A Batch-Arrival Queue with Multiple Servers and Fuzzy Parameters: Parametric Programming Approach

International Journal of Science and Research (IJSR), India Online ISSN: 2319-7064
Volume 2 Issue 9, September 2013
www.ijsr.net
A Batch-Arrival Queue with Multiple Servers and
Fuzzy Parameters: Parametric Programming
Approach
1
R. Ramesh, 2
S. Kumara G Ghuru
1
Department of Mathematics, Arignar Anna Government Arts College, Musiri, Tamilnadu, India
2
Department of Mathematics, Chikkanna Government Arts College, Tiruppur, Tamilnadu, India
Abstract: This work constructs the membership functions of the system characteristics of a batch-arrival queuing system with multiple
servers, in which the batch-arrival rate and customer service rate are all fuzzy numbers. The  -cut approach is used to transform a
fuzzy queue into a family of conventional crisp queues in this context. By means of the membership functions of the system
characteristics, a set of parametric nonlinear programs is developed to describe the family of crisp batch-arrival queues with multiple
servers. A numerical example is solved successfully to illustrate the validity of the proposed approach. Because the system characteristics
are expressed and governed by the membership functions, the fuzzy batch-arrival queues with multiple servers are represented more
accurately and the analytic results are more useful for system designers and practitioners.
Keywords: Fuzzy sets, Membership function, Multiple server, Nonlinear programming
1. Introduction
Queuing models with multiple servers are effective methods
for performance analysis of computer and
telecommunication systems, manufacturing/production
systems and inventory control (Kleinrock [11], Buzacott and
Shanthi Kumar [1], Gross and Harris [7], Trivedi [19]). In
general, these analyses consider a queuing system where
requests for service arrive in units, one at a time (single-unit
arrival). In many practical situations, however, requests for
service usually arrive in batches. For example, in
manufacturing systems of the job-shop type, each job order
often requires the manufacture of more than one unit; in
computer communication systems, messages which are to
be transmitted could consist of a random number of packets.
If the usual crisp batch-arrival queues with multiple servers
can be extended to fuzzy batch-arrival queues, such queuing
models would have wider applications.
For queuing models with multiple servers under various
considerations, the M/M/c vacation systems with a single-
unit arrival have attracted much attention from numerous
researchers since Levy and Yechiali [12]. The extensions of
this model can be referred to Vinod [20], Igaki [8], Tian et
al. [17], Tian and Xu [18], and Zhang and Tian [23, 24].
Zhang and Tian [23, 24] studied the M/M/c vacation
systems with a single-unit arrival and a “partial server
vacation policy”. They proved several conditional stochastic
decomposition results for the queue length and waiting time.
Chao and Zhao [3] investigated the GI/M/c vacation models
with a single-unit arrival and provided iterative algorithms
for computing the stationary probability distributions.
In the literature described above, customer inter-arrival
times and customer service times are required to follow
certain probability distributions with fixed parameters.
However, in many real-world applications, the parameter
distributions may only be characterized subjectively; that is,
the arrival and service are typically described in everyday
language summaries of central tendency, such as “the mean
arrival rate is around 5 per day”, or “the mean service rate is
about 10 per hour”, rather than with complete probability
distributions. In other words, these system parameters are
both possibilistic and probabilistic. Thus, fuzzy queues are
potentially much more useful and realistic than the
commonly used crisp queues (see Li and Lee [13] and
Zadeh [22]). By extending the usual crisp batch-arrival
queues to fuzzy batch-arrival queues in the context of
multiple servers, these queuing models become appropriate
for a wider range of applications.
Li and Lee [13] investigated the analytical results for two
typical fuzzy queues (denoted M/F/1/  and FM/FM/1/  ,
where F represents fuzzy time and FM represents fuzzified
exponential distributions) using a general approach based on
Zadeh’s extension principle (see also Prade [15] and Yager
[21]), the possibility concept and fuzzy Markov chains (see
Stanford [16]). A useful modeling and inferential technique
would be applied their approach to general fuzzy queuing
problems (see Stanford [16]). However, their approach is
complicated and not suitable for computational purposes;
moreover, it cannot easily be used to derive analytic results
for other complicated queuing systems (see Negi and Lee
[14]). In particular, it is very difficult to apply this approach
to fuzzy queues with more fuzzy variables or multiple
servers. Negi and Lee [14] proposed a procedure using α-
cuts and two-variable simulation to analyze fuzzy queues
(see also Chanas and Nowakowski [2]). Unfortunately, their
approach provides only crisp solutions; i.e., it does not fully
describe the membership functions of the system
characteristics. Using parametric programming, Kao et al.
[9] constructed the membership functions of the system
characteristics for fuzzy queues and successfully applied
them to four simple fuzzy queue models: M/F/1/  ,
F/M/1/  , F/F/1/  and FM/FM/1/  . Recently, Chen
[4,5] developed FM/FM/1/L and FM/FM[K]
/1/  fuzzy
systems using the same approach.
All previous researches on fuzzy queuing models are
focused on ordinary queues with a single server. In this
Paper ID: 02091301 135

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paper, we develop an approach that provides system
characteristics for batch-arrival queues with multiple servers
and fuzzy parameters: fuzzified exponential batch-arrival
and service rates. Through  -cuts and Zadeh’s extension
principle, we transform the fuzzy queues to a family of crisp
queues. As  varies, the family of crisp queues is described
and solved using parametric nonlinear programming (NLP).
The NLP solutions completely and successfully yield the
membership functions of the system characteristics,
including the expected number of customers in the system
and the expected waiting time in the queue.
The remainder of this paper is organized as follows. Section
2 presents the system characteristics of standard and fuzzy
batch-arrival queuing models with multiple servers. In
Section 3, a mathematical programming approach is
developed to derive the membership functions of these
system characteristics. To demonstrate the validity of the
proposed approach, one realistic numerical example is
described and solved. Discussion is provided in Section 4,
and conclusions are drawn in Section 5. For notational
convenience, our model in this paper is hereafter denoted
FM[x]
/FM/c.
2. Fuzzy Batch Queue With Multiple Servers
We consider a batch-arrival queuing system with c servers
where the customers arrive in batches to occur according to
a compound Poisson process with batch-arrival rate  . Let
kA denote the number of customers belonging to the kth
arrival batch, where ,kA ,,3,2,1 k are with a
common distribution ,3,2,1,]Pr[ nanA nk  ,
and




1
][
n
nnaAE
. Customers arriving at the service facility
(servers) form a single-file queue and are served in order.
The service time for each of all c servers is exponentially
distributed with rate  and each server can serve only one
customer at a time. Customers who upon entry the service
facility find that all servers are busy have to wait in the
queue until any one server is available. Let sN and qW
represents the expected number of customers in the system
and the expected waiting time in the queue, respectively.
Through a Markov process, we can easily obtain sN and
qW in terms of system parameters
])[(2
),()(2)])1([][2(
1
0
AEc
PncnAAEAE
N
c
n
n
s



 


 , (1)


1
])[]([2
),()(2)])1([][2(
1
0


 



AEcAE
PncnAAEAE
W
c
n
n
q
, (2)
Where ),( nP represents the probability that there are n
customers in the system. And the probability depends on 
and  . In steady-state, it is necessary that we
have 1
][
0 


c
AE .
To extend the applicability of the batch-arrival queuing
model with multiple servers, we allow for fuzzy
specification of system parameters. Suppose the batch-
arrival rate  for customers and service rate  for each
server are approximately known and can be represented by
the fuzzy sets 
~
and ~ . Let )(~ x
 and )(~ y denote
the membership functions of 
~
and ~ . We then have the
following fuzzy sets:
 Xxxx  ))(,(
~
~

 , (3a)
 Yyyy  ))(,(~ ~ , (3b)
where X and Y are the crisp universal sets of the batch-
arrival and service rates.
Let ),( yxf denote the system characteristic of interest.
Since 
~
and ~ are fuzzy numbers, )~,
~
( f is also a
fuzzy number. Following Zadeh’s extension principle (see
Yager [21] and Zadeh [22]), the membership function of the
system characteristic )~,
~
( f is defined as:
 ),()(),(minsup)( ~~
1/][0,,
)~,
~
(
yxfzyxz
cyAxEYyXx
f



 , (4)
Assume that the system characteristic of interest is the
expected number of customers in the system. It follows
from (1) that the expected number of customers in the
system is:
),( yxf
])[(2
),()(2)])1([][2(
1
0
AxEcy
yxPncnyAAEAEx
c
n
n

 

 . (5)
The membership function for the expected number of
customers in the system is:
Unfortunately, the membership function is not expressed in
the usual form, making it very difficult to imagine its shape.
In this paper we approach the representation problem using
a mathematical programming technique. Parametric NLPs
are developed to find the  -cuts of )~,
~
( f based on the
extension principle.
3. Parametric Nonlinear Programming
To re-express the membership function )(~ zsN
 of sN
~
in
an understandable and usable form, we adopt Zadeh’s
approach, which relies on  -cuts of sN
~
. Definitions for
the  -cuts of 
~
and ~ as crisp intervals are as follows:
The constant batch-arrival and service rates are shown as
intervals when the membership functions are no less than a
Paper ID: 02091301 136

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given possibility level for  . As a result, the bounds of
these intervals can be described as functions of  and can
be obtained as: ,)(min 1
~ 

L
x
,)(max 1
~ 

U
x )(min 1
~ 

L
y
&
.)(max 1
~ 

U
y
Therefore, we can use the  -cuts of sN
~
to construct its
membership function since the membership function
defined in (6) is parameterized by  .
Using Zadeh’s extension principle, )(~ zsN
 is the minimum
of )(~ x
 and )(~ y . To derive the membership function
)(~ zsN
 , we need at least one of the following cases to hold
such that z =
])[(2
),()(2)])1([][2(
1
0
AxEcy
yxPncnyAAEAEx
c
n
n

 


satisfies  )(~ zsN
:
Case (i): ( 
)(~ x ,  )(~ y ),
Case (ii): ( 
)(~ x ,  )(~ y ),
This can be accomplished using parametric NLP techniques.
The NLP to find the lower and upper bounds of the  -cut
of )(~ zsN
 for Case (i) are:
min)( 1
L
sN 
])[(2
),()(2)])1([][2(
1
0
AxEcy
yxPncnyAAEAEx
c
n
n

 

 , (8a)
max)( 1
U
sN 
])[(2
),()(2)])1([][2(
1
0
AxEcy
yxPncnyAAEAEx
c
n
n

 

 , (8b)
and for Case (ii) are:
min)( 2
L
sN 
])[(2
),()(2)])1([][2(
1
0
AxEcy
yxPncnyAAEAEx
c
n
n

 

 , (8c)
max)( 2
U
sN 
])[(2
),()(2)])1([][2(
1
0
AxEcy
yxPncnyAAEAEx
c
n
n

 

 . (8d)
From the definitions of )( and )( in (7),
)(x and )(y can be replaced by
],[ UL
xxx  and ],[ UL
yyy  . The  -cuts form a
nested structure with respect to  (see Kaufmann [10] and
Zimmermann [25]); i.e., given 10 12   , we have
],[],[ 2211
ULUL
xxxx   and ],[],[ 2211
ULUL
yyyy   .
Therefore, (8a) and (8c) have the same smallest element and
(8b) and (8d) have the same largest element. To find the
membership function )(~ zsN
 , it suffices to find the left and
right shape functions of )(~ zsN
 , which is equivalent to
finding the lower bound
L
sN )( and upper bound
U
sN )(
of the  -cuts of sN
~
, which can be rewritten as:
min)( L
sN 
])[(2
),()(2)])1([][2(
1
0
AxEcy
yxPncnyAAEAEx
c
n
n

 

 (9a)
s.t.
UL
xxx   and
UL
yyy   ,
max)( U
sN 
])[(2
),()(2)])1([][2(
1
0
AxEcy
yxPncnyAAEAEx
c
n
n

 

 (9b)
s.t.
UL
xxx   and
UL
yyy   ,
At least one of x and y must hit the boundaries of their
 -cuts to satisfy )(~ zsN
 = . This model is a set of
mathematical programs with boundary constraints and lends
itself to the systematic study of how the optimal solutions
change with
L
x ,
U
x ,
L
y , and
U
y as  varies over
(0,1]. The model is a special case of parametric NLPs (see
Gal [6]).
The crisp interval [
L
sN )( ,
U
sN )( ] obtained from (9)
represents the  -cuts of sN
~
. Again, by applying the
results of Kaufmann [10] and Zimmermann [25] and
convexity properties to sN
~
, we have
L
s
L
s NN 21
)()(  
and
U
s
U
s NN 21
)()(   , where 10 12   . In other
words,
L
sN )( increases and
U
sN )( decreases as 
increases. Consequently, the membership function )(~ zsN

can be found from (9).
If both
L
sN )( and
U
sN )( in (9) are invertible with respect
to  , then a left shape function
1
])[()( 
 L
sNzL  and a
right shape function
1
])[()( 
 U
sNzR  can be derived,
from which the membership function )(~ zsN
 is
constructed:












.)()(),(
,)()(,1
,)()(),(
)(
0s1s
1s1s
1s0s
~
U
α
U
α
U
α
L
α
L
α
L
α
sN
NzNzR
NzN
NzNzL
z (10)
In most cases, the values of
L
sN )( and
U
sN )( cannot be
solved analytically. Consequently, a closed-form
membership function for )(~ zsN
 cannot be obtained.
However, the numerical solutions for
L
sN )( and
U
sN )( at
different possibility levels can be collected to approximate
the shapes of )(zL and )(zR . That is, the set of intervals
]}1,0[|])(,){[( 
U
s
L
s NN shows the shape of
)(~ zsN
 , although the exact function is not known
explicitly.
Note that the membership functions for the expected waiting
time in the queue can be expressed in a similar manner.
4. Numerical Example
This section we present one example motivated by real-life
systems to demonstrate the practical use of the proposed
approach, which is based on
http://www.macaudata.com/macauweb/book175/html/19301
.htm
Paper ID: 02091301 137

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Example: Considering one sewerage treatment system
collects sewage from the urban areas and sends them to the
sewerage treatment plant. The sewerage treatment plant has
three supply pipes (referred to 3-servers). Each pipe can
settle the larger solids and put the settled into the chemical
process tank. After the chemical process, the treated water is
discharged to the sea. We assume that the number of
arriving sewage solids each time follows a geometric
distribution with parameter 0.5p = ; i.e., the size of
arriving sewage solids A is
,2,1,)5.01(5.0)Pr( 1
 
kkA k
. Clearly, this problem
can be described by FM[x]
/FM/3 system. For efficiency, the
management wants to get the system characteristics such as
the expected number of sewage solids in the system and the
expected waiting time in the queue.
Suppose the batch-arrival rate and service rate are
trapezoidal fuzzy numbers represented by 4],3,2,1[
~

and ]14,13,12,11[~  . First, it is easy to find that
]4,1[],[  UL
xx and
]41,11[],[  UL
yy . Next, it is obvious that
when
U
xx  and
L
yy  , the expected number of
sewage solids in the system attains its maximum value, and
when
L
xx  , and
U
yy  , the expected number of
sewage solids in the system attains its minimum value.
According to (9), the  -cuts of sN
~
are:
32
32
2512752100099200
11036302346027200
)(





L
sN ,
32
32
2510501402547000
1102640465067880
)(





U
sN .
With the help of MATLAB®
7.0.4, the membership function
is:













1175
1697
115
112
),(
115
112
7945
4714
,1
7945
4714
62
17
,)(
)(~
zzR
z
zzL
zsN

where:
3
1
23
1
3
2
)225(2
)268428025)(31(3)24285(2)31(3
)(
Pz
zziPzPi
zL



3
1
23
1
3
2
)225(
)268428025(3)8835(23
)(
Qz
zzQzQ
zR



,
with:
39569269988226351100500)22050(135522706039001125 23423
 zzzzzzzzP ,
39569269988226351100500)22050(135522706039001125 23423
 zzzzzzzzQ ,
as shown in Fig. 1. The overall shape turns out as expected.
The membership functions )(zL and )(zR have complex
values with their imaginary parts approaching zero when
7945
4714
62
17
 z for )(zL and
1175
1697
115
112
 z for )(zR . Hence,
the imaginary parts of these two functions have no influence
on the computational results and can be disregarded.
Next, we perform  -cuts of batch-arrival and service rates
and fuzzy expected number of sewage solids in the system
at eleven distinct  values: 0, 0.1, …, 1. Crisp intervals for
fuzzy expected number of sewage solids in the system at
different possibilistic  levels are presented in Table 1.
The fuzzy expected number of sewage solids in the system
sN
~ has two characteristics to be noted. First, the support of
sN
~ ranges from 0.2742 to 1.4443; this indicates that,
though the expected number of sewage solids in the system
is fuzzy, it is impossible for its values to fall below 0.2742
or exceed 1.4443. Second, the  -cut at 1 contains the
values from 0.5933 to 0.9739, which are the most possible
values for the fuzzy expected number of sewage solids in
the system.
Figure 1: The membership function for fuzzy expected
number of sewage solids in the system
Table 1:  -cuts of batch-arrival and service rates and
expected number of sewage solids in the system
 L
x
U
x
L
y
U
y
L
sN )( U
sN )(
0.00 1.00 4.00 11.00 14.00 0.2742 1.4443
0.10 1.10 3.90 11.10 13.90 0.3039 1.3919
0.20 1.20 3.80 11.20 13.80 0.3340 1.3410
0.30 1.30 3.70 11.30 13.70 0.3646 1.2912
0.40 1.40 3.60 11.40 13.60 0.3957 1.2427
0.50 1.50 3.50 11.50 13.50 0.4273 1.1953
0.60 1.60 3.40 11.60 13.40 0.4594 1.1491
0.70 1.70 3.30 11.70 13.30 0.4920 1.1038
0.80 1.80 3.20 11.80 13.20 0.5252 1.0596
0.90 1.90 3.10 11.90 13.10 0.5590 1.0163
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1.00 2.00 3.00 12.00 13.00 0.5933 0.9739
Similarly, the membership function for the fuzzy expected
waiting time in the queue ( qW
~
) is obtained as shown in Fig.
2. Crisp intervals for the fuzzy expected waiting time in the
queue at different possibilistic  levels are given in Table
2. For the fuzzy expected waiting time qW
~
, the range of qW
~
at 1 is [0.0714, 0.0790], indicating that expected
waiting time for any sewage solids definitely falls between
0.0714 and 0.0790. Moreover, the range of qW
~
at 0 is
[0.0657, 0.0896], indicating that the expected waiting time
in the queue will never exceed 0.0896 or fall below 0.0657.
Figure 2: The membership function for fuzzy expected
waiting time in the queue
Table 2:  -cuts of batch-arrival and service rates and
expected waiting time
 L
x
U
x
L
y
U
y
L
qW )( U
qW )(
0.00 1.00 4.00 11.00 14.00 0.0657 0.0896
0.10 1.10 3.90 11.10 13.90 0.0662 0.0884
0.20 1.20 3.80 11.20 13.80 0.0667 0.0872
0.30 1.30 3.70 11.30 13.70 0.0672 0.0860
0.40 1.40 3.60 11.40 13.60 0.0678 0.0849
0.50 1.50 3.50 11.50 13.50 0.0684 0.0838
0.60 1.60 3.40 11.60 13.40 0.0689 0.0828
0.70 1.70 3.30 11.70 13.30 0.0695 0.0818
0.80 1.80 3.20 11.80 13.20 0.0701 0.0808
0.90 1.90 3.10 11.90 13.10 0.0708 0.0799
1.00 2.00 3.00 12.00 13.00 0.0714 0.0790
5. Conclusions
This paper applies the concepts of  -cuts and Zadeh’s
extension principle to a batch-arrival queuing system with
multiple servers and constructs membership functions of the
expected number of customers and the expected waiting
time using paired NLP models. Following the proposed
approach,  -cuts of the membership functions are found
and their interval limits inverted to attain explicit closed-
form expressions for the system characteristics. Even when
the membership function intervals cannot be inverted,
system designers or managers can specify the system
characteristics of interest perform numerical experiments to
examine the corresponding  -cuts and then use this
information to develop or improve system processes.
For example, in Example, a designer (manager) can set the
range of the number of sewage solids to be [0.5252, 1.0596]
to reflect the desired service and find that the corresponding
 level is 0.8 with L
y 11.80 and U
y 13.20. In other
words, the designer can determine that the service rate is
between 11.80 and 13.20. Similarly, a designer can also set
the expected waiting time with “rounder” numbers like
[0.0678, 0.0849] to reflect the desired service, and the
corresponding  level is 0.4 with L
y 11.40 and
U
y 13.60. As this example demonstrates, the approach
proposed in this paper provides practical information for
system designers and practitioners.
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Paper ID: 02091301 140

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