QoS Provisioning in a Scalable Wireless Mesh
Network for Intelligent Transportation Systems
Jane-Hwa Huang, Li-Chun Wang, Senior Member, IEEE, and Chung-Ju Chang, Fellow, IEEE
Abstract—The wireless mesh network (WMN) is an economical
low-power solution to provide wireless broadband access in intel-ligent transportation systems (ITSs). This paper investigates how to deploy access points (APs) to improve throughput and quality-of-service (QoS), when the multihop communication is used to extend the coverage of wireless ITS network. Frequency planning is suggested to improve the capacity with QoS provisioning and to make the system more scalable. To investigate the overall tradeoffs among QoS, throughput, and coverage, we develop an analyt-ical model to evaluate throughput, frame delay, and jitter for the considered ITS WMN using the carrier-sense multiple-access (CSMA) medium access control (MAC) protocol. Then, we apply an optimization approach to determine the optimal separation distances between APs with the delay requirement. To provide a guideline for network planning, we compare the uniform-spacing and increasing-spacing AP deployment strategies. The uniform-spacing strategy is to make all the APs with the same separa-tion distance. In the increasing-spacing strategy, the separasepara-tion distances between APs are increased from the central AP to the outer APs. Because of the shorter separation distance, the AP closer to the central AP can deliver higher traffic. It is shown that the increasing-spacing strategy outperforms the uniform-spacing strategy in terms and a profit function, considering the capacity and the total cost for deploying APs.
Index Terms—Intelligent transportation system (ITS),
quality-of-service (QoS), wireless mesh network (WMN).
I. INTRODUCTION
D
UE to the advantages of easy deployment and lower cost, the wireless mesh network (WMN) is a promising infor-mation dissemination technology for intelligent transportation systems (ITSs) [1]–[5]. The ITS development aims to integrate communication, information, and control technologies to im-prove the safety and convenience of transportation systems. In wireless-enabled ITS systems, the users can conveniently retrieve public safety warning, traveller and traffic information, and rich media content [6], [7]. Therefore, widely deploying ITS wireless networks has become a critical topic. Since a Manuscript received February 23, 2007; revised July 4, 2007, November 4, 2007, and December 31, 2007. This work was supported in part by the MoE ATU Plan, by the Program for Promoting Academic Excellence of Universities (Phase II), and by the National Science Council under Grant 95W803C, Grant NSC 95-2752-E-009-014-PAE, Grant NSC 95-2221-E-009-148, and Grant NSC 95-2221-E-009-155. This work was presented in part at the IEEE Conference on Systems, Man, and Cybernetics (SMC) 2006, Taipei, Taiwan, R.O.C., October 2006. The review of this paper was coordinated by Prof. X. Shen.The authors are with the Department of Communication Engineering, National Chiao Tung University, Hsinchu 300, Taiwan, R.O.C. (e-mail: hjh@mail.nctu.edu.tw; lichun@cc.nctu.edu.tw; cjchang@cc.nctu.edu.tw).
Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TVT.2008.918701
multihop architecture can avoid costly and difficult cabling engineering, the WMN is an economical low-power solution to support ubiquitous ITS broadband applications.
Fig. 1 illustrates a cluster-based ITS WMN for the dense-urban scenario. To provide reliable broadband communications, the access points (APs) are deployed along the main streets as in [8]. The APs are connected via wireless links to facilitate deployment. Moreover, the APs are grouped into clusters. In each cluster, only the central AP has a wireline connection to the Internet, whereas other APs will relay the neighboring APs’ traffic toward/from the central AP. By this multihop architecture, ITS WMNs can rapidly be deployed on a large scale with less cabling engineering.
Nevertheless, WMNs face scalability issues since cov-erage extension, throughput enhancement, and delay im-provement are usually contradictory goals [1]–[3], [9], [10]. Specifically, multihop communications can extend coverage by more hops and longer hop distance. Maximizing the coverage of AP can reduce infrastructure cost. However, the repeatedly relayed traffic with more hops may rapidly exhaust radio re-sources and degrade quality-of-service (QoS), e.g., longer de-lay. The longer hop distance will also lower the data rate in the relay link between APs. Meanwhile, the increasing collisions due to more users in a larger coverage of AP will further lower user throughput. Therefore, while multihop communication is used to extend coverage, how to improve throughput and QoS is a major concern for deploying ITS WMNs.
In the literature, the AP deployment issues for wireless local area networks (WLANs) are studied in [11]–[14]. In [11], an optimization approach is applied to minimize the area with poor signal quality. The authors in [12] and [13] have developed optimization algorithms to minimize the bit error rate. In [14], an optimal AP deployment problem is formulated, which aims to minimize the maximum of channel utilization to achieve load balancing. In [11]–[14], all the APs are connected via cables. The performance of WMNs is investigated in [1], [3], [9], [10], and [15]–[17]. The simulation results in [9] and the experimental results in [10] show that even if only one user is sending data, the user throughput rapidly drops as the number of hops increases from one. Then, it stabilizes at a very low throughput as the number of hops becomes larger (e.g., larger than seven [9]). This phenomenon is due to the fact that adjacent hop nodes have to contend for the channel to relay traffic. The authors in [15] and [16] point out that with k users in an ad hoc network, the user throughput is scaled like O(1/√k log k). It is also shown in [3] that the user throughput in a WMN sharply decreases as O(1/k) due to the throughput bottleneck at the central AP. A few papers have considered the overall 0018-9545/$25.00 © 2008 IEEE
Fig. 1. System architecture for ITS WMN. (This is an example for the uniform-spacing deployment strategy.)
Fig. 2. Cluster of APs in ITS WMN, where only the single side of a cluster is shown since the APs in the other side are symmetrically deployed. (This is an example for the increasing-spacing deployment strategy.)
throughput and coverage performance issues when deploying APs in ITS WMNs. The work in [1] studies the throughput and coverage of WMN in a single-user case. In [17], the tradeoff between throughput and coverage for a multiuser ITS WMN is investigated. However, the QoS (delay and jitter) issues are not considered in [1] and [17].
This paper investigates the AP deployment issue in a scalable ITS WMN, as shown in Fig. 1. We suggest allocating each AP with different channels. This simple frequency planning can reduce contention collisions to improve throughput and make the WMN more scalable. The cluster-based structure also facil-itates the management of QoS and throughput in a WMN since we can adjust the separation distances between APs to control the data rate of relay link and the number of users served by an AP. To investigate the overall tradeoffs among throughput, coverage, and delay, we develop an analytical model to evaluate the throughput and delay for the considered multihop network employing the carrier-sense multiple-access (CSMA) medium access control (MAC) protocol. The developed model considers a practical case where all nodes have bidirectional asymmetric traffic load and operate in the unsaturated situation. That is, the node does not always have a frame waiting for transmission. On top of the analytical model, we apply an optimization approach to determine the best number of APs in a cluster and the optimal separation distances between APs. The objective aims to maximize the profit function defined as the ratio of the capacity to the total cost for deploying a cluster of APs with the QoS requirement. To provide a useful guideline for network planning, we compare the uniform-spacing and increasing-spacing AP deployment strategies. In the uniform-increasing-spacing strat-egy (see Fig. 1), all APs have the same separation distance. In the increasing-spacing strategy (in Fig. 2), the separation distances between APs are increased from the central AP to the
outer APs. We respectively determine the optimal deployment parameters for these strategies and compare their performances. The analytical model is validated by simulations. It is shown that although there are some assumptions and approximations, the analytical model nicely estimates the link capacity. More-over, the analytical model can estimate the upper bound of traffic load subject to delay requirement. With the help of this analytical model, we can analytically determine the optimal deployment parameters to maximize the profit of a cluster with the QoS requirement.
The rest of this paper is organized as follows. Section II describes the system architecture and the traffic model for an ITS WMN. In Section III, we formulate an optimization problem to determine the optimal deployment parameters with the delay constraint. Sections IV and V describe the analytical throughput and delay model for the considered ITS WMN. Performance evaluations are shown in Section VI. Concluding remarks are given in Section VII.
II. SCALABLEITS WMN A. Network Architecture and Assumptions
Fig. 1 shows a scalable cluster-based ITS WMN for dense-urban coverage. The APs are deployed along the main streets to avoid heavy attenuation due to buildings and walls. In a cluster, the ith AP (APi) from the central AP (AP0) connects to AP0
via an i-hop communication, and only the central AP is Internet connected through cables. Clearly, this WMN is very suitable for wireless-enabled ITS systems thanks to easy deployment and less cost.
This ITS WMN operates in a multichannel and multiinterface fashion. The APs are allocated with different channels to reduce the number of contending users. Moreover, each AP has three
Fig. 3. Example of frequency assignment in ITS WMN, where seven nonoverlapping channels ensure two buffer cells. radio interfaces: one for data access and two for data
forward-ing. By doing so, an AP can concurrently provide data access for local users at channel fi and receive/deliver relay traffic from/to APi−1 and APi+1 at different channels fi and fi+1,
respectively, thereby reducing delay.
In general, spectrum and hardware costs are the major con-cerns in the multichannel and multiinterface systems. However, there are multiple channels available for wireless public access. For example, there are 12 nonoverlapping channels for IEEE 802.11a WLAN, three channels for IEEE 802.11b/g WLAN, and 75 MHz of spectrum reserved for the dedicated short-range communication (DSRC)1 [6], [7]. The price of interface also
rapidly goes down since the WLAN has become an off-the-shelf product. Indeed, many WLAN equipment vendors have developed the outdoor IEEE 802.11 a/b/g multimode APs with multiple interfaces [18].
Frequency planning in this ITS WMN is simple since it only needs to design the separation distances between APs to en-sure a sufficient cochannel reuse distance without interference. Fig. 3 shows an example of frequency assignment. Let the term cell represent the coverage of an AP. In Fig. 3, seven channels can ensure two buffer cells between two cochannel cells. We also assume that the interference from adjacent streets is very minor since it is heavily attenuated by buildings. In practice, other WLAN users may interfere with this ITS WMN operating at the unlicensed band. In this situation, we suggest allocating the channels in the licensed DSRC band for the relay links between APs to ensure throughput. To understand performance bounds, this interference issue is not considered in this paper. B. Uniform-Spacing and Increasing-Spacing
AP Deployment Strategies
To provide a useful insight into AP deployment, we com-pare the uniform-spacing and increasing-spacing strategies. The uniform-spacing strategy is to make all the APs have the same separation distance as shown in Fig. 1. According to the increasing-spacing strategy as in Fig. 2, the separation
1DSRC is a short-to-medium-range communication system for ITS wireless
applications, which supports both public safety and private operations in vehicle-to-roadside and vehicle-to-vehicle environments [7].
distances dibetween APs follow the order d1≤ d2≤ · · · ≤ dn.
In a WMN, the AP closer to the central AP aggregates more traffic. With a shorter separation distance, the AP near the central AP can deliver higher traffic load for a cluster.
C. Traffic Model for User and AP in an ITS WMN
This paper considers bidirectional asymmetric traffic. The demanded uplink and downlink traffic of a user are rU and
rD (in megabits per second), respectively. Suppose that all
the APs have the same transmission power. We also assume an ideal handoff mechanism, and the user always associates with the closer AP by estimating the average signal strength from APs. As in Fig. 2, the separation distance between APi−1 and APi is di, and the distance from APn to the outermost
AP in the neighboring cluster is dn+1. Then, the coverage
of APiis defined as li= (di+ di+1)/2, and clearly, l0= d1.
Let DM (users/m) be the user density. The aggregated uplink
and downlink access traffic of APi are R(A)U,i = liDMrU and
R(A)D,i= liDMrD, respectively.
Fig. 4 depicts the traffic model for the AP and users in an ITS WMN. Let L be the frame payload size. The frame arrival rate and the service rate of a user are λk= rU/L and
μk (in frames per second), respectively. The traffic load of an
AP includes the access traffic for local users and the relay traffic for other APs. To facilitate management, each AP has three queues dedicated for downlink access traffic, uplink relay traffic, and downlink relay traffic, respectively. The aggregated downlink access frame arrival rate of APiis λ(A)D,i= R
(A)
D,i/L =
liDMrD/L. The uplink and downlink relay frame arrival rates
of APiare equal to λ(R)U,i =R (R) U,i L =
R(A)U,i + R(R)U,i+1 L = n i=iliDMrU L (1) λ(R)D,i=R (R) D,i L = n i=i+1R (A) D,i L = n i=i+1liDMrD L (2)
where n is the number of APs in the single side of a cluster; R(R)U,i =ni=iR(A)U,i is the aggregated uplink relay traffic from
the users served by APi, APi+1, . . . , APn; and R(R)D,i is the
Fig. 4. Traffic models for users and the AP. For management simplicity, each AP has three isolated queues for downlink access traffic, uplink relay traffic, and downlink relay traffic, respectively.
TABLE I LIST OFNOTATIONS
μ(A)D,i, μ(R)U,i, and μ(R)D,i, respectively. For clarity, the notations in the traffic model are summarized in Table I.
D. Scalability and QoS
Most traditional WMNs are not scalable. As the number of users increases, user throughput and QoS (delay) sharply degrade due to the increasing collisions [3], [9], [10], [15], [16]. By contrast, the presented ITS WMN is more scalable to accommodate more users because frequency planning can reduce the number of contending nodes to improve contention situation. Furthermore, delay and throughput can be guaranteed by properly designing the number of APs in a cluster and the separation distances of APs. The remaining challenge lies in
the way to determine the optimal separation distances between APs to achieve the optimal tradeoff among QoS, throughput, and coverage.
III. OPTIMALAP DEPLOYMENT
A. Problem Formulation
All the performance issues of coverage, throughput, and QoS are essential factors in designing a scalable ITS WMN. From the coverage viewpoint, a larger separation distance between APs can lower the infrastructure cost due to fewer APs. From the throughput standpoint, however, a shorter separation dis-tance can achieve a higher relay link capacity between APs. Meanwhile, a smaller cell with fewer contending users also improves the access link capacity between users and the AP. We also consider the frame delay consisting of contention delay and queuing delay. From the queuing delay perspective, a longer separation distance of AP is better due to fewer hops. From the contention delay viewpoint, a shorter separation distance and then a smaller cell may be preferred due to higher data rate and fewer contending users. In the following, an optimization problem is formulated to determine the best number of APs in a cluster and the optimal separation distances between APs subject to the constraints on throughput and delay.
First, we discuss the constraints in the optimization problem for the considered ITS WMN.
1) The frame arrival rate for each user and AP should be less than the frame service rate to guarantee minimum throughput. Therefore, for each user, it is required that
λk ≤ μk (3)
and for each AP
λ(A)D,i ≤ μ(A)D,i, λ(R)U,i ≤ μ(R)U,i, λ(R)D,i≤ μ(R)D,i. (4) These constraints also mean that the link capacity of each node is large enough to accommodate its traffic load.
2) The overall frame delay DT(i) for the user in the
cell of APi should meet the delay requirement Dreq.
Accordingly
DT(i)≤ Dreq. (5)
3) The cell coverage of an AP should be designed to ensure an acceptable data rate in the access link. Recall that the coverage of APiis li= (di/2 + di+1/2). The maximum
possible separation distance between a user and the AP should be less than a threshold rMAX. That is
max(di/2, di+1/2)≤ rMAX. (6)
4) The separation distance between APs should be designed from two folds. The separation distance di should be
less than the maximal reception range dMAXwhile larger
than dMIN to lower the handoff probability and ensure
sufficient cochannel reuse distance without interference. Hence,
dMIN≤ di ≤ dMAX, for i = 1, 2, . . . , n
dMIN≤ dn+1 (7)
where dn+1is the separation distance from APn to the
outermost AP in the neighboring cluster, as in Fig. 2. According to the above considerations, the AP deployment issue can be formulated as a mixed-integer nonlinear program-ming problem with the following decision variables: n (the number of APs in the single side of one cluster) and di (the
separation distances between APs). In this scalable WMN, frequency planning reduces the number of contending users to relieve the collision issue as the coverage of a cluster in-creases. The optimal coverage and capacity can simultaneously be achieved since more users in a cluster can also lead to higher capacity of a cluster. It is noteworthy that due to the limitation of relay link capacity between APs, deploying more APs in a cluster may not significantly improve the capacity of a cluster. Therefore, the optimization problem aims to maximize the profit of a cluster defined as the ratio of the capacity to the cost for deploying a cluster of APs.
The capacity of a cluster is defined as the total aggregated traffic of the central AP (AP0), including the relay traffic
from/to other APs and the local access traffic for the users served by AP0. According to the constraints in (3) and (4),
the link capacity of each node is large enough to accommodate the carried traffic load. Hence, the aggregated traffic of AP0
is equal to the sum of demanded traffic of all users. Consider that a cluster includes AP0and 2n APs deployed symmetrically
with respect to AP0. The coverage of a cluster is (2
n i=1di+
dn+1). Let DM (users/m) be the user density, and let rU and
rDbe the demanded uplink and downlink traffic of each user,
respectively. Then, the capacity of a cluster is 2 n i=1 di+ dn+1 DM(rU + rD). (8)
The deployment cost means the investment for deploying a cluster of APs, including the expense of devices and that of
cabling engineering work. The more APs in a cluster, the higher the deployment cost of a cluster. Suppose that cAPis the cost
of an AP and that ρl is the wireline overhead for connecting
AP0 to the Internet. Since a cluster consists of (2n + 1) APs,
the deployment cost of a cluster is (2n + 1)cAP+ ρl. For
convenience, the cost of a cluster is normalized to the cost of an AP. Then, the normalized deployment cost of a cluster can be expressed as (2n + 1 + ρ), where ρ = ρl/cAP.
In the following, we discuss the optimal AP deployment problems for the increasing-spacing and uniform-spacing strategies.
B. Increasing-Spacing Deployment Strategy
In the increasing-spacing strategy, the separation distances of APs follow the order d1≤ d2≤ · · · ≤ dn. That is, the heavier
the load of AP, the shorter the separation distance. The opti-mal deployment parameters can be determined by solving the following optimization problem:
MAX
n,d1,d2,...,dn+1
Capacity of a cluster of APs Total cost for deploying a cluster of APs
= MAX n,d1,d2,...,dn+1 (2ni=1di+dn+1) DM(rU+rD) (2n+1+ρ) (9) subject to λk≤ μk, λ(A)D,i≤ μ (A) D,i, λ (R) U,i ≤ μ (R) U,i, λ (R) D,i≤ μ (R) D,i (10) DT(i)≤ Dreq (11)
max(di/2, di+1/2)≤ rMAX (12)
dMIN≤ di≤ dMAX, for i = 1, 2, . . . , n (13)
dMIN≤ dn+1. (14)
C. Uniform-Spacing Placement Strategy
According to the uniform-spacing strategy, all the APs have the same separation distance, i.e., di= d. Thus, the
optimiza-tion problem for this strategy is modified as
MAX n,d (2n + 1)d DM(rU+ rD) (2n + 1 + ρ) (15) subject to λk≤ μk, λ(A)D,i≤ μ (A) D,i, λ (R) U,i ≤ μ (R) U,i, λ (R) D,i≤ μ (R) D,i (16) DT(i)≤ Dreq (17) d/2≤ rMAX (18) dMIN≤ d ≤ dMAX. (19)
IV. THROUGHPUTANALYSIS
This section suggests a throughput model for cluster-based ITS WMN considering bidirectional asymmetric traffic for users operating in the unsaturated situation. The IEEE 802.11a WLAN is used as an example here.
A. Throughput
We calculate the throughput (frame service rate) for each node on top of the channel activity concept. From the viewpoint of a particular node, there are five types of channel activities in this wireless network, including the following:
1) successful frame transmission; 2) unsuccessful frame transmission;
3) empty slot, where all nodes are in backoff or idle; 4) successful frame transmission from other nodes; 5) unsuccessful frame transmission from other nodes. The channel activities can be described by a sequence of activity time slots [19]–[21]. Subject to the backoff procedure, the duration Ttfor channel activity type t is defined as
T
1= T4= TS
T2= T5= TC
T3= σ
(20)
where σ is the duration of an empty slot, and TSand TCare the
successful frame transmission time and collision duration, re-spectively. From the viewpoint of node k, the average duration Tv,kof activity time slot can be expressed as
Tv,k=
5
t=1
νt,k, Tt. (21)
Here, νt,kis the probability of channel activity type t observed
by node k, and clearly, 5t=1νt,k= 1. It is noteworthy that
ν1,kalso represents the average number of frames successfully
delivered by node k in an activity time slot. Therefore, the frame service rate of node k is equal to
μk=
ν1,k Tv,k
(in frames per second). (22) Now, we calculate the channel activity type probability νt,k.
We consider a wireless network with K nodes employing the CSMA MAC protocol to share the channel. Suppose that the frame arrival rate and the service rate of node k are λkand μk
(in frames per second) for k = 0, 1, . . . , K− 1, respectively. The probability that one node is busy in contending for the channel and sending data is ρk= λk/μk [22], [23]. Then,
the channel activity type probability νt,k can be calculated as
follows.
1) Successful/Unsuccessful Transmission: One node can successfully deliver a frame only if no other node is sending at the same time. Let τk be the conditional
trans-mission probability of node k given that node k is busy due to a nonempty queue. The unsuccessful transmission probability pkfor the frame from node k is equal to
pk = 1− K−1 j=0,j=k
(1− τjρj) (23)
where the second term represents the probability that all the other nodes are in backoff or idle due to an empty queue. Hence, given that the considered node is busy,
the probability that this node successfully/unsuccessfully sends a frame in an activity slot can be written as
ν1,k= τk(1− pk) (24)
ν2,k= τkpk. (25)
2) Empty Slot: One node observes an empty slot when all the nodes are silent. From the viewpoint of the considered node, the empty-slot probability can be computed by
ν3,k= (1− τk) K−1 j=0,j=k
(1− τjρj) (26)
where the first term is the probability of the considered node being in backoff, and the second term denotes the probability that all the other nodes are in backoff or idle. 3) Successful/Unsuccessful Transmission From Other
Node: Suppose that the considered node is in backoff at the current activity time slot. The probability that at least one node sends a data frame is potr,k=
1− K−1j=0,j=k(1− τjρj). Then, the conditional
success-ful transmission probability pos,k that one frame is
successfully delivered from the other node is
pos,k= K−1 j=0,j=kτjρj K −1 j=0,j=j,j=k (1− τjρj) potr,k (27) where the numerator is the probability that exactly one frame is transmitted from the other node. Hence, from the viewpoint of the considered node, the probability of an activity time slot containing a successful/unsuccessful frame transmission from the other node can be given as
ν4,k= (1− τk)potr,kpos,k (28) ν5,k= (1− τk)potr,k(1− pos,k). (29)
According to the CSMA MAC protocol, the throughput is influenced by the backoff time. Consider a binary exponential backoff procedure with the initial backoff window size of W . Let pk be the unsuccessful frame transmission probability of
node k as defined in (23), and let mbkbe the maximum backoff
stage. The average backoff time of node k is expressed as Bk = (1− pk) W − 1 2 + pk(1− pk) 2W− 1 2 +· · · + pmbk k (1− pk) 2mbkW− 1 2 + pk(mbk+1)(1− pk) 2mbkW− 1 2 +· · · =[1− pk− pk(2pk) mbk] W− (1 − 2p k) 2(1− 2pk) . (30)
Since a busy node transmits data frames every (Bk+ 1) slots on
average [24], the transmission probability τkfor the considered
node can be written as τk = 1 Bk+ 1 = 2 1 + W + pkW mbk−1 i=0 (2pk)i . (31)
From (23) and (31), we can numerically derive the unique solution of τ = (τ0, τ1, . . . , τK−1) and p = (p0, p1, . . . , pK−1). Then, from (21)–(31), we can obtain ν1= (ν1,0, ν1,1, . . . , ν1,K−1) and μ = (μ0, μ1, . . . , μK−1) for the given
λ = (λ0, λ1, . . . , λK−1).
The hop distance also affects the throughput and the trans-mission PHY mode. Generally, the radio signal suffers from path loss, shadowing, and multipath fading. Considering these radio channel effects along with a proper fading margin, we assume that the average reception ranges for eight PHY modes in the IEEE 802.11a WLAN are Dκ, κ = 1, 2, . . . , 8, and
D1> D2>· · · > D8. In principle, the node with a shorter
separation distance can transmit at a higher data rate. Therefore, the transmission PHY mode ma for the data frame can be
determined according to the separation distance di, i.e.,
ma= κ, if Dκ≥ di> Dκ+1. (32)
Once the transmission PHY modes for data frame ma and
control frame mc are given, we can calculate the successful
frame transmission time TSand the collision duration TCby
TS = TDATA(L, ma) + δ + SIFS
+ TACK(mc) + δ + DIFS (33)
TC= TDATA(L, ma) + δ + EIFS (34)
where δ is the propagation delay. The durations of the short interframe space (SIFS), distributed interframe space (DIFS), and extended interframe space (EIFS = SIFS + TACK(mc) +
DIFS) are defined in [25]. TDATA(L, ma) is the transmission
time of a data frame with payload size L using PHY mode ma,
and TACK(mc) is that of an acknowledgement (ACK) frame
using PHY mode mc. TDATA(L, ma) and TACK(mc) can be
calculated as in [25].
B. Wireless Access Link Capacity Between AP and User In the cell of APi, there are K = 1 + liDM nodes
(in-cluding one AP and liDM users) contending for the channel.
Suppose that all the users have the same uplink frame arrival rate and service rate, i.e., λk = λ1= rU/L and μk = μ1 for k = 1, 2, . . . , K− 1. In addition, the downlink frame arrival rate and the service rate of APi are λ0= λ(A)D,i= liDMrD/L
and μ0, respectively. Thus, the busy probability of user k is ρk= λ1/μ1, and that of the AP is ρ0= λ0/μ0. For simplicity,
we consider a worst case where all the nodes in a cell transmit at the same data rate. In the cell of APi with coverage of
li= (di/2 + di+1/2), the transmission PHY mode mafor the
data frame is determined by the maximum possible separation distance between a user and the AP, i.e., max(di/2, di+1/2).
We evaluate the access frame service rates as follows. Let τ0 and τ1 be the transmission probabilities for the AP and
a user, respectively. By (23), the unsuccessful transmission probabilities for AP (p0) and that for a user (p1) are equal to
p0= 1− (1 − τ1ρ1)K−1 (35) p1= 1− (1 − τ0ρ0)(1− τ1ρ1)K−2. (36)
The channel activity type probability νt,k for the AP can be
expressed as ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ν1,0 = τ0(1− p0) ν2,0 = τ0p0 ν3,0 = (1− τ0)(1− τ1ρ1)K−1 ν4,0 = (1− τ0)potr,0pos,0 ν5,0 = (1− τ0)potr,0(1− pos,0) (37)
and that for a user ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ν1,1 = τ1(1− p1) ν2,1 = τ1p1 ν3,1 = (1− τ1)(1− τ0ρ0)(1− τ1ρ1)K−2 ν4,1 = (1− τ1)potr,1pos,1 ν5,1 = (1− τ1)potr,1(1− pos,1) (38)
where potr,0= 1− (1 − τ1ρ1)K−1, and potr,1= 1− (1 − τ0ρ0)(1− τ1ρ1)K−2. In addition pos,0= (K− 1)τ1ρ1(1− τ1ρ1)K−2 potr,0 (39) pos,1= 1 potr,1 τ0ρ0(1− τ1ρ1)K−2 + (K− 2)τ1ρ1(1− τ0ρ0)(1− τ1ρ1)K−3 . (40) Then, according to the developed throughput model in Section IV-A, we can obtain the access frame service rates for the AP and the users, i.e., μ(A)D,i = μ0and μk= μ1, respectively. C. Wireless Relay Link Capacity Between APs
In the relay link, APi−1with downlink traffic and APiwith
uplink traffic contend for the channel. The transmission PHY mode ma for APi−1 and APi is determined by the separation
distance dibetween APi−1and APi. Let λD= λ(R)D,i−1be the
downlink relay frame arrival rate of APi−1, and let λU = λ(R)U,i
be the uplink relay frame arrival rate of APi. Suppose that
the relay frame service rates of APi−1 and APi are μD and
μU, respectively. Then, the busy probability of APi−1is ρD=
λD/μD, and that of APiis ρU = λU/μU.
Now, we evaluate the relay frame service rates of APi−1and APi. Let τDand τU be the transmission probabilities of APi−1
and APi. From (23), the unsuccessful transmission probability
of APi−1(pD) and that of APi(pU) can be expressed as
pD= 1− (1 − τUρU) = τUρU (41)
pU = τDρD. (42)
Referring to (24)–(29), the channel activity type probability for APi−1can be calculated by ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ν1,D = τD(1− pD) = τD(1− τUρU) ν2,D = τDpD= τDτUρU ν3,D = (1− τD)(1− τUρU) ν4,D = (1− τD)τUρU ν5,D = 0 (43)
Fig. 5. State transition diagram of the queue for the uplink traffic of an AP, where the state variable s is the number of frames queued in the AP. and that of APiare
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ν1,U = τU(1− pU) = τU(1− τDρD) ν2,U = τUpU = τUτDρD ν3,U = (1− τU)(1− τDρD) ν4,U = (1− τU)τDρD ν5,U = 0. (44)
Then, the relay frame service rate for APi−1 (i.e., μ(R)D,i−1= μD) and that for APi(i.e., μ(R)U,i = μU) can be obtained by the
developed throughput model in Section IV-A.
V. DELAYANALYSIS
This section presents an analytical method to evaluate frame delay and jitter (delay variance) in the considered ITS WMN. The p-persistent CSMA MAC protocol is considered, which nicely approximates the standard protocol when the average backoff times are the same [26], [27]. In addition, the parame-ter p is equal to the transmission probability τkfor a busy node,
as defined in (31). Due to the memoryless backoff time, the p-persistent protocol is more tractable.
The considered multihop ITS network can be described by the series queue model [22], where the output frames from a hop node are the input frames to the next hop node. Suppose that the frame arrivals for each end user follow a Poisson process. Following the p-persistent CSMA protocol, the frame contention delay (i.e., the frame service time) at each hop is geometrically distributed. Referring to [22, ch. 4.1 and 4.2], the frame interdeparture time at each hop is also geometrically distributed. More importantly, the whole multihop network can be decomposed into individual hops, and each one can be modeled by an independent Markov chain. Hence, we can evaluate the delay and jitter hop by hop.
We consider the queue for the uplink relay traffic of APi
as an example. By (44), we can obtain the probability ν1 that APisuccessfully sends one relay frame to APi−1in an activity
slot. In APi, the interface to receive relay frames from APi+1
and the interface to receive access frames from local users independently operate at different channels. Due to the geo-metric frame service time of APi+1, we assume the geometric
interarrival time for the relay frames from APi+1[22]. We also
assume the geometric interarrival time for the access frames from local users. Consider the example in Section IV-B, where one AP and (K− 1) users contend for the channel. Following the p-persistent CSMA protocol, one busy node transmits its frame independently in each activity slot with probability τk.
The probability that one uplink access frame is successfully delivered to the AP is pSU = (1− ρ0τ0)[(K− 1)τ1ρ1(1− τ1ρ1)K−2], which is also independent in each activity slot.
TABLE II
NOTATIONS INDELAYANALYSIS
Thus, it is reasonable to assume the geometric interarrival time for the uplink access frames to APi.
The uplink relay frame and the uplink access frame may concurrently arrive in an activity slot, as shown in Fig. 4. Let Tv be the average duration of activity slot for the relay link
between APi−1 and APi. Suppose that the relay traffic from
APi+1to APiis R(R)U,i+1and the aggregated access traffic from
local users to APi is R(A)U,i. From the viewpoint of APi, the
arrival probability that one uplink relay frame arrives in an activity slot (with average duration Tv) is αR= (R(R)U,i+1/L)Tv,
where L is the frame payload size. The arrival probability of the uplink access frame in an activity slot is αL= (R(A)U,i/L)Tv.
Thus, the probability that two frames concurrently arrive in a slot is αLαR.
Fig. 5 shows the bulk-arrival queuing model for the uplink relay traffic of APi, where the state variable s is the number of
frames queued in the node. The state-transition probabilities are expressed as ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ps,s+2= χ2= αLαR(1− ν1) ps,s+1= χ1= αLαRν1+ αL(1− αR)(1− ν1) + (1− αL)αR(1− ν1) ps,s−1= μ = (1− αL)(1− αR)ν1 ps,s= 1− χ1− χ2− μ (45)
where ps,s+2represents the probability that two frames
simul-taneously arrive, and no queued frame is successfully delivered. We can obtain the steady-state probability Ps as detailed in
Appendix I. For clarity, Table II summarizes the notations used in the delay analysis.
Now, we evaluate the delay and jitter (delay variance). Con-sider a frame being placed at the hth position of the first-come–first-serve (FCFS) queue of a node. The state-transition
Fig. 6. State transition diagram for the considered frame in a node, where the state variable h denotes the considered frame being placed at the hth position of the queue.
diagram for the considered frame is illustrated in Fig. 6. In the figure, state h = 1 represents the state that the considered frame is contending for the channel. State h = φ is defined as the state that the considered frame has been successfully delivered. Let Ih= h be the state transition from state h to state h
in an activity slot. The state-transition probability can be ex-pressed as
Pr [Ih= h] =
ν1, for h= (h− 1)
(1− ν1), for h= h. (46)
First, we deal with the time Dh spent for a frame to enter
state φ (i.e., be successfully transmitted), given that this frame is now at state h. Clearly
E[Dh|Ih] =
1 + E[Dh−1], for Ih= (h− 1)
1 + E[Dh], for Ih= h
(47) where Dhis expressed in activity slots. Therefore, the mean of
Dhis equal to E[Dh] = Ih Pr[Ih]E[Dh|Ih] = 1 + ν1E[Dh−1] + (1− ν1)E[Dh]. (48)
Since E[D1] = 1/ν1, from (48), we can obtain E[Dh] = 1 ν1 + E[Dh−1] = h ν1 . (49)
By the conditional variance formula [28], the variance of Dh
can be expressed as
Var(Dh) = Var (E[Dh|Ih]) + E [Var(Dh|Ih)] . (50)
From (47), it follows that Var (E[Dh|Ih]) =
Ih
Pr[Ih]E[Dh|Ih]2− (E[Dh])2
= 1
ν1 − 1.
(51) In addition, it is obvious that
E [Var(Dh|Ih)] = ν1Var(Dh−1)+(1−ν1)Var(Dh). (52)
From (50) to (52), we can obtain Var(Dh) = 1− ν1 ν2 1 + Var(Dh−1) = h(1ν− ν2 1) 1 (53) where the initial condition is Var(D1) = (1− ν1)/ν12.
Therefore, the mean and variance of sojourn time for a frame spent in a node can be calculated by
Mean = [E[Dh]] = ∞
h=1
QhE[Dh] (54)
Variance = Var (E[Dh]) + E [Var(Dh)]
= ∞ h=1 Qh(E[Dh])2− (E [E[Dh]])2 + ∞ h=1 QhVar(Dh). (55)
In (54), Qh represents the probability of an incoming frame
being placed at the hth position of the FCFS queue at the instant of frame arrival, as detailed in (68) in Appendix II. By (54) and (55), we can obtain the delay TU,R(i) and variance σ2U,R(i) for
the uplink relay frames of APi, as detailed in Appendix II. By
the same method, we also obtain the delay T (k) and variance σ2(k) for the uplink access frames of a user, T
D,A(i) and
σ2
D,A(i) for the downlink access frames of APi, and TD,R(i)
and σD,R2 (i) for the downlink relay frames of APi.
This paper considers a two-way overall frame delay between the source user and the central AP. The average overall frame delay DT(i) and variance σ2T(i) for a user in the cell of APiare
defined as
DT(i) = T (k) + TD,A(i)
+
i
i=1
(TU,R(i) + TD,R(i− 1)) (56)
σT2(i) = σ2(k) + σD,A2 (i) + i i=1 σ2U,R(i) + σD,R2 (i− 1). (57)
VI. PERFORMANCEEVALUATIONS
In this section, we compare the performances of the increasing-spacing and uniform-spacing strategies. Moreover, we investigate the interactions among delay, capacity, and coverage in ITS WMN. We consider the IEEE 802.11a WLAN with bidirectional asymmetric traffic for each user. The system parameters are summarized in Table III. The demanded uplink and downlink traffic of each user are rU = 0.1 Mb/s and rD=
0.4 Mb/s, respectively. The ACK control frames are transmitted with PHY mode mc= 1 for reliability. Assume that all the
APs and the users have the same transmission power. Referring to the measured results in [29], we assume that the average reception ranges for eight PHY modes in the IEEE 802.11a WLAN are Dκ={290, 282, 267, 244, 213, 167, 107, 52} (in
meters) for the given transmission power. These reception ranges may vary for different environments and transmission power. However, the proposed optimization approach is general enough to evaluate the performances for different wireless ITS networks with various reception ranges.
The maximum reception range in this WMN is assumed to be max(Dκ) = 290 m. Therefore, the maximum
TABLE III
SYSTEMPARAMETERS FORPERFORMANCEEVALUATIONS
Fig. 7. Achieved profit of a cluster for the increasing-spacing and uniform-spacing AP deployment strategies with various wireline overhead ρ, where the delay requirement Dreq= 0.1 s.
rMAX = 290 m. In addition, the maximum hop distance of
AP is limited to dMAX= 290 m. This paper also assumes that
the minimum reuse distance to avoid cochannel interference is 400 (m), which is much larger than the maximum reception range. As shown in Fig. 3, there are two buffer cells between two cochannel cells. To ensure a sufficient reuse distance, the minimum separation distance between APs is limited to dMIN= 200 m. The minimum reuse distance to avoid
cochan-nel interference will vary with propagation environment, trans-mission power, and interference tolerance of devices. In other cases, if necessary, we can adopt a greater dMIN to achieve a
larger cochannel reuse distance to avoid interference. A. Comparison of Increasing-Spacing and
Uniform-Spacing Strategies
Fig. 7 compares the achieved profit of a cluster for the increasing-spacing and uniform-spacing deployment strategies with various wireline overhead ρ. The delay requirement is Dreq= 0.1 s. Fig. 7 demonstrates the advantage of the
increasing-spacing deployment strategy over the uniform-spacing strategy in terms of the achieved profit. In the increasing-spacing strategy, the AP near the central AP uses a shorter hop distance to deliver more traffic load of a cluster, whereas the AP far away from the central AP adopts a longer
Fig. 8. Capacity of a cluster for the increasing-spacing and uniform-spacing deployment strategies, where the delay requirement Dreq= 0.1 s is set for the
cases with optimal separation distances.
hop distance to extend the coverage of a cluster. By the proposed optimization approach to determine the optimal hop distances of APs, the increasing-spacing strategy can achieve better capacity (coverage) performance and higher profit of a cluster.
Fig. 7 also shows that the achieved profit of a cluster is a concave function of the number of APs in a cluster (n). More-over, there exists an optimal solution of n to maximize the profit of a cluster. For example, with the wireline overhead ρ = 5, n = 3 can achieve the optimal profit for both deployment strate-gies. The corresponding separation distances between APs are di={200, 220, 250, 496} (in meters) for the
increasing-spacing strategy and d = 224 m for the uniform-increasing-spacing strategy. In this example, the increasing-spacing strategy can achieve 17% higher profit of a cluster than the uniform-spacing strategy. The optimal number of APs in a cluster may vary for different strategies. For ρ = 10, n = 3 can achieve the optimal profit for the increasing-spacing strategy, whereas n = 4 for the uniform-spacing strategy. In this case, the achieved profit of a cluster for the increasing-spacing strategy is about 13% better than that of the uniform-spacing strategy.
Fig. 8 shows the capacity of a cluster for the increasing-spacing and uniform-increasing-spacing strategies. The optimal separa-tion distances between APs are analytically determined by the optimization approach. The simulation results conform with the analysis results. In Fig. 8, as the number of APs (n) in a cluster increases, the optimal capacity with the increasing-spacing strategy increases faster than that with the uniform-spacing strategy. However, the increment of capacity gradually diminishes, and even the capacity remains unchanged (see n = 3 to n = 4 with the increasing-spacing strategy). Since the profit of a cluster is proportional to the capacity and inversely proportional to the total cost for deploying a cluster as defined in (9), the achieved profit is a concave function of n, as shown in Fig. 7.
For comparison, Fig. 8 also shows the achieved capacity for the cases where the separation distances between APs are
Fig. 9. Coverage of a cluster of APs versus the number of APs (n) in a clus-ter under different delay requirements, where the increasing-spacing strategy is used.
fixed at d = 200 and 240 m, respectively. In Fig. 8, since the case with d = 200 m has a sufficient relay link capacity to accommodate the increasing traffic, the achieved capacity is proportional to the number of APs in a cluster. In addition, due to the larger coverage with more users, the case with d = 240 m can achieve a higher capacity than the case with d = 200 m when n≤ 3. However, if the number of APs increases from n = 3 to n = 4, the achieved capacity for d = 240 m remains unchanged. This is because the relay link between AP0 and AP1 is fully utilized. Meanwhile, deploying more APs to
extend the coverage with more served users cannot increase the capacity. Noteworthy, although with a smaller coverage, the case with d = 224 m at n = 3 has a larger relay link capacity, and thus, it achieves a higher capacity than the case with d = 240 m. This example explains that to achieve the optimal capacity, we should properly design the separation distances between APs to obtain the best tradeoff between the relay link capacity and the coverage.
In these figures, it is demonstrated that the increasing-spacing strategy can achieve higher capacity and better profit of a cluster. In addition, the best number of n and the optimal sepa-ration distances between APs can analytically be determined by the proposed optimization approach subject to the constraints on delay and throughput.
B. Interactions Among Delay, Coverage, and Capacity Figs. 9–11 investigate the interactions among delay, cover-age, and capacity, where the increasing-spacing deployment strategy is used. Fig. 9 shows the coverage of a cluster under different delay requirements. In Fig. 9, the optimal coverage at n = 3 slightly diminishes from 1874 to 1835 m to meet the delay requirement Dreq= 0.1 s. If setting a more stringent
delay requirement Dreq= 0.05 s, the optimal coverage at n =
3 will further decrease to 1770 m. In this multihop network, slightly reducing the hop distances of APs can improve the relay link capacity. Meanwhile, the smaller coverage of a cluster also lowers the traffic load. Due to higher link capacity and lower
Fig. 10. Overall frame delay versus capacity of a cluster of APs under different delay requirements, where the increasing-spacing strategy is used.
Fig. 11. Overall delay variance (jitter) versus the number of APs (n) in a cluster under different delay requirements, where the increasing-spacing strategy is used.
traffic load, the delay in each link and the overall delay can significantly be improved at the cost of smaller coverage.
Fig. 9 also illustrates that the number of APs (n) in a cluster has a maximum value. Generally, the coverage of a cluster increases as n increases. To accommodate the increasing relay traffic as n increases, we can reduce the hop distances between APs to improve the link capacity. However, the hop distance should be larger than the threshold dMINto ensure a sufficient
cochannel reuse distance, as defined in (7). Therefore, there exists a maximum value of n. In Fig. 9, the maximum number of APs is n = 3 for the case with Dreq= 0.05 s, and n = 4 for
the other cases. In this example, to meet the delay requirement Dreq= 0.05 s, the separation distance between AP0and AP1
at n = 4 will be less than dMIN. Since the constraint of (7)
cannot be met, no feasible solution exists at n = 4. In addition, Fig. 9 shows that the coverage remains unchanged for n = 3 and n = 4. The coverage of a cluster is actually constrained by the relay link capacity between AP0 and AP1 since the
link between AP0and AP1carries more traffic than others. In
this example, the separation distance between AP0 and AP1
for n = 3 and n = 4 is equal to the minimum threshold. With the same relay link capacity between AP0and AP1, both cases
have the same coverage.
Fig. 10 depicts the overall frame delay versus the capacity of a cluster under different delay requirements. At n = 3, the frame delay can dramatically be improved from 76.4 to 0.1 s, whereas the capacity of a cluster merely decreases from 46.9 to 45.9 Mb/s. The phenomenon of high delay is mainly due to the fact that the relay link between APs is fully utilized if without any delay constraint. Thus, the sojourn time of data frame at the AP will grow toward a large value [22]. However, by prop-erly shortening the hop distances between APs, the relay link capacity and delay can be improved at the expense of a smaller coverage, as shown in Fig. 9. In this example, at n = 3, the opti-mal separation distances between APs for the case without any delay requirement are di={200, 229, 257, 504} (in meters).
To guarantee the delay requirement Dreq = 0.1 s, the optimal
separation distances are reduced to di={200, 220, 250, 496}
(in meters). For both cases, the separation distances between AP0 and AP1 are equal to dMIN= 200 m. However, smaller
coverage and lower traffic load in the case with Dreq = 0.1 s
lead to lower overall frame delay. Fig. 9 also shows that, at n = 3, the delay requirement Dreq = 0.05 s can be fulfilled at
the cost of the optimal capacity decreasing to 44.2 Mb/s. From Fig. 11, one can observe the variance of overall frame delay against the number of APs in a cluster subject to various delay requirements. In Fig. 11, the case without delay require-ment has a relatively high delay variance. However, if the delay requirement Dreq= 0.1 s is set, the delay variance can also
be improved to about 5× 10−3. For Dreq= 0.05 s, the delay
variance can further be controlled to about 6× 10−4.
In Figs. 9–11, we investigate the interactions among delay, capacity, and coverage. It is shown that the optimal capacity and coverage of a cluster can simultaneously be achieved. Moreover, QoS (delay and jitter) can be guaranteed at the expense of lower capacity and coverage.
C. Model Validation
The analytical model is validated by simulations. In the sim-ulations, the legacy IEEE 802.11 CSMA protocol with binary exponential backoff scheme is used. We consider a general case where Kusers and one AP contend for the channel. We also consider the asymmetric traffic for each user. In addition, assume that the time between frame generation in each node is exponentially distributed as in [24].
Fig. 12 illustrates the downlink traffic arrival rate and the achieved service rate in a WLAN with K users, where the transmission rate is 12 Mb/s. Fig. 12 shows the downlink case as an example since the downlink traffic is much higher than the uplink traffic and thereby dominates the network performance. As shown in Fig. 12, the analytical model nicely estimates the service rate if the number of users is larger (e.g., greater than 15 in this example) and slightly underestimates the service rate for the cases with fewer users. This is because the analytical model assumes that the busy probability of each user and the
Fig. 12. Downlink traffic arrival rate and achieved service rate in a WLAN with Kusers and one AP contending for the channel. The transmission rate is 12 Mb/s. For convenience, both the arrival rate and the service rate are expressed in megabits per second.
average number of busy users are fixed all the time. However, in practice, when the AP is idle, the users have more opportunities to successfully deliver their frames and then enter the idle state. Therefore, when the AP becomes busy due to frame arrival, the average number of busy users is actually less than that assumed in the analytical model. With an overestimated collision prob-ability, the analytical model underestimates the service rate. Nevertheless, as the number of users (K) increases, most of the time, the AP is busy in sending traffic. In this situation, the analytical model can accurately estimate the average number of busy users and the service rate.
Fig. 12 also shows that the analytical model can estimate the accurate upper bound of traffic load in a wireless network with asymmetric traffic. This information is a useful design guideline for network management to guarantee the minimum throughput of each user. In this example, the analysis results suggest that the number of users (K) should be no more than 20. Obviously, if the number of users increases to K= 21, the service rate is less than the arrival rate. Meanwhile, the AP operates in the saturated situation, and the number of frames queued in the AP is ever increasing.
Fig. 13 shows the downlink frame delay against the number of users (K). As shown in Fig. 13, the analytical model well captures the delay behavior if the AP operates near the saturated situation. For example, if the transmission rate is 9 Mb/s, the frame delay will rapidly increase as K≥ 14. The analysis results provide a useful insight for network management to avoid intolerable frame delay. According to the analysis results, we can suggest that to meet the frame delay requirement of 0.1 s, the number of users should be no more than 13 if the transmission rate is 9 Mb/s.
In Figs. 12 and 13, it is demonstrated that the analytical model can provide useful information for network management and design. The analytical model nicely estimates the link capacity (i.e., the service rate) in a WLAN with bidirectional
Fig. 13. Downlink frame delay against the number of users Kin WLAN, where the transmission rates are 9, 12, and 18 Mb/s, respectively.
asymmetric traffic, although there are some assumptions and approximations. Moreover, the developed model accurately estimates the upper bound of traffic load under the delay requirement. The upper bound of traffic load is an important guideline for network management to determine the system parameters to ensure the QoS of users.
VII. CONCLUSION
In this paper, we have investigated the AP deployment issue for a scalable ITS WMN considering the overall performances of throughput, coverage, and QoS. The presented mesh network architecture is very suitable for ITS applications due to easy network deployment and lower infrastructure cost. With the QoS requirement, an optimization approach has been proposed to maximize the ratio of total capacity to total cost for deploying a cluster of APs in the considered ITS WMN.
From the system architecture perspective, the proposed WMN has two main advantages. First, the suggested frequency planning can relieve the collision issue and make the WMN more scalable. Second, the proposed network architecture also facilitates the management of QoS and throughput of a WMN. From the system design perspective, this paper has three im-portant elements. First, we have proposed an analytical model to evaluate the throughput, which has considered bidirectional asymmetric traffic for users operating in the unsaturated situa-tion. Second, we have developed a queuing model to analyze the frame delay and jitter. Third, we have applied an opti-mization approach to determine the best number of APs in a cluster and the optimal separation distances between APs for the proposed ITS WMN. Numerical results have shown that the optimal deployment parameters can analytically be determined, and the goals of throughput enhancement and QoS provisioning can be fulfilled at a slight cost of coverage. In addition, the increasing-spacing strategy outperforms the uniform-spacing strategy in terms of a profit function, considering the capacity, the cost of APs, and the wireline overhead.
APPENDIXI
STEADY-STATEPROBABILITYPs, MEANE[s],AND
VARIANCEVar(s)
The steady-state probability Ps is derived by using the
probability generating function approach [22]. The generating functionP(z) for the steady-state probability Psis written as
P(z) =∞ s=0 Pszs= μP0 μ− (χ1+ χ2)z− χ2z2 . (58)
With the condition thatP(1) =∞s=0Ps= 1, we have
P0=
μ− (χ1+ 2χ2)
μ = 1−
χ1+ 2χ2
μ . (59)
Then, the generating functionP(z) can be rearranged as P(z) = μ− χ1− 2χ2 μ− (χ1+ χ2)z− χ2z2 = μ− χ1− 2χ2 (χ1+ χ2)2+ 4μχ2 × 1 z1 1 1− z/z1 − 1 z2 1 1− z/z2 (60) where z1= −(χ1+ χ2) + (χ1+ χ2)2+ 4μχ2 2χ2 (61) z2= −(χ1+ χ2)− (χ1+ χ2)2+ 4μχ2 2χ2 . (62)
Since∞s=0zs= 1/(1− z), it follows that P(z) = ∞ s=0 Pszs = μ− χ1− 2χ2 (χ1+ χ2)2+ 4μχ2 × ∞ s=0 1 z1 z z1 s − 1 z2 z z2 s . (63)
By inspecting the coefficient of zs, the steady-state probability
is obtained as Ps= μ− χ1− 2χ2 (χ1+ χ2)2+ 4μχ2 1 z1s+1− 1 z2s+1 . (64)
Therefore, the average number E[s] of queued frames in a node can be calculated by
E[s] =P(1) = lim z→1 ∞ s=0 sPszs−1= χ1+ 3χ2 μ− χ1− 2χ2 . (65)
SinceP(1) = limz→1∞s=0s(s− 1)Pszs−2= E[s2]− E[s],
the variance Var(s) is expressed as Var(s) = E[s2]− (E[s])2
=P(1) + E[s]− (E[s])2 =χ1(μ− χ2) + (5μ− χ2)χ2
(u− χ1− 2χ2)2
. (66)
APPENDIXII
AVERAGEDELAY ANDVARIANCE
A. Delay and Variance for Uplink Relay Frames of AP First, we evaluate the initial probability Qh of an incoming
frame being placed at the hth position of the FCFS queue at the arrival instant. In an activity slot, at most two frames will arrive at the queue for the uplink traffic of AP. Let βx be the
probability that there are x incoming frames at an arbitrary arrival instant. Clearly
βx= αL(1−αR)+αR(1−αL) 1−(1−αL)(1−αR) = αL+αR−2αLαR αL+αR−αLαR, x = 1 1−β1, x = 2. (67) At an arbitrary arrival instant, the average number of in-coming frames is equal to β1+ 2β2. Consider that there are s frames queued at the AP with the probability Ps. On average,
(β1+ β2) = 1 incoming frames will be placed at the (s + 1)th
position of the queue and β2incoming frames at the (s + 2)th
position at an arrival instant. Let αi be the probability of an
incoming frame being placed at the (s + i)th position of the queue. Obviously, α1= (β1+ β2)/(β1+ 2β2), and α2= 1− α1= β2/(β1+ 2β2). Therefore, the initial probability Qh of
an incoming frame being placed at the hth position of the queue can be calculated by
Qh=
α1Ps|s=h−1+ α2Ps|s=h−2, h≥ 2
α1Ps|s=h−1, h = 1. (68)
Then, we evaluate the average delay TU,R(i) and variance
σ2
U,R(i) for the uplink relay frames of APi. From (54), we
obtain TU,R(i) = ∞ h=1 QhE[Dh] = 1 ν1 ∞ h=1 hQh = 1 ν1 ∞ h=1 hα1Ps|s=h−1+ ∞ h=2 hα2Ps|s=h−2 = 1 ν1 α1 ∞ s=0 (s + 1)Ps+ α2 ∞ s=0 (s + 2)Ps = 1 ν1 (E[s] + 1 + α2) . (69)
Here, E[Dh] = h/ν1 is the average delay for a frame to be
successfully transmitted, given that the frame is now at state h, as in (49). Ps is the probability of s frames being queued at
the node. E[s] =∞s=0sPsis the average number of queued
frames at the node, as in (65). According to (55), we have σ2U,R(i) =
∞
h=1
Qh(E[Dh])2− [TU,R(i)]2+ ∞ h=1 QhVar(Dh) =Var(s) + α2− α22 ν2 1 +(1− ν1) (E[s] + 1 + α2) ν2 1 (70) where Var(Dh) = h(1− ν1)/ν12 is the variance of Dh, as in
(53), and the variance Var(s) is detailed in (66).
B. Delay and Variance for Uplink Access Frames of User For a user, only one uplink frame will arrive at any ar-rival instant. Accordingly, αR= 0, χ2= 0, and α2= 0. From
(64) and (68), the initial probability is equal to Qh= Ph−1=
ρh−1c (1− ρc), where ρc= χ1/μ. By (69) and (70), the average
delay T (k) and variance σ2(k) for the uplink access frames of
a user can be calculated by T (k) = 1 ν1 (E[s] + 1) = 1 ν1(1− ρc) (71) σ2(k) = 1 ν2 1 Var(s) +1− ν1 ν2 1 (E[s] + 1) =1− ν1(1− ρc) ν2 1(1− ρc)2 (72) where, from (65) and (66), E[s] = ρc/(1− ρc), and Var(s) =
ρc/(1− ρc)2.
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