Heterogeneity Increases Multicast Capacity in Clustered Network

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1 Heterogeneity Increases Multicast Capacity in Clustered Network Qiuyu Peng Xinbing Wang Huan Tang Department of Electronic Engineering Shanghai Jiao Tong University April 15, 2010 Infocom / 32

2 Outline 1 Background Motivation Objectives 2 Network Topology Transmission Protocol Capacity Definition 3 Main results Intuitions 4 5 Infocom / 32

3 Outline Background Motivation Objectives 1 Background Motivation Objectives 2 Network Topology Transmission Protocol Capacity Definition 3 Main results Intuitions 4 5 Infocom / 32

4 Capacity of Ad Hoc Network Background Motivation Objectives Capacity of wireless ad hoc network not scalable: in a static ad hoc wireless network with n nodes, the per-node ( ) 1 [1]. capacity is limited as O n log n Interference is the main reason behind. [1] P. Gupta and P. R. Kumar, The capacity of wireless networks, in IEEE Trans. on Information Theory, Infocom / 32

5 Background Motivation Objectives Multicast Capacity of Ad Hoc Network Multicast traffic pattern is a generalized version of unicast traffic in ad hoc network: Each source sends identical packets to multiple destinations. ( ) [2] 1 The per-node throughput is limited as O nk log n if each multicast session composes of 1 source and k destinations. [2] X.-Y. Li, S.-J. Tang, and O. Frieder. Multicast capacity for large scale wireless ad hoc networks, in Proc. ACM Mobicom Infocom / 32

6 Clustered Network Background Motivation Objectives The network models studied in previous works are non-clustered and uniformly distributed ones. Most realistic networks are characterized by various clustered heterogeneity. Spatial Heterogeneity: The nodes are clustered according to some specified distributions [3]. Pattern Heterogeneity: More than one type of traffic patterns exist in the network [4]. [3] G. Alfano, M. Garetto, E. Leonardi, Capacity Scaling of Wireless Networks with Inhomogeneous Node Density: Upper Bounds, [4] M. Ji, Z. Wang, H. Sadjadpour, J. J. Garcia-Luna-Aceves, The Capacity of Ad Hoc Networks with Heterogeneous Traffic Using Cooperation Infocom / 32

7 Motivation Background Motivation Objectives Network with multicast traffic pattern can also be regarded as clustered network since nodes of the same multicast session composes of a cluster. The network heterogeneities investigated in prior works are inadequate for exploring the clustering behavior of such network. Infocom / 32

8 Background Motivation Objectives Features of Clustered Heterogeneities Heterogeneous Cluster Traffic (HCT): Clients of the same cluster (data flow) are likely to be deployed around a cluster head specified by an inhomogeneous poison process (IPP) Heterogeneous Cluster Size (HCS): Clusters may have different size (cardinality) and HCS is employed to describe the population variation for each multicast data flow Infocom / 32

9 Main Question Background Motivation Objectives Main Question What are the impacts of heterogeneous cluster traffic and size on multicast capacity in clustered network? Heterogeneous Cluster Traffic increases network capacity for all the clusters. Heterogeneous Cluster Size does not influence the network capacity. Infocom / 32

10 Outline Network Topology Transmission Protocol Capacity Definition 1 Background Motivation Objectives 2 Network Topology Transmission Protocol Capacity Definition 3 Main results Intuitions 4 5 Infocom / 32

11 General Assumption Network Topology Transmission Protocol Capacity Definition There are n s clusters and each with C j p number of clients. Both n s and p scales with n and n s p = n. The edge of the deployed region O is L = n β, which also scales with n. Infocom / 32

12 How to model HCT (I) Network Topology Transmission Protocol Capacity Definition Each cluster Client is distributed according to an inhomogeneous poisson process around their cluster head specified by a probability density function φ( ). H: cluster head C: cluster client Infocom / 32

13 How to model HCT (II) Network Topology Transmission Protocol Capacity Definition The inhomogeneous poisson process is specified by a probability density function φ( ). O φ(ξ)dξ = 1 Infocom / 32

14 How to model HCT (III) Network Topology Transmission Protocol Capacity Definition Given a probability density function φ( ), we must provide a quantitative value of its degree of heterogeneity. The expectation describes average node density: E[φ(ξ)] = 1/L 2. The variance can describe HCT and a novel variable distribution variance σ O is proposed. σ O = O ( O φ(ξ) φ(ξ )dξ ) 2 L 2 dξ = O φ 2 (ξ)dξ 1 L 2 Infocom / 32

15 How to model HCS Network Topology Transmission Protocol Capacity Definition The size of these n s clusters is not identical and for each cluster C j (1 j n s ), its size C j Θ(n 1 α ). (n s = n α ) Infocom / 32

16 Protocol Model Network Topology Transmission Protocol Capacity Definition Definition Let d ij denotes the distance between node i and node j, and R T the common transmission range, then a transmission from i to j at rate W is successful if: { d ij < R T d kj > (1 + )R T for any other k transmitting simultaneously. Infocom / 32

17 Asymptotic Capacity Network Topology Transmission Protocol Capacity Definition Definition Definition of Asymptotic Capacity: Let λ j (1 j n s ) denote the sustainable rate of data flow for cluster C j. Assume that λ = min{λ 1, λ 2,..., λ ns 1, λ ns }. Then λ = Θ(f (n)) is defined as the asymptotic network capacity if there exist constants c > c > 0, such that lim Pr(λ = cf (n) is achievable) < 1, n lim Pr(λ = n c f (n) is achievable) = 1. Infocom / 32

18 Outline Main results Intuitions 1 Background Motivation Objectives 2 Network Topology Transmission Protocol Capacity Definition 3 Main results Intuitions 4 5 Infocom / 32

19 Main results (I) Main results Intuitions Given the distribution variance σ O, λ is bounded as follows: { ( )} max{1, LσO }W λ min O(W), O. ns Infocom / 32

20 Main results (II) Main results Intuitions Given a distribution variance σ O, a set of probability density functions φ( ) can satisfy the requirement. Uniform Cluster Random Model is the right point process that can achieve the capacity upper bound in order sense. Uniform Cluster Random Model The dispersion density function is as follows: φ u (ξ) = { 1 πr 2 ξ R 0 otherwise L where R = is defined as cluster radius. It means π(1+(lσo ) 2 ) clients of each cluster are randomly and uniformly distributed in a disk of radius R centered at its cluster head. Infocom / 32

21 Fully Cluster Overlapping Main results Intuitions HCT is relative slight and each cluster is fully overlapped with other clusters. Each node is required to serve for approximately Θ( n s ) clusters and it is identical to uniform case. Infocom / 32

22 Trivial Cluster Overlapping Main results Intuitions HCT is relative severe and each cluster can be viewed as an isolated one. Each node is required to serve for a constant number of cluster so Θ(1) capacity can be achieved. Infocom / 32

23 Partial Cluster Overlapping Main results Intuitions The degree of HCT is neither too severe nor slight therefore each cluster is overlapped with only some of the clusters. Each node is required to serve for a smaller number of clusters than the case of fully cluster overlapping. The network capacity is larger than Θ( n s ). Infocom / 32

24 Outline 1 Background Motivation Objectives 2 Network Topology Transmission Protocol Capacity Definition 3 Main results Intuitions 4 5 Infocom / 32

25 Why for UCRM UCRM, which is a special type of node distribution function, can achieve maximized capacity given a fixed σ O theoretically. Can such results applicable for real world scenario? is required to approach such upper bound. Infocom / 32

26 Case σ O = Ω( n s L ) In this case, R = L = O( L π(1+l 2 (σ O ) 2 ) ns ). There are at most a constant number of clusters inside D(ξ, R) for ξ O and a simple TDMA scheme can achieve Θ(W) capacity for each cluster. Trivial Cluster Overlapping Infocom / 32

27 Case σ O = o( n s L ) In this case, R = Θ( 1 σ O ) = ω( L ns ) and the traffics in each cluster is not so aggregated because σ O is relative smaller. Scheduling policy becomes relative complicated. Infocom / 32

28 Case σ O = o( n s L ) Illustration of information highway, access point, routing protocol. Infocom / 32

29 Outline 1 Background Motivation Objectives 2 Network Topology Transmission Protocol Capacity Definition 3 Main results Intuitions 4 5 Infocom / 32

30 Result We provide a close formula of the relationship between the achievable capacity λ and the distribution variance σ O and the corresponding scheduling policy to achieve such capacity. Based on the formula, we find that HCT can increase the network capacity while HCS does not have impact on the network capacity. Infocom / 32

31 Future Work For HCS, we find it can not increase network capacity. However, it may increase the achievable capacity for small cluster because small cluster may rely on the information highway constructed by the bigger clusters. We can investigate the lower bound of the network capacity given a specified σ O. We can also discuss the impact of base stations on the network capacity in our heterogeneous cases. Infocom / 32

32 Questions? Thanks for listening. Infocom / 32