1. Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective Yun Wang, Xiaoyu Chu, Xinbing Wang Department of Electronic Engineering Shanghai Jiao Tong University, China Yu Cheng Department of Electrical and Computer Engineering Illinois Institute of Technology, USA

2. Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 2 Outline Introduction  Motivations  Objectives Models and Definitions Two-Dimensional I.I.D. Mobility Model One-Dimensional I.I.D. Mobility Model Hybrid Random Walk Mobility Model Conclusion and Future Works

3. Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 3 Motivation  The Capacity of Wireless Networks, [1, Gupta&Kumar].  Introduce mobility into the unicast network, [2], [3], [4], [5], [6], [7].  As the demand of information sharing increases rapidly, multicast flows are expected to be predominant in many of the emerging applications.  Hu et al. [15] introduced mobility into the multicast traffic pattern.  Zhou and Ying [16] studied the two-dimensional i.i.d. mobility model and provided an optimal tradeoff under their network assumptions.

4. Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 4 Objectives  In our work, we give a global perspective of multicast capacity and delay analysis in Mobile Ad-hoc Networks (MANETs).  Specifically, we consider four node mobility models: 1. two-dimensional i.i.d. mobility; 2. two-dimensional hybrid random walk; 3. one-dimensional i.i.d. mobility; 4. one-dimensional hybrid random walk.  Two mobility time-scales are included: 1. Fast mobility; 2. Slow mobility.

5. Objectives For each of the eight types of mobility models:  Given a delay constraint D, we first characterize the optimal multicast capacity for each of the eight types of mobility models.  Then we develop a scheme that can achieve a capacity- delay tradeoff close to the upper bound up to a logarithmic factor. 5 Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective

6. 6 Outline Introduction Models and Definitions Two-Dimensional I.I.D. Mobility Model One-Dimensional I.I.D. Mobility Model Hybrid Random Walk Mobility Model Conclusion and Future Works

7. Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 7 Models and Definitions – I/VII  Multicast Traffic Pattern:  n nodes move within a unit suquare.  n s = n s nodes are selected as sources, and each has n d = n α distinct destination nodes.  We group each source and its n d destinations as a multicast session. Thus, n s multicast sessions are formed. Note: a particular node may be included by different multicast sessions as either source or destination.  Protocol Model:

8. Models and Definitions – II/VII Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 8  Mobility Models:  Two-dimensional i.i.d. mobility: I.Nodes uniformly randomly positioned in the unit square; II.Node positions independent of each other, and independent from time slot to time slot.  Two-dimensional hybrid random walk: I.The unit square is divided into 1/B 2 RW-cells; II.A node which is in one RW-cell at a time slot moves to one of its eight adjacent RW-cells or stays in the same RW-cell in the next time slot equally likely.

9.  Mobility Models:  One-dimensional i.i.d. mobility: I.Among the mobile nodes n/2 nodes (including n s /2 source nodes), named H-nodes, move horizontally; and the other n/2 nodes (including n s /2 source nodes), named V-nodes, move vertically. II.Let (x i, y i ) denote the position of a node i. If node i is an H-node, y i is fixed and x i is a value randomly uniformly chosen from [0, 1]. We also assume that H-nodes are evenly distributed vertically. V-nodes have similar properties. Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 9 Models and Definitions – III/VII

10.  Mobility Models:  One-dimensional hybrid random walk: I.Each orbit is divided into 1/B RW-intervals; II.At each time slot, a node moves into one of two adjacent RW- intervals or stays at the current RW interval. III.The node position in the RW-interval is randomly, uniformly selected. Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 10 Models and Definitions – IV/VII

11.  Mobility Time Scales:  Fast mobility: The mobility of nodes is at the same time scale as the transmission of packets, i.e., in each time slot, only one-hop transmission is allowed;  Slow mobility: The mobility of nodes is much slower than the transmission of packets, i.e., multi-hop transmission may happen within a single time slot. Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 11 Models and Definitions – V/VII

12.  Scheduling Policies: We assume there exists a scheduler that has all the information about the current and past status of the network, and can schedule any radio transmission in the current and future time slots, [9].  Capture: The scheduler needs to decide whether to deliver the packet p to the destination k. If yes, then choose one relay node (possibly the source node itself), and schedule radio transmissions to forward the packet to the destination.  Duplication: For a packet p that has not been successfully multicast, the scheduler needs to decide whether to duplicate packet p to other nodes. If yes, then decide which nodes to relay from and to, and how. Models and Definitions – VI/VII Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 12

13. Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 13 Models and Definitions – VII/VII  Def. of Capacity:  We assume the same packet arrival rate per time-slot for each source, say λ.  The network is said stable if and only if there exists a certain scheduling scheme which can guarantee the finite length of queue in each node as time goes to infinity.  Capacity, which is short for per-session capacity, is defined as the maximum arrival rate λ that the stable network can support.  Def. of Delay:  Average time it takes for a bit to reach its n d destination nodes.

14. Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 14 Outline Introduction Models and Definitions Two-Dimensional I.I.D. Mobility Model  2-D I.I.D. Fast Mobility Model  2-D I.I.D. Slow Mobility Model One-Dimensional I.I.D. Mobility Model Hybrid Random Walk Mobility Model Conclusion and Future Works

15. Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 15 2-D I.I.D. Fast Mobility Model  Notations:  : the capture range for packet p and destination k  : the capture range for packet p and its last destination  : the number of time slots it takes to reach the last destination of packet p  : # of mobiles relays holding packet p when the packet reaches its last destination

16. Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 16 Upper Bound  [Lem.]: Under two-dimensional i.i.d. mobility model and concerning successful encounter, the following inequality holds for any causal scheduling policy (c 1 is some positive constant).  [Lem.]: Under fast mobility model and concerning network radio resources consumption, the following inequality holds for any causal scheduling policy (c 2 is some positive constant).

17. Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 17 Upper Bound  [The.]: Under two-dimensional i.i.d. fast mobility model, the following upper bound holds for any causal scheduling policy,

18. Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 18 Achievable Lower Bound  Choosing Optimal Values for Key Parameters   By studying the conditions under which the inequalities in the proof are tight, we identify the optimal choices of various key parameters of the scheduling policy.   The scheduling policy should use the same parameters for all packets and all destinations.

19. Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 19 Capacity Achieving Scheme I We group every D time slots into a super-slot:  Step 1. At each odd super-slot: we schedule transmissions from the sources to the relays in every time slot. We divide the unit square into cells (duplication cell). Each cell can be active for 1/c 4 amount of time, [9]. When a cell is scheduled to be active, each source node in the cell broadcasts a new packet to all other nodes in the same cell for amount of time.  Step 2. At each even super-slot: we schedule transmissions from the mobile relays to the destinations in every time slot. We divide the unit square into cells (capture cell).  Remarks: Our scheme uses different cell-partitioning in the odd super-slot than that in the even super-slot.

20. Capacity Achieving Scheme I Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective  [Pro.]: With probability approaching one, as, the above scheme allows each source to send D packets of size, the above scheme allows each source to send D packets of size to their respective destinations within 2D time slots. to their respective destinations within 2D time slots. 20

21. Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 21 2-D I.I.D. Slow Mobility Model  Upper Bound  : # of hops packet p takes from the last mobile relay to destination k  : the length of hth hop [Lem.]: Under slow mobility model, the following inequality holds for any causal scheduling policy, [The.]: Under two-dimensional i.i.d. slow mobility model,

22. Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 22 2-D I.I.D. Slow Mobility Model  Achievable Lower Bound  Choosing Optimal Values of Key Parameters

23.  Step 2: At each even super-slot: We divide the unit square into cells. We then schedule multi-hop transmissions in the following fashion. Further divide each capture cell into hop-cells. hop-cells. Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 23  Capacity Achieving Scheme II Similar to Scheme I, we group every D time slots into a super-slot.  Step 1: At each odd super-slot: We divide the unit square into cells. When a cell is scheduled to be active, each node in the cell broadcasts for amount of time. cells. When a cell is scheduled to be active, each node in the cell broadcasts for amount of time. 2-D I.I.D. Slow Mobility Model

24. Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 24 Outline Introduction Models and Definitions Two-Dimensional I.I.D. Mobility Model One-Dimensional I.I.D. Mobility Model Hybrid Random Walk Mobility Model Conclusion and Future Works

25. 1-D I.I.D. Fast Mobility Model Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 25  Upper Bound [The.]: Under one-dimensional i.i.d. fast mobility model, when, the following upper bound holds for any causal scheduling policy,  Achievable Lower Bound Choosing Optimal Values of Key Parameters:

26.  Capacity Achieving Scheme III  We assume capture only happens within two parallel nodes, defined as H(V) capture. 1-D I.I.D. Fast Mobility Model Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 26  The transmission of a packet in the H(V) multicast session will go through H(V)-V(H) duplication, V(H)-H(V) duplication and H(V)- H(V) capture, three procedures, sequentially.

27.  Capacity Achieving Scheme III  We propose a flexible rectangle-partition scheme, similar to [10], which divides the unit square into multiple horizontal rectangles, named as H-rectangles; and multiple vertical rectangles, named as V-rectangles.  Each H-rectangle and V-rectangle cross to form a cell, and transmissions only happen within the same crossing cell. 1-D I.I.D. Fast Mobility Model Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 27

28. 1-D I.I.D. Slow Mobility Model  Similarly, we get the result of one-dimensional i.i.d. slow mobility model.  [The.]: Under one-dimensional i.i.d. slow mobility model, Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 28

29. Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 29 Outline Introduction Models and Definitions Two-Dimensional I.I.D. Mobility Model One-Dimensional I.I.D. Mobility Model Hybrid Random Walk Mobility Model Conclusion and Future Works

30. Hybrid R.W. Mobility Models Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 30  [The.]: Under two-dimensional hybrid random walk fast mobility model,  [The.]: Under two-dimensional hybrid random walk slow mobility model,  [The.]: Under one-dimensional hybrid random walk fast mobility model,

31. Hybrid R.W. Mobility Models  [The.]: Under one-dimensional hybrid random walk slow mobility model, Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 31

32. Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 32 Outline Introduction Models and Definitions Two-Dimensional I.I.D. Mobility Model One-Dimensional I.I.D. Mobility Model Hybrid Random Walk Mobility Model Conclusion and Future Works

33. Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 33 Conclusion and Future Works  Our results of optimal multicast capacity-delay tradeoffs in MANETs give a global perspective on the multicast traffic pattern:  It generalizes the optimal delay-throughput tradeoffs in unicast traffic pattern in [10], when taking n s = n and n d =1.  It generalizes the multicast capacity result under delay constraint in [16], which is better than [15], when considering the two-dimensional i.i.d. fast mobility model and taking n s n d =n.

34. Conclusion and Future Works  We summarize our results here. Setting and, our results are shown in the second column. Setting and, our results are shown in the third column. Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 34

35. Conclusion and Future Works  We would like to mention that, similar to the unicast case, [5], our one-dimensional mobility models achieve a higher capacity than two-dimensional models under the multicast traffic pattern.  This motivates us to propose a hybrid dimensional model, [20], and we plan to study its capacity improvement in the future. Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 35

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