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

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

Yun Wang, Xiaoyu Chu, Xinbing WangDepartment of Electronic Engineering

Shanghai Jiao Tong University, China

Yu ChengDepartment of Electrical and Computer Engineering

Illinois Institute of Technology, USA

Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective 2

OutlineOutline IntroductionIntroduction

MotivationsMotivations ObjectivesObjectives

Models and DefinitionsModels and Definitions

Two-Dimensional I.I.D. Mobility ModelTwo-Dimensional I.I.D. Mobility Model

One-Dimensional I.I.D. Mobility ModelOne-Dimensional I.I.D. Mobility Model

Hybrid Random Walk Mobility ModelHybrid Random Walk Mobility Model

Conclusion and Future WorksConclusion and Future Works

MotivationMotivation

The Capacity of Wireless Networks, The Capacity of Wireless Networks, [[11, Gupta&Kumar, Gupta&Kumar].].

Introduce mobility into the unicast network, [2], [3], [4], [5], [6], [7].Introduce mobility into the unicast network, [2], [3], [4], [5], [6], [7].

As the demand of information sharing increases rapidly, multicast As the demand of information sharing increases rapidly, multicast flows are expected to be predominant in many of the emerging flows are expected to be predominant in many of the emerging applications.applications.

Hu et al. [15] introduced mobility into the multicast traffic pattern.Hu et al. [15] introduced mobility into the multicast traffic pattern.

Zhou and Ying [16] studied the two-dimensional i.i.d. mobility Zhou and Ying [16] studied the two-dimensional i.i.d. mobility model and provided an optimal tradeoff under their network model and provided an optimal tradeoff under their network assumptions.assumptions.

ObjectivesObjectives

In our work, we give a global perspective of multicast capacity In our work, we give a global perspective of multicast capacity and delay analysis in Mobile Ad-hoc Networks (MANETs).and delay analysis in Mobile Ad-hoc Networks (MANETs).

Specifically, we consider four node mobility models:Specifically, we consider four node mobility models:

1. two-dimensional 1. two-dimensional i.i.d. i.i.d. mobility;mobility;

2. two-dimensional hybrid random walk;2. two-dimensional hybrid random walk;

3. one-dimensional 3. one-dimensional i.i.di.i.d. mobility;. mobility;

4. one-dimensional hybrid random walk.4. one-dimensional hybrid random walk.

Two mobility time-scales are included:Two mobility time-scales are included:

1. Fast mobility;1. Fast mobility;

2. Slow mobility.2. Slow mobility.

ObjectivesObjectives

For each of the eight types of mobility models:For each of the eight types of mobility models:

Given a delay constraint Given a delay constraint DD, we first characterize the , we first characterize the optimal multicast capacity for each of the eight types of optimal multicast capacity for each of the eight types of mobility models.mobility models.

Then we develop a scheme that can achieve a capacity-Then we develop a scheme that can achieve a capacity-delay tradeoff close to the upper bound up to a logarithmic delay tradeoff close to the upper bound up to a logarithmic factor. factor.

5Optimal Multicast Capacity and Delay Tradeoffs in MANETs: A Global Perspective

Models and Definitions – I/Models and Definitions – I/VIIVII Multicast Traffic Pattern:Multicast Traffic Pattern: nn nodes nodes move withmove within a unit suquarein a unit suquare..

nn s s = = nnss nodes are selected as sources, and each has nodes are selected as sources, and each has nn d d = = nnαα

distinct destination nodes.distinct destination nodes. We group each source and its We group each source and its nn d d destinations as a multicast session. destinations as a multicast session.

Thus, Thus, nn s s multicast sessions are formed.multicast sessions are formed.

Note: Note: a particular node may be included by different multicast sessions a particular node may be included by different multicast sessions as either source or destination.as either source or destination.

ProtocolProtocol Model: Model:

Models and Definitions – IModels and Definitions – III//VIIVII

Mobility Models:Mobility Models: Two-dimensional Two-dimensional i.i.d. i.i.d. mobility:mobility:

I.I. Nodes uniformly randomly positioned in the unit square;Nodes uniformly randomly positioned in the unit square;

II.II. Node positions independent of each other, and Node positions independent of each other, and independent from time slot to time slot.independent from time slot to time slot.

Two-dimensional hybrid random walk:Two-dimensional hybrid random walk:

I.I. The unit square is divided into The unit square is divided into 1/B1/B22

RW-cells;RW-cells;

II.II. A node which is in one RW-cell at a time A node which is in one RW-cell at a time

slot moves to one of its eight adjacent slot moves to one of its eight adjacent

RW-cells or stays in the same RW-cell RW-cells or stays in the same RW-cell

in the next time slot equally likely.in the next time slot equally likely.

Mobility Models:Mobility Models: One-dimensional One-dimensional i.i.d. i.i.d. mobility:mobility:

I.I. Among the mobile nodes Among the mobile nodes n/2n/2 nodes (including nodes (including nn s s/2 /2 source source

nodes), named H-nodes, move horizontally; and the othernodes), named H-nodes, move horizontally; and the other n/2 n/2 nodes (including nodes (including nn s s/2 /2 source nodes), named V-nodes, source nodes), named V-nodes,

move vertically.move vertically.

II.II. Let Let (x(xii, y, yii) denote the position of a node) denote the position of a node i i. .

If node If node ii is an H-node, is an H-node, yyii is fixed and is fixed and xxii is is

a value randomly uniformly chosen from a value randomly uniformly chosen from

[0, 1]. [0, 1]. We also assume that H-nodes are We also assume that H-nodes are

evenly distributed vertically. V-nodes evenly distributed vertically. V-nodes

have similar properties.have similar properties.

Models and Definitions – IModels and Definitions – IIIII//VIIVII

Mobility Models:Mobility Models: One-dimensional hybrid random walk:One-dimensional hybrid random walk:

I.I. Each orbit is divided into Each orbit is divided into 1/B1/B RW-intervals; RW-intervals;

II.II. At each time slot, a node moves into one of two adjacent RW-At each time slot, a node moves into one of two adjacent RW-intervals or stays at the current RW interval.intervals or stays at the current RW interval.

III.III. The node position in the RW-interval is randomly, uniformly The node position in the RW-interval is randomly, uniformly selected.selected.

Models and Definitions – IModels and Definitions – IVV//VIIVII

Mobility Time Scales:Mobility Time Scales:

Fast mobilityFast mobility: The mobility of nodes is at the same time : The mobility of nodes is at the same time scale as the transmission of packets, i.e., in each time slot, scale as the transmission of packets, i.e., in each time slot, only one-hop transmission is allowed;only one-hop transmission is allowed;

Slow mobilitySlow mobility: The mobility of nodes is much slower than : The mobility of nodes is much slower than the transmission of packets, i.e., multi-hop transmission the transmission of packets, i.e., multi-hop transmission may happen within a single time slot.may happen within a single time slot.

Models and Definitions – Models and Definitions – VV//VIIVII

Scheduling Policies: Scheduling Policies:

We assume there exists a scheduler that has all the We assume there exists a scheduler that has all the information about the current and past status of the information about the current and past status of the network, and can schedule any radio transmission in the network, and can schedule any radio transmission in the current and future time slots, [9]. current and future time slots, [9].

CaptureCapture: The scheduler needs to decide whether to deliver : The scheduler needs to decide whether to deliver the packet the packet p p to the destination to the destination k. k. If yes, then choose one If yes, then choose one relay node (possibly the source node itself), and schedule relay node (possibly the source node itself), and schedule radio transmissions to forward the packet to the destination.radio transmissions to forward the packet to the destination.

DuplicationDuplication: For a packet : For a packet p p that has not been successfully that has not been successfully multicast, the scheduler needs to decide whether to multicast, the scheduler needs to decide whether to duplicateduplicate packet packet p p to other nodes. If yes, then decide which to other nodes. If yes, then decide which nodes to relay from and to, and how.nodes to relay from and to, and how.

Models and Definitions – Models and Definitions – VIVI//VIIVII

Models and Definitions – Models and Definitions – VIIVII//VIIVII Def. of Def. of CapacityCapacity: : We assume the same packet arrival rate per time-slot for each We assume the same packet arrival rate per time-slot for each

source, say source, say λλ.. The network is said The network is said stablestable if and only if there exists a certain if and only if there exists a certain

scheduling scheme which can guarantee the finite length of scheduling scheme which can guarantee the finite length of queue in each node as time goes to infinity.queue in each node as time goes to infinity.

CapacityCapacity, which is short for per-session capacity, is defined as , which is short for per-session capacity, is defined as the maximum arrival rate the maximum arrival rate λλ that the stable network can support. that the stable network can support.

Def. of Delay:Def. of Delay:

Average time it takes for a bit to reach itsAverage time it takes for a bit to reach its nn d d destination nodesdestination nodes..

Two-Dimensional I.I.D. Mobility ModelTwo-Dimensional I.I.D. Mobility Model 2-D I.I.D. Fast Mobility Model2-D I.I.D. Fast Mobility Model 2-D I.I.D. Slow Mobility Model2-D I.I.D. Slow Mobility Model

2-D I.I.D. Fast Mobility Model2-D I.I.D. Fast Mobility Model

Notations: Notations:

: the capture range for packet : the capture range for packet pp and destination and destination kk : the capture range for packet : the capture range for packet pp and its last destination and its last destination : the number of time slots it takes to reach the last : the number of time slots it takes to reach the last

destination of packet destination of packet pp : # of mobiles relays holding packet : # of mobiles relays holding packet pp when the packet when the packet

reaches its last destination reaches its last destination

Upper BoundUpper Bound

[Lem.]: Under [Lem.]: Under two-dimensional i.i.d. mobility model two-dimensional i.i.d. mobility model and and concerning successful encounter, the following inequality holds concerning successful encounter, the following inequality holds for any causal scheduling policy (cfor any causal scheduling policy (c11 is some positive constant). is some positive constant).

[Lem.]: Under [Lem.]: Under fast mobility model fast mobility model and concerning network radio and concerning network radio resources consumption, the following inequality holds for any resources consumption, the following inequality holds for any causal scheduling policy (ccausal scheduling policy (c22 is some positive constant). is some positive constant).

1log [ ]

1[ ] [ ]( )

c n DL R

, 21 1 1

[ ][ ] log

s s dn T n T np d

p kp p k

R nL c WT n

[The.]: Under [The.]: Under two-dimensional i.i.d. fast mobility model, two-dimensional i.i.d. fast mobility model, the the following upper bound holds for any causal scheduling policy,following upper bound holds for any causal scheduling policy,

min (1), ,{ ( ) ( )}d

s d s d

n Dn n

n n n n n

Achievable Lower BoundAchievable Lower Bound

Choosing Optimal Values for Key ParametersChoosing 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.

Capacity AchievingCapacity Achieving Scheme Scheme I I

We group every We group every DD time slots into a super-slot: time slots into a super-slot:

Step Step 11. . At each odd super-slotAt each odd super-slot:: we schedule transmissions from we schedule transmissions from the sources to the relays in every time slot. We divide the unit the sources to the relays in every time slot. We divide the unit square into cells (square into cells (duplication cellduplication cell). Each cell can be ). Each cell can be active for 1/cactive for 1/c44 amount of time, [9]. When a cell is scheduled to amount of time, [9]. When a cell is scheduled to

be active, each source node in the cell broadcasts a new packet be active, each source node in the cell broadcasts a new packet to all other nodes in the same cell for amount of time.to all other nodes in the same cell for amount of time.

Step 2Step 2.. At each even super-slot: At each even super-slot: we schedule transmissions we schedule transmissions from the mobile relays to the destinations in every time slot. We from the mobile relays to the destinations in every time slot. We divide the unit square into cells (capture cell). divide the unit square into cells (capture cell).

Remarks: Our scheme uses Remarks: Our scheme uses differentdifferent cell-partitioning in the odd cell-partitioning in the odd super-slot than that in the even super-slot. super-slot than that in the even super-slot.

(1 )/2

log( )

(1 )/2( )dc n C

(2 1 )/2

2log( )

Capacity Achieving Scheme ICapacity Achieving Scheme I

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

[Pro.]: With probability [Pro.]: With probability approaching one, as approaching one, as

, , the above scheme the above scheme allows each source to allows each source to send send D packets of size D packets of size

to their to their respective destinations respective destinations within 2D within 2D time slots. time slots.

(2 1 )/2

2log( )

2-D I.I.D. Slow Mobility Model2-D I.I.D. Slow Mobility Model

Upper BoundUpper Bound :: # of hops packet # of hops packet p p takes from the last mobile relay to takes from the last mobile relay to

destination destination kk : the length of : the length of hth hth hophop

,hp kS

[Lem.]: Under [Lem.]: Under slow mobility model, the following inequality slow mobility model, the following inequality holds for any causal scheduling policy,holds for any causal scheduling policy,

, 51 1 1 1

[ ]( ) log .

4 4[ ]

p ks s dhn T n T n

p d hp k

p p k h

R nS c WT n

[The.]: Under [The.]: Under two-dimensional i.i.d. slow mobility model, two-dimensional i.i.d. slow mobility model,

Choosing Optimal Values of Key ParametersChoosing Optimal Values of Key Parameters

Step 2: At each even super-slot: Step 2: At each even super-slot: We We divide the unit square into divide the unit square into

cells. We then schedule multi-hop cells. We then schedule multi-hop transmissions in the following fashion. transmissions in the following fashion. Further divide each Further divide each capture cell capture cell into into

hop-cells.hop-cells.

Capacity Achieving Scheme IICapacity Achieving Scheme II

Similar to Scheme I, we group every Similar to Scheme I, we group every D D time slots into a super-slot.time slots into a super-slot. Step 1: At each odd super-slot: Step 1: At each odd super-slot: We divide the unit square intoWe divide the unit square into

cells. When a cell is scheduled to be active, each node cells. When a cell is scheduled to be active, each node in the cell broadcasts for amount of time.in the cell broadcasts for amount of time.

(2 2 )/3

log( )

C(3 2 2 )/3

2log( )

(1 2 2 )/3( )dc n C

(2 2 2 )/3

log( )

[The.]: Under [The.]: Under one-dimensional i.i.d. fast mobility model, one-dimensional i.i.d. fast mobility model, when when , , the following upper bound holds for any the following upper bound holds for any causal scheduling policy,causal scheduling policy,

Choosing Optimal Values of Key Parameters:Choosing Optimal Values of Key Parameters:

Capacity Achieving Scheme IIICapacity Achieving Scheme III

We assume capture only happens within two parallel We assume capture only happens within two parallel nodes, defined as nodes, defined as H(V) capture.H(V) capture.

The transmission of a packet in The transmission of a packet in the H(V) multicast session will go the H(V) multicast session will go through H(V)-V(H) duplication, through H(V)-V(H) duplication, V(H)-H(V) duplication and H(V)-V(H)-H(V) duplication and H(V)-H(V) capture, three procedures, H(V) capture, three procedures, sequentially.sequentially.

Capacity Achieving Scheme IIICapacity Achieving Scheme III

We propose a flexible We propose a flexible rectangle-partition rectangle-partition scheme, similar scheme, similar to [10], which divides the unit square into multiple to [10], which divides the unit square into multiple horizontal rectangles, named as H-rectangles; and horizontal rectangles, named as H-rectangles; and multiple vertical rectangles, named as V-rectangles.multiple vertical rectangles, named as V-rectangles.

Each H-rectangle and V-rectangle cross to form a cell, Each H-rectangle and V-rectangle cross to form a cell, and transmissions only happen within the same crossing and transmissions only happen within the same crossing cell.cell.

Similarly, we get the result of Similarly, we get the result of one-dimensional i.i.d. one-dimensional i.i.d. slow mobility modelslow mobility model..

[The.]: Under [The.]: Under one-dimensional i.i.d. slow mobility one-dimensional i.i.d. slow mobility model,model,

Hybrid R.W. Mobility ModelsHybrid R.W. Mobility Models

[The.]: Under [The.]: Under two-dimensional hybrid random walk fast two-dimensional hybrid random walk fast mobility model,mobility model,

[The.]: Under [The.]: Under two-dimensional hybrid random walk slow two-dimensional hybrid random walk slow mobility model,mobility model,

[The.]: Under [The.]: Under one-dimensional hybrid random walk fast one-dimensional hybrid random walk fast mobility model,mobility model,

Hybrid R.W. Mobility ModelsHybrid R.W. Mobility Models

[The.]: Under [The.]: Under one-dimensional hybrid random walk slow one-dimensional hybrid random walk slow mobility model,mobility model,

Our results of optimal multicast capacity-delay tradeoffs in Our results of optimal multicast capacity-delay tradeoffs in MANETs give a global perspective on the multicast traffic MANETs give a global perspective on the multicast traffic pattern:pattern:

It generalizes the optimal delay-throughput tradeoffs in It generalizes the optimal delay-throughput tradeoffs in unicast traffic pattern in [10], when taking unicast traffic pattern in [10], when taking nnss = n = n and and nndd =1. =1.

It generalizes the multicast capacity result under It generalizes the multicast capacity result under delay constraint in [16], which is better than [15], when delay constraint in [16], which is better than [15], when considering the two-dimensional i.i.d. fast mobility model considering the two-dimensional i.i.d. fast mobility model and taking and taking nns s nnd d =n.=n.

( / )sO D n

We summarize our results here. Setting and ,We summarize our results here. Setting and ,

our results are shown in the second column. Settingour results are shown in the second column. Setting

and , our results are shown in the third column.and , our results are shown in the third column.

sn n 1dn

We would like to mention that, similar to the unicast case, We would like to mention that, similar to the unicast case, [5], our one-dimensional mobility models achieve a higher [5], our one-dimensional mobility models achieve a higher capacity than two-dimensional models under the multicast capacity than two-dimensional models under the multicast traffic pattern.traffic pattern.

This motivates us to propose a hybrid dimensional model, This motivates us to propose a hybrid dimensional model, [20], and we plan to study its capacity improvement in the [20], and we plan to study its capacity improvement in the future.future.

Thank you !Thank you !

ReferenceReference

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

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