Throughput-coverage tradeoff in a scalable wireless mesh network

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Throughput-coverage tradeoff in a scalable wireless mesh network

夡

Jane-Hwa Huang

∗

, Li-Chun Wang, Chung-Ju Chang

Department of Communication Engineering, National Chiao Tung University, Taiwan, ROC

Received 11 November 2006; received in revised form 1 August 2007; accepted 19 October 2007 Available online 1 November 2007

Abstract

The wireless mesh network (WMN) is an economical and low-power solution to support ubiquitous broadband services. However, mesh networks face scalability and throughput bottleneck issues as the coverage and the number of users increase. Specifically, if the coverage is extended by multiple hops, the repeatedly relayed traffic will exhaust the radio resource and degrade user throughput. Meanwhile, as the traffic increases because of more users, the throughput bottleneck will occur at the users close to the gateway. The contention collisions among these busy users near the gateway will further reduce user throughput. In this paper, a newly proposed scalable multi-channel ring-based WMN is employed. Under the ring-based cell structure, multi-channel frequency planning is used to reduce the number of contending users at each hop and overcome the throughput bottleneck issue, thereby making the system more scalable to accommodate more users and facilitate coverage extension. This paper mainly focuses on investigating the overall tradeoffs between user throughput and cell coverage in the ring-based WMN. An analytical throughput model is developed for the ring-based WMN using the carrier sense multiple access (CSMA) medium access control (MAC) protocol. In the analysis, we also develop a bulk-arrival semi-Markov queueing model to describe user behavior in a non-saturation condition. On top of the developed analytical model, a mixed-integer nonlinear optimization problem is formulated, aiming to maximize cell coverage and capacity. Applying this optimization approach, we can obtain the optimal number of rings and the associated ring widths of the ring-based WMN.

Keywords: Wireless mesh network (WMN); Scalability issue; Throughput-coverage tradeoff; Multi-channel and multi-radio operations; Frequency planning

1. Introduction

With the abilities of enhancing coverage and capacity by low transmission power, wireless mesh networks (WMNs) play a signiﬁcant role in providing ubiquitous broadband access [1,7,18,21,22,25]. Fig. 1 shows a multi-hop WMN, where each user relays other users’ trafﬁc toward the central gateway and only the gateway directly connects to the Internet. In general, the advantages of WMN can be summarized into four folds. First, WMN can combat shadowing and path loss to extend 夡_{Expanded version of a talk presented at the IEEE WirelessCom (Hawaii,}

June 2005). This work was supported in part by the MoE ATU Plan, the Program for Promoting Academic Excellence of Universities (Phase II), and the National Science Council under Grant 95W803C, Grant NSC 95-2752-E-009-014-PAE, Grant NSC E-009-148, and Grant NSC 95-2221-E-009-155.

∗_{Corresponding author. Fax: +886 3 571 0116.}

E-mail addresses:hjh@mail.nctu.edu.tw(J.-H. Huang),

service coverage. Second, WMN can be rapidly deployed in a large-scale area with less cabling engineering work and in-frastructure costs[7,18,21,25]. Third, WMN can concurrently support various wireless radio and access technologies such as 802.16 (WiMAX), 802.11 (WiFi), and 802.15 (Bluetooth and Zigbee), thereby providing the ﬂexibility to integrate different radio access networks [1]. Fourth, WMN can be managed in a self-organization and self-recovery fashion [1,22]. If some nodes malfunction, the trafﬁc can be forwarded by alternative nodes.

However, WMNs face scalability issue because throughput enhancement and coverage extension are usually two con-tradictory goals in WMNs [1,11,17,18,21]. Specifically, the multi-hop communications can extend the coverage of gateway to serve more users by more hops and longer hop distance. However, the repeatedly relayed traffic with more hops will exhaust the radio resource and thus degrade the user through-put [11,17]. The longer hop distance will lower the data rate in the relay link between users. Moreover, increasing traffic from

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r1 r₂ r3 r4 A1 A2 A4 A3 Gateway (Central AP) Mesh Cell 0 Mesh Cell 1 Mesh Cell 2 f1 f2 f3 f4 Switch/Router Internet

Fig. 1. Ring-based cell architecture for a scalable wireless mesh network, where each ring is allocated with different channels.

more users will induce the throughput bottleneck at the users near the gateway, thereby further degrading user throughput. Therefore, while multi-hop communication is used to extend coverage with more users, how to improve user throughput is a key challenge in designing a scalable WMN.

In the literature, the performances of WMNs have been stud-ied mainly from two directions[9,18–21]. On one hand, the au-thors in [20] demonstrated the advantage of a multi-hop WMN over a single-hop network in terms of coverage by simulations. On the other hand, the results in [9,19] showed that with k users in an ad hoc network, the throughput per user is scaled like

O(1/√k log k). Moreover, the authors in [18] pointed out that

the user’s throughput in the WMN decreases sharply as O(1/k) because of the throughput bottleneck at the gateway. To resolve the scalability issue, our previous work [13] proposed a ring-based WMN. The work in [13] investigated the delay and cell capacity tradeoff in a WMN. To our knowledge, a few papers have studied the overall performances of user throughput and cell coverage of the WMN [21]. However, the work [21] con-sidered the single-user case.

To resolve the scalability and throughput bottleneck issues, this paper employs the newly proposed ring-based WMN in [13], where the rings in a cell are allocated with different chan-nels as shown in Fig. 1. This WMN is scalable due to the fol-lowing two factors. First, multi-channel frequency planning can reduce the number of users contending for the same channel, and overcome the throughput bottleneck issue at the gateway.

Second, with the capability to adjust the ring width to control the contention level, the ring structure can facilitate managing throughput in a WMN.

This paper investigates the optimal tradeoff between user throughput and cell coverage in the scalable ring-based WMN. We develop an analytical throughput model by considering the impacts of ring-based cell structure and frame contentions in the carrier sense multiple access (CSMA) medium access control (MAC) protocol. In the throughput analysis, we also develop a bulk-arrival semi-Markov model to describe user behavior under the non-saturation condition. On top of the developed analytical model, we formulate an optimization problem aiming to improve the performance tradeoff between throughput and coverage. With the optimization technique, we can determine the optimal number of rings and the associated ring widths in a mesh cell.

The rest of this paper is organized as follows. Section2 dis-cusses the considered scalable ring-based WMN. In Section 3, we formulate an optimization problem to maximize coverage and capacity of a mesh network. Section 4 investigates the channel activity in the ring-based WMN, with considering the impact of ring structure on frame contentions. On top of the channel activity concept, in Section 5 we develop a MAC throughput model for the considered WMN. Numerical exam-ples are shown in Section 6. Concluding remarks are given in Section 7.

2. Scalable ring-based WMN

2.1. Network architecture and assumptions

Fig. 1 shows the scalable ring-based WMN, where stationary mesh users form a multi-hop network to extend cell coverage. The mesh cell is divided into several rings Ai, i = 1, 2, . . . , n, determined by n concentric circles centered at the gateway with radii r1< r2 < · · · < r_n. The user in ring A_i connects to the gateway by an i-hop communication. The users in the inner rings will relay data for users in the outer rings toward the gateway and only the gateway connects to the Internet directly. Clearly, this WMN can be rapidly deployed in a large-scale area with less cabling engineering work.

The ring-based WMN operates in a multi-channel with multi-interface fashion. In a mesh cell, the rings are allocated with different channels to avoid ring co-channel inter-ference and reduce contention collisions. As shown in Fig. 1, the user in ring Ai communicates with the users in rings Ai−1 and Ai+1at different channels fi and fi+1, respectively. This frequency planning is simple because it only needs to design each ring width to ensure a sufﬁcient co-channel reuse distance without interference. We also assume that each user is equipped with two radio interfaces as in [1]. With multiple interfaces independently operating at different channels, each user can concurrently receive and deliver the relay trafﬁc, thereby improving the throughput and delay performances. In addi-tion, the WMN can work well even if employing the legacy CSMA MAC protocol, which in turns avoids complexity and compatibility issues.

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5 6 4 7 8 10 CH-3 1 2 5 6 4 9 7 12 1 2 CH-2 5 6 4 8 9 11 1 2 CH-1 3 3 3

Fig. 2. Example of a three-cell WMN with 12 available channels. Four buffer rings between two co-channel rings are ensured, and the congested inner rings (A1.A2) are sectorized.

Spectrum and hardware costs are the major concerns in the multi-channel with multi-interface systems. However, there are multiple channels available in wireless local area networks (WLANs), for example, 12 channels assigned for the IEEE 802.11a WLAN[16,23]. The price of radio interface also goes down rapidly, since the WLAN has become an off-the-shelf product.

2.2. Ring-based frequency planning

Now, we explain the ring-based frequency assignment by a three-cell WMN as shown in Fig. 2. In this example, channels 1.3 and 4.6 are assigned to the sectors in the innermost rings

A1and A2of each cell. Channels 7.9 are repeatedly allocated to the middle rings A3 and A4 of the cells with four buffer rings. Channels 10.12 are allocated to rings A5 of the cells, respectively. Then, with four buffer rings, the channels 1.3 are reused in the outer rings A6. This example shows that 12 available channels can ensure four buffer rings between two co-channel rings. Besides, the channels allocated to the inner rings can be spatially reused in the outer rings with a sufﬁcient reuse distance.

Referring to Fig. 2, we also suggest sectorizing the congested inner rings and allocating a different channel to each sector, to overcome the throughput bottleneck issue near the gateway. In a WMN, the users in the inner rings near the gateway will re-lay more trafﬁc than the users in the outer rings. By partition-ing the inner rpartition-ings into several sectors to reduce the number of contending users, the throughput can be further improved. In this example, the innermost rings of each cell are divided into three sectors. Apparently, if more non-overlapping chan-nels are available, more inner rings can be sectorized without

inter-ring co-channel interferences to enhance cell capacity and coverage.

In practice, the WLAN users may interfere with this ring-based WMN operating at the unlicensed band, and thus the throughput of each user will decrease. In this situation, we suggest allocating the channels with less interference to the congested inner rings to ensure throughput. To understand per-formance bound of this multi-hop network, this interference issue is not considered in this paper.

2.3. Scalability

Most traditional WMNs are not scalable to cell coverage be-cause the user throughput is not guaranteed with increasing col-lisions. By contrast, the employed ring-based WMN is scalable to coverage since the ring-based frequency planning can re-duce the number of contending users to resolve the contention issue. Then, the user throughput can be ensured by properly designing the ring widths in a mesh cell. The remaining im-portant problem lies in the way to determine the optimal ring widths to achieve the optimal performance tradeoff between user throughput and cell coverage.

3. Coverage and capacity maximization 3.1. Problem formulation

Both user throughput and cell coverage performance issues will impact the design of WMN. From a deployment cost per-spective, a larger coverage per cell is better since fewer gate-ways are needed. From a user throughput viewpoint, however, a smaller cell is preferred since fewer users contend for the same channel. In the following, we formulate an optimization prob-lem to ﬁnd out the best number of rings and the optimal ring widths in a cell subject to the tradeoff between user throughput and cell coverage.

To begin with, we discuss the constraints in the considered optimization problem:

• The capacity HC(i) of the lowest-rate link in ring Ai should be greater than the carried trafﬁc load Ri of one mesh user. That is,

HC(i) = Hi(ri− ri−1)Ri, (1) where (ri−ri−1) is the width of ring Aiand Hi(d) represents the link capacity between two users at a separation distance

d. This constraint guarantees the minimum throughput for

each user. Fig.3 shows some examples of lowest-rate links, for example, the link between users PC,iand QC,iat the ring boundaries with a separation distance d = (ri − ri−1). • The ring width (ri− ri−1) should be less than the maximum

reception range. Therefore,

(ri− ri−1)dmax. (2)

• The ring width should be greater than the average distance

dminbetween two neighboring users. Hence,

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Examples of lowest-rate links-r4-r3 r3-r2 PC4 QC4 r1 r3 r2 r4 A4 A3 A2 A1 PC3 QC3 PC2 QC2 PC1

Fig. 3. Examples of the lowest-rate links for a mesh cell with n = 4.

where dmin = 1/√ (m) is dependent on the user density

(users/m2

). This constraint also represents the limit on the

hop distance due to user density.

3.2. Optimization approach

From the above considerations, the optimal cell coverage and capacity issues in a WMN can be formulated as a mixed-integer nonlinear programming (MINLP) problem with the nonlinear objective function (4). The decision variables includes the num-ber of rings in a mesh cell, n (which is an integer) and the radii

r1, r2, . . . , rn. The objective is to maximize the coverage of a mesh cell. In this scalable ring-based WMN, the ring-based frequency planning resolves the collision issue as cell overage increases. Therefore, the optimal coverage and capacity will be achieved simultaneously, since more users in a mesh cell can also lead to higher cell capacity. The optimal system parame-ters for the ring-based WMN can be determined by solving the following optimization problem:

MAX

n,r1,r2,...,rn rn (cell coverage),

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subject to HC(i)Ri, (5)

dmax(ri− ri−1)dmin. (6) In this paper, cell coverage is deﬁned as cell radius rn, and cell capacity is the overall throughput of a cell, that is,r_n2RD, where is the user density, and RDis the trafﬁc load generated by each user.

4. Channel activity in the ring-based WMN

This section discusses the channel activity seen by an indi-vidual user employing the CSMA MAC protocol in the ring-based WMN. On top of the channel activity concept, we will develop an analytical throughput model for the considered WMN in Section5.

From a particular user’s viewpoint, there are ﬁve types of channel activities in the WMN:

(1) successful frame transmission; (2) unsuccessful frame transmission;

(3) empty slot, where all users are in backoff or idle; (4) successful frame transmission from other users; (5) unsuccessful frame transmission from other users. For clarity, the channel activity is described by a sequence of

activity time slots [2,3,6]. Subject to the backoff procedures, the slot duration Tj for the channel activity type j is equal to

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ T1= T4= TS, T2= T5= TC, T3= , (7)

where is the duration of an empty slot, TSand TCare the suc-cessful transmission time and collision duration, respectively. Therefore, the average duration Tvof activity time slot can be written as Tv= 5 j =1 jTj. (8)

Here,_jis the corresponding probability for the channel activity type as calculated in the following, and5_{j =1}j = 1. 4.1. Assumptions

In the following, we consider the case where the traffic is unidirectional from the users to the gateway. The developed an-alytical method can be extended straightforwardly for the case with bidirectional traffic. We need to consider the contentions from the users with downlink traffic, and thus the number of contending users increases. It may also need to consider the asymmetric traffic load in downlink and uplink. Therefore, the users contending for the same channel may have different traf-fic loads. To clarify the developed analytical approach, this paper focuses on a simplified case with uplink traffic as an example.

To understand the coverage and capacity performance bounds in a ring-based WMN, we also assume that all the traf-ﬁc is forwarded in the centripetal direction toward the gateway. Moreover, there always exists an intermediate relay node at the appropriate position. In a real WMN, the next-hop node may be too far away from the current node and therefore user throughput may degrade. In this situation, it may be needed to deploy a pure relay station to help forward data as discussed in [21]. In other cases, if the trafﬁc is not forwarded in the cen-tripetal direction, the throughput and coverage performances may degrade due to longer hop distance required.

4.2. Frame contention under ring-based cell structure

To investigate the channel activity in the ring-based WMN, we should consider the impacts of ring-based cell structure on frame contentions. At ﬁrst, we deﬁne the mutually interfered

region as an area in which any two users can sense the

activ-ity of each other. In Fig. 4, the area including users C and D is an example of a mutually interfered region. Since each ring

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ri Gateway (Central AP) lRC Ai Ai-1 A B W,i S,i P ri-1 ri-2 VR (_r_i_{+ r}_i-1)/2 C D S,i

Wireless Collision Domain, W,i in Ai Mutually-Interfered Region, S,i in Ai Mutually-Interfered Region, _S,i-1 in A_i-1 Working-in-Vain Region V_R of User P

S,i-1 Q

S,i-1

Fig. 4. Examples of wireless collision domain and mutually interfered region.

is allocated with a different channel, a mutually interfered re-gion is the intersection of two circles and the associated ring, depending on the locations of considered users and the inter-ference distance. For simplicity, we assume that the mutually interfered region can be approximated as an annulus sector as shown in the ﬁgure. Suppose that all the users transmit at the same power and the interference distance is lRC. Referring to Fig.4, the central angle_S,i of a mutually interfered region in ring Ai is equal to S,i= 2 sin−1 lRC ri + ri−1 for lRC < (ri+ ri−1). (9) If lRC(ri + ri−1), we deﬁne S,i = 2. This means that the whole ring is in the same mutually interfered region. Clearly, the area of a mutually interfered region is AS,i = (S,i/2)ai and ai = (r_i2− r_i−12 ) is the area of ring Ai.

Then, we deﬁne the wireless collision domain as the area in which at any instant at most one user can successfully deliver data trafﬁc at a particular frequency. As shown in Fig.4, the wireless collision domain in ring Ai is also approximated as an annulus sector with a central angle of_W,i = _S,i−1, and its area is AW,i = (W,i/2)ai. The phenomenon of W,i =

S,i−1 is due to the fact that the request-to-send/clear-to-send (RTS/CTS) mechanism is employed to avoid the hidden node problem. Referring to the example in Fig. 4, user A in ring

Ai is sending data to user B in ring Ai−1. Meanwhile, since users P and A are not in the same mutually interfered region, user P in ring Ai can send an RTS request to users Q in ring

Ai−1. However, user Q will not reply the CTS to P, because it has overheard the CTS of B and determined that the channel is busy. This example shows that users P and A are in the same wireless collision domain even though they are not in the same mutually interfered region. Furthermore, the central angle_W,i of wireless collision domain in ring Ai is determined by the

angle _S,i−1 of mutually interfered region in the inner ring

Ai−1, that is,W,i = S,i−1.

The example in Fig.4 also shows that the transmission form the user in region VRinvalidates the RTS request of P. Hence, we deﬁne the region VRwith a central angle of (W,i− S,i) as the working-in-vain region of P. These effects of the ring structure on frame contentions will be incorporated into the throughput model later.

Note that the innermost ring A1is in the same wireless col-lision domain and _W,1 = 2 since all users in ring A1 can overhead the CTS from the gateway. By sectorizing ring A1as shown in Fig. 2, the number of contending users is decreased by a factor of three since_W,1 = 2/3. Thus, the contention collisions can be also reduced to resolve the throughput bottle-neck issue in the WMN.

4.3. Successful/unsuccessful transmission

As shown in Fig. 5, user P can successfully send data as long as no other user is transmitting in the adjacent wireless collision domains of P. Consider user P and its two wireless collision domains inﬂuenced by the closest two neighboring transmitters

PLand PR, which are out of the mutually interfered regions of P as shown in the ﬁgure. Note that the considered area of angle

2_W,i will be inﬂuenced by at most two neighboring transmit-ters (for example, users PL and PR). Other transmitters (for example, users PL and PR) are too far away and will not affect the considered area. LetL andRrepresent the positions of PLand PR, respectively. If one of the transmitters PL and PR is within the working-in-vain regions of P, that is, _L,_R ∈ [S,i, W,i], user P can still send the RTS request to user Q, but user Q cannot reply the CTS, as discussed in Section 4.2. Sup-pose that ZW,i is the average probability (average fraction of time) of a wireless collision domain in which a user is sending

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X X_R W,i P Q W,i S,i W,i Gateway (Central AP) X_L QR PR QL PL S,i Working-in-Vain Region L R Ai Ai-1 ri-2 ri-1 ri S,i-1 S,i-1 S,i P,L P,R S,i S,i Considered user P and area with 2_W,i

Fig. 5. The considered user P and two adjacent wireless collision domains, where user P is contending for the radio channel.

data as detailed in (30). Then, the working-in-vain probability

pvof user P can be expressed as pv= 1 − Pr L, R∈ [/ S,i, W,i] = 1 −

1− Z_W,iW,i− S,i

W,i 2

, (10)

where ZW,iaccounts for the existence probability of transmitter PL (PR) which is affecting the considered area.

Now, we consider the case where both transmitters PL and

PR are not in the working-in-vain regions of user P, that is,

L, R∈ [0, S,i]. In the considered area of angle 2W,i, only the users in the area{2A_W,i−(XL+XR)} can send RTS frames as shown in Fig.5. Those users in regions XL and XR will not send their requests since they can sense the transmissions of PL and PR. Let _X be the average central angle for re-gion XL, and AW,i be the area of a wireless collision domain of user P. Therefore, the average number of contending users

C1,i in the considered area of angle 2W,i is equal to the av-erage number of users in the area of {2A_W,i− (XL+ XR)}. Consequently, c1,i=ai 22(W,i− ZW,iX) =ai W,i−ZW,i W,i _S,i 0 LdL = (r2 i − rr−12 ) W,i− ZW,i2S,i 2_W,i , (11)

where is the user density; a_i = (r_i2−r_i−12 ) is the area of ring Ai;S,i is the central angle of the mutually interfered region

as deﬁned in (9);_X= (L+ S,i) − S,i= Lis the central angle of region XLandLis uniformly distributed in[0, W,i] as shown in Fig. 5. Subject to the RTS/CTS procedures, the frame collisions may only occur when the contending users concurrently deliver their RTS requests. Let be the average probability of an active user sending the RTS request at the beginning of an activity slot. Suppose that 0 is the average probability of a user being idle due to empty queue. Incor-porating the impacts of ring structure on frame contention, the unsuccessful transmission probability pu can be com-puted by

pu= pv+ (1 − pv)[1 − (1 − (1 − 0))C1,i−1]. (12) In (12), the ﬁrst term is the probability that at least one trans-mitter is inside the working-in-vain regions of P. That is, user

P will not receive the CTS response. The second term

repre-sents the probability that the RTS request from P is collided with other RTS frames.

Thus, given that the considered user has a non-empty queue, the probability that this user successfully/unsuccessfully sends data frame in an activity slot can be expressed as

1= (1 − pu), (13)

2= pu. (14)

4.4. Empty slot

As shown in Fig.6, user P observes an empty slot if all the users in the adjacent mutually interfered regions of user P are silent. In the ﬁgure, the users in regions YLand YRwill not send RTS since PL and PRare transmitting. Let_Y be the average central angle of region YL, and AS,i be the area of a mutually interfered region of user P. The average number of contending

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QR PR Ai Ai-1 QL PL S,i S,i-1 ri-2 ri-1 ri P Q Gateway (Central AP) P, Q, Y L R YL YR W,i W,i S,i S,i S,i S,i-1 L R

Considered user _{P and area with 2}_S,i

Fig. 6. The considered user P and its two mutually interfered regions, where user P is in backoff at the current slot.

users C2,i in the considered area of angel 2S,i is equal to the average number of users in the area of{2A_S,i−(YL+YR)}, and

c2,i=ai 22(S,i− ZW,iY) =ai S,i−ZW,i W,i _W,i

0 max(0, L+S,i−W,i) dL = (r2 i − r 2 r−1) S,i− ZW,i2S,i 2_W,i , (15)

where _Y = max(0, L + S,i − W,i) is the central angle of region YL. Therefore, from the viewpoint of the considered user, the empty-slot probability is

3= (1 − )[1 − (1 − 0)]c2,i−1, (16) where the ﬁrst term is the probability of the considered user being in backoff, and the second term represents the probability that all the other users are in backoff or idle.

4.5. Successful/unsuccessful transmission from other users

To calculate the probability of successful transmission from other users, we consider user P and its adjacent mutually-interfered regions, as shown in Fig.6. In the considered area of angle 2_S,i, the average number of contending users is c2,i as derived in (15). Given that user P is in backoff at the cur-rent slot, the probability that at least one user sends RTS is equal to potr = 1 − [1 − (1 − 0)]c2,i−1. Suppose that Xj is the probability of the considered area being inﬂuenced by j neighboring transmitters. In the considered area of angle 2_S,i, the conditional probability that there is at least one successful

transmission from other users is equal to

pos= 2 j =0(2s1,j − s2,j)Xj potr , (17) where Xj = 2 j

Z_W,ij (1 − ZW,i)2−j, s1,j is the probability that there is a successful transmission in the left-side mutually interfered region of user P, and s2,j is the probability that there is a successful transmission in each mutually interfered region of P. Then, the probability that the considered user P observes successful/unsuccessful transmission(s) from other users in an activity slot can be expressed as

4= (1 − )potrpos, (18)

5= (1 − )potr(1 − pos), (19)

where the term (1 − ) accounts for the probability of the con-sidered user being in backoff. The successful probabilities s1,j and s2,j will be derived in Appendix A.

5. Throughput analysis

On top of the channel activity concept, we suggest an an-alytical throughput model for the ring-based WMN using the CSMA MAC protocol with RTS/CTS. In the throughput analy-sis, we also develop a bulk-arrival queueing model to describe user behavior under the non-saturation condition, considering the case where the forwarded frame and local frame may ar-rive at one user simultaneously. Although the 802.11a WLAN is used as an example here, the modeling framework can be applied to various wireless systems using different variation of CSMA protocol.

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5.1. Background

Now we calculate the durations of a successful frame trans-mission and a collision in the IEEE 802.11a network. Let l be the payload size of data frame, maand mcbe the transmission PHY mode for data frame and that for control frame, respec-tively. Subject to the IEEE 802.11 CSMA MAC protocol with RTS/CTS, the successful frame transmission time TS and col-lision time TCare expressed as

TS= TRTS(mc) + + SIFS + TCTS(mc) + + SIFS + TDATA(l, ma) + + SIFS

+ TACK(mc) + + DIFS, (20)

TC= TRTS(mc) + + EIFS, (21)

where is the propagation delay; the durations of short inter-frame space (SIFS), distributed interinter-frame space (DIFS) and ex-tended interframe space (EIFS = SIFS+TCTS(mc)+DIFS) are speciﬁed in[14,15]. TDATA(l, ma) is the transmission time for a data frame with payload size l using PHY mode ma. TRTS(mc), TCTS(mc), and TACK(mc) are the transmission durations of RTS, CTS and acknowledgment (ACK) control frames using PHY mode mc, respectively. According to the IEEE 802.11a WLAN standard [15], the values of TDATA(l, ma), TRTS(mc), TCTS(mc) and TACK(mc) can be speciﬁed.

5.2. Carried trafﬁc load of a mesh user

The carried traffic load of each mesh user includes its own traffic and the forwarded traffic from other users. Assume that an ideal load-balancing path selection is employed to avoid congested links [1,12]. Hence, all the users in the inner ring

Ai evenly share the forwarded trafﬁc from the outer ring Ai+1. Suppose that the users are uniformly distributed with the density

. The average number of users ki in ring Ai is ki = (r_i2−

r_i−12 ) and (ri − ri−1) is the width of ring Ai. Let RD and RF,i be the average traffic load generated by one user and the forwarded traffic load per user in ring Ai, respectively. With the load-balancing path selection, the carried traffic load Ri of a mesh user in ring Ai can be expressed as

Ri= RF,i+ RD=ki+1 ki Ri+1+ R D = _n j =i+1kj ki + 1 RD, (22)

and the forwarded trafﬁc load per user is RF,i = (n_{j =i+1}

kj/ki)RD. For the outermost ring An, Ri = RD and RF,i= 0.

5.3. MAC throughput

To evaluate the MAC throughput in the ring-based WMN, we should consider the impacts of the ring-based cell structure

on frame contentions. Consider a binary exponential backoff procedure with the initial backoff window size of W . Let mbk be the maximum backoff stage. The average backoff time can be calculated by Bk= (1 − pu)W − 1 2 + pu(1 − pu) 2W − 1 2 + · · · +pumbk(1 − pu) 2mbk_{W − 1} 2 +pu(mbk+1)(1 − pu) 2mbk_{W − 1} 2 + · · · =[1 − pu− pu(2pu)mbk]W − (1 − 2pu) 2(1 − 2pu) , (23)

where puis the unsuccessful transmission probability with con-sidering the effects of ring structure on frame contentions, as deﬁned in (12). Since an active user sends RTS requests every

(Bk+ 1) slots on average [24], the transmission probability for an active user can be written as

= 1 Bk+ 1 = 2 1+ W + puW _mbk₋₁ i=0 (2pu)i . (24)

From (12) and (24), we can obtain the unique solution of and

pufor a given idle probability0of a user. The idle probability 0will be derived by the following queueing model.

Fig. 7 illustrates the proposed bulk-arrival discrete-time queueing model for a mesh user, where the state variable s represents the number of frames queued at the user. Let l be the data frame payload size. In this WMN, the total traffic to a mesh user includes the forwarded traffic from other users with mean arrival rate F = RF,i/ l (frames/s) and the local traffic generated by user with mean rate L = RL/ l. The forwarded frame and the local frame may arrive at one user si-multaneously. According to the CSMA MAC protocol, in each activity time slot one user can successfully receive at most one forwarded frame. Therefore, we assume that the average arrival probability of forwarded traffic in an activity time slot is F = FTv. Since the duration of activity slot is relatively short, we also assume that the probability of one local frame generated in a slot is L = LT_v. In addition, we note that a mesh user can successfully send one data frame in a slot with probability1, as discussed in Section 4.3.

Therefore, from above considerations, the number of frames queued at a user can be modeled by a semi-Markov model as shown in Fig. 7. The state-transition probabilities for the semi-Markov model can be expressed as

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ps,s+2= 2= LF(1 − 1), ps,s+1= 1= LF1+ L(1 − F)(1 − 1) +(1 − L)F(1 − 1), ps,s−1= = (1 − L)(1 − F)1, ps,s = 1 − 1− 2− , (25)

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... ... 0 1 2 s-1 s s+1 χ2 χ2 χ2 χ2 χ2 χ1 χ1 χ1 1_−χ1 −χ2 χ1 1_−χ1−χ2− 1−χ1−χ2− 1_−χ1−χ2− 1− χ1−χ2− 1− χ1−χ2−

Fig. 7. State transition diagram for a user, where the state variable s is the number of frames queued at the user.

where ps,s+2 represents the probability that two frames (one local frame and one forwarded frame) are simultaneously ar-rived in an activity slot, and no queued frame of the node is successfully delivered. Let s be the steady-state probability of s frames being queued at the node. Referring to Fig.7, the global-balance equations for the considered queueing model can be written as ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 0= −(1+ 2+ )s+ 2 i=1 _is−i+ s+1, s 2, 0= −(1+ 2+ )1+ 10+ 2, 0= −(₁+ ₂)0+ 1. (26)

Then, by the probability generating function approach [8] with some manipulations, the generating functionP(z) for the steady-state probabilitys can be expressed as

P(z) =∞ s=0 szs = 0  − (1+ 2)z − 2z2 . (27)

With the condition P(1) =∞s₌₀s= 1, we can ﬁnd the idle probability0of a mesh user

0=  − (1+ 22)

= 1 −

1+ 22

. (28)

Now we evaluate the MAC throughput of one user. With the activity slot concept, the average busy probability (average fraction of time) ZO,i of one user being sending data and the channel utilization ZW,i of a wireless collision domain can be expressed as

ZO,i = 51T1

j =1jTj

(1 − 0) = 1T1

T (1 − 0), (29)

ZW,i = AW,iZO,i, (30)

where 1 is the probability that one user successfully sends a frame in an activity slot, T1 = TS is the time duration for successful frame transmission, Tvis the average duration of an activity slot, andA_W,i is the number of users in a wireless collision domain. From (8), (13) and (28)–(30),1, Tvand0 can be calculated by an iterative method. Then, the capacity

Hi(d) of a mesh link between two users at a separation distance

d can be calculated by Hi(d) = 1T1 Tv · l TS = 1l Tv, (31)

where l is the payload size of data frame. It is noteworthy that the payload size l of data frame is affected by the separation distance d and the PHY mode ma, which will be discussed in the following.

5.4. Impact of hop distance on transmission rate

In a multi-hop network, the hop distance will also affect the throughput of relay link. Generally, the radio signal is affected by path loss, shadowing as well as multi-path fading. With all these radio channel effects, we assume that for a given transmission power the average reception ranges for eight PHY modes are dj, j = 1, 2, . . . , 8, where d1 > d2 > · · · > d8. In principle, two users with a shorter separation distance can transmit at a higher data rate. Therefore, the transmission PHY mode ma is determined according to the separation distance d between two users. That is,

ma= j if dj +1< d dj. (32) Furthermore, we suggest that all data frames have the same transmission time TDATA(l, ma). That is, the payload size l of data frame is determined by the adopted PHY mode ma. As in[4,10], the same transmission time for each data frame can achieve fairness and avoid throughput degradation due to low-rate transmissions.

6. Numerical results

In this section, we investigate the tradeoff between user throughput and cell coverage for the ring-based WMN. The analytical results are obtained by means of the proposed ana-lytical throughput model and the optimization approach. The system parameters are summarized in Table 1. In this paper, the 802.11a WLAN is used as an example. The transmission PHY mode mafor data frame is determined by the hop distance. Be-sides, the control frames (RTS/CTS/Acknowledge frames) are transmitted with PHY mode mc = 1 for reliability. The user density is assumed to be = 10−4(users/m2). We assume

that each user transmits at the same power and the interference range is lRC = Idmax, whereIis 1.3. By using the ring-based frequency planning scheme as in Section 2.2, four buffer rings can be achieved to avoid inter-ring co-channel interference. As in [4], the chosen data frame payload sizes for eight PHY modes are{425, 653, 881, 1337, 1793, 2705, 3617, 4067} (bytes). Re-ferring to the measured results in [5], at a given transmis-sion power the corresponding average reception ranges are

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Table 1

System parameters for numerical examples

Symbol Item Nominal value

User density (100)−2(users/m2₎

RD Demanded trafﬁc of each user 0.4, 0.8 Mb/s dmin Min. ring width, i.e., (1/√) 100 m dmax Max. reception range 300 m lRC Interference range (IDmax) 1.3 ∗ dmax(m) ma PHY mode for data frame 1–8 (6–54 Mb/s) mc PHY mode for control frame 1 (ﬁxed at 6 Mb/s)

1 2 3 4 5 6 200 250 300 350 400 450 500 550 600 650 700

Number of rings in a cell, n

Cell coverage (cell radius),

rn

(m)

User demanded traffic, RD = 0.2 Mbps User demanded traffic, RD = 0.4 Mbps

Fig. 8. Cell coverage (cell radius rn) versus the number of rings n in a cell,

for various user demanded trafﬁc RD.

dj = {300, 263, 224, 183, 146, 107, 68, 30} (m). These recep-tion ranges may vary for different environments. However, the proposed optimization approach is general enough for different WMNs with various reception ranges.

6.1. Tradeoff between user throughput and cell coverage

Fig. 8 illustrates cell coverage (deﬁned as cell radius rn) against the number of rings n in a mesh cell for various demanded trafﬁc per user RD. The optimal ring widths are determined by the proposed optimization approach. In general, as the number of rings n in a cell increases, cell coverage also increases. However, because of the limit of link capacity of users in the innermost ring near the gateway, cell coverage remains the same for a larger n (see n4 in the case with

RD= 0.2 Mb/s). Besides, it is shown that the number of rings n in a cell has a maximum value. For accommodating the

in-creasing trafﬁc as n increases, the ring width will be shortened to reduce the number of contending users and improve the link capacity. For example, if n = 3, the optimal ring widths for RD = 0.4 Mb/s are {113, 119, 211} (m). As the number of rings increases to n = 4, the optimal ring widths are reduced to {100, 103, 126, 136} (m). However, since the minimum

1 2 3 4 5 6 5 10 15 20 25 30

Number of rings in a cell, _n

Cell capacity (Mbps)

User demanded traffic, R_D = 0.2 Mbps User demanded traffic, RD = 0.4 Mbps

Fig. 9. Cell capacity versus the number of rings n in a cell, for various user demanded trafﬁc RD.

allowable ring width is constrained by the node density as in (3), no feasible solution can be found for a larger n and thus the maximum value of n exists. In this example, the maximum number of rings in a cell is n = 6 for RD= 0.2 and n = 4 for RD= 0.4 Mb/s.

From Fig. 8, we can observe the tradeoff between user throughput and cell coverage. To guarantee the throughput for each user, the number of users in a cell and cell coverage should be properly decreased when the user demanded trafﬁc

RDincreases. In this example, when the user demanded trafﬁc RD increases from 0.2 to 0.4 Mb/s, the optimal cell cover-age is reduced from 659 (m) at n = 6 to 465 (m) at n = 4. The corresponding optimal ring widths are {100, 100, 100, 100, 109, 150} (m) for RD= 0.2 and {100, 103, 126, 136} (m) for RD= 0.4 Mb/s, respectively.

In Fig. 9, cell capacity (defined as the overall throughput of a cell,r_n2RD) against the number of rings n in a cell for various user demanded traffic RD is shown. Because each user gener-ates more traffic, a larger RD can achieve higher cell capacity for n3, although with a smaller cell coverage. However, con-strained by the link capacity of users in the innermost ring, the optimal cell capacity for various user demanded traffic RD are almost the same at about 27 Mb/s as shown in the figure.

In the above ﬁgures, we investigate the interaction between user throughput and cell coverage. The optimal solution is de-termined by the proposed optimization approach, subject to the constraints on the link capacity and the ring width. These ﬁg-ures show that by ring-based frequency planning and properly designing the ring widths, the optimal cell capacity and cover-age can be simultaneously achieved with a guaranteed through-put for each user.

6.2. Impact of ring sectorization

Fig. 10 compares the impact of ring sectorization on cell coverage, for RD = 0.4 Mb/s. As shown in the ﬁgure, since

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1 2 3 4 5 6 200 300 400 500 600 700 800

Number of rings in a cell, _n

Cell coverage (cell radius),

rn

(m)

With sectorized rings A1 and A2 With sectorized rings A1 Without sectorization

Fig. 10. Effect of ring sectorization on cell coverage (cell radius rn), for the

user demanded trafﬁc RD= 0.4 Mb/s.

1 2 3 4 5 6 0 10 20 30 40 50 60 70

Number of rings in a cell, n

Cell capacity (Mbps)

With sectorized rings _{A1 and A2} With sectorized rings _A1 Without sectorization

Fig. 11. Effect of ring sectorization on cell capacity, for the user demanded trafﬁc RD= 0.4 Mb/s.

sectorizing the inner rings can reduce the number of contending users to overcome the throughput bottleneck issue near the gateway, cell coverage can be extended to serve more users. One can observe from the ﬁgure that if only ring A1is sectorized, the optimal cell coverage can increase from 464 to 508 (m) at

n = 4. If sectorizing both rings A1 and A2, the optimal cell coverage increases to 703 (m) at n = 6.

In Fig.11, the effect of ring sectorization on cell capacity for

RD= 0.4 Mb/s is shown. In the ﬁgure, by sectorizing ring A1, the optimal cell capacity can be improved by 20% over the case without sectorization. If the congested inner rings A1and A2 are sectorized, the throughput bottleneck issue near the gateway can be overcome. By doing so, the optimal cell capacity will be further improved by 90% over the case with only sectorizing ring A1.

Clearly, sectoring more inner rings can improve cell cover-age and capacity. For sectorizing more rings, however, the sys-tem requires more available non-overlapping channels[16,23], since each mesh cell should be allocated with more channels to ensure sufﬁcient buffer rings and reuse distance.

7. Conclusions

This paper has investigated the tradeoff between user throughput and cell coverage in the WMN. To overcome the scalability and throughput bottleneck issues in the WMN, a scalable multi-channel ring-based WMN has been employed. An optimization approach has been applied to maximize cov-erage and capacity for the considered WMN, subject to the user throughput requirement.

In the ring-based WMN, a simple ring-based frequency planning scheme has been employed to reduce collisions, and to make the network more scalable in terms of coverage. We have also suggested sectorizing the congested inner rings to resolve the throughput bottleneck issue of the WMN. From the system design perspective, this paper has three impor-tant components. First, an analytical throughput model has been developed, which considers the effects of ring-based cell structure and frame contentions in the CSMA MAC proto-col. Second, we have developed a bulk-arrival semi-Markov queueing model to describe user behavior in the non-saturation condition. Third, to investigate the optimal tradeoff between user throughput and cell coverage, we have applied an opti-mization approach to determine the optimal number of rings and the associated ring widths in a mesh cell. Numerical re-sults have demonstrated that the optimal system parameters (that is, the number of rings and ring widths) can be deter-mined analytically. In addition, both the capacity enhancement and coverage extension can be achieved with a guaranteed throughput for each user.

Appendix A. Successful probabilities, s1,j and s2,j

Now we derive the probabilities s1,j and s2,j mentioned in Section 4.5. As shown in Fig. 6, suppose that the considered area of angle 2_S,i is inﬂuenced by two neighboring transmit-ters PLand PR. Let represent the position of the contend-ing user P, L and R be the positions of the neighboring transmitters PL and PR. Accordingly, the central angles for re-gions {A_S,i − YL} and {A_S,i − YR} can be written as L = S,i− max(0, L+ S,i− W,i) and R= S,i− max(0, R+ S,i− W,i), respectively. Suppose that e= (1 − P0) is the effective transmission probability for one user.

Then, given the positions , _L and _R, the conditional probability that there is a successful transmission in the left-side mutually interfered region of user P can be expressed as

s1,j(L, R, ) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ _ai 2L−1 1 e(1− e) ai 2(L+L)−2

for max(0, R−S,i) max(0, S,i−L), 0 otherwise.

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In (33), the term

(ai/2)L−1 1

represents the probability that only user P sends an RTS request in the left-side mutually interfered region of user P. The term (1 − e)(ai/2)(L+

L)−2 accounts for the probability that all the users except for P and P in the adjacent wireless collision domains of Pare in backoff or idle, where L = min( + W,i, (S,i + W,i) − (L + S,i)) = min( + W,i, W,i− L) and in the same way L = min(W,i− , W,i− R). In addition, the constraint for means that both the neighboring transmitters PL and PR are not inside the working-in-vain regions of P.

By the same method, the conditional probability that there is a successful transmission in each mutually interfered region of user P can be obtained from

s2,j(L, R, ) = s1,j(L, R, ) × _a_i 2R 1 e(1 − e) ai 2R−1 . (34)

Here, the term within brackets represents the probability that there is also a successful transmission in the right-side mutually interfered region of user P, whereR= max(0, (S,i+W,i) − (R+S,i−1)−(W,i−)) = max(0, (S,i−R)−(W,i−)) and_R= max(0, (_S,i+ _W,i) − (R+ S,i) − (W,i− )) = max(0, − R).

By averaging over the positions, _Land_R, the probabil-ities s1,j and s2,j for j = 2 can be computed by

st,j = 1 2 W,i _W,i 0 _W,i 0 _L 0 st,j(L, R, ) L d dRdL, for t = 1, 2. (35)

In this section, we take the case of j = 2 as an example to explain how to evaluate the successful probability st,j. By the same reasoning, one can also calculate the probabilities st,jfor j = 0, 1. Thus, the detailed derivations are omitted here.

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Jane-Hwa Huang received the B.S., M.S., and

Ph.D. degrees in electrical engineering from the National Cheng-Kung University, Taiwan, ROC, in 1994, 1996, and 2003, respectively. He joined the Department of Communication Engineering, National Chiao-Tung University, Taiwan, as a Postdoctoral Researcher from 2004 to January 2006, and a Research Assistant Pro-fessor since January 2006. His current research interests are in the areas of wireless networks, wireless multi-hop communications, vehicular communication networks, and radio resource management.

Li-Chun Wang received the B.S. degree in

electrical engineering from the National Chiao-Tung University, Hsinchu, Taiwan, ROC, in 1986, the M.S. degree in electrical engineering from the National Taiwan University, Taipei, Taiwan, in 1988, and the M.Sc. and Ph.D. degrees in electrical engineering from Georgia Institute of Technology, Atlanta, in 1995 and 1996, respectively.

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From 1990 to 1992, he was with Chunghwa Telecom. In 1995, he was afﬁliated with Northern Telecom in Richardson, Texas. From 1996 to 2000, he was with AT&T Laboratories, where he was a Senior Technical Staff Member in the Wireless Communications Research Department. Since August 2000, he has joined the Department of Communication Engineering of National Chiao-Tung University in Taiwan as an Associate Professor and has been promoted to a full professor since August 2005. Dr. Wang was a corecipient of the Jack Neubauer Best Paper Award from the IEEE Vehicular Technology Society in 1997. His current research interests are in the areas of cellular architectures, radio network resource management, cross-layer optimization for cooperative and cognitive wireless networks. He is the holder of three US patents with three more pending.

Chung-Ju Chang was born in Taiwan, ROC,

in August 1950. He received the B.E. and M.E. degrees in electronics engineering from Na-tional Chiao-Tung University (NCTU), Hsinchu, Taiwan, in 1972 and 1976, respectively, and the Ph.D. degree in electrical engineering from National Taiwan University (NTU), Taiwan in 1985.

From 1976 to 1988, he was with Telecommuni-cation Laboratories, Directorate General of Tele-communications, Ministry of Communications,

Taiwan, as a Design Engineer, Supervisor, Project Manager, and then Division Director. In the meantime, he also acted as a Science and Technical Advisor for the Minister of the Ministry of Communications from 1987 to 1989. In 1988, he joined the Faculty of the Department of Communication Engineering, College of Electrical Engineering and Computer Science, National Chiao-Tung University as an Associate Professor. He has been a Professor since 1993. He was Director of the Institute of Communication Engineering from August 1993 to July 1995, Chairman of Department of Communication Engineering from August 1999 to July 2001, and the Dean of the Research and Development Ofﬁce from August 2002 to July 2004. Also, he was an Advisor for the Ministry of Education to promote the education of communication science and technologies for colleges and universities in Taiwan during 1995–1999; he is acting as a Committee Member of the Telecommunication Deliberate Body, Taiwan. He serves as Editor for IEEE Communications Magazine and Associate Editor for IEEE Transactions on Vehicular Techonology. His research interests include performance evaluation, wireless communication networks, and broadband networks. Dr. Chang is a member of the Chinese Institute of Engineers (CIE) and an IEEE Fellow.