(µj − λj) + λj/(µj − λj)2 (internal plus external effects), at all open facilities j. In terms of α, the individually optimal allocation can be written as λe j(α) = (µj − 1/α)+ , j = 1, . . . , n . Figure 1.8 illustrates the behavior of λe j(α), assuming µ1 ≥ µ2 ≥ · · · ≥ µn. Now α must be chosen so that Pn j=1 λe j(α) = λ, in order to find λe j(λ) , j = 1, . . . , n. This can be done in the same way as for socially optimal allocations. (The details are left to the reader.) In general, define sk := Pk i=1(µi − µk) , k = 1, . . . , n , sn+1 := Pn i=1 µi. Then the individually optimal allocation is as
- 43. 26 INTRODUCTION TO DESIGN MODELS Figure 1.9 Arrival Control to Parallel Queues: Explicit Individually Optimal Solu- tion follows: for k = 1, . . . , n, if sk ≤ λ ≤ sk+1, then λe j(λ) = µj − [ k X i=1 µi − λ]/k , j = 1, . . . , k , = 0 , j = k + 1, . . . , n . Figure 1.9 illustrates the behavior of the individually optimal facility arrival rates as a function of the total arrival rate. Note that the positive λe j(λ) are piecewise linear in λ, with nonincreasing slope. The slopes of all positive λe j(λ) are equal in this case. In Figure 1.10, the individually optimal allocation is superimposed on the socially optimal allocation, for purposes of comparison. As a general observa- tion, we can say that the individually optimal allocation assigns more (fewer) customers to faster (slower) servers than the socially optimal allocation. More specifically, for the example in Figure 1.10, the individually optimal allocation always assigns more arrivals to facility 1, the fastest one, and fewer arrivals to facility 3, the slowest one, than the socially optimal allocation does. As λ increases, facility 2 first receives fewer, then more, arrivals in the individually optimal than in the socially optimal allocation. Thus, facility 2 plays the role of a “slower” server in light traffic and a “faster” server in heavy traffic. 1.6 Endnotes Over the past forty years, there have been a number of survey papers and books that discuss optimal control of queues, including Sobel [181], Stid-
- 44. ENDNOTES 27 Figure 1.10 Arrival Control to Parallel Queues: Comparison of Socially and Indi- vidually Optimal Solutions ham and Prabhu [191], Crabill, Gross, and Magazine [46], Serfozo [174], Stid- ham [184], [185], [186], Kitaev and Rykov [111], and Hassin and Haviv [86]. Optimal design is touched on in some of these references but, to the best of my knowledge, the present book is the first to provide a comprehensive treatment of optimal design of queues. Section 1.1 The model and results in this section were introduced in a pioneering paper by Hillier [93]. Indeed, the emergence of optimization of queueing systems (both design and control) as a legitimate subject for research owes a great deal to Hillier and his PhD students in operations research at Stanford University, beginning in the mid 1960s. Section 1.2 Edelson and Hildebrand [59] introduced the basic model of this section. They compared the socially optimal toll with the facility optimal toll, that is, the toll that maximizes revenue to the toll collector (e.g., the facility oper- ator). They showed that the two are equal when all customers received the same reward, r, from joining and receiving service. When customers are het- erogeneous – that is, the reward is a random variable, R – the facility optimal toll (arrival rate) is in general larger (smaller) than the socially optimal toll (arrival rate) (see Chapter 2). Section 1.3 Surprisingly, I could find no published reference in which exactly this model
- 45. 28 INTRODUCTION TO DESIGN MODELS is considered. The material in this section is taken largely from my class notes for a course on Optimization of Queueing Systems which I have taught, in various versions, since the early 1970s. Dewan and Mendelson [54] consid- ered a model for combined choice of the arrival and service rate, but with heterogeneous rewards only. They examined only the solution to the neces- sary first-order optimality conditions, without considering the possibility that these conditions might not be sufficient. (In the examples they presented, the conditions were, fortuitously, always sufficient.) Stidham [187] considered es- sentially the same model as Dewan and Mendelson [54] and pointed out the possible failure of the objective function to be jointly concave and the resulting insufficiency of the first-order conditions. Section 1.4 The model and results of this section come primarily from Stidham [189], which considered a more general model for a multiclass network of queues. Chapter 4 expands on the material in this section and Chapter 8 considers the extension to networks. Section 1.5 This model was introduced in an unpublished paper (Stidham [182]) and then elaborated and extended in Bell and Stidham [18]. We return to the topic of parallel queues in Chapter 6.
- 46. CHAPTER 2 Optimal Arrival Rates in a Single-Class Queue The model we study in this chapter is a generalization of the model introduced in Section 1.2 of the introductory chapter. We observed there that many of the salient features of the optimal arrival-rate model with deterministic reward and linear waiting cost do not depend on the system being an M/M/1 queue operating in steady state. For example, the individually optimal arrival rate λe is an upper bound on the socially optimal arrival rate λs for any queueing system satisfying the following conditions: 1. W(λ) is strictly increasing in 0 ≤ λ µ ; 2. W(λ) ↑ ∞ as λ ↑ µ ; 3. W(0) = 1/µ . Moreover, an individually optimizing customer who enters the system should be charged a toll equal to the external effect in order to render its behavior optimal for the system as a whole. To what extent do properties like these continue to hold when one relaxes the assumptions that the system is operating in steady state and that all entering customers earn the same reward r and incur a waiting cost at the same constant rate h while in the system? We shall address these questions, and many others, in this chapter. 2.1 A Model with General Utility and Cost Functions We consider a service facility operating over a finite or infinite time interval. At this stage, rather than specify a particular queueing model (we shall later consider specific examples), we prefer to describe the system in general terms, keeping structural and stochastic assumptions at a minimum. The essential ingredients are: • the arrival rate λ – the average number of customers entering the system per unit time during the period (the decision variable); • the average (gross) utility per unit time, U(λ), during the period; • the average waiting cost per customer, G(λ), during the period; • the admission fee or toll, δ, paid by each entering customer. The meaning of the word “average” depends on the specific model context. For example, it may mean a sample-path time average or (in the case of an 29
- 47. 30 OPTIMAL ARRIVAL RATES IN A SINGLE-CLASS QUEUE infinite time period) the expectation of a steady-state random variable. These ingredients are now discussed in more detail. The arrival rate λ measures the average number of customers arriving and joining the system per unit time during the period of interest. As indicated, λ is a decision variable. The set of feasible values for λ is denoted A. Our default assumption will be that A = [0, ∞). Alterations to our model and results to allow for more general feasible sets are usually straightforward and will be left to the reader. To capture the benefit of having a higher throughput, there is a utility function, U(λ), which measures the average gross value received per unit time as a function of the arrival rate λ. For example, in the model considered in Section 1.2 with a deterministic reward, r, per entering customer, U(λ) = r·λ, so that the utility function is linear. Our default assumption is that U(λ) is nondecreasing, differentiable, and concave in λ ≥ 0. We allow U0 (0) = ∞. In Section 2.3 we show how a value function of this form can arise when there is a renewal process of potential arriving customers (with rate Λ ∞) and a probabilistic joining rule is followed. We can accommodate a finite upper bound, Λ, on λ by defining U(λ) = U(Λ) for λ Λ. But this definition may not be compatible with the differentiability assumption, unless U0 (Λ) = 0. Later (in Section 2.2) we shall examine the effects of relaxing some of the regularity conditions satisfied by U(λ), including differentiability. Balanced against the benefit of throughput is the cost to customers caused by the time they spend in the system. For a given λ, G(λ) denotes the average waiting cost of a job, averaged over all customers who arrive during the period in question. Our default assumption is that G(λ) takes values in [0, ∞] and is strictly increasing and differentiable in λ ≥ 0. We allow G(λ) to equal ∞ in order to accommodate, for example, a single-server system with service rate µ, in which we typically have G(λ) = ∞ for λ ≥ µ. This convention makes it unnecessary to include the constraint λ µ explicitly in our formulation. For example, in the case of an infinite time period, it might be that G(λ) = E[h(W (λ))] , (2.1) where h(t) is the waiting cost incurred by a job that spends a length of time t in the system and, for each λ ≥ 0, W (λ) is the steady-state random waiting time in the system for the queueing system induced by λ. In the case of a linear waiting cost, h(t) = h · t, G(λ) = h · W(λ) , where W(λ) := E[W (λ)]. For the example of an M/M/1 queue operating in steady state considered in Section 1.2 of Chapter 1, we have G(λ) = h µ − λ . Let H(λ) = λG(λ). Then H(λ) is a measure of the average waiting cost incurred by the system per unit time, inasmuch as it equals the product of the average number of customers arriving per unit time and the average waiting
- 48. A MODEL WITH GENERAL UTILITY AND COST FUNCTIONS 31 cost per customer.∗ We shall assume that H(λ) is a convex function of λ ≥ 0. (Note that the assumption that G(λ) is strictly increasing and differentiable implies that H(λ) is also strictly increasing and differentiable.) All these properties are weak and common in the queueing literature. A sufficient condition for H(λ) to be convex is that G(λ) is convex, which is simply an assumption that each customer’s marginal cost of waiting does not decrease as the arrival rate increases. As an illustration this property holds in our canonical example: an M/M/1 queue with linear waiting costs and FIFO queue discipline. Remark 1 Customers might be sensitive to losses rather than (or in addition to) delays. (This situation can arise in a system with a finite buffer, in which an arriving customer who finds the buffer full is lost.) In this case, G(λ) might measure the cost incurred if a customer is lost because of buffer overflow. In the special case in which G(λ) = h · P(λ), P(λ) might measure the steady- state probability that a customer is lost (or the fraction of customers lost) and h the sensitivity of customers to such a loss. Although we shall continue to refer to “delay sensitivity” or “waiting costs” throughout the discussion of this model, the reader should keep in mind that the results also apply to other measures of congestion, such as losses. In addition to incurring the waiting cost G(λ), an entering customer may have to pay an admission fee (or toll) δ. In the present model, the sum of the toll and the waiting cost constitutes the full price of admission, which we denote in general by π, or π(λ), when we want to emphasize its dependence on λ for a fixed δ. Thus we have π(λ) = δ + G(λ) . Remark 2 The concept of the full price of admission is common to many models for arrival-rate selection, as we shall see. In more complicated systems, such as a set of parallel facilities (Chapter 6) or a network of queues (Chap- ters 7 and 8), the derivation of the full price is more complicated, as it may involve choices among alternate facilities or routes. But the analysis of the arrival-rate selection problem is basically the same as in the single-facility, single-class model considered in this chapter. Consequently we shall develop much of the theory for the present model in a general framework that will allow our results to be carried over to the more complicated models in subse- quent chapters without unnecessary repetition. When we are operating in this general framework, we shall simply assume that π = π(λ) is a given strictly increasing and differentiable function of λ. As noted, the arrival rate λ is a decision variable. The solution to the ∗ In queueing terms, the relation – waiting cost per unit time = (arrival rate) × (waiting cost per customer) – is a special case of H = λG, the generalization of L = λW, which holds under weak assumptions. (See El-Taha and Stidham [60], Chapter 6. In the case of linear waiting cost, it just follows from L = λW itself.)
- 49. 32 OPTIMAL ARRIVAL RATES IN A SINGLE-CLASS QUEUE decision problem depends on who is making the decision. The decision may be made by the individual customers, each concerned only with its own net utility (individual optimality), or by a system operator, who might be interested in maximizing the aggregate net utility to all customers (social optimality) or in maximizing profit (facility optimality). 2.1.1 Individually Optimal (Equilibrium) Arrival Rate We first consider the decision problem from the point of view of an arriving customer concerned only with its own net utility, which it wishes to maxi- mize (individual optimality). Suppose we are given the full price of admission, π(λ), as a function of λ ≥ 0. We assume that π(·) is strictly increasing and differentiable. For a particular value π of the full price of admission, an arriving customer concerned only with maximizing its own net utility will join if the value it receives from joining exceeds π, balk if it is lower, and be indifferent between joining and balking if it equals π. The marginal utility, U0 (λ), may be inter- preted as the value received by the marginal user when the arrival rate is λ. (See below for more discussion and motivation of this interpretation.) At an individually optimal arrival rate, the marginal user will be indifferent between joining and balking, so that U0 (λ) = π , (2.2) if this equation has a solution in A = [0, ∞). If U0 (0) π, then there is no solution to (2.2) in A; in this case no user has any incentive to join and we set λ = 0. If U0 (0) ≥ π, then since U0 (λ) is continuous and nonincreasing in λ, there is a solution to (2.2) in A. (We assume that limλ→∞ U0 (λ) π, in order to avoid trivialities.) Thus for a fixed price π an individually optimal arrival rate is characterized by the following equations: U0 (λ) ≤ π , and λ ≥ 0 (2.3) U0 (λ) = π , if λ 0 (2.4) Now if π = π(λ) for a value of λ satisfying these conditions, then the system is in equilibrium: no individually optimizing customer acting unilaterally will have any incentive to deviate from its current action. In this case we have U0 (λ) ≤ π(λ) , and λ ≥ 0 (2.5) U0 (λ) = π(λ) , if λ 0 (2.6) These are the equilibrium conditions which uniquely define the individually optimal arrival rate, which we shall denote by λe . (To avoid trivialities we shall assume that limλ→∞ U0 (λ) limλ→∞ π(λ). The equilibrium conditions then have a unique solution since π(λ) is strictly increasing and continuous.) Equivalent equilibrium conditions are the following: π(λ) − U0 (λ) ≥ 0 λ ≥ 0
- 50. A MODEL WITH GENERAL UTILITY AND COST FUNCTIONS 33 Figure 2.1 Characterization of Equilibrium Arrival Rate λ(π(λ) − U0 (λ)) = 0 Note that the equality constraint takes the form of a complementary-slackness condition. This form of the equilibrium conditions will facilitate comparison of individual optimization with social optimization. For the present case, in which π(λ) = δ + G(λ), the equilibrium conditions (2.5) and (2.6) take the form U0 (λ) ≤ δ + G(λ) , and λ ≥ 0 (2.7) U0 (λ) = δ + G(λ) , if λ 0 (2.8) Figure 2.1 illustrates the case of a solution to the equilibrium condition (2.8). Equivalent equilibrium conditions are the following: δ + G(λ) − U0 (λ) ≥ 0 λ ≥ 0 λ(δ + G(λ) − U0 (λ)) = 0 Note that if U0 (0) = ∞, then the equilibrium conditions reduce to the single equation, U0 (λ) = δ + G(λ) . In this case the individually optimal arrival rate, λe , is the unique solution to this equation and λe 0. We have characterized a positive individually optimal arrival rate by equat- ing the marginal utility to the full price of entering the system. Here is an
- 51. 34 OPTIMAL ARRIVAL RATES IN A SINGLE-CLASS QUEUE informal motivation for this definition. (A formal justication in terms of the Nash equilibrium is contained in Section 2.3 on probabilistic joining rules.) Suppose the current arrival rate is λ and one must decide whether to increase it to λ + ∆. Think of this decision from the perspective of the increment in flow, ∆. An additional value per unit time, U(λ + ∆) − U(λ), will result from this increment in flow. On the other hand, the increment in flow will pay a price per unit time approximately equal to ∆·π. It will therefore be profitable for this increment in flow to add itself to the total flow if and only if U(λ + ∆) − U(λ) ≥ ∆ · π . From this perspective, increments of flow will continue to add themselves to the total flow until an equilibrium, that is, a point of indifference, is reached, at which U(λ + ∆) − U(λ) = ∆ · π . Dividing both sides of this equality by ∆ and letting ∆ → 0 leads to (2.2). Of course, this characterization depends on the assumption that the decision about increasing the flow is based on the benefits and costs to the increment in flow, without taking account of the effect of the increment on the costs incurred by the existing flow, λ. It is this effect – the external effect – that must be considered in order to find a value of the arrival rate that is optimal from the perspective of the total flow, that is, from the perspective of the collective of all customers. We turn our attention to this socially optimal flow in the next subsection. 2.1.2 Socially Optimal Arrival Rate Now consider the decision problem of the system operator, who wishes to select an arrival rate that maximizes the average net benefit earned per unit time by the collective of all customers. We call such an arrival rate socially optimal. The problem is formulated as follows: max {λ≥0} U(λ) := U(λ) − λG(λ) (2.9) (Since the toll is simply a transfer fee, it does not appear in the objective function for social optimality.) The first-order necessary condition for an interior maximum is: U0 (λ) = G(λ) + λG0 (λ) (2.10) Let λs denote the optimal arrival rate for this problem. The assumed concavity of U(λ) and convexity of H(λ) = λG(λ) imply that the maximum net benefit occurs at λs = 0, if U0 (0) G(0). Otherwise, λs is the solution to the first- order necessary condition (2.10) and λs ≥ 0. Now suppose that the facility operator wishes to implement the socially optimal arrival rate by charging a toll and allowing the arriving customers, who are individual optimizers, to decide whether or not to enter the system.
- 52. A MODEL WITH GENERAL UTILITY AND COST FUNCTIONS 35 In this case it follows from (2.7) and (2.10) that δs = λs G0 (λs ) . (2.11) That is, the optimal toll equals the external effect. If the system operator charges entering customers the toll δs , then λe = λs : the individually optimal arrival rate is also socially optimal. Note that, by requiring that G(0) U0 (0), one can guarantee the existence of a (unique) solution to (2.10) in A = [0, ∞), so that the constraint λ ≥ 0 can effectively be ignored. In particular, this is the case if U0 (0) = ∞. 2.1.3 Facility Optimal Arrival Rate Now consider the system from the point of view of a facility operator whose goal is to set a toll δ that will maximize its revenue. We call such a toll (and the associated arrival rate) facility optimal. Assuming that the arriving customers are individual optimizers, choosing a value δ for the toll will result in an arrival rate λ that (uniquely) satisfies the equilibrium conditions, U0 (λ) ≤ δ + G(λ) , and λ ≥ 0 U0 (λ) = δ + G(λ) , if λ 0 , or equivalently, λ(δ + G(λ) − U0 (λ)) = 0 δ + G(λ) − U0 (λ) ≥ 0 λ ≥ 0 Thus, the facility optimization problem may be written in the following form: max {δ,λ} λδ s.t. λ(δ + G(λ) − U0 (λ)) = 0 δ + G(λ) − U0 (λ) ≥ 0 λ ≥ 0 Subtracting the term λ(δ + G(λ) − U0 (λ)) (which equals zero by the first constraint) from the objective function and simplifying leads to the following equivalent formulation: max {δ,λ} λU0 (λ) − λG(λ) s.t. δ ≥ U0 (λ) − G(λ) δ = U0 (λ) − G(λ) , if λ 0 λ ≥ 0 To avoid technical difficulties we shall assume for now that limλ→0 λU0 (λ) = 0. (See below for further discussion of this point.) The first two constraints now serve simply to define δ (nonuniquely), given
- 53. 36 OPTIMAL ARRIVAL RATES IN A SINGLE-CLASS QUEUE λ. Therefore, it suffices to solve the following problem with λ as the only decision variable: max {λ≥0} Ũ(λ) := λU0 (λ) − λG(λ) , and then choose δ to satisfy δ ≥ U0 (λ) − G(λ) δ = U0 (λ) − G(λ) , if λ 0 . Now if G(0) ≥ U0 (0), then G(λ) U0 (λ), for all λ 0. In this case the facility- optimal arrival rate, λf , equals zero (along with the individually optimal ar- rival rate, λe , and the socially optimal arrival rate, λs ). On the other hand, if G(0) U0 (0), then the first-order necessary condition for a positive value of λ to be optimal for this problem is U0 (λ) + λU00 (λ) − G(λ) − λG0 (λ)) = 0 . (2.12) Note that (2.12) may not be sufficient for λ 0 to be optimal, since the objective function may not be concave (or even unimodal), because λU0 (λ) may not be concave. Remark 3 Note that the objective function for facility optimization, Ũ(λ) = λU0 (λ) − λG(λ), takes the same form as the objective function for social optimization, but with a modified utility function, Ũ(λ) := λU0 (λ). This ob- servation suggests that we can directly apply the results from our analysis of social optimization to the facility optimization problem (at least under the technical assumption that limλ→0 λU0 (λ) = 0). A crucial difference, however, is that the modified utility function, Ũ(λ), need not be concave (as we just observed). Indeed, it need not even be nondecreasing (see Section 2.1.3.5). So, to treat the facility optimal problem as a special case of the socially optimal problem, we would first have to extend the formulation and analysis of the latter to allow for utility functions that fail to be concave and nondecreas- ing. Instead of doing this, however, we prefer to deal directly with the facility optimization problem. Recall that we allow U0 (0) = ∞. In this case, λU0 (λ) is undefined at λ = 0. However, the only reasonable value for the objective function, Ũ(λ) = λU0 (λ) − λG(λ), to assume at λ = 0 is zero, since λ = 0 corresponds to a decision on the part of the facility operator not to operate the facility at all. Indeed, it was to ensure the continuity of the objective function at λ = 0 that we made the technical assumption above that limλ→0 λU0 (λ) = 0. Other limits are possible, however, including limλ→0 λU0 (λ) = κ, 0 κ ≤ ∞. (We shall consider this situation in detail when we return to the topic of facility optimality in Section 2.3.3.) Let us briefly consider each of these possibilities in turn. Case 1. Suppose λU0 (λ) → 0 as λ → 0. Then the value of λ that maximizes Ũ(λ) may be λ = 0. This will be the case if and only if Ũ(λ) ≤ 0 = Ũ(0) for all λ 0 (which is true if and only if U0 (0) ≤ G(0)). If this is not the
- 54. A MODEL WITH GENERAL UTILITY AND COST FUNCTIONS 37 case, then the maximum will occur at a positive value of λ which must satisfy the first-order necessary condition, (2.12). As noted above, if λU0 (λ) is not concave, then this equation may have multiple solutions, some of which are local maxima or minima, and only one of which can be the global maximum. Case 2. Suppose λU0 (λ) → κ as λ → 0, where 0 κ ∞. In this case, Ũ(λ) has a discontinuity at λ = 0, since (by convention) Ũ(0) = 0, whereas Ũ(0+) = κ 0. Now the maximum can no longer occur at λ = 0, but it may occur at λ = 0+. More precisely, if Ũ(λ) κ for all λ 0, then supλ0 Ũ(λ) = κ, but this supremum is not attained. Instead, the facility operator can attain a profit arbitrarily close to κ by choosing an arbitrarily small positive arrival rate λ. We shall use the notation, λf = 0+, as a shorthand for this property. On the other hand, if there exists a positive value of λ such that Ũ(λ) ≥ κ, then (as in Case 1) the (positive) maximizing value of λ is a solution to the first-order necessary condition, (2.12). Case 3. Suppose limλ→0 λU0 (λ) = ∞. In this case the facility optimization problem has an unbounded objective function, Ũ(λ), which approaches ∞ as λ approaches zero. In other words a profit-maximizing facility operator can earn an arbitrarily large profit by charging an arbitrarily large toll, resulting in an arbitrarily small arrival rate. So again we have λf = 0+. Note that, by contrast, the individually and socially optimal arrival rates, λe and λs , still exist and are positive and the associated values of the objective function are still finite. In Section 2.3.3 we analyze facility optimization in the con- text of probabilistic joining rules, and there we are able to give a behavioral interpretation of the property, limλ→0 λU0 (λ) = ∞. 2.1.3.1 Comparison of Facility Optimal and Socially Optimal Arrival Rates What is the relationship between λf and the socially optimal arrival rate, λs ? The following theorem shows that λf ≤ λs . In other words, a facility operator concerned only with maximizing the revenue received from tolls will choose a toll that results in fewer customers joining the system than is optimal from the point of view of the total welfare of all customers. This result is a direct consequence of the concavity of U(λ) and is consistent with classical results from welfare economics. The proof of this result depends on the following lemma, which is of indepen- dent interest. (In this section we shall again assume that limλ→∞ λU0 (λ) = 0, unless otherwise noted.) Lemma 2.1 Let d(λ) := U(λ) − λU0 (λ), λ ≥ 0. The function d(λ) is nonde- creasing in λ ≥ 0 if (and only if) U(λ) is concave in λ ≥ 0. Proof Differentiating d(λ) yields d0 (λ) = U0 (λ) − U0 (λ) − λU00 (λ) = −λU00 (λ) . Since λ ≥ 0, we conclude that d0 (λ) ≥ 0 if and only if U00 (λ) ≤ 0.
- 55. Another Random Document on Scribd Without Any Related Topics
- 56. she felt no envy in regard to the boa, and indeed never envied any girl the tenth-rate--no, nor the second-rate! Her desire was for the best or nothing; she could not compromise. The neighbouring shop- windows had effectively educated her because she was capable of self-education. Millicent and Gertie actually preferred the inferior displays of Oxford Street. She gazed in froward insolence at the workroom full of stitching girls on the opposite side of the street. They were toiling as though they had been toiling for hours. Customers had not yet begun to be shown into the elegant apartment on the floor below the workrooms. Customers were probably still sipping tea in bed with a maid to help them, and some of them had certainly never been in a Tube in their lives. Yet the workgirls, seen broadly across the street, were on the average younger, prettier, daintier and more graceful than the customers. Why then...? Etc. The upper floors of all the surrounding streets were studded with such nests of heads bent over needles. There were scores and scores of those crowded rooms, excruciatingly feminine. Modes et Robes--a charming vocation! You were always seeing and touching lovely stuff, laces, feathers and confections of stuffs. A far more attractive occupation than typewriting, Lilian thought. Sometimes she had dreamt of a change, but not seriously. To work on other women's attire, knowing that she could never rise to it herself, would have broken her heart. Quickly she turned away from the window, still uplifted-- passionately determined that one day she would enter the most renowned and exclusive arcana in Hanover Square, and not as an employee either! Then, on that day, would she please with the
- 57. virtuosity of a great pianist playing the piano, then would she exert charm, then would she be angelic and divine; and when she departed there should be a murmur of conversation. She smiled her best in anticipation; her fingers ran smoothingly over her blouse. Gertie Jackson came in and transformed the rehearsed smile into an expression of dissatisfaction and hostility far from divine; the fingers dropped as it were guiltily; and Lilian remembered all her grievances and her tragedy. Gertie Jackson's bright, pleasant, clear, drawn face showed some traces of fatigue, but no sign at all of being a martyr to the industrial system or to the despotism of individual employers. She was a tall, well-made girl of twenty-eight, and she held herself rather nicely. She was kindly, cheerful and of an agreeable temper--as placid as a bowl of milk. She loved her work, regarding it as of real importance, and she seemed to be entirely without ambition. Apparently she would be quite happy to go on altruistically typing for ever and ever, and to be cast into a typist's grave. Lilian's attitude towards her senior colleague was in various respects critical. In the first place, the poor thing did not realize that she was growing old--already approaching the precipice of thirty! In the second place, though possessed of a good figure and face, she did nothing with these great gifts. She had no desire to be agreeable; she was agreeable unconsciously, as a bird sings; there was no merit in it. She had no coquetry, and not the slightest inclination for chic. Her clothes were good, and bought in Upper Street, Islington; her excellent boots gave her away. She was not uninterested in men; but she did not talk about them, she twittered about them. To Lilian she had the soul of an infant. And she was too
- 58. pure, too ingenuous, too kind, too conscientious; her nature lacked something fundamental, and Lilian felt but could not describe what it was--save by saying that she had no kick in either her body or her soul. In the third place, there was that terrible absence of ambition. Lilian could not understand contentment, and Gertie's contentment exasperated her. She admitted that Gertie was faultless, and yet she tremendously despised the paragon, occasionally going so far as to think of her as a cat. And now Gertie straightened herself, stuck her chest out bravely, according to habit, and smiled a most friendly greeting. Behind the smile lay concealed no resentment against Lilian for having failed to appear on the previous evening, and no moral superiority as a first-class devotee of duty. What lay behind it, and not wholly concealed, was a grave sense of responsibility for the welfare of the business in circumstances difficult and complex. Have you seen Miss Grig? she asked solemnly. Yes, said Lilian, with a touch of careless defiance; she supposed Gertie to be delicately announcing that Miss G. had been lying in wait for her, Lilian. Doesn't she look simply frightfully ill? She does, admitted Lilian, who in her egotism had quite forgotten her first impression that morning of Miss G.'s face. What is it? Gertie mentioned the dreadful name of one of those hidden though not shameful maladies which afflict only women--but the majority of women. The crude words sounded oddly on Gertie's prim lips. Lilian was duly impressed; she was as if intimidated. At intervals the rumour of a victim of that class of diseases runs whisperingly
- 59. through assemblages of women, who on the entrance of a male hastily change the subject of talk and become falsely bright. Yet every male in the circle of acquaintances will catch the rumour almost instantly, because some wife runs to inform her husband, and the husband informs all his friends. Who told you? Lilian demanded. Oh! I've known about it for a long time, said Gertie without pride. I told Milly just now, before I went out. Everybody will know soon. Lilian felt a pang of jealousy. It means a terrible operation, Gertie added. But she oughtn't to be here! Lilian exclaimed. No! Gertie agreed with a surprising sternness that somewhat altered Lilian's estimate of her. No! And she isn't going to be here, either! Not if I know it! I shall see that she gets back home at lunch- time. She's quarrelled already with Mr. Grig this morning about her coming up. Do you mean at home they quarrelled? Yes. He got so angry that he said if she came he wouldn't. He was quite right to be angry, of course. But she came all the same. Miss G. must have told Gertie all that herself, Lilian reflected. She'd never be as confidential with me. She'd never tell me anything! And she had a queer feeling of inferiority. We must do all we can to help things, said Gertie. Of course! agreed Lilian, suddenly softened, overcome by a rush of sympathy and a strong impulse to behave nobly, beautifully, forgivingly towards Miss G. Nevertheless, though it was Gertie's attitude that had helped to inspire her, she still rather disdained the virtuous senior. Lilian
- 60. appreciated profoundly--perhaps without being able to put her feeling into words--the heroic madness of Miss G. in defying common sense and her brother for the sake of the beloved business. But Gertie saw in Miss G.'s act nothing but a piece of naughty and sick foolishness. To Lilian Miss G. in her superficial yearning softness became almost a terrible figure, a figure to be regarded with awe, and to serve as an exemplar. But in contemplating Miss G. Lilian uneasily realized her own precariousness. Miss G. was old and plain (save that her eyes had beauty), and yet was fulfilling her great passion and was imposing herself on her environment. Miss G. was doing. Lilian could only be; she would always remain at the mercy of someone, and the success which she desired could last probably no longer than her youth and beauty. The transience of the gifts upon which she must depend frightened her--but at the same time intensified anew her resolves. She had not a moment to lose. And Gertie, standing there close to her, sweet and reliable and good, in the dull cage, amid the daily circumstances of their common slavery, would have understood nothing of Lilian's obscure emotion. III Shut The two girls had not settled to work when the door of the small room was pushed cautiously open and Mr. Grig came in--as it were by stealth. Milly, prolonging her sweet hour of authority in the large room, had not yet returned to her mates. By a glance and a gesture
- 61. Mr. Grig prevented the girls from any exclamation of surprise. Evidently he was secreting himself from his sister, and he must have entered the office without a sound. He looked older, worn, worried, captious--as though he needed balm and solace and treatment at once firm and infinitely soft. Lilian, who a few minutes earlier had been recalcitrant to Miss Grig's theory that women must protect men, now felt a desire to protect Mr. Grig, to save him exquisitely from anxieties unsuited to his temperament. He shut the door, and in the intimacy of the room faced the two girls, one so devoted, the other perhaps equally devoted but whose devotion was outshone by her brilliant beauty. For him both typists were very young, but they were both women, familiar beings whom the crisis had transformed from typists into angels of succour; and he had ceased to be an employer and become a man who demanded the aid of women and knew how to rend their hearts. Is she in there? he snapped, with a movement of the head towards the principals' room. Yes, breathed Lilian. Yes, said Gertie. Oh! Mr. Grig, she ought never to have come out in her state! Well, God damn it, of course she oughtn't! retorted Mr. Grig. His language, unprecedented in that room, ought to have shocked the respectable girls, but did not in the slightest degree. To judge from their demeanour they might have been living all their lives in an environment of blasphemous profanity. Didn't I do everything I could to keep her at home? Oh! I know you did! Gertie agreed sympathetically. She told me.
- 62. I made a hades of a row with her about it in the hope of keeping her in the house. But it was no use. I swore I wouldn't move until she returned. But of course I've got to do something. Look here, one of you must go to her and tell her I'm waiting in a taxi downstairs to take her home, and that I shall stick in it till she gives way, even if I'm there all day. That ought to shift her. Tell her I've arranged for the doctor to be at the house at a quarter to eleven. You'd better go and do it, Miss Jackson. She's more likely to listen to you. Yes, do, Gertie! You go, Lilian seconded the instruction. Then: What's the matter, Gertie? What on earth's the matter? The paragon had suddenly blanched and she seemed to shiver: first sign of acute emotion that Lilian had ever observed in the placid creature. It's nothing. I'm only---- It's really nothing. And Gertie, who had not taken off her street-things, rose resolutely from her chair. She, who a little earlier had seemed quite energetic and fairly fresh after her night's work, now looked genuinely ill. You go along, Mr. Grig urged her, ruthlessly ignoring the symptoms which had startled Lilian. And mind how you do it, there's a good creature. I'll get downstairs first. And he stepped out of the room. The door opening showed tall, thin Millicent returning to her own work. Mr. Grig pushed past her on tiptoe. As soon as Gertie had disappeared on her mission into the principals' room, Lilian told Millicent, not without an air of superiority, as of an Under-secretary of State to a common member of Parliament, what was occurring.
- 63. Millicent, who loved incidents, bit her lips in a kind of cruel pleasure. (She had a long, straight, absolutely regular nose, and was born to accomplish the domestic infelicity of some male clerk.) She made an excuse to revisit the large room in order to spread the thrilling news. Lilian stood just behind the still open door of the small room. A long time elapsed. Then the door of the principals' room opened, and Lilian, discreetly peeping, saw the backs of Miss Grig and Gertie Jackson. They seemed to be supporting each other in their progress towards the outer door. She wondered what the expressions on their faces might be; she had no clue to the tenor of the scene which had ended in Gertie's success, for neither of the pair spoke a word. How had Gertie managed to beat the old fanatic? After a little pause she went to the window and opened it and looked out at the pavement below. The taxi was there. Two foreshortened figures emerged from the building. Mr. Grig emerged from the taxi. Miss Grig was induced into the vehicle, and to Lilian's astonishment Gertie followed her. Mr. Grig entered last. As the taxi swerved away, a little outcry of voices drew Lilian's attention to the fact that both windows of the large room were open and full of clusters of heads. The entire office, thanks to that lath, Millicent, was disorganized. Lilian whipped in her own head like lightning. At three o'clock she was summoned to the telephone. Mr. Grig was speaking from a call-office. Miss Jackson's got influenza, the doctor says, he announced grimly. So she has to stay here. A nice handful for me. You'd better carry on. I'll try to come up later. Miss Grig said something about some accounts--I don't know.
- 64. Lilian, quite unable to check a feeling of intense, excited happiness, replied with soothing, eager sympathy and allegiance, and went with dignity into the principals' room, now for the moment lawfully at her mercy. The accounts of the establishment were always done by Miss Grig, and there was evidence on the desk that she had been obdurately at work on bills when Gertie Jackson enticed her away. In the evening Lilian, after a day's urgent toil at her machine, was sitting in Miss Grig's chair in the principals' room, at grips with the day-book, the night-book, the ledger and some bill- forms. Although experiencing some of the sensations of a traveller lost in a forest (of which the trees were numerals), she was saturated with bliss. She had dismissed the rest of the staff at the usual hour, firmly refusing to let anybody remain with her. Almost as a favour Millicent had been permitted to purchase a night's food for her. Just as the clock of St. George's struck eight, it occurred to her that to allow herself to be found by Mr. Grig in the occupation of Miss Grig's place might amount to a grave failure in tact; and hastily- -for he might arrive at any moment---she removed all the essential paraphernalia to the small room. She had heard nothing further from Mr. Grig, who, moreover, had not definitely promised to come, but she was positive that he would come. However late the hour might be, he would come. She would hear the outer door open; she would hear his steps; she would see him; and he would see her, faithfully labouring all alone for him, and eager to take a whole night-watch for the second time in a week. For this hour she had made a special toilette, with much attention to her magnificent hair. She looked spick-and-span and enchanting.
- 65. Nor was she mistaken. Hardly had she arranged matters in her own room when the outer door did open, and she did hear his steps. The divine moment had arrived. He appeared in the doorway of the room. Rather to her regret he was not in evening dress. (But how could he be?) Still, he had a marvellous charm and his expression was less worried. He was almost too good to be true. She greeted him with a smile that combined sorrow and sympathy and welcome, fidelity and womanly comprehension, the expert assistant and the beautiful young Eve. She was so discomposed by the happiness of realization that at first she scarcely knew what either of them was saying, and then she seemed to come to herself and she caught Mr. Grig's voice clearly in the middle of a sentence:' ... with a temperature of 104. The doctor said it would be madness to send her to Islington. This sort of influenza takes you like this, it appears. I shall have it myself next.... What are you supposed to be doing? Bills, eh? He looked hard at her, and her eyes dropped before his experienced masculine gaze. She liked him to be wrinkled and grey, to be thirty years older than herself, to be perhaps even depraved. She liked to contrast her innocent freshness with his worn maturity. She liked it that he had not shown the slightest appreciation of her loyalty. He spoke only vaguely of Miss Grig's condition; it was not a topic meet for discussion between them, and with a few murmured monosyllables she let it drop. I do hope you aren't thinking of staying, Mr. Grig, she said next. I shall be perfectly all right by myself, and the bills will occupy me till something comes in.
- 66. I'm not going to stay. Neither are you, replied Mr. Grig curtly. We'll shut the place up. Her face fell. But---- We'll shut up for to-night. But we're supposed to be always open! Supposing some work does come in! It always does---- No doubt. But we're going to shut up the place--at once. There was fatigue in his voice. Tears came into Lilian's eyes. She had expected him, in answer to her appeal to him to depart, to insist on staying with her. She had been waiting for heaven to unfold. And now he had decided to break the sacred tradition and close the office. She could not master her tears. Don't worry, he said in tones suddenly charged with tenderness and sympathetic understanding. It can't be helped. I know just how you feel, and don't you imagine I don't. You've been splendid. But I had to promise Isabel I'd shut the office to-night. She's in a very bad state, and I did it to soothe her. You know she hates me to be here at nights--thinks I'm not strong enough for it. That's not her reason to-night, said Lilian to herself. I know her reason to-night well enough! But she gave Mr. Grig a look grateful for his exquisite compassion, which had raised him in her sight to primacy among men. Obediently she let herself be dismissed first, leaving him behind, but in the street she looked up at her window. The words Open day and night on the blind were no longer silhouetted against a light
- 67. within. The tradition was broken. On the way to the Dover Street Tube she did not once glance behind her to see if he was following. IV The Vizier Late in the afternoon of the following day Mr. Grig put his head inside the small room. Just come here, Miss Share, he began, and then, seeing that Millicent was not at her desk, he appeared to decide that he might as well speak with Lilian where she was. He had been away from the office most of the day, and even during his presences had seemingly taken no part in its conduct. Much work had been received, some of it urgent, and Lilian, typing at her best speed, had the air of stopping with reluctance to listen to whatever the useless and wandering man might have to say. He merely said: We shall close to-night, like last night. Oh, but, Mr. Grig, Lilian protested--and there was no sign of a tear this time--we can't possibly keep on closing. We had one complaint this morning about being closed last night. I didn't tell you because I didn't want to worry you. Now listen to me, Mr. Grig protested in his turn, petulantly. Nothing worries me more than the idea that people are keeping things from me in order that I shan't be worried. My sister was always doing that; she was incurable, but I'm not going to have it
- 68. from anyone else. If you hide things, why are you silly enough to let out afterwards that you were hiding them and why you were hiding them? That's what I can't understand. Sorry, Mr. Grig, Lilian apologized briefly and with sham humility, humouring the male in such a manner that he must know he was being humoured. His petulancy charmed her. It gave him youth, and gave her age and wisdom. He had good excuse for it--Miss Grig had been moved into a nursing home preparatory to an operation, and Gertie was stated to be very ill in his house--and she enjoyed excusing him. It was implicit in every tone of his voice that they were now definitely not on terms of employer and employee. That's all right! That's all right! he said, mollified by her discreet smile. But close at six. I'm off. I really don't think we ought to close, she insisted, with firmness in her voice followed by persuasion in her features, and she brushed back her hair with a gesture of girlishness that could not be ineffective. He hesitated, frowning. She went on: If it gets about that we're closing night after night, we're bound to lose a lot of customers. I can perfectly well stay here. Yes! And be no use at all to-morrow! I should be here to-morrow just the same. If other girls can do it, why can't I? (A touch of harshness in the question.) Oh, Milly! she exclaimed, neglecting to call Milly Miss Merrislate, according to the custom by which in talking to the principals everybody referred to everybody else as Miss. Oh, Milly!--Millicent appeared behind Mr. Grig at the door and he nervously made way for her--here's Mr. Grig wants to close again to-night! I'm sure we really oughtn't to.
- 69. I've told Mr. Grig I'll stay--and be here to-morrow too. Don't you agree we mustn't close? Millicent was flattered by the frank appeal as an equal from one whom she was already with annoyance beginning to regard as a superior. From timidity in Mr. Grig's presence she looked down her too straight nose, but she nodded affirmatively her narrow head, and as soon as she had recovered from the disturbing novelty of deliberately opposing the policy of an employer she said to Lilian: I'll stay with you if you like. There's plenty to do, goodness knows! You are a dear! Lilian exclaimed, just as if they had been alone together in the room. Oh, well, have it as you like! Mr. Grig rasped, and left, defeated. Is he vexed? Milly demanded after he had gone. Of course not! He's very pleased, really. But he has to save his face. Milly gave Lilian a scarcely conscious glance of admiration, as a woman better versed than herself in the mysteries of men, and also as a woman of unsuspected courage. And she behaved like an angel through the whole industrious night--so much so that Lilian was nearly ready to admit to an uncharitable premature misjudgment of the girl. And now what are you going to do about keeping open? inquired Mr. Grig, with bland, grim triumph the next afternoon to the exhausted Lilian and the exhausted Millicent. I thought I'd let you have your own way last night. But you can't see any further than your noses, either of you. You're both dead.
- 70. I can easily stay up another night, said Lilian desperately, but Millicent said nothing. No doubt! Mr. Grig sneered. You look as if you could! And supposing you do, what about to-morrow night? The whole office is upset, and, of course, people must go and choose just this time to choke us with work! Well, anyhow, we can't close, Lilian stoutly insisted. No! Mr. Grig unexpectedly agreed. Miss Merrislate, you know most about the large room. You'd better pick two of 'em out of there, and tell 'em they must stay and do the best they can by themselves. But that won't carry us through. I certainly shan't sit up, and I won't have you two sitting up every second night in turn. There's only one thing to do. I must engage two new typists at once--that's clear. We may as well face the situation. Where do we get 'em from? But neither Lilian nor Milly knew just how Miss Grig was in the habit of finding recruits to the staff. Each of them had been taken on through private connexions. Gertie Jackson would probably have known how to proceed, but Gertie was down with influenza. I'll tell you what I shall do, said Mr. Grig at last. I'll get an advertisement into to-morrow's Daily Chronicle. That ought to do the trick. This affair's got to be handled quickly. When the applicants come you'd better deal with 'em, Miss Share--in my room. I shan't be here to-morrow. He spoke scornfully, and would not listen to offers of help in the matter of the advertisement. He would see to it himself, and wanted no assistance, indeed objected to assistance as being merely troublesome. The next day was the day of Miss Grig's operation, and
- 71. the apprehension of it maddened this affectionate and cantankerous brother. Millicent left the small room to bestow upon two chosen members of the rabble in the large room the inexpressible glory of missing a night's sleep. On the following morning, when Lilian, refreshed, arrived zealously at the office half an hour earlier than usual, she found three aspirants waiting to apply for the vacant posts. The advertisement had been drawn up and printed; the newspaper had been distributed and read, and the applicants, pitifully eager, had already begun to arrive from the ends of London. Sitting in Miss Grig's chair, Lilian nervously interviewed and examined them. One of the three gave her age as thirty-nine, and produced yellowed testimonials. By ten o'clock twenty-three suitors had come, and Lilian, frightened by her responsibilities, had impulsively engaged a couple, who took off hats and jackets and began to work at once. She had asked Millicent to approve of the final choice, but Millicent, intensely jealous and no longer comparable to even the lowest rank of angel, curtly declined. You're in charge, Millicent said acidly. Don't you try to push it on to me, Miss Lilian Share. Aspirants continued to arrive. Lilian had the clever idea of sticking a notice on the outer door: All situations filled. No typists required. But aspirants continued to enter, and all of them averred positively that they had not seen the notice on the door. Lilian told a junior to paste four sheets of typing paper together, and she inscribed the notice on the big sheet in enormous characters. But aspirants continued to enter, and all of them averred positively that they had not seen the notice on the door. It was dreadful, it was
- 72. appalling, because Lilian was saying to herself: I may be like them one day. Millicent, on the other hand, disdained the entire procession, and seized the agreeable rôle of dismissing applicants as fast as they came. In the evening Mr. Grig appeared. The operation had been a success. Gertie Jackson was, if anything, a little worse; but the doctor anticipated an improvement. Mr. Grig showed not the least interest in his business. Lilian took the night duty alone. Thenceforward the office settled gradually into its new grooves, and, though there was much less efficiency than under Miss Grig, there was little friction. Everybody except Millicent regarded Lilian as the grand vizier, and Millicent's demeanour towards Lilian was by turns fantastically polite and fantastically indifferent. A fortnight passed. The two patients were going on well, and it was stated that there was a possibility of them being sent together to Felixstowe for convalescence. Mr. Grig's attendance grew more regular, but he did little except keep the books and make out the bills; in which matter he displayed a facility that amazed Lilian, who really was not a bit arithmetical. One day, entering the large room after hours, Lilian saw Millicent typing on a machine not her own. As she passed she read the words: My darling Gertie. I simply can't tell you how glad I was to get your lovely letter. And it flashed across her that Millicent would relate all the office doings to Gertie, who would relate them to Miss Grig. She had a spasm of fear, divining that Millicent would misrepresent her. In what phrases had Millicent told that Lilian had sat in Miss Grig's chair and interviewed applicants for situations! Was it not strange that Gertie had not written to her, Lilian, nor she even
- 73. thought of writing to Gertie? Too late now for her to write to Gertie! A few days later Mr. Grig said to Lilian in the small room:' You're very crowded here, aren't you? The two new-comers had been put into the small room, being of a superior sort and not fitted to join the rabble. Oh, no! said Lilian. We're quite comfortable, thank you. You don't seem to be very comfortable. It occurs to me it would be better in every way if you brought your machine into my room. An impulse, and an error of judgment, on Felix's part! But he was always capricious. I should prefer to stay where I am, Lilian answered, not smiling. What a letter Millicent would have written in order to describe Lilian's promotion to the principals' room! Often, having made a mistake, Felix would persist in it from obstinacy. Oh! As you like! he muttered huffily, instead of recognizing by his tone that Lilian was right. But the next moment he repeated, very softly and kindly: As you like! It's for you to decide. He had not once shown the least appreciation of, or gratitude for, Lilian's zeal. On the contrary, he had been in the main querulous and censorious. But she did not mind. She was richly rewarded by a single benevolent inflection of that stirring voice. She seemed to have forgotten that she was born for pleasure, luxury, empire. Work fully satisfied her, but it was work for him. The mere suggestion that she should sit in his room filled her with deep joy.
- 74. V The Martyr Miss Grig came back to the office on a Thursday, and somewhat mysteriously. Millicent, no doubt from information received through Gertie Jackson, had been hinting for several days that the return would not be long delayed; but Mr. Grig had said not one word about the matter until the Wednesday evening, when he told Lilian, with apparent casualness, as she was leaving for the night, that his sister might be expected the next morning. As for Miss Jackson, she would resume her duties only on the Monday, having family affairs to transact at Islington. Miss Jackson, it seemed, had developed into the trusted companion and intimate--almost ally, if the term were not presumptuous--of the soul and dynamo of the business. Miss Grig and she had suffered together, they had solaced and strengthened each other; and Gertie, for all her natural humility, was henceforth to play in the office a rôle superior to that of a senior employee. She had already been endowed with special privileges, and among these was the privilege of putting the interests of Islington before the interests of Clifford Street. The advent of Miss Grig, of course, considerably agitated the office and in particular the small room, two of whose occupants had never seen the principal of whose capacity for sustained effort they had heard such wonderful and frightening tales. At nine-thirty that Thursday morning it was reported in both rooms that Miss Grig had re-entered her fortress. Nobody had seen her, but ears had heard her, and, moreover, it was mystically known by certain signs, as, for example, the reversal of a doormat which
- 75. had been out of position for a week, that a higher presence was immanent in the place and that the presence could be none other than Miss Grig. Everybody became an exemplar of assiduity, amiability, and entire conscientiousness. Everybody prepared a smile; and there was a universal wish for the day to be over. Shortly after ten o'clock Miss Grig visited the small room, shook hands with Lilian and Millicent, and permitted the two new typists to be presented to her. Millicent spoke first and was so effusive in the expression of the delight induced in her by the spectacle of Miss Grig and of her sympathy for the past and hope for the future of Miss Grig's health, that Lilian, who nevertheless did her best to be winning, could not possibly compete with her. Miss Grig had a purified and chastened air, as of one detached by suffering from the grossness and folly of the world, and existing henceforth in the world solely from a cold, passionate sense of duty. Her hair was greyer, her mild equable voice more soft, and her burning eyes had a brighter and more unearthly lustre. She said that she was perfectly restored, let fall that Mr. Grig had gone away at her request for a short, much-needed holiday, and then passed smoothly on to the large room. After a while a little flapper of a beginner came to tell Millicent that Miss Grig wanted her. Millicent, who had had charge of the petty cash during the interregnum, was absent for forty minutes. When she returned, flushed but smiling, to her expectant colleagues, she informed Lilian that Miss Grig desired to see her at twelve o'clock. I notice there's an account here under the name of Lord Mackworth, Miss Grig began, having allowed Lilian to stand for a few seconds before looking up from the ledger and other books in
- 76. which she was apparently absorbed. She spoke with the utmost gentleness, and fixed her oppressive deep eyes on Lilian's. Yes, Miss Grig? It hasn't been paid. Oh! Lilian against an intense volition began to blush. Didn't you know? I didn't, said Lilian. But you've been having something to do with the books during my absence. I did a little at first, Lilian admitted. Then Mr. Grig saw to them. Miss Merrislate tells me that you had quite a lot to do with them, and I see your handwriting in a number of places here. I've had nothing to do with them for about three weeks--I should think at least three weeks, and--and of course I expected the bill would be paid by this time. But you never asked? No. It never occurred to me. This statement was inaccurate. Lilian had often wondered whether Lord Mackworth had paid his bill, but, from some obscurely caused self-consciousness, she had not dared to make any inquiry. She felt herself to be somehow mixed up with Lord Mackworth, and had absurdly feared that if she mentioned the name there might appear on the face or in the voice of the detestable Milly some sinister innuendo. Miss Merrislate tells me that she didn't trouble about the account as she supposed it was your affair.
- 77. My affair! exclaimed Lilian impulsively. It's no more my affair than anybody else's. She surmised in the situation some ingenious malevolence of the flat-breasted mischief-maker. But you did the work? Yes. It came in while I was on duty that night, and I did it at once. There was no one else to do it. Who brought it in? Lord Mackworth. Did you know him? Certainly not. I didn't know him from Adam. Never mind Adam, Miss Share, observed Miss Grig genially. Has Lord Mackworth been in since? If he has I've not seen him, Lilian answered defiantly. Miss Grig's geniality exasperated her because it did not deceive her. I'm only asking for information, Miss Grig said with a placatory smile. I see the copies were delivered at six-thirty in the morning. Who delivered the job? I did. Where? At his address. I dropped it into the letter-box on my way home after my night's work. I stayed here because somebody had to stay, and I did the best I could. I'm quite sure of that, Miss Grig agreed. And, of course, you've been paid for all overtime--and there's been quite a good deal. We all do the best we can. At least, I hope so.... And you've never seen Lord Mackworth since? No.
- 78. And you simply dropped the envelope into the letter-box? Yes. Didn't see Lord Mackworth that morning? Certainly not. By this time Lilian was convinced that Miss Grig's intention was to provoke her to open resentment. She guessed also that Milly must have deliberately kept silence to her, Lilian, about the Mackworth account in the hope of trouble on Miss Grig's return, and that Milly had done everything she could that morning to ensure trouble. The pot had been simmering in secret for weeks; now it was boiling over. She felt helpless and furious. You know, Miss Grig proceeded, there's a rule in this office that night-work must only be delivered by hand by the day-staff the next day. If it's wanted urgently before the day-staff arrives the customer must fetch it. Excuse me, Miss Grig, I never heard of that rule. Miss Grig smiled again: Well, at any rate, it was your business to have heard of it, my dear. Everybody else knows about it. I told Mr. Grig I was going to deliver it myself, and he didn't say anything. Please don't attempt to lay the blame on my brother. He is far too good-natured. Miss Grig's gaze burned into Lilian's face as, with an enigmatic intonation, she uttered these words. You did wrong. And I suppose you've never heard either of the rule that new customers must always pay on or before delivery? Yes, I have. But I couldn't ask for the money at half-past six in the morning, could I? And I couldn't tell him how much it would be before I'd typed it.
- 79. Yes, you could, my dear, and you ought to have done. You could have estimated it and left a margin for errors. That was the proper course. And if you know anything about Lord Mackworth you must know that his debts are notorious. I believe he's one of the fastest young men about town, and it's more than possible that that account's a bad debt. But can't we send in the account again? Lilian weakly suggested; she was overthrown by the charge of fast-living against Lord Mackworth, yet she had always in her heart assumed that he was a fast liver. I've just telephoned to 6a St. James's Street, and I needn't say that Lord Mackworth is no longer there, and they don't know where he is. You see what comes of disobeying rules. Lilian lifted her head: Well, Miss Grig, the bill isn't so very big, and if you'll please deduct it from my wages on Saturday I hope that will be the end of that. It was plain that the bewildered creature had but an excessively imperfect notion of how to be an employee. She had taken to the vocation too late in life. Miss Grig put her hand to the support of her forehead, and paused. I can tolerate many things, said she, with great benignity, but not insolence. I didn't mean to be insolent. You did. And I think you had better accept a week's notice from Saturday. No. On second thoughts, I'll pay your wages up to Saturday week now and you can go at once. She smiled kindly. That will give you time to turn round.
- 80. Oh! Very well, if it's like that! Miss Grig unlocked a drawer; and while she was counting the money Lilian thought despairingly that if Mr. Grig, or even if the nice Gertie, had been in the office, the disaster could not have occurred. Miss Grig shook hands with her and wished her well. Where are you going to? It's not one o'clock yet, asked Millicent in the small room as Lilian silently unhooked her hat and jacket from the clothes-cupboard. Out. What for? For Miss G., if you want to know. And she left. Except her clothes, not a thing in the office belonged to her. She had no lien, no attachment. The departure was as simple and complete as leaving a Tube train. No word! No good- bye! Merely a disappearance.
- 81. VI The Invitation She walked a mile eastwards along Oxford Street before entering a teashop, in order to avoid meeting any of the girls, all of whom, except the very youngest and the very stingiest, distributed themselves among the neighbouring establishments for the absurdly insufficient snack called lunch. Every place was full just after one o'clock, and crammed at one-fifteen. She asked for a whole meat pie instead of a half, for she felt quite unusually hungry. A plot! That was what it was! A plot against her, matured by Miss G. in a few minutes out of Milly's innuendoes written to Gertie and spoken to Miss G. herself. And the reason of the plot was Miss G.'s spinsterish, passionate fear of a friendship between Felix Grig and Lilian! Lilian was ready to believe that Miss G. had engineered the absence of both her brother and Gertie so as to be free to work her will without the possibility of complications. If Miss G. hated her, she hated Miss G. with at least an equal fierceness--the fierceness of an unarmed victim. The injustice of the world staggered her. She thought that something ought to be done about it. Even Lord Mackworth was gravely to blame, for not having paid his bill. Still, that detail had not much importance, because Miss G., deprived of one pretext, would soon have found another. After all that she, Lilian, had done for the office, to be turned off at a moment's notice, and without a character--for Miss G. would never give a reference, and Lilian would
- 82. never ask for a reference! Never! Nor would she nor could she approach Felix Grig; nor Gertie either. Perhaps Felix Grig might communicate with her. He certainly ought to do so. But then, he was very casual, forgetful and unconsciously cruel. All the men and girls in the packed tea-shop had work behind them and work in front of them. They knew where they were; they had a function on the earth. She, Lilian, had nothing, save a couple of weeks' wages and perhaps a hundred pounds in the Post Office Savings Bank. Resentment against her father flickered up anew from its ashes in her heart. How could she occupy herself after lunch? Unthinkable for her to go to her lodging until the customary hour, unless she could pretend to be ill; and if she feigned illness the well-disposed slavey would be after her and would see through the trick at once, and it would be all over the house that something had happened to Miss Share. The afternoon was an enormous trackless expanse which had to be somehow traversed by a weary and terribly discouraged wayfarer. Her father had been in the habit of conducting his family on ceremonial visits to the public art galleries. She went to the Wallace Collection, and saw how millionaires lived in the 'seventies, and how the unchaste and lovely ladies were dressed for whom entire populations were sacrificed in the seventeenth and eighteenth centuries. Thence to a cinema near the Marble Arch, and saw how virtue infallibly wins after all. When, after travelling countless leagues of time and ennui, she reached home she received a note from Mr. Pladda inviting her to the Hammersmith Palais de Danse for the following night. Mr. Pladda was the star lodger in the house--a man of forty-five, legally
- 83. separated from his wife but of impeccable respectability and decorum. His illusion was that he could dance rather well. Mr. Pladda was evidently coming on. The next morning, which was very fine, Lilian spent in Hyde Park, marshalling her resources. Beyond her trifling capital she had none. Especially she had no real friends. She had unwisely cut loose from her parents' acquaintances, and she could not run after them now that she was in misfortune. Her former colleagues? Out of the question! Gertie might prove a friend, but Gertie must begin; Lilian could not begin. Lord Mackworth? Silly idea! She still thought of Lord Mackworth romantically. He was an unattainable hero at about the same level as before in her mind, for while his debts had lowered him his advertised dissoluteness had mysteriously raised him. (Yet in these hours and days Mr. Pladda himself was not more absolutely respectable and decorous, in mind and demeanour, than Lilian.) She went to two cinemas in the afternoon, and, safe in the darkness of the second one, cried silently. But with Mr. Pladda at the Palais de Danse she was admirably cheerful, and Mr. Pladda was exceedingly proud of his companion, who added refined manners to startling beauty. She delicately praised his dancing, whereupon he ordered lemon squashes and tomato sandwiches. At the little table she told him calmly that she was leaving her present situation and taking another. Back in her room she laughed with horrid derision. And as soon as she was in bed the clockwork mice started to run round and round in her head. A plot! A plot! What a burning shame! What a burning shame! ... A few weeks earlier she had actually been bestowing situations on pitiful applicants. Now she herself had no
- 84. situation and no prospect of any. She had never had to apply for a situation. She had not been educated to applying for situations. She could not imagine herself ever applying for a situation. She had not the least idea how to begin to try to get a situation. She passed the greater part of Sunday in bed, and in the evening went to church and felt serious and good. On Monday morning she visited the Post Office and filled up a withdrawal form for forty pounds. She had had a notion of becoming a companion to a rich lady, or private secretary to a member of Parliament. She would advertise. Good clothes, worn as she could wear them, would help her. (She could not face another situation in an office. No, she couldn't.) The notion of a simpleton, of course! But she was still a simpleton. The notion, however, was in reality only a pretext for obtaining some good clothes. All her life she had desired more than anything a smart dress. There was never a moment in her life when she was less entitled to indulge herself; but she felt desperate. She was taking to clothes as some take to brandy. On the Wednesday she received the money: a colossal, a marvellous sum. She ran off with it and nervously entered a big shop in Wigmore Street; the shop was a wise choice on her part, for it combined smartness with a discreet and characteristic Englishness. Impossible to have the dangerous air of an adventuress in a frock bought at that shop! The next few days were spent in exactly fitting and adapting the purchases to her body. She had expended the forty pounds and drawn out eight more. Through the medium of the slavey she borrowed a mirror, and fixed it at an angle with her own so that she could see her back. She was so interested and absorbed that she
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