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792 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 30, NO. 4, MAY 2012
DORA: Dynamic Optimal Random Access for
Vehicle-to-Roadside Communications
Man Hon Cheung, Student Member, IEEE, Fen Hou, Member, IEEE,
Vincent W.S. Wong, Senior Member, IEEE, and Jianwei Huang, Senior Member, IEEE
Abstract—In this paper, we study random access in a drive-
thru scenario, where roadside access points (APs) are installed on
a highway to provide temporary Internet access for vehicles. We
consider vehicle-to-roadside (V2R) communications for a vehicle
that aims to upload a file when it is within the APs’ coverage
ranges, where both the channel contention level and transmission
data rate vary over time. The vehicle will pay a fixed amount
each time it tries to access the APs, and will incur a penalty if
it cannot finish the file uploading when leaving the APs. First,
we consider the problem of finding the optimal transmission
policy with a single AP and random vehicular traffic arrivals.
We formulate it as a finite-horizon sequential decision problem,
solve it using dynamic programming (DP), and design a general
dynamic optimal random access (DORA) algorithm. We derive
the conditions under which the optimal transmission policy has
a threshold structure, and propose a monotone DORA algorithm
with a lower computational complexity for this special case. Next,
we consider the problem of finding the optimal transmission
policy with multiple APs and deterministic vehicular traffic
arrivals thanks to perfect traffic estimation. We again obtain the
optimal transmission policy using DP and propose a joint DORA
algorithm. Simulation results based on a realistic vehicular traffic
model show that our proposed algorithms achieve the minimal
total cost and the highest upload ratio as compared with some
other heuristic schemes. In particular, we show that the joint
DORA scheme achieves an upload ratio 130% and 207% better
than the heuristic schemes at low and high traffic densities,
respectively.
Index Terms—Random access, medium access control, vehic-
ular ad hoc networks, dynamic programming, Markov decision
processes, threshold policy.
I. INTRODUCTION
VEHICULAR ad hoc networks (VANETs) enable au-
tonomous data exchanges among vehicles and roadside
access points (APs), and are essential to various intelligent
transportation system (ITS) applications. For example, safety
applications (such as cooperative forward collision warning,
lane change warning, and left turn assistant [1], [2]) have been
Manuscript received 29 April 2011; revised 10 October 2011.This work
was supported by the AUTO21, a member of the Network of Centres of
Excellence of Canada program, and the Natural Sciences and Engineering
Research Council (NSERC) of Canada.
M. H. Cheung and V. W. S. Wong are with the Department of Electrical
and Computer Engineering, University of British Columbia, Vancouver, BC,
Canada, V6T 1Z4 (e-mail: {mhcheung, vincentw}@ece.ubc.ca).
F. Hou was with the Department of Information Engineering, the Chi-
nese University of Hong Kong, Hong Kong, China. She is now with
the Joint Research Center in Ubiquitous Computing, Computing/Computer
Studies Programme, Macao Polytechnic Institute, Macao, China (e-mail:
fhou@ipm.edu.mo).
J. Huang is with the Department of Information Engineering,
the Chinese University of Hong Kong, Hong Kong, China (e-mail:
jwhuang@ie.cuhk.edu.hk).
Digital Object Identifier 10.1109/JSAC.2012.120513.
proposed to improve the safety of the passengers by informing
the vehicles of potential dangers ahead of time. Non-safety
applications (such as traffic management, instant messaging,
and media content delivery) have been designed to avoid traffic
congestion and improve the experience of driving. Clearly, the
quality of service (QoS) requirements of various applications
are different.
VANETs support various ITS applications through different
types of communication mechanisms, including vehicle-to-
roadside (V2R) and vehicle-to-vehicle (V2V) communications
[3]. V2R communications involve data transmissions between
vehicular nodes and roadside APs. V2V communications only
involve data exchanges among vehicular nodes. For both types,
we can further classify the communications as either single-
hop or multi-hop. In this paper, we focus on analyzing V2R
single-hop uplink transmissions from vehicles to APs. Due
to the limited communication opportunities between vehicles
and APs, efficient resource allocation (either centralized or
distributed) is crucial for the successful deployment of V2R
ITS applications.
In the centralized setting, the AP schedules the trans-
missions from different vehicles based on some predefined
criteria. Hadaller et al. in [4] proposed a scheduling protocol
that grants channel access to a vehicle that achieves the
maximum transmission rate. Analytical and simulation results
showed significant overall system throughput improvement
over a benchmark scheme. Zhang et al. in [5] considered the
case where roadside APs only store the data uploaded by the
vehicles. Scheduling priority is determined by two factors:
data size and deadline. A request with either a smaller data
size or an earlier deadline will be served first. Alcaraz et al. in
[6] considered both uplink and downlink scheduling of non-
real-time traffic for non-safety applications. The scheduling
problem was formulated as a constrained linear quadratic reg-
ulator design problem that aims to reduce the residual queue
backlog for each user. However, because centralized resource
allocation is not scalable due to its computational complexity,
we focus on distributed resource allocation scheme in this
paper.
In the distributed setting, the vehicles contend for the chan-
nel for transmission based on the applications’ QoS require-
ments. Shrestha et al. in [7] considered the scenario where
the data packets are first distributed from the roadside units
(RSUs) to the onboard units (OBUs). The OBUs then bargain
with each other for the missing data packets, and exchange
them using BitTorrent protocol. Jarupan et al. in [8] proposed
a cross-layer protocol for V2R multi-hop communication. The
medium access control (MAC) module collects information of
0733-8716/12/$25.00 c 2012 IEEE
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CHEUNG et al.: DORA: DYNAMIC OPTIMAL RANDOM ACCESS FOR VEHICLE-TO-ROADSIDE COMMUNICATIONS 793
local data traffic, and the routing module finds a path with the
minimum delay. Niyato et al. in [9] proposed a hierarchical
optimization framework for downlink data streaming in V2R
communications. The optimal pricing and bandwidth reserva-
tion of a service provider is obtained using game theory, and
the optimal download policy of an OBU is obtained using
constrained Markov decision processes. Tan et al. in [10]
analyzed the performance of a downlink resource allocation
scheme in a V2R communication system with one AP on a
road. The distribution of the number of bytes downloaded
per drive-thru was derived using Markov reward processes.
Roman et al. in [11] proposed a cross-layer protocol in the
physical and MAC layers that addresses the issues of channel
fading, synchronization, and channel contention. Performance
analysis was presented for the channel contention scheme, and
a testbed was used to evaluate the proposed protocol.
In this paper, we aim to design a uplink random access
algorithm that is distributed in nature, so that it is compatible
with the IEEE 802.11p standard that is developed to facilitate
the provision of wireless access in vehicular environment
[12], [13]. Different from most previous works on heuristic
distributed uplink V2R communication algorithm design, we
aim at designing an optimal uplink resource allocation scheme
in VANETs analytically in this paper.
In this work, we consider the drive-thru scenario [14], where
vehicles pass by several APs located along a highway and
obtain Internet access for only a limited amount of time. We
assume that a vehicle wants to upload a file when it is within
the coverage ranges of the APs, and needs to pay for the
attempts to access the channel. As both the channel contention
level and achievable data rate vary over time, the vehicle needs
to decide when to transmit by taking into account the required
payment, the application’s QoS requirement, and the level of
contention in current and future time slots. Because of the
dynamic nature of the problem, we formulate it as a finite-
horizon sequential decision problem and solve it using the
dynamic programming (DP).
The main contributions of our work are as follows:
• Optimal Access Policy Design: In the case of a single
AP with random vehicular traffic, we propose a general
dynamic optimal random access (DORA) algorithm to
compute the optimal access policy. We further extend
the results to the case of multiple consecutive APs and
propose a joint DORA (JDORA) algorithm to compute
the optimal policy.
• Low Complexity Algorithm: We consider a special yet
practically important case of a single AP with constant
data rate. We show that the optimal policy in this case
has a threshold structure, which motivates us to propose a
low complexity and efficient monotone DORA algorithm.
• Superior Performance: Extensive simulation results show
that our proposed algorithms achieve the minimal total
cost and the highest upload ratio as compared with three
other heuristic schemes. In the multi-AP scenario, the
performance improvements in upload ratio of the JDORA
scheme are 130% and 207% at low and high traffic
densities, respectively.
The rest of the paper is organized as follows. We describe
our system model in Section II and formulate the DP problem
Fig. 1. Drive-thru vehicle-to-roadside (V2R) communications with multiple
APs.
in Section III. The general and monotone DORA algorithms
for single AP are proposed in Section IV-A, and the JDORA
algorithm for multiple APs is discussed in Section IV-B.
Simulation results are given in Section V, and the paper is
concluded in Section VI.
II. SYSTEM MODEL
We consider a drive-thru scenario on a highway as shown
in Fig. 1, where multiple APs are installed and connected to
a backbone network to provide Internet services to vehicles
within their coverage ranges. We focus on a vehicle that wants
to upload a single file of size S when it moves through a
segment of this highway with a set of APs J = {1, . . . , J},
where the vehicles pass through the ith
AP before the jth
AP
for i < j with i, j ∈ J . We assume that the jth
AP has a
transmission radius Rj. We also assume that the vehicle is
connected to at most one AP at a time. If the coverage areas
of the APs are overlapping, then proper handover between the
APs will be performed [15]. For the ease of exposition, we
assume that the APs are set up in a way that any position in
this segment of highway is covered by an AP. Our work can
easily be extended to consider the settings where the coverage
areas of adjacent APs are isolated from each other.
A. Traffic Model
Let λ denote the average number of vehicles passing by a
fixed AP per unit time. We assume that the number of vehicles
moving into this segment of the highway follows a Poisson
process [16] with a mean arrival rate λ. Let ρ denote the
vehicle density representing the number of vehicles per unit
distance along the road segment, and ν be the speed of the
vehicles. From [17], we have
λ = ρν. (1)
The relation between the vehicle density ρ and speed ν is
given by the following equation [17]:
ν = νf (1 − ρ/ρmax), (2)
where νf is the free-flow speed when the vehicle is moving on
the road without any other vehicles, and ρmax is the vehicle
density during traffic jam.

ζ(1,1) ζ(2,1)ζ(1,10) ζ(3,1)ζ(2,15) ζ(3,12)ζ(1,2)
Fig. 2. An example of the time line representation for the events happened
with three APs (i.e., J = {1, 2, 3}). Here, we assume that T1 = 10, T2 =
15, and T3 = 12. With respect to the time line, we have T1 = {1, . . . , 10},
T2 = {11, . . . , 25}, and T3 = {26, . . . , 37}. It is clear from the figure that
ζ(j, τ) =
Pj−1
i=0 Ti + τ, ∀ τ ∈ {1, . . . , Tj }, where T0 = 0.
As we are studying the traffic flow in steady state, all the
vehicles within the coverage range are assumed to move with
the same speed ν in (2). Let · denote the floor function.
The maximum number of vehicles that can be accommodated
within the coverage range of the jth
AP is given by
Nmax,j = 2Rjρmax , ∀ j ∈ J . (3)
B. Channel Model
Wireless signal propagations suffer from path loss, shadow-
ing, and fading. Since the distance between the vehicle and the
AP varies in the drive-thru scenario, we focus on the dominant
effect of channel attenuation due to path loss. The data rate
at time slot t is given by
wt = W log2 1 +
P
N0Wdγ
t
, (4)
where W is the channel bandwidth, P is the transmit power
of the vehicle, dt is the distance between the vehicle and the
closest AP at time slot t, and γ is the path loss exponent.
We assume that the additive white Gaussian noise has a zero
mean and a power spectral density N0/2. In addition, we also
consider a special case with fixed data rate in Section IV-A1.
C. Distributed Medium Access Control (MAC)
We consider a slotted MAC protocol, where time is divided
into equal time slots of length Δt. We assume that there is
perfect synchronization between the APs and the vehicles with
the use of global positioning system (GPS) [18]. The total
number of time slots that the vehicle stays within the coverage
range of the jth
AP is Tj =
2Rj
νΔt . We use the notation ζ(j, τ)
to denote the τth
time slot when the vehicle is in the coverage
area of the jth
AP, i.e.,
ζ(j, τ) =
j−1
i=0
Ti + τ, ∀ τ ∈ {1, . . . , Tj}, (5)
where T0 = 0. The set of time slots in the jth
AP with respect
to this time line representation is Tj = {ζ(j, 1), . . . , ζ(j, Tj)}.
An example of the time line representation is given in Fig. 2.
When the vehicle first enters the coverage range of the jth
AP, it declares the type of its application to the AP. In return,
the jth
AP informs the vehicle the channel contention in the
coverage range (λ and psucc
t , ∀ t ∈ Tj), data rate in all the
time slots in the jth
coverage range (i.e., wt, ∀ t ∈ Tj), the
price qj, and the estimated number of vehicle departures from
Fig. 3. The structure of a time slot of the jth AP.
the coverage range in all the time slots in the jth
coverage
range (i.e., lt, ∀ t ∈ Tj). We further elaborate these system
parameters as follows:
• psucc
t represents the probability that the vehicle can
successfully obtain access in time slot t ∈ Tj after
contending with all the vehicles in the jth
coverage range.
psucc
t is estimated by the AP based on the level of system
contention and it varies over time. Since psucc
t is related
to the number of vehicles nt currently in the jth
coverage
range at time slot t, we define psucc
t = gj(nt), where
gj is a strictly decreasing function. An AP knows the
value of nt, since vehicles need to establish and terminate
their connections when they enter and leave the coverage
range, respectively.
• qj ≥ 0 denotes the amount a vehicle needs to pay the AP
for each time slot that it sends a transmission request in
the jth
coverage range, even it fails to access the channel.
The value of qj does not change over time.
• lt represents the number of vehicle departing at time
slot t ∈ Tj from the jth
coverage range. Since all the
vehicles move with constant speed ν in the traffic model,
we assume that (lt, ∀ t ∈ Tj) are accurately known by
the jth
AP, and are sent to the vehicle when it enters the
coverage range.
In each time slot t ∈ Tj in the jth
coverage range, the jth
AP first broadcasts the value of psucc
t to all the vehicles in
its coverage range. If a vehicle decides to transmit within this
time slot, it sends a request to the jth
AP at its scheduled mini-
slot, where Nmax,j mini-slots are reserved for transmission
requests. The transmissions of requests are thus collision-free.
After collecting the requests from all vehicles in its coverage
range, the jth
AP assigns the time slot to one of these vehicles.
The vehicle, which receives the acknowledgement (ACK), can
transmit the data packets in the remaining time Δtdata of this
time slot, where Δtdata < Δt. The structure of a time slot is
shown in Fig. 3.
Meanwhile, regardless of which vehicle is granted the time
slot, each vehicle which requested to transmit in the time slot
needs to pay qj to the jth
AP. Without such pricing, each
vehicle would send a request in every time slot, which unnec-
essarily increases the contention level and prevents efficient
allocation of time slots to the most needed application.
The vehicle aims to achieve a good tradeoff between the
total uploaded file size and the total payment to the APs
according to the QoS requirement of the application. For
example, a higher priority may be placed on the total uploaded
file size for safety applications, but on the total payment for
non-safety applications. The problem is further complicated
by the time-varying data rate wt and channel contention level.
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Therefore, it is a challenge for the vehicle to decide when to
request for data transmission.
III. PROBLEM FORMULATION
In this section, we formulate the optimal transmission
problem of a single vehicle as a finite-horizon sequential
decision problem [19]. The decision epochs of the vehicle are
t ∈ T =
j∈J
Tj =
j∈J
ζ(j, 1), . . . , ζ(j, Tj) , (6)
where T is the set of all the time slots in the total of J coverage
ranges.
The system state of a vehicle is defined as (s, psucc
), where
the state element s ∈ S = [0, S] represents the remaining size
(in bits) of the single file to be uploaded. If we denote the
number of vehicles in the coverage range of the jth
AP as
n ∈ Nj = {1, . . ., Nmax,j}, then psucc
∈ Pj = {gj(n) : n ∈
Nj}.
At any state (s, psucc
), the vehicle has two possible actions:
a ∈ A = {0, 1}, (7)
where action a = 1 implies that the vehicle decides to request
to transmit, and action a = 0 otherwise.
The cost at state (s, psucc
) with action a ∈ A at time slot
t ∈ Tj in the jth
coverage range is
ct(s, psucc
, a) = aqj, ∀ t ∈ Tj. (8)
After the vehicle has left the Jth
coverage range at time
ζ(J, TJ + 1), we define a self-incurred penalty of the vehicle
for not being able to complete the file uploading as
ˆcζ(J,TJ +1)(s, psucc
) = h(s), (9)
where h(s) ≥ 0 is a nondecreasing function of s with
h(0) = 0. The function depends on the QoS requirement of
the application. To sum up, each vehicle is incurred with two
costs: the transmission cost in each time slot in (8) and the
penalty after leaving the Jth
coverage range in (9).
The state transition probability
pt (s , psucc
) | (s, psucc
), a is the probability that the
system will go into state (s , psucc
) if action a is taken at
state (s, psucc
) at time slot t ∈ T. Since the transition from
psucc
to psucc
is independent of the value of s but depends
on time t, we have
pt (s , psucc
) | (s, psucc
), a
= pt s | (s, psucc
), a pt psucc
| psucc
.
(10)
With action a = 1, we have
pt s | (s, psucc
), 1 =
⎧
⎪⎨
⎪⎩
psucc
, if s = [s − wtΔtdata]+
,
1 − psucc
, if s = s,
0, otherwise,
(11)
where [x]+
= max{0, x}. The first and second cases corre-
spond to the scenarios of successful and unsuccessful packet
transmissions, respectively. With action a = 0, we have
pt s | (s, psucc
), 0 =
1, if s = s,
0, otherwise,
(12)
where the remaining size of the file to upload does not change.
The derivation of pt psucc
| psucc
will be discussed in detail
in Section IV.
Let δt : S × Pj → A be the decision rule that specifies
the transmission decision of the vehicle at state (s, psucc
)
at time slot t ∈ Tj in the jth
coverage range. We define a
policy as a set of decision rules covering all the states as
π = (δt(s, psucc
), ∀ s ∈ S, psucc
∈ Pj, t ∈ Tj, ∀ j ∈ J ). We
denote (sπ
t , psucc,π
t ) as the state at time slot t if policy π is
used, and we let Π be the feasible set of π. The vehicle aims
to find an optimal policy that minimizes the total expected
cost, which can be formulated as the following optimization
problem
min
π∈Π
Eπ,(S,psucc
1 )
J
j=1
Tj
τ=1
cζ(j,τ) sπ
ζ(j,τ), psucc,π
ζ(j,τ) ,
δζ(j,τ)(sπ
ζ(j,τ), psucc,π
ζ(j,τ) ) +ˆcζ(J,TJ +1)(sπ
ζ(J,TJ +1), psucc,π
ζ(J,TJ +1)) ,
(13)
where Eπ,(S,psucc
1 ) denotes the expectation with respect to
the probability distribution by policy π with an initial state
(S, psucc
1 ) at time slot t = ζ(1, 1) = 1. In the following
section, we will study two scenarios: single AP with random
vehicular traffic and multiple APs with traffic pattern estima-
tion.
IV. FINITE-HORIZON DYNAMIC PROGRAMMING
In this section, we describe how to obtain the optimal
transmission policies in both the single-AP and multiple-AP
scenarios using finite-horizon dynamic programming. We first
study the single-AP scenario with random vehicular traffic
arrival in Section IV-A. In particular, we consider a special
case that the optimal policy has a threshold structure in
Section IV-A1. When the traffic pattern can be estimated
accurately, we consider a joint AP optimization in Section
IV-B.
A. Single AP Optimization with Random Vehicular Traffic
Since we are considering one AP (i.e., J = {1}) in this
subsection, we drop the subscript j for simplicity. Although
the exact traffic pattern (i.e., the exact number of vehicles
in the coverage range of the AP in each time slot) is not
known, the vehicles arrive according to a Poisson process
with parameter λ. Meanwhile, the parameters lt (∀ t ∈ T ),
ρmax, Δt, R, and the function g(·) are available. The transition
probability of psucc
is given by
pt psucc
| psucc
= pt g(n ) | g(n) = pt n | n
=
(λΔt)n −n+lt+1
(n −n+lt+1)! φt(n) , if n − lt+1 ≤ n ≤ Nmax,
0, otherwise,
(14)
where φt(n) =
Nmax−n+lt+1
y=0
(λΔt)y
y! is a normalization
factor. Because psucc
= g(n) is a strictly decreasing function
of n, there is a one-to-one mapping between psucc
and n as
shown in the first two equalities in (14). The expression after
the third equalities describes the probability with n −n+lt+1

arrivals due to the Poisson process and lt+1 deterministic
departures at time t + 1. n is lower-bounded by n − lt+1 ≥ 0
when there is no vehicle arrival, and is upper-bounded by
Nmax.
In this subsection, since we consider J = {1}, we can
simplify problem (13) as
min
π∈Π
Eπ,(S,psucc
1 )
T
t=1
ct sπ
t , psucc,π
t , δt(sπ
t , psucc,π
t )
+ˆcT +1(sπ
T +1, psucc,π
T +1 ) .
(15)
Let vt(s, psucc
) be the minimal expected total cost that the
vehicle has to pay from time t to time T + 1 when it is in the
coverage range, given that the system is in state (s, psucc
)
immediately before the decision at time slot t ∈ T . The
optimality equation [19, pp.83] relating the minimal expected
total cost at different states for t ∈ T is
vt(s, psucc
) = min
a∈A
{ψt(s, psucc
, a)}, (16)
where
ψt(s, psucc
, a) = ct(s, psucc
, a)+
s ∈S psucc ∈P
pt (s , psucc
) | (s, psucc
), a vt+1(s , psucc
) (17)
= aq+
psucc ∈P
pt psucc
|psucc
apsucc
vt+1 [s − wtΔtdata]+
, psucc
+ 1 − apsucc
vt+1(s, psucc
) .
(18)
The first and second terms on the right hand side of (17)
are the immediate cost and the expected future cost in the
remaining time slots in the coverage range for choosing action
a, respectively. Equation (18) follows directly by evaluating
(17) using (10) - (12). For time t = T + 1, we have the
boundary condition that
vT +1(s, psucc
) = ˆcT +1(s, psucc
) = h(s). (19)
Lemma 1: The value of ψt(s, psucc
, a), ∀ t ∈ T , can be
obtained as
ψt(s, psucc
, a) = aq +
Nmax−n+lt+1
m=0
(λΔt)m
m!φt(n)
× apsucc
vt+1 [s − wtΔtdata]+
, g(n + m − lt+1)
+ 1 − apsucc
vt+1 s, g(n + m − lt+1) ,
(20)
where n = g−1
(psucc
) is the number of vehicles in the
coverage range of the AP.
Proof: The result follows directly by evaluating (18)
using (14).
Intuitively, the minimal expected cost vt(s, psucc
) should
be smaller when the remaining file size s to be uploaded is
smaller. It is confirmed by the following lemma:
Lemma 2: vt(s, psucc
) is a nondecreasing function in s,
∀ psucc
∈ P, t ∈ T .
The proof of Lemma 2 is given in Appendix A.
Algorithm 1 General DORA Algorithm for single AP opti-
mization (i.e., problem (15)).
1: Planning Phase:
2: Input the traffic parameters: ν, λ, ρmax, lt (∀ t ∈ T );
3: Input the system parameters: h(·), S, R, wt (∀ t ∈ T ), q, Δt,
Δtdata, σ, g(·);
4: Set the boundary condition vT +1(s, psucc
), ∀ s ∈ S, ∀ psucc
∈
P using (19);
5: t := T;
6: while t ≥ 1
7: for psucc
∈ P
8: s := 0;
9: while s ≤ S
10: Calculate ψt(s, psucc
, a), ∀ a ∈ A = {0, 1} using (20);
11: δ∗
t (s, psucc
) := arg min
a∈A
{ψt(s, psucc
, a)};
12: vt(s, psucc
) := ψt
`
s, psucc
, δ∗
t (s, psucc
)
´
;
13: s := s + σ;
14: end while
15: end for
16: t := t − 1;
17: end while
18: Output the optimal policy π∗
for use in the transmission phase;
19: Transmission Phase:
20: t := 1 and s := S;
21: while t ≤ T
22: Receive the information of psucc
from the AP;
23: Set action a := δ∗
t (s, psucc
) based on the policy π∗
;
24: If action a = 1
25: Send a request to the AP;
26: If ACK is received from the AP
27: Transmit packets with total size wtΔtdata;
28: s := [s − wtΔtdata]+
;
29: end if
30: end if
31: t := t + 1;
32: end while
Using the optimality equation and backward induction [19,
pp. 92], we propose the general dynamic optimal random
access (DORA) algorithm in Algorithm 1 to obtain the optimal
policy π∗
= (δ∗
t (s, psucc
∈ P, t ∈ T ), where
δ∗
t (s, psucc
) = arg min
a∈A
{ψt(s, psucc
, a)}. (21)
Theorem 1: The policy π∗
obtained from Algorithm 1 is
the optimal solution of problem (15).
Proof: Using the principle of optimality [20, pp. 18], we
can show that π∗
is the optimal solution of problem (15).
The proposed DORA algorithm consists of two phases:
Planning phase and transmission phase. The planning phase
starts when the vehicle enters the coverage range. The vehicle
then obtains information from the AP and computes the
optimal policy π∗
offline using dynamic programming. In
fact, π∗
is a contingency plan that contains information about
the optimal decisions at all possible states (s, psucc
) in the
coverage range. In the transmission phase, the transmission
decision in each time slot is made according to the optimal
policy π∗
, and s is updated depending on whether the time
slot is granted to the vehicle for transmission or not. We let
σ > 0 be the granularity of the discrete state element s in the
algorithm.
1) Special Case: Convex Penalty Function and Fixed Data
Rate: In this subsection, we further investigate a special yet
MAY 2013

practically important case with convex penalty function and
non-adaptive data rate [10]. Specifically, if the self-incurred
penalty function h(s) is convex and the data rate wt is fixed
within the coverage range (i.e., wt = w, ∀ t ∈ T ), we can
show in Appendix B that ψt(s, psucc
, a) is subadditive [19,
pp. 103] on S × A, ∀ t ∈ T , which is defined as follows.
Definition 1: Given psucc
, the function ψt(s, psucc
, a) is
subadditive on S × A if for ˆs, ˇs ∈ S and â, ˇa ∈ A, where
ˆs ≥ ˇs and â ≥ ˇa, we have
ψt(ˆs, psucc
, â) + ψt(ˇs, psucc
, ˇa)≤ψt(ˆs, psucc
, ˇa) + ψt(ˇs, psucc
, â).
(22)
Furthermore, with δ∗
t (s, psucc
) as defined in (21), we can
establish the threshold structure of the optimal policy [19,
pp. 104, 115], [21].
Theorem 2: If h(s) is a convex and nondecreasing func-
tion in s, and the data rate wt is fixed such that wt =
w, ∀ t ∈ T , then we have a threshold optimal policy π∗
=
(δ∗
t (s, psucc
∈ P, t ∈ T ) in s as follows:
δ∗
t (s, psucc
) =
1, if s > s∗
t (psucc
),
0, otherwise,
(23)
where s∗
t (psucc
) is the threshold that depends on both psucc
and t.
The proof of Theorem 2 is given in Appendix C. By
modifying Algorithm 1, we are ready to propose the monotone
DORA algorithm with a lower computational complexity
in Algorithm 2 using monotone backward induction [19,
pp. 111]. Let Ã ⊆ A be the set of actions that we need
to consider in the minimization in line 11 in Algorithm 2.
When δ∗
t (s, psucc
) = 1 and flag = 0 are satisfied (line 13),
which means that the threshold s∗
t (psucc
) is reached, set Ã
is reduced from {0, 1} to {1} and the threshold s∗
t (psucc
) is
recorded (line 14). Then the minimization in line 11 is readily
known, since set Ã = {1} is a singleton. The computational
complexity is thus reduced. Moreover, memory can be saved,
because we do not need to store the complete optimal policy
π∗
= (δ∗
t (s, psucc
∈ P, t ∈ T ). We just need
to store the thresholds (s∗
t (psucc
), ∀ psucc
∈ P, t ∈ T ), which
completely characterize the optimal policy π∗
as shown in
(23).
B. Joint AP Optimization with Deterministic Vehicular Traffic
In the previous subsection, we consider the optimization
problem in a single AP. In this subsection, we extend the
result to the case of multiple APs, where we assume that
the traffic pattern (i.e., the exact number of vehicles in the
coverage ranges of the APs in each time slot) can be estimated
accurately. The traffic pattern can be estimated in various
ways, such as by installing a traffic monitor at a place before
the first AP to observe the actual traffic pattern when the
vehicles pass by (e.g., using computer vision [22] and pattern
recognition [23]). If the traffic flow reaches the steady state
(as discussed in Section II-A), the estimation of the number
of vehicles nt at time t ∈ T can be reasonably accurate. As a
result, the values of psucc
t = gj(nt), ∀ t ∈ T can be obtained
accurately. As an example, we consider that the traffic model
is as described in Section II-A, and all the APs have the
same transmission radii. After the traffic monitor has estimated
the values of psucc
τ , ∀ τ ∈ T1 for the first coverage range, it
Algorithm 2 Monotone DORA Algorithm for single AP opti-
mization (i.e., problem (15)) for the special case with convex
penalty function h(s) and fixed data rate wt.
1: Planning Phase:
2: Input the traffic parameters: ν, λ, ρmax, lt (∀ t ∈ T );
3: Input the system parameters: h(·), S, R, wt (∀ t ∈ T ), q, Δt,
Δtdata, σ, g(·);
4: Set the boundary condition vT +1(s, psucc
), ∀ s ∈ S, ∀ psucc
∈
P using (19);
5: t := T;
6: while t ≥ 1
7: for psucc
∈ P
8: Set s := 0, flag := 0, and Ã := {0, 1};
9: while s ≤ S
, a), ∀ a ∈ Ã using (20);
11: δ∗
t (s, psucc
) := arg min
a∈ Ã
{ψt(s, psucc
, a)};
12: vt(s, psucc
) := ψt
`
s, psucc
, δ∗
t (s, psucc
)
´
;
13: if δ∗
t (s, psucc
) = 1 and flag = 0
14: Set Ã := {1}, s∗
t (psucc
) = s, and flag = 1;
15: end if
16: s := s + σ;
17: end while
18: end for
19: t := t − 1;
20: end while
21: Output the thresholds (s∗
t (psucc
), ∀ psucc
∈ P, t ∈ T ) for use
in the transmission phase;
23: t := 1 and s := S;
24: while t ≤ T
25: Receive the information of psucc
from the AP;
26: If s > s∗
t (psucc
)
;
31: end if
32: end if
33: t := t + 1;
34: end while
can set psucc
ζ(j,τ) := psucc
τ for the remaining coverage ranges
j ∈ J {1}.
The optimality equations relating the minimal expected total
cost at different time t ∈ T for problem (13) are similar to
that described in Section IV-A, but are simplified because we
assume that psucc
t , ∀ t ∈ T are known. At time t ∈ Tj, we
have
vt(s, psucc
t ) = min
a∈A
{ψt(s, psucc
t , a)}, (24)
where
ψt(s, psucc
t , a)
= ct(s, psucc
t , a)+
s ∈S
pt (s , psucc
t+1 )|(s, psucc
t ), a vt+1(s , psucc
t+1 )
= aqj + apsucc
t vt+1 [s − wtΔtdata]+
, psucc
t+1
+ 1 − apsucc
t vt+1(s, psucc
t+1 ). (25)
The second line in (25) is obtained by using (11) and
(12). After the vehicle has left the Jth coverage range at
t = ζ(J, TJ + 1), the boundary condition is
vζ(J,TJ +1)(s, psucc
ζ(J,TJ +1)) = ˆcζ(J,TJ +1)(s, psucc
ζ(J,TJ +1)) = h(s).
(26)

Algorithm 3 JDORA Algorithm for joint AP optimization (i.e.,
problem (13)).
1: Planning Phase:
2: Input the traffic parameters: ν, psucc
t (∀ t ∈ T);
3: Input the system parameters: h(·), S, Rj (∀ j ∈ J ), wt (∀ t ∈
T), qj(∀ j ∈ J ), Δt, Δtdata, σ;
4: Set the boundary condition vζ(J,TJ +1)(s, psucc
ζ(J,TJ +1)), ∀ s ∈ S
using (26);
5: t := ζ(J, TJ );
6: while t ≥ 1
7: s := 0;
8: while s ≤ S
t , a), ∀ a ∈ A = {0, 1} using (25);
10: δ∗
t (s, psucc
t ) := arg min
a∈A
{ψt(s, psucc
t , a)};
11: vt(s, psucc
t ) := ψt
`
s, psucc
t , δ∗
t (s, psucc
t )
´
;
12: s := s + σ;
13: end while
14: t := t − 1;
15: end while
16: Output the optimal policy π∗
for use in the transmission phase;
18: t := 1 and s := S;
19: while t ≤ ζ(J, TJ )
20: Set action a := δ∗
t (s, psucc
t ) based on the policy π∗
;
21: If action a = 1
;
26: end if
27: end if
28: t := t + 1;
29: end while
The JDORA algorithm for joint AP optimization is given in
Algorithm 3. In Algorithm 3, the vehicle first needs to obtain
the values of psucc
t , ∀ t ∈ T, from the traffic monitor. In the
planning phase, for each s ∈ S and t ∈ T, the optimal decision
rule δ∗
t (s, psucc
t ) is the action that minimizes the expected total
cost (line 10), where the expected total cost ψt(s, psucc
t , a)
for all possible actions is calculated (line 9) based on vt+1
obtained (line 11) in the previous iteration t + 1. After the
process is repeated for all t ∈ T (line 6) and s ∈ S (line 8),
we obtain the optimal policy π∗
. In the transmission phase,
the transmission decision in each time slot is made according
to the optimal policy π∗
, and it follows the MAC protocol
described in Section II-C.
Theorem 3: The policy π∗
= (δ∗
t (s, psucc
t ), ∀ s ∈ S, t ∈ T)
obtained from Algorithm 3 is the optimal solution of problem
(13) when psucc
t , ∀ t ∈ T are accurately known.
Proof: The result follows directly from the principle of
optimality [20, pp. 18].
V. PERFORMANCE EVALUATIONS
In this section, we first compare Algorithms 1 and 3 with
three heuristic schemes using the traffic model described in
Section II-A in both the single-AP and multiple-AP scenarios.
In particular, we study the performance of Algorithm 3 under
imperfect estimations of the psucc
t in the multiple-AP scenario.
We then study the threshold policies obtained by Algorithm
2.
TABLE I
SIMULATION PARAMETERS
Parameters Values
Number of APs J 1, 5
AP’s transmission radius R 100 m
Free-flow speed νf 110 km/hr
Vehicle jam density ρmax 100 veh/km
Duration of a time slot Δt 0.02 sec
Duration for data transmission in 0.018 sec
a time slot Δtdata
Channel bandwidth W 20 MHz
Transmit signal-to-noise ratio P
N0W
60 dB
Path loss exponent γ 3
Payment per time slot q 1
Contention window cw ∈ [cwmin, cwmax] [1, 8]
MCBC parameter K (used in [11]) 3
MCBC parameter [p1p2, p3] (used in [11]) [0.12, 0.77, 0.86]
MCBC parameter F (used in [11]) 15
The three heuristic schemes that we consider are as follows.
The first heuristic scheme is a greedy algorithm, in which
each vehicle sends transmission requests at all the time slots
if its file upload is not complete. That is, the greedy algorithm
aims to maximize the total uploaded file size. The second
heuristic scheme is the exponential backoff algorithm that
is similar to the one used in the IEEE 802.11. We have
slightly modified it for the system that we consider as follows.
Each vehicle has a counter, which randomly and uniformly
chooses an initial integer value cnt from the interval [0, cw),
where cw is the contention window size. The value of cnt
is decreased by one after each time slot. When cnt = 0, the
vehicle will send a request. If the vehicle has sent a request
in a time slot, the size of cw ∈ [cwmin, cwmax] will change
according to the response from the AP: If an ACK is received
from the AP, cw is set to cwmin. Otherwise, cw is doubled
until it reaches cwmax. For the DORA, JDORA, greedy,
and exponential schemes, we assume that the APs allow
the vehicles to share the channel with an equal probability.
Therefore, psucc
t = 1/nt. The third heuristic scheme is the
MAC protocol in the multi-carrier burst contention (MCBC)
scheme [11]. Similar to the greedy scheme, a vehicle will send
a request if it has data to send in each time slot. However,
the vehicles need to undergo K rounds of contention in each
time slot. First, in round r, a vehicle survives the contention
with probability pr. Each of these vehicles will choose a
random integer in {1, . . . , F}. Vehicles that have chosen the
largest number can proceed to round r + 1. The transmission
is successful if there is only one vehicle left in round K.
Otherwise, packet collision will occur.
For the evaluations of all the schemes, we use the convex
self-incurred penalty function
h(s) = bs2
, ∀ s ∈ S, (27)
where b ≥ 0 is a constant. The three heuristic schemes are
evaluated using a similar transmission phase as in Algorithms
1 and 3, but with π∗
in Algorithms 1 and 3 replaced by the
corresponding policies. The simulation parameters are listed
in Table I.
We first study the impact of penalty parameter b on the total
uploaded file size for S = 100 Mbits and ρ = 20 veh/km in
one AP. As shown in Fig. 4, by increasing b, a larger penalty
NO.4,MAY 2013

0 0.5 1 1.5 2
0
10
20
30
40
50
60
70
80
90
100
Penalty b
TotalUploadedFileSize(Mbits)
Greedy
MCBC
DORA
Exponential Backoff
Fig. 4. Total uploaded file size against the penalty parameter b for S = 100
Mbits and ρ = 20 veh/km with a single AP. As b increases, a larger file size
is uploaded for the DORA scheme.
20 30 40 50 60 70 80
500
1000
1500
2000
Density ρ (veh/km)
TotalCost
Exponential Backoff
MCBC
Greedy
DORA
48%
24%
Fig. 5. Total cost versus traffic density ρ for file size S = 200 Mbits with
a single AP. The DORA scheme has the minimal total cost.
is incurred on the size of the file not yet uploaded by using
Algorithm 1. As a result, a larger file size is uploaded to
reduce the penalty. Depending on the QoS requirements of
different applications, different values of b should be chosen
that tradeoff the total uploaded file size and total payment
to the AP by a different degree. Taking safety application
as an example, it may be more important to maximize the
uploaded file size than to reduce the total payment to the APs,
so a large value of b should be be chosen. Also, since the
transmission policies of the greedy, MCBC, and exponential
backoff schemes do not consider the self-incurred penalty in
(27), their total uploaded file size are independent of b.
Next, we plot the total cost against the traffic density ρ for
S = 200 Mbits with b = 0.1 for the case of one AP in Fig.
5. It is clear that the DORA scheme in Algorithm 1 achieves
the minimal total cost as stated in Theorem 1, with 48% and
24% cost reduction as compared with the exponential backoff
scheme at low and high ρ, respectively. To measure the cost
effectiveness of the file uploading for the four schemes, we
propose a metric called the upload ratio, which is defined
20 30 40 50 60 70 80
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
Density ρ (veh/km)
UploadRatio
DORA
Exponential Backoff
Greedy
MCBC
17%
77%
Fig. 6. Upload ratio (i.e., total uploaded file size / total payment to the APs)
versus traffic density ρ for file size S = 200 Mbits with a single AP. The
DORA scheme achieves the highest upload ratio.
20 30 40 50 60 70 80
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Density ρ (veh/km)
TotalCost
MCBC
Greedy
Exponential Backoff
JDORA (θ = 2)
JDORA (θ = 1)
JDORA (Perfect Estimation)
71%
53%
Fig. 7. Total cost versus traffic density ρ for file size S = 500 Mbits with five
APs. The JDORA scheme with perfect estimation of psucc
t has the minimal
total cost. Moreover, a higher total cost is required when the precision of the
estimation reduces (i.e., when the variance of the estimation θ increases).
as the total uploaded file size divided by the total payment
to the APs. In other words, it represents the size of the
file uploaded per unit payment. As shown in Fig. 6, since
the DORA algorithm takes into account the varying channel
contention level and data rate in determining the transmission
policy, it is cost effective and achieves the highest upload ratio.
In particular, the performance gains in upload ratio over the
exponential backoff scheme are 17% and 77% at low and high
ρ, respectively.
Furthermore, we consider the case with five APs, where
we assume that all of them have the same transmission radii
R and price q. For the JDORA scheme in Algorithm 3, we
consider that the estimated number of vehicles ñt at time t ∈ T
is obtained by rounding off a normally distributed random
variable with a mean nt and a variance θ to the nearest non-
negative integer. Thus, the lower the variance θ, the higher is
the precision of the estimation. The value of psucc
t is obtained
by setting psucc
t = gj(ñt), ∀ t ∈ Tj, j ∈ J . We plot the total
cost and upload ratio in Figs. 7 and 8 for S = 500 Mbits with

20 30 40 50 60 70 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Density ρ (veh/km)
UploadRatio
JDORA (Perfect Estimation)
JDORA (θ = 1)
JDORA (θ = 2)
Exponential Backoff
Greedy
MCBC
130%
207%
Fig. 8. Upload ratio versus traffic density ρ for file size S = 500
Mbits with five APs. The JDORA scheme with perfect estimation of psucc
t
achieves the highest upload ratio as compared with three other heuristic
schemes. Moreover, a lower upload ratio is achieved when the precision of
the estimation reduces (i.e., when the variance of the estimation θ increases).
b = 0.01, respectively. In Fig. 7, we can see that the JDORA
scheme with perfect estimation (i.e., θ = 0) of psucc
t , ∀ t ∈ T
achieves the minimal total cost as stated in Theorem 3, where
it achieves 53% and 71% cost reduction as compared with the
exponential backoff scheme at low and high traffic density ρ,
respectively. In Fig. 8, we can see that the JDORA scheme
with perfect estimation achieves the highest upload ratio. In
particular, it achieves an upload ratio 130% and 207% better
than the exponential backoff scheme at low and high traffic
density ρ, respectively. As shown in Figs. 7 and 8, the total
cost is increased and the upload ratio is reduced, respectively,
when the estimation precision decreases. However, this result
based on equal share of bandwidth that psucc
t = 1/ñt, ∀ t ∈ T
is less sensitive to the estimation error when the traffic density
ρ is high. It suggests that the JDORA algorithm is suitable
especially for VANETs with high traffic densities.
Finally, we study the threshold policy in a single AP
obtained by Algorithm 2 when the penalty function h(s) is
convex and data rate wt is fixed. We consider that S = 100
Mbits, ν = 100 km/hr, wt = 54 Mbps, ∀ t ∈ T , and h(s)
is defined as in (27). From Theorem 2, we know that the
optimal policy has a threshold structure. In Fig. 9, we plot
the thresholds s∗
t (psucc
) of the optimal policy against the
decision epoch t for different values of psucc
. With the use
of the convex penalty function, we can see that the threshold
increases with t. In Fig. 9(a), for b = 0.1, we can observe that
the threshold increases when psucc
decreases. It is because
a small penalty parameter is chosen, which places a higher
priority on the total payment than on the uploaded file size.
When psucc
is small, the chance of successful transmission is
low, so the vehicle chooses a higher threshold and transmits
less aggressively to reduce the amount of payment. In Fig.
9(b), we choose a larger penalty parameter b = 10 such that a
higher priority is placed on the uploaded file size than on the
total payment. We can observe that the threshold decreases
when psucc
decreases. It is because when psucc
is small, the
vehicle needs to transmit more aggressively (i.e., with a lower
0 20 40 60 80 100
70
72
74
76
78
80
82
84
86
88
Decision Epoch t
Thresholdst
*
(p
succ
)
psucc
= 0.071
psucc
= 0.083
psucc
= 0.100
(a) b = 0.1
0 20 40 60 80 100
9
10
11
12
13
14
15
16
17
18
Decision Epoch t
Thresholdst
*
(p
succ
)
p
succ
= 0.100
psucc
= 0.083
p
succ
= 0.071
(b) b = 10
Fig. 9. The thresholds s∗
t (psucc) of the optimal policy against the decision
epoch t for different penalty parameters b.
threshold) to prevent a large penalty. Moreover, we can see
that the thresholds presented in Fig. 9(b) is lower than that
in Fig. 9(a) due to the higher incentive to transmit when the
penalty is large.
VI. CONCLUSION
In this paper, we studied the V2R uplink transmission from
a vehicle to the APs in a dynamic drive-thru scenario, where
both the channel contention level and data rate vary over
time. Depending on the applications’ QoS requirements, the
vehicle can achieve different levels of tradeoff between the
total uploaded file size and the total payment to the APs
by tuning the self-incurred penalty. For a single AP with
random vehicular traffic, we proposed a DORA algorithm
based on DP to obtain the optimal transmission policy for
the vehicle in a coverage range. We prove that if the self-
incurred penalty function h(s) is convex and the data rate
wt is non-adaptive and fixed, then the optimal transmission
policy has a threshold structure. A monotone DORA algorithm
with a lower computational complexity was proposed for this
special case. Next, for multiple APs with known vehicular
patterns, we considered the transmission policy in multiple
MAY 2013

coverage ranges jointly and proposed an optimal JDORA
algorithm. Simulation results showed that our schemes achieve
the minimal total cost and the highest upload ratio as compared
with three other heuristic schemes. An interesting topic for
future work is to consider joint AP optimization without traffic
pattern estimation.
APPENDIX
A. Proof of Lemma 2
We prove it by induction. From (19), since vT +1(s, psucc
) =
h(s), vT +1(s, psucc
) is a nondecreasing function in s,
∀ psucc
∈ P. Assume that vt+1(s, psucc
) is a nondecreasing
function in s, ∀ psucc
∈ P. Since pt psucc
| psucc
≥ 0
and 0 ≤ apsucc
≤ 1, it can be inferred from (18) that
ψt(s, psucc
, a) is a nondecreasing function in s, ∀ psucc
∈ P.
Thus, vt(s, psucc
) in (16) is a nondecreasing function in s,
∀ psucc
∈ P.
B. Subadditivity of ψt(s, psucc
, a)
Because wt = w, ∀ t ∈ T , we let ω = wΔtdata. Since
δ∗
t (s, psucc
) is defined as in (21), we can establish the thresh-
old policy if we can prove that ψt(s, psucc
, a) is subadditive on
S ×A, ∀ t ∈ T [19, pp.104, 115]. The following results from
Lemma 3 and 4 establish the subadditivity of ψt(s, psucc
, a).
First, Lemma 3 shows that vt(s, psucc
) has a nondecreasing
difference in s if h(s) is a convex and nondecreasing function.
Lemma 3: If h(s) is a convex and nondecreasing function
in s, then
vt(s, psucc
) − vt [s − ω]+
, psucc
≥ vt [s − σ]+
, psucc
−vt [s − σ − ω]+
, psucc
, ∀ s∈S, psucc
∈P, t∈T ∪{T + 1}.
(28)
Proof: We prove it by induction. Since h(s) is a nonde-
creasing convex function, we have
h(s)−h([s−ω]+
) ≥ h([s−σ]+
)−h([s−σ−ω]+
), ∀ s ∈ S.
(29)
Let s ∈ S, psucc
∈ P be given. For t = T + 1, we have
vT +1(s, psucc
)−vT +1 [s − ω]+
, psucc
=h(s) −h [s − ω]+
≥ h([s − σ]+
) − h([s − σ − ω]+
)
= vT +1([s − σ]+
, psucc
) − vT +1 [s − σ − ω]+
, psucc
,
(30)
where the equalities are due to (19) and the inequality is due
to (29).
Assume that for a given t ∈ T , we have
vt+1(s, psucc
) − vt+1 [s − ω]+
, psucc
≥ vt+1 [s − σ]+
, psucc
−vt+1 [s − σ − ω]+
, psucc
, ∀ s ∈ S, psucc
∈ P.
(31)
From (16), let actions a1, a2, a3, and a4 be defined such
that
vt(s, psucc
) = min
a∈A
{ψt(s, psucc
, a)} = ψt(s, psucc
, a1), (32)
vt [s − ω]+
, psucc
= min
a∈A
{ψt [s − ω]+
, psucc
, a }
= ψt [s − ω]+
, psucc
, a2 , (33)
vt [s − σ]+
, psucc
= min
a∈A
{ψt [s − σ]+
, psucc
, a }
= ψt [s − σ]+
, psucc
, a3 , and (34)
vt [s − σ − ω]+
, psucc
= min
a∈A
{ψt [s − σ − ω]+
, psucc
, a }
= ψt [s − σ − ω]+
, psucc
, a4 .
(35)
We thus have
vt(s, psucc
) − vt [s − ω]+
, psucc
− vt [s − σ]+
, psucc
+ vt [s − σ − ω]+
, psucc
= ψt(s, psucc
, a1) − ψt [s − ω]+
, psucc
, a2
− ψt [s − σ]+
, psucc
, a3 + ψt [s − σ − ω]+
, psucc
, a4
=
A
ψt(s, psucc
, a1) − ψt [s − σ]+
, psucc
, a1
+
B
ψt [s − σ]+
, psucc
, a1 − ψt [s − σ]+
, psucc
, a3
C
− ψt [s − ω]+
, psucc
, a2 + ψt [s − ω]+
, psucc
, a4
−
D
ψt [s − ω]+
, psucc
, a4 − ψt [s − σ − ω]+
, psucc
, a4
= A + B + C − D. (36)
We have
A =
psucc ∈P
pt psucc
| psucc
a1psucc
vt+1([s − ω]+
, psucc
)
− vt+1([s − σ − ω]+
, psucc
) + 1 − a1psucc
× vt+1(s, psucc
) − vt+1([s − σ]+
, psucc
)
≥
psucc ∈P
pt psucc
| psucc
vt+1([s − ω]+
, psucc
)
− vt+1([s − σ − ω]+
, psucc
)
≥
psucc ∈P
pt psucc
| psucc
a4psucc
vt+1([s − 2ω]+
, psucc
)
− vt+1([s − σ − 2ω]+
, psucc
) + 1 − a4psucc
× vt+1([s − ω]+
, psucc
) − vt+1([s − σ − ω]+
, psucc
)
= D, (37)
where the two equalities are obtained by using (18) and the
two inequalities are due to the induction hypothesis in (31).
From (34) and (33), we have B ≥ 0 and C ≥ 0, respectively.
Overall, from (36), we obtain
vt(s, psucc
) − vt [s − ω]+
, psucc
−vt [s − σ]+
, psucc
+ vt [s − σ − ω]+
, psucc
≥ 0,
(38)
which completes the proof.
Lemma 4 shows that ψt(s, psucc
, a) is subadditive if
vt(s, psucc
) has a nondecreasing difference in s.
Lemma 4: If ∀ ˆs, ˇs ∈ S, psucc
∈ P, t ∈ T with ˆs ≥ ˇs,
where
vt+1(ˆs, psucc
) − vt+1 [ˆs − ω]+
, psucc
≥ vt+1(ˇs, psucc
) − vt+1 [ˇs − ω]+
, psucc
,
(39)

then ψt(s, psucc
, a) is subadditive on S × A, ∀ t ∈ T .
Proof: Let ˆs, ˇs ∈ S, â, ˇa ∈ A, psucc
∈ P, and t ∈ T be
given, where ˆs ≥ ˇs and â ≥ ˇa. Then
−ψt(ˆs, psucc
, â)+ψt(ˇs, psucc
, ˇa)−ψt(ˆs, psucc
, ˇa)−ψt(ˇs, psucc
, â)
= −
psucc ∈P
pt psucc
| psucc
psucc
(â − ˇa) vt+1(ˆs, psucc
)−
vt+1 [ˆs − ω]+
, psucc
−vt+1(ˇs, psucc
)+vt+1 [ˇs − ω]+
, psucc
≤ 0, (40)
where the equality is obtained by using (18). The inequality at
the end is due to the fact that pt psucc
| psucc
≥ 0, psucc
≥ 0,
â ≥ ˇa, and the given condition in Lemma 4. From Definition
1 and (22), the result follows.
C. Proof of Theorem 2
Let ˆs, ˇs ∈ S, ω ≥ 0, psucc
∈ P, and t ∈ T be given.
Moreover, let ˇs = [ˆs − kσ]+
, where k > 0. If the condition
of the theorem is satisfied, by repetitively applying Lemma 3,
we have
vt(ˆs, psucc
) − vt [ˆs − ω]+
, psucc
≥ vt [ˆs − σ]+
, psucc
− vt [ˆs − σ − ω]+
, psucc
≥ · · ·
≥ vt [ˆs − kσ]+
, psucc
− vt [ˆs − kσ − ω]+
, psucc
= vt ˇs, psucc
− vt [ˇs − ω]+
, psucc
. (41)
From Lemma 4, ψt(s, psucc
, a) is subadditive on S × A,
∀ t ∈ T . From [19, pp. 104,115], δ∗
t (s, psucc
) is a monotone
nondecreasing function in s. Since δ∗
t (s, psucc
) ∈ A = {0, 1},
δ∗
t (s, psucc
) is in the form of (23).
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Man Hon Cheung (S’06) received the B.Eng. and
M.Phil. degrees in Information Engineering from
the Chinese University of Hong Kong (CUHK) in
2005 and 2007, respectively. He is currently working
towards the Ph.D. degree in the Department of Elec-
trical and Computer Engineering at the University of
British Columbia (UBC). As a graduate student, he
received the IEEE Student Travel Grant for attending
IEEE ICC 2009. He was awarded the Graduate
Student International Research Mobility Award by
UBC, and the Global Scholarship Programme for
Research Excellence by CUHK. His research interests include the design
and analysis of medium access control protocols in wireless networks using
optimization theory, game theory, and dynamic programming.
Fen Hou (M’10) has been a lecturer in the Joint
Research Center in Ubiquitous Computing, Com-
puting/Computer Studies Programme at the Macao
Polytechnic Institute from 2011. She received the
Ph.D. degree in electrical and computer engineering
from the University of Waterloo, Waterloo, Canada,
in 2008. She worked as a postdoctoral fellow in
the Electrical and Computer Engineering at the
University of Waterloo and in the Department of
Information Engineering at the Chinese University
of Hong Kong from 2008 to 2009 and from 2009
to 2011, respectively. Her research interests include resource allocation and
protocol design in vehicular sensor network and participatory sensor networks,
QoS provisioning for multimedia communications in cognitive radio network
and broadband wireless networks.
MAY 2013

Vincent W.S. Wong (SM’07) received the B.Sc.
degree from the University of Manitoba, Winnipeg,
MB, Canada, in 1994, the M.A.Sc. degree from the
University of Waterloo, Waterloo, ON, Canada, in
1996, and the Ph.D. degree from the University of
British Columbia (UBC), Vancouver, BC, Canada,
in 2000. From 2000 to 2001, he worked as a
systems engineer at PMC-Sierra Inc. He joined the
Department of Electrical and Computer Engineering
at UBC in 2002 and is currently an Associate Pro-
fessor. His research areas include protocol design,
optimization, and resource management of communication networks, with
applications to the Internet, wireless networks, smart grid, RFID systems,
and intelligent transportation systems. Dr. Wong is an Associate Editor of the
IEEE Transactions on Communications and IEEE Transactions on Vehicular
Technology, and an Editor of KICS/IEEE Journal of Communications and
Networks. He has served as the Symposium Co-chair of IEEE Globecom’11,
Wireless Communications Symposium.
Jianwei Huang (S’01-M’06-SM’11) is an Assistant
Professor in the Department of Information Engi-
neering at the Chinese University of Hong Kong. He
received Ph.D. in Electrical and Computer Engineer-
ing from Northwestern University (Evanston, IL,
USA) in 2005, and worked as a Postdoc Research
Associate in the Department of Electrical Engineer-
ing at Princeton University during 2005-2007. He
was a visiting scholar in EPFL in June 2009 and
University of California-Berkeley in August 2010.
Dr. Huang currently leads the Network Commu-
nications and Economics Lab (ncel.ie.cuhk.edu.hk), with main research focus
on nonlinear optimization and game theoretical analysis of communication
networks, especially on network economics, cognitive radio networks, and
smart grid. He is the recipient of the IEEE Marconi Prize Paper Award in
Wireless Communications in 2011, the ICST WiCON Best Paper Award 2011,
the IEEE GLOBECOM Best Paper Award in 2010, the IEEE ComSoc Asia-
Paciﬁc Outstanding Young Researcher Award in 2009, and the Asia-Paciﬁc
Conference on Communications Best Paper Award in 2009.
Dr. Huang has served as Editor of IEEE Journal on Selected Areas in
Communications - Cognitive Radio Series, Editor of IEEE Transactions on
Wireless Communications, and Guest Editor of several special issues related
to Network Economics in IEEE Journal on Selected Areas in Communications
and IEEE Commun. Mag.. He has also served as Vice Chair of IEEE MMTC
(Multimedia Communications Technical Committee), as well as the TPC Co-
Chair of IEEE WiOpt 2012, GameNets 2009, IEEE GlOBECOM Wireless
Communications Symposium, and IWCMC Mobile Computing Symposium
2010.

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