Analysis of Bottleneck Delay and Throughput in Wireless Mesh Networks

Analysis of Bottleneck Delay and Throughput in Wireless Mesh Networks Xiaobing Wu 1, Jiangchuan Liu 2, Guihai Chen 1 1 State Key Laboratory for Novel Software Technology, Nanjing University, China wuxb@dislab.nju.edu.cn, gchen@nju.edu.cn 2 School of Computing Science, Simon Fraser University, Canada jcliu@cs.sfu.ca Abstract Wireless mesh networking has emerged as a promising technology in providing economical and scalable broadband Internet accesses. A wireless mesh network consists of mesh routers and mesh clients, connected in an ad hoc manner via wireless links. A subset of the mesh routers, referred to as gateway nodes, are capable of external Internet connections. Since other mesh routers and clients have to access the Internet through the gateway nodes, these nodes can easily become bottlenecks in the network. In this paper, we present a novel queuing model based analysis of the delay and throughput of the gateway nodes. Our simulation results suggest that the analytical results are quite accurate, which provides an effective guideline for gateway-related design and optimization in wireless mesh networks. I. INTRODUCTION Recently, Wireless Mesh Networks (WMNs) have emerged as a promising technology in providing economical and scalable broadband Internet accesses. In WMNs, there are mesh routers and mesh clients. The mesh routers, which are generally stationary, are connected in an ad hoc manner to form a network backbone. A subset of them could be equipped with gateway functions so as to link the mesh networks to the Internet. The mesh clients can therefore access the Internet through the mesh routers. The clients can roam freely from one mesh router to another as well. Moreover, other wireless networks such as WLANs, wireless sensor networks (WSNs), cellular networks, and wireless networks running WiMax can also be connected to the Internet via the mesh routers [1]. Providing quality Internet access for the clients is a critical issue in designing WMNs. Given the presence of gateway nodes, data requests from mesh clients generally follow a many-to-one or many-to-few model, that is, data requests flow through the only or a few gateway nodes which are directly connected to the wired networks. As a result, these gateway nodes could be a performance bottleneck of the whole networks. In this paper, we try to answer the following two questions: Given the number of gateway nodes and mesh routers, what is the delay before a data request could be processed at a gateway node? And what is the amount of the data requests that the WMN can accommodate? We analyze this delay and throughput in WMNs by using queuing theory. We first assume a linear network and then extend the results to a more general topology. We have the following contributions: First, to the best of our knowledge, our work is the first to address the bottleneck delay and throughput in wireless mesh networks. Second, we model the gateway nodes as independent M/D/1 queue stations and have derived closedform solutions for delay and throughput. The results serve as baselines for further capacity analysis in WMNs. Third, the analytical results provide useful guidelines for gateway-related design and optimization in wireless mesh networks. In particular, given the data request frequency and the per-packet service time at the gateway nodes, we can determine the relationship between the number of gateway nodes and the number of pure mesh routers in the network. The rest of this paper is organized as follows: Sec. II surveys the related work. Sec. III describes the network model. Sec. IV presents our analysis of bottleneck delay and throughput in wireless mesh networks. Performance evaluation is given in Sec. V. We discuss related issues in Sec. VI. Finally, Sec. VII concludes this paper. II. RELATED WORK Much research on network capacity has been done for wireless ad hoc networks. Gupta and Kumar [2] calculate the capacity upper bound and lower bound of wireless ad hoc networks. They adopt an asymptotic analysis. In order to increase the capacity, nodes in the network 1-4244-0507-6/06/$20.00 2006 IEEE 765

Fig. 1. Gateway Mesh Router Data Request Many-to-few data flow model in WMNs. would better communicate only with nearby nodes. In [4], Li et al. further validate the conclusions in [2] with simulations. Grossglauser and Tse draw a similar conclusion as Gupta and Kumar in [3]. They state that if nodes communicate with their targets only when they move to a place near them, the network capacity can be improved. Yi, Pei and Kalyanaraman [6] investigate the effectiveness of using directional antennas in wireless ad hoc networks. They present the capacity improvement when both the source node and destination node use directional antennas. In [7], Gastpar and Vetterli consider the network capacity under the condition when there is only one active data request in the network and all other nodes behave as relay nodes. Although capacity analyses in wireless ad hoc network provide us some hints on the related research in WMNs, they have some limitations to be applied directly to WMNs. For example, the authors use an asymptotic analysis in [2]. Thus their results do not match the case when there are limited number of nodes in WMNs [1], which could be very common in practice. Likewise, conclusions in [3] are based on quite different assumptions on node mobility in the network. Literature [6] and [7] address orthogonal issues as in WMNs. In summary, it is not straightforward to apply the capacity results for wireless ad hoc networks in the context of wireless mesh networks [1]. Recently, Jun and Sichitiu [5] propose a method to compute the capacity of wireless mesh networks. This method adopts a conception of bottleneck collision domain. Capacity is computed through identifying bottleneck collision domains. However, the method only applies to a given topology. In [8], Roy et al. use OPNET to simulate the capacities of single-radio and multi-radio WMNs, respectively. They examine how the carrier sensing range influences the capacity of single-radio WMNs and how the channel assignment affects the capacity of multi-radio WMNs. Kodialam and Nandagopal [9] propose an algorithm to answer the feasibility problem of a data request in multi-radio multi-channel WMNs. In [10], Kyasanur, Yang, and Vaidya consider the effectiveness of multi-channel in WMNs and introduce a spatial back-off scheme to improve the capacity. Kyasanur and Vaidya [11] derive the lower and upper bound on the capacity of static multi-channel wireless networks in the asymptotic case. They investigate how the numbers of interfaces and channels affect network capacity. Wang and Liu [12] try to answer the following questions: given the numbers of radios and channels that can be equipped with mesh routers, what is the maximum achievable capacity? How many radios are needed in order to fully exploit a given number of channels? They propose a framework to model these two problems using Integer Linear Programming (ILP). Our work is inspired by these existing studies, with a focus on the delay and throughput performance analysis for general WMNs. A closely related work is done by Bisnik and Abouzeid [13]. They consider delay and throughput issues where mesh clients communicating with each other. We however observe that in WMNs, data requests are mainly for Internet accesses, which is consistent with the traffic model suggested in [5]. III. NETWORK MODEL AND ASSUMPTIONS In this section we describe our network model and basic assumptions. In our discussion, a wireless mesh network consists of mesh routers and mesh clients. For ease of exposition, we call mesh routers with gateway functions gateway nodes. Other mesh routers are referred to as pure mesh routers. We assume that each router has a fixed transmission range. Thus two routers can set up a link between them and communicate when they are within the transmission range of each other. As for the mesh clients, Each of them is attached to a mesh router. Data requests originating from them are relayed by the intermediate mesh routers hop by hop and delivered to the wired networks by gateway nodes in the end. This is the many-to-few data model we have mentioned. It is depicted in Fig. 1. We assume that each mesh router is equipped with multiple radios and there are multiple channel available in the network. Because of the intrinsic broadcasting nature of wireless medium, links using the same channel would interfere with each other. So we further assume that there is a static channel assignment to resolve the channel interference in the network [14]. If two mesh routers are within the interfering range of each other, we 766

1λ 2λ 3λ 4λ 1 2 3 4 5 Fig. 2. Linear wireless mesh topology. Gateway assign non-overlapped channels to them. On the other hand, we can reuse channels to the links if they do not interfere with each other. Note that channel reuse is possible because the interfering range of mesh routers is limited. After the static channel assignment, routers in the network can send and receive data simultaneously. IV. ANALYSIS OF BOTTLENECK DELAY AND THROUGHPUT In this section, we present our analysis of bottleneck delay and throughput in WMNs using the queuing theory. We define the bottleneck delay as the average delivery delay of data requests at the gateway nodes. Besides, we define the bottleneck throughput as the maximum feasible data requesting frequency from all the mesh clients attached to a mesh router. We first present a generic queuing analysis. Then we perform our analysis in a linear wireless mesh network. We extend our results to more general network topology at last. A. A Generic Queuing Analysis We observe that data requests from mesh clients mainly aim to the gateway nodes. Thus we treat each gateway node as a service station in a queuing system. We assume the gateway nodes process data requests independently. We further assume that data requests are of constant length. We consider the case that data requests arrive at the gateway node in Poisson process. Let λ denote the mean arrival rate and T denote the service time. Note that the bandwidth between the gateway nodes and wired networks is much more bigger comparing to the bandwidths between mesh routers, because the gateway nodes are always wired into the Internet. Therefore we can assume there is an infinite buffer in each gateway node. We further assume a data request can be processed by the gateway node in a constant time s. ThatisT = s. For a gateway with an infinite buffer, data requests arrive at a Poisson process, and the service time of a data request is constant, we model the data incoming and outgoing as in an M/D/1 queue. According to the Pollaczek-Khintchine (P-K) Formula, the number of packets at the gateway nodes in a steady state is L s = ρ + ρ2 + λ 2 Var[T ] 2(1 ρ) (1) where ρ = λe[t ]. ρ should be smaller than one in order to arrive at a steady state, The mean time of T is also s and the variation of T is zero because the service time for a data packet is a constant. Thus the Eq.1 can be rewritten into L s = λs + λ2 s 2 2(1 λs). (2) To this end, we can derive the average data request delivery delay W s = L s λ = s + λs 2 2(1 λs). (3) B. Bottleneck Delay and Throughput of WMNs: A Linear Topology In this section we derive values of bottleneck delay and throughput in a linear WMN based on the results of Sec. IV-A. A WMN may appear to be a linear topology if it covers a long and narrow terrain (e.g., a street in a city). Fig. 2 shows such a topology. The grey node is a gateway and others are pure mesh routers. Since we focus our discussion on the bottleneck delay related to the gateway nodes, we do not consider the delays incurred by the intermediate pure mesh routers. Similar to [13], we assume that each mesh router generate data requests in a Poisson distribution with the same mean value of λ. For simplicity and without loss of generality, we assume gateway nodes do not generate data requests. In Fig. 2, data requests originated from node 1 would be relayed to the gateway node 5 by the nodes 2, 3, and 4. In the same way, data requests from node 2 would be routed to node 5 by nodes 3 and 4. According to the additivity of Poisson distribution, data requests rate at node 2 will be of a Poisson distribution with a mean value of 2λ. Similarly, data rate of pure mesh router node i wouldbeinaniλ-poisson distribution. Theorem 4.1: For a linear mesh network with total N pure mesh routers, if the inputting data flow of each mesh router follows a λ-poisson distribution and the gateway node is the only one exit, the inputting data flow at the gateway node would follow an Nλ-Poisson distribution. Proof: Because of the bottleneck attribute of the gateway node and the additivity of Poisson distribution, the theorem is self-evident. 767

Gateway Fig. 3. Wireless mesh network with grid topology. For a linear mesh network with one gateway and N other pure mesh routers, replacing λ with Nλ in Eq. 1, we have the result of the bottleneck delay Ws N Nλs 2 = s + 2(1 Nλs). (4) We use the data request arrival rate to measure the bottleneck throughput. Then it can be calculated like this. For a gateway with infinite buffer, the value of ρ should be smaller than one in order to arrive at a steady state. So the data request arrival rate should be smaller than 1 s. As a result, we can take 1 s as an upper bound for an Nλ Poisson input. Thus the upper bound of data request frequency for each mesh router should be 1 Ns. C. Bottleneck Delay and Throughput of WMNs: A Grid Topology We consider the bottleneck delay and throughput of WMNs under a more general topology in this section. As shown in Fig. 3, 16 pure mesh routers form a 4 4 grid. The mesh router grid is connected to wired networks via 4 independent gateway nodes. Since in our discussion the 4 gateway nodes are the only exits of all the data requests, we extend the results of Sec. IV-B to a grid topology. Suppose that there are N pure mesh routers and M gateway nodes in the network. We assume there is an underlying routing protocol which select next relaying mesh router with equal probability. We assume in the long run the data traffic are distributed equally among these M gateway nodes. That is we consider the average case. So we can derive bottleneck delay and throughput as follows. W s (M,N) Nλs 2 = s + 2(M Nλs) λ (M,N) upper bound = M Ns. (5) When M becomes 1, the Eqs. in 5 are reduced to the results for a linear network. Note that we have assumed an equal traffic distribution among the gateway nodes in the network. It might be different in practice, and it remains difficult to well model the traffic distribution in a WMN given the many factors as routing, channel assignment and congestion control. Nonetheless, Our results can be used as reference values in the average case, when the mean data request arrival rate, the perpacket service time at the gateway node, and the number of pure mesh routers and gateway nodes are known. V. PERFORMANCE EVALUATION We perform simulations to validate our bottleneck delay and throughput analysis. First, we assume a linear network with one gateway node. We simulate the bottleneck delay performance when the number of pure mesh routers increases from 1 to 20. The results are depicted in Fig. 4. We observe that the simulation results are quite close to the analysis results. We also see that for a specific data request arrival rate λ, when the number of pure mesh routers increases, the bottleneck delay increases as well. This is because the only one gateway has to handle more data requests when there are more pure mesh routers. On the other hand, for a specific number of pure mesh routers, the bottleneck delay increases when the data request arrival rate increases. This is also easy to understand because the higher data request arrival rate, the busier the gateway node would be. In Fig. 5, we give the simulation results when the number of routers is fixed but data request arrival rate changes. The results also match analysis results quite well. Note that when the data request arrival rate λ approaches 1 Ns, the bottleneck delay increases dramatically. This is in accordance with our analysis. Then we perform our simulation with 16 pure mesh routers forming a 4 4 grid. The number of gateway nodes varies from 2 to 8. Equal number of pure mesh routers on the edge of the grid are chose randomly and connected to the gateway nodes. Fig. 3 is a case when there are 4 gateway nodes. Fig. 6 shows that the bottleneck delay decreases when the number of gateway nodes increases. This can be explained as that each gateway node receives less traffic with more gateway nodes. Fig. 7 shows that the bottleneck delay increases to a value of much more higher order when data request arrival rate is close to its upper bound. The results are also consistent with our analysis. 768

Analysis result 0.0106 0.0106 Analysis result 0.0105 0.1 λ = 4.7619 0.0104 0.0104 0.0104 0.0103 0.05 λ = 4.3478 0.0103 0.0102 λ = 4 0.0101 0 2 4 6 8 10 12 14 16 18 20 # of pure mesh routers. 0.0101 1 2 3 4 5 6 7 8 # of gateway nodes. Fig. 4. Bottleneck delay with different number of pure mesh routers, s = 0.01. Fig. 6. Bottleneck delay with different number of gateway nodes, λ =1.2, s = 0.01, N = 16. 0.5 0.45 Analysis result 0.5 0.45 Analysis result 0.4 0.4 0.35 0.35 0.3 0.25 0.2 0.3 0.25 0.2 0.15 0.15 0.1 0.1 0.05 0.05 0 1 2 3 4 5 6 7 8 9 10 Value of λ 0 10 15 20 25 Value of λ Fig. 5. Bottleneck delay with different data arrival rates λ in a linear topology, s = 0.01, N = 10. When λ = 1 =10, the analysis value Ns of bottleneck delay is infinity. The simulation value grows drastically. VI. DISCUSSION A. The Locations of Gateway Nodes In our study, we have ignored the delay incurred by the intermediate pure mesh routers, because we focus on the bottleneck delay and throughput related to the gateway nodes. As such, the locations of the gateway nodes will not affect the results in our analysis. We plan to address this type of delay in our future study. B. The Number of Pure Mesh Routers and Gateway Nodes In our analysis, we have suggested that the value of ρ should be less than one in order to arrive at a steady state. Therefore, the number of gateway nodes should be bigger than λns, given the number of pure mesh routers N, mean data arrival rate of each pure mesh router λ, and the per-packet server time s at the gateway node. Equivalently, we can figure out how many pure mesh Fig. 7. Bottleneck delay with different data arrival rate λ in a grid topology, s = 0.01, N = 16, M = 4. When λ = M = 25,the Ns analysis value of bottleneck delay is infinity. The simulation value grows drastically. routers M gateway nodes are able to support if we know the values of M,λ and s. These results give us guideline for planning a wireless mesh network with gateways. VII. CONCLUSIONS We observed that in wireless mesh networks the limited number of gateway nodes could be the bottleneck of the entire networks. In this paper, we presented a formal study on the delay and throughput of the gateway nodes. We modeled the gateway nodes as independent M/D/1 queue stations, and derived closed-form solutions for the bottleneck delay and throughput with linear and grid topologies. s were given to validate our analysis. These results can serve as baselines for further capacity analysis in wireless mesh networks. 769

ACKNOWLEDGMENTS The work is supported by China NSF grant (60573131), China Jiangsu Provincial NSF grant (BK2005208), and China 973 projects (2006CB303000, 2002CB312002). The Conference Participation is supported by Nokia Bridging the World Program. X. Wu would like to thank Prof. Wanyang Dai for helpful discussion. REFERENCES [1] I. F. Akyildiz and X. Wang, A survey on wireless mesh networks, IEEE Communications Magazine, vol. 43, no. 9, pp. 23-30, Sept. 2005. [2] P. Gupta and P. R. Kumar, The capacity of wireless networks, IEEE Trans. on Information Theory, vol. 46, no. 2, pp. 388-404, Mar. 2000. [3] M. Grossglauser and D. Tse, Mobility increases the capacity of Ad Hoc wireless networks, IEEE/ACM Trans. on Networking, vol. 10, no. 4, pp. 477-86, Aug. 2002. [4] J. Li, C. Blake, D. S. J. De Couto, H. I. Lee, and R. Morris. Capacity of Ad Hoc wireless networks, Proc. of ACM Mobi- Com, July 2001. [5] J. Jun and M. L. Sichitiu, The nominal capacity of wireless mesh networks, Wireless Communications Magazine, Oct. 2003. [6] S. Yi, Y. Pei and S. Kalyanaraman, On the capacity improvement of ad hoc wireless networks using directional antennas, Proc. of MobiHoc, June 2003. [7] M. Gastpar and M. Vetterli, On the capacity of wireless networks: the relay case, Proc. of IEEE INFOCOM, June 2002. [8] S. Roy, A. K. Das, R. Vijayakumar, H. M. K. Alazemi, H. Ma, and E. Alotaibi, Capacity scaling with multiple radios and multiple channels in wireless mesh networks, Proc. of The First IEEE Workshop on Wireless Mesh Networks (WiMesh), Sept. 2005. [9] M. Kodialam and T. Nandagopal, Characterizing the capacity region in multi-radio multi-channel wireless mesh networks, Proc. of ACM MobiCom, Aug. 28 - Sept. 2, 2005. [10] P. Kyasanur, X. Yang, and N. H. Vaidya, Mesh networking protocols to exploit physical layer capabilities, (Invited) Proc. of The First IEEE Workshop on Wireless Mesh Networks (WiMesh), Sept. 2005. [11] P. Kyasanur and N. Vaidya, Capacity of Multi-Channel Wireless Networks: Impact of Number of Channels and Interfaces, Proc. of ACM Mobicom, Aug. 28 - Sept. 2, 2005. [12] W. Wang and X. Liu, A framework for maximum capacity in multi-channel multi-radio wireless networks, (Invited) Proc. of IEEE Consumer Communications and Networking Conference (CCNC), Jan. 2006. [13] N. Bisnik and A. Abouzeid, Delay and throughput in random access wireless mesh networks, Proc. of ICC, Istanbul, Turkey, June 2006. [14] K. N. Ramachandran, E. M. Belding, K. C. Almeroth, and M. M. Buddhikot, Interferce-aware channel assignment in multiradio wireless mesh networks, Proc. of INFOCOM, Barcelona, Spain, Apr. 2006. 770