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

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
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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 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.
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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 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.
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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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 capacitydelay tradeoff close to the upper bound up to a logarithmic factor.
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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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
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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 Multicast Traffic Pattern:
 n nodes move within a unit suquare.
 n s
= n s nodes are selected as sources, and each has distinct destination nodes.
n d
= n
α
 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:
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 7

 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.
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 8

 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

 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

 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

 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.
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 12

 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.
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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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
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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 Notations:
 L : the capture range for packet p and destination k
 L p
: the capture range for packet p and its last destination
 D p
: the number of time slots it takes to reach the last destination of packet p

R p
: # of mobiles relays holding packet p when the packet reaches its last destination
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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 [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).
1 c
1 log n E [ D p
]

(
E [ L p
]

1 n
2
) 2
E [ R p
]
 [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).
 p


1

2
4
E [ R p
]
 n d n


  d p

1 k

1
 
2
E L
2
[ ]
4
 c WT n
2 log
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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 [The.]: Under two-dimensional i.i.d. fast mobility model, the following upper bound holds for any causal scheduling policy,
  min
{ 
(1),
 ( n n n s d
) ( n n D d n n n s d
)}
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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 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.
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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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
C d n
(1
 d square into cells ( log n
)/2
) 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

(2 s

1 d )/2
 ( n
2
) log n
 Step 2 . At each even super-slot: we schedule transmissions from the mobile relays to the destinations in every time slot. We
C c
  ( n
(1
 d
 Remarks: Our scheme uses different cell-partitioning in the odd super-slot than that in the even super-slot.
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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 [Pro.]: With probability approaching one, as n
 , the above scheme allows each source to send D packets of size

(2 s

1 d )/2
   ( n
2 n
) to their log respective destinations within 2D time slots.
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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 Upper Bound
 h : # of hops packet p takes from the last mobile relay to destination k

S h : the length of hth hop
[Lem.]: Under slow mobility model, the following inequality holds for any causal scheduling policy,
 p


1
 2
4
E [ R p
]
 n d n

 d h
  p

1 k

1 h

1
  2
E
[
S h
( )
2 ] 
4 c WT log .
[The.]: Under two-dimensional i.i.d. slow mobility model,
 
O
( n n n s d
3 n
)
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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 Achievable Lower Bound
 Choosing Optimal Values of Key Parameters
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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 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 n
(2 2
  d )/3
C d
  ( ) cells. When a cell is scheduled to be active, each node log n
 (

(3 s

2

2 d )/3 in the cell broadcasts for amount of time.
log
2 n
)
 Step 2: At each even super-slot: We divide the unit square into
C c
  ( n
  
2 )/3 ) cells. We then schedule multi-hop transmissions in the following fashion.
Further divide each capture cell into
C h
  ( n
  
2 )/3
) log n hop-cells.
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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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
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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 Upper Bound
[The.]: Under one-dimensional i.i.d. fast mobility model, when
D
 o
( n
) n d
, the following upper bound holds for any causal scheduling policy,
 
O
( n n n s d
3
2 2 n D d
) n
 Achievable Lower Bound
Choosing Optimal Values of Key Parameters:
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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 Capacity Achieving Scheme III
 We assume capture only happens within two parallel nodes, defined as H(V) capture.
 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.
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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 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.
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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 Similarly, we get the result of one-dimensional i.i.d. slow mobility model .
 [The.]: Under one-dimensional i.i.d. slow mobility model,
 
O
( n n n s d
4
2 2 n D d
) n
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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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
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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 [The.]: Under two-dimensional hybrid random walk fast mobility model,
 
O
( n n n n s d
)
 [The.]: Under two-dimensional hybrid random walk slow mobility model,
 
O
( n
3 n D d n n n s d
)
 [The.]: Under one-dimensional hybrid random walk fast mobility model,
 
O
( n n n s d
3
2 2 n D d
) n
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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 [The.]: Under one-dimensional hybrid random walk slow mobility model,
 
O
( n n n s d
4
2 2 n D d
) n
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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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
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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 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.
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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 We summarize our results here. Setting and , our results are shown in the second column. Setting n s
 n
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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 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
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Reference
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective
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Reference
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 38

Reference
Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 39