Mobility Weakens the Distinction between Multicast and Unicast Xinbing Wang Dept. of Electronic...

Mobility Weakens the Distinction betweenMulticast and Unicast

Xinbing Wang

Dept. of Electronic EngineeringShanghai Jiao Tong University

Shanghai, China

2

Outline Introduction

Previous works & Motivation

System model and main idea

The impact of mobility on capacity for restricted

mobility model

The impact of mobility on delay for restricted

mobility model

Discussion

Conclusion and future direction

Previous Works & MotivationWhat is multicast?

One source to m destinations

3

Data copy is necessary

One copy may be sent to multiple destinations

The number of flows is reduced comparing with unicast

Xiangyang Li [1]

[1] X. Li, “Multicast Capacity of Large Scale Wireless Ad Hoc Networks”, IEEE/ACM Trans. Networking, Vol.17, No. 3, pp. 950-961, Jan. 2008.(citation:234)

Previous Works & MotivationThe multicast uses

4

Any Applications with multiple receivers

(1-to-many)

Collaborative groupware

Reducing Network/Resource Overhead

Live Video distribution

Server/Web-site replication

Resource Discovery

Periodic Data Delivery –

"Push" technology

Stock quotes, sports scores, magazines, newspapers, adverts

more than multiple point-to-point flows

Distributed Interactive

Simulation (DIS)Wargames, virtual reality

Video sources

Internet and mobile networks

Video clients

Hub site

Branch office

Branch office

Branch office

Previous Works & MotivationThe essential difference between multicast and unicast

The flow aggregation in multicast case (multiple flows with different destinations can be aggregated, and therefore only one flow is enough)

Due to the flow aggregation, the number of flows is reduced (11→5 in this example) by the multicast scheduling comparing with the multi-unicast case.

5

multi-unicast multicast

Number of flows

Previous Works & MotivationThe mobility

Everything is going mobile Currently, more than 70% of Facebook users access the service via a mobile

device at least some of the time [2]. 65% growth in mobile data traffic between Q1 2013 and Q1 2014 [3].

6

[2] Christopher Penn, “State of Facebook: 70% use a mobile device to access Facebook”, http://www.shiftcomm.com/2013/07/state-of-facebook-70-use-a-mobile-device-to-access-facebook[3] Ericsson, “Ericsson Mobility Report-June 2014”, http://www.ericsson.com/ericsson-mobility-report

Previous Works & MotivationThe study of mobility

7

Static networks

[4]

Random I.I.D.

mobility[5]

Restricted mobility[7]

Correlative mobility

Markov mobility

[8]

Heterogenous mobility[6]

……

A large number of studies focus on the mobility, including the modeling, measurement, scheduling design, performance analysis and etc.

[4] P. Gupta and P. R. Kumar, "The capacity of wireless networks", IEEE Trans. Inform. Theory, vol. 46, no. 2, pp.388 -404 2000. (citation:7678)[5] A. Gamal, J. Mammen, B. Prabhakar and D. Shah, “Optimal Throughput-Delay Scaling in Wireless Networks-Part I: The Fluid Model,” in IEEE Transactions on Information Theory, vol. 52, no. 6, pp. 2568-2592, 2006. (citation:249)[6] Y. Tao, B. Ye, X. Wang, et al.,“Capacity and delay of heterogeneous wireless networks with correlated mobility,” in Wireless Communications and Networking Conference (WCNC) 2013, Shanghai, China, Apr. 2013. [7] M. Garetto, E. Leonardi, “Restricted Mobility Improves DelayThroughput Tradeoffs in Mobile Ad Hoc Networks,” in IEEE Transactions on Information Theory, vol. 56, no. 10, pp. 5010-5029, 2010. [8] C. Zhang, X. Zhu and Y. Fang, “On the improvement of scaling laws for large-scale MANETs with network coding,” in IEEE Journal on Selected Areas in Communications, vol. 27, no. 5, pp. 662-672, 2009.

Previous Works & MotivationThe impact of mobility(1)

The mobility helps deliver the packet

8

S D

S

Static

Random i.i.d.

R

R

RR

R

R

D S

R D

Time slot 1 Time slot 2

R

RR

R

Previous Works & MotivationThe impact of mobility(2)

The mobility may reduce the probability of flow aggregation

9

S

D1D2

D3

The aggregated flow

S

D1D2

D3

Time slot 1 Time slot 2

Static

Random i.i.d.

S

D1D2

D3

S

D1

D2

D3

S

D1

D2

D3

Time slot 3

Cannot be aggregated with other flow(s) due to

the random mobility

Previous Works & MotivationOur view on the impact of mobility

According to the mentioned impacts above, we can conclude that:

The mobility helps deliver the packet → The mobility improves the capacity.

The mobility may reduce the probability of flow aggregation → The mobility weakens the distinction between multicast and unicast.

10


The mobility improves the capacity

11

Static1

( )logn n

Random i.i.d.

(1)Mobility

nn log:Gain

Static Random i.i.d.

nnm log

1

m

1Mobility

nnm log:Gain 1

Unicast

Multicast

[9] Z. Wang, H. Sadjadpour, J.J. Garcia Luna Aceves, “A Unifying Perspective on the Capacity of Wireless Ad Hoc Networks,” in Proc. of IEEE INFOCOM 2008, Phoenix, AZ, USA, Apr. 2008.[10] X. Wang, Q. Peng, Y. Li, “Cooperation Achieves Optimal Multicast Capacity-Delay Scaling in MANET,” in IEEE Transactions on Communications, vol. 60, no. 10, pp. 3023-3031, 2012.

[9] [10]


The mobility weakens the distinction between multicast and unicast

12

We verify this observation in our paper

13




mobility model


mobility model

Discussion


System modelThe restricted mobility model

14

,

,

| |

| |

| |

i j i j

Hi j i j

Hi i i

X X

H H

X H

Each node i has a corresponding home point Hi. Defining:

The PDF of satisfies Hi

This model includes many important mobility models such as static model, random i.i.d. mobility model etc.

The node speed is a decreasing function of α. Therefore, by adjusting the α, the impact of mobility on the network capacity can be well studied.

Os

sf

)(

)()(

where . Consequently, the can be further expressed as

2( ) 0 2

( )( ) 2

log

( ) 2

s n

sf

n

s

)(f},1min{)( s

iX jX

iH

jH

Hi

Hj

ji ,

Hji ,

System modelThe network model

15

Restricted mobility networks

Total number of nodes : n

Unicast, Multicast

Protocol model

Transmission range:

Bandwidth of each hop: constant

Destinations are randomly selected

Number of destinations is m

n

n)1()( nr

System modelCapacity definition

16

Per-node Throughput: For a given scheme, we define the per-node throughput as the maximum achievable transmission rate. In t time slots, we assume that there are M(i,t) packets transmitted from node i to its destination(s). Firstly, the long term per-node throughput is defined as

Afterwards, the per-node throughput of this model for a given scheme is defined by the maximum T(n) satisfying

t

tiMn

ti

),(inflim)(

1)allfor)()(Pr(lim

inTnin

Per-node Capacity: For a given network, the per-node capacity of it is defined as

where is a scheme for the network, is the set of all possible schemes, and is the per-node throughput of scheme .

)(nT

)(max)( nTnC

System modelDelay definition

17

Delay: For a given scheme, assuming that the source sends the packet to the network at time slot ts and the destination receives the packet at time slot td, the delay is defined as the average value of ts - td , i.e.,

It should be noted that the queuing delay at source is not considered here, which is the same as in many important works. Moreover, for wireless networks, we assume that the operation time spent in coding/decoding is negligible compared to the transmission time.

( ) ( )s dD n t t E

Main idea of this paper

18

Restricted mobility model

Unicast Capacity, Delay Multicast Capacity, Delay

Multicast gain(The capacity and delay gain of

multicast comparing with unicast)

More general case(The upper-bound and lower-

bound of multicast gain)

Main contribution of this paper

19

Multi-hop transmission scheme[7]

Throughput upper-bound

Upper-bound achieving scheme Unicast capacity

Delay Lower-bound

Lower-bound achieving schemeUnicast delay

Throughput upper-bound (Based on a round cut)

Upper-bound achieving scheme

Delay Lower-bound

Lower-bound achieving scheme

Multicast capacity

Multicast delay

No related work

Multicast capacity gain

Multicast Delay gain

Unicast

Multicast

Related work Our contribution

[7] M. Garetto, E. Leonardi, “Restricted Mobility Improves DelayThroughput Tradeoffs in Mobile Ad Hoc Networks,” in IEEE Transactions on Information Theory, vol. 56, no. 10, pp. 5010-5029, 2010.

20




mobility model

The capacity of unicast case

The capacity of multicast case

The multicast capacity gain


mobility model

Discussion


21

The capacity of unicast caseThe throughput upper-bound

In order to derive the per-node capacity, a contact graph is considered, in which the nodes are allocated at their home-points respectively. Moreover, we put an edge between any two-nodes, whose weight is defined as the probability that they happen to be within distance of each other.

Considering a cut dividing the contact graph into two parts with the same size, there are nodes in each part in average. Therefore, the sum per-node throughput of these pairs is bounded by the sum weight of the edges across the cut.

1O

2O

The cut

( )r n

)(n

Based on the contact graph

22


The sum weight of the edges across the cut can be expressed as

where

Further computation

12

The cut

iNode i

1is ini

i

W W O

2

,is in

( )Hi i j

j

W p O

arccos( )

0

2 2

2

2

2

2 ( )

2 arccos( ) ( )

2 ( )

( ) ( )

(1) 0 2

( ) 2

i

i

i

i

i

i i

n

i

n

i

n

i

n

i i i

i

W p d d

p d

p d

p d p d

23


After some mathematical manipulations, the throughput upper-bound can be obtained

12

(1) 0 2

12

log

( )2 3

13

n

C nn

n

24

The capacity of unicast caseThe optimal throughput achieving scheme

Source

Relay

Destination

3

0 3

The 2-hop relay scheme: the relay is selected from the nodes with home-point in the circle centered at the middle point of the source’s and destination’s home-points. The radius is 1/3 of the distance between the source’s and destination’s home-points.

1

1

1( ) liminf ( , )

(Pr (source meets a relay))

(Pr (destination meets a relay))

i tn M i t

t

25

Source

Relays

Destination

3

The multi-hop relay scheme: the relays are selected in the cells that the line between source’s and destination’s home-points lines across.


Pr(the node meets the relay of the next hop) (1)

26




mobility model





mobility model

Discussion


27

The capacity of multicast caseThe throughput upper-bound

8

nm

Node i

1O

2O

The cut

Each source selects m destinations. The contact graph is also considered.

Considering a circle cut dividing the contact graph into two parts as in the figure. The radius of the circle is .

The numbers of packets transmitting into the circle and out of the circle are both . Therefore, the sum per-node throughput is bounded by the sum weight of the edges across the cut.

m

n

8

)(n

Based on the contact graph

28


Similar to the unicast case, the sum weight of the edges across the cut can be expressed as

8

n

m

i

8

n

m

Node i

1O

2O

The cut

where

Further computation

where

1is ini

i

W W O

2

,is in

( )Hi i j

j

W p O

2

,is in

0

( )

2 ( )i

Hi i j

j

n

W p

p d d

O

22

i

n

ms

29

After some mathematical manipulations, the throughput upper-bound can be obtained

1 22 2

10 2

12

log( )

2 3

13

m

nmC n m

n m

nm


30


Case 1: α<2, similar to random i.i.d. mobility model.(2-hop, random relay selection)

Case 2:α≥2,

Step1: build the Euclidean Minimum Spanning Tree (EMST) among the home points of the destinations and source.

Step 2: Each edge of the EMST is treated as a unicast transmission.

31




mobility model





mobility model

Discussion


32

The multicast capacity gainDefinition of multicast capacity gain

Multicast Capacity Gain: For a given network, we assume that the per-node capacity of multicast is . Moreover, if each node has m destinations, each multicast session can be treated as m unicast sessions (multi-unicast), and the corresponding sum per-node capacity is denoted as . Comparing the capacity of multicast and multi-unicast, we define the multicast capacity gain as

The multicast capacity gain indicates the enhancement of per-node capacity by multicast transmission. Specially for the restricted mobility model, we use to represent the multicast capacity gain instead of since it is mainly related with and m.

)(nCmulti

)(_ nC unim

)(

)()(

_ nC

nCn

unim

multi

)(n),( mg

)(n

33

The multicast capacity gainThe multicast capacity gain of restricted mobility model

According to the theoretical results of unicast and multicast, the multicast capacity gain can be expressed as

12

log

log

1 0 2

2

( , )

2 3

3

n

m

n

g m

m

m

34

The multicast capacity gainThe multicast capacity gain of restricted mobility model

There is a gap when α=2 since there is a gap in the sum of p-series.

1

1

( ) 0 1

(log ) 1

(1) 1

p

np

i

n p

i n p

p

35




mobility model


mobility model

The delay of unicast case

The delay of multicast case

The multicast delay gain

Discussion


36

The delay of unicast caseOptimal delay achieving scheme: the flooding scheme

In the restricted mobility model, the PDF of packet-holding nodes in the next time slot is determined by the packet-holding nodes in current time slot. Therefore, the transmission process in restricted mobility model can be treated as a Markov chain with 2n-1 states. Moreover, there are 2n-2 target states and 1 initial state. Hence, it is too complex to obtain the exact order of delay.

Source’s home-point

Destination’s home-point

37

The upper-bound of the optimal delay is analyzed in this paper. In particular, we divide the transmissions into two groups, i.e., long distance transmission (LDT) and short distance transmission (SDT).

LDT: the distance between the two transmission nodes’ home-points is .

SDT: the distance between the two transmission nodes’ home-points is .

( )n

( )o n

We calculate the following two kinds of delay:Delay 1: the delay of the packet transmitted from source to destination only through LDT.Delay 2: the delay of the packet transmitted from source to destination only through SDT.

Consequently, the total delay of flooding scheme is upper-bounded by the minimum value of Delay 1 and 2.


38

Delay 1 (LDT delay)

The event of LDT happens with probability

For any nodes i and j, the event that i transmits a packet to j within two hops happens with probability , which is the same as in random i.i.d. mobility model in order sense. Thus, the Delay 1 equals to timing the delay of random i.i.d. mobility model. Hence, the Delay 1 satisfies

1( )n1

LDTp

2

2 1

log 0 2

log 2

log 2

LDT

O n

D O n

O n n

,, , ,( )

1 2

min 1, ( )

1 0 2

12

log

2

Hi k

H H HLDT i k i k i kn

P p d

O

On

O n


39

Delay 2 (SDT delay)

To calculate Delay 1, we consider the condition that there is an region of radius centered at the home-point of source, and each node with home-point in holds a packet from i with probability . After time slots, there is a region

of radius centered at the home-point of source, and each node with home-point in holds a packet from i with probability . This process is called region extension, which is illustrated in the figure, and is the region extension time.

0O

00

O(1)

0 1t

0 1 O 0 1

0 1 O (1)

0 1t

0

0O

0 1

Region extension

0 1 OSource


40

Delay 2 (SDT delay)

It should be noted that we ignore the transmissions out of as while as the relay to relay transmissions within the ring during the region extension.

After some manipulations, the optimal relation between and is derived in our paper to minimize the number of ignored transmissions.

0 1 O

0 1 0 O O

0 1

01

3

(1) 3

0

0O

0 1

Regi on extensi on

0 1 O

Source


41

Delay 2 (SDT delay)

It should be noted that the delay of case is mainly determined by Delay 1. 2

3

2

2

log 2

log 2 3

log 3

SDT

O n

D O n n

O n n


42

The upper-bound of the delay for flooding scheme (The minimum value of Delay 1 and Delay 2)

2

2

2

log 0 2

log 2

( )log 2 3

log 3

O n

O n

D nO n n

O n n


43




mobility model


mobility model




Discussion


44

The delay of multicast caseOptimal delay achieving scheme: the flooding scheme

To obtain the optimal delay for multicast case, we also adopt the flooding scheme under the same assumption that the transmission range is constant. When the flooding scheme is adopted, all of the nodes in the network will receive a replica of the packet from the source within the same time scale of the delay for unicast case. Therefore, for multicast case, the optimal delay is of the same order of unicast case, which is also proved in random i.i.d. mobility model in [10].

[10] X. Wang, Q. Peng, Y. Li, “Cooperation Achieves Optimal Multicast Capacity-Delay Scaling in MANET,” in IEEE Transactions on Communications, vol. 60, no. 10, pp. 3023-3031, 2012.

45




mobility model


mobility model




Discussion


46

The multicast delay gainDefinition of multicast delay gain

Multicast Delay Gain: For a given network, we assume that the network delay of multicast is . Moreover, if each node has m destinations, the sum delay of the transmissions from the source to them by unicast is denoted as . Comparing the delay of multicast and multi-unicast, we define the multicast delay gain as

The multicast delay gain indicates the enhancement of delay performance by multicast transmission.

( )multiD n

_ ( )m uniD n

_

( )( )

( )multi

m uni

D nn

D n

( )n

47

The multicast gainThe multicast delay gain of restricted mobility model

( ) ( )n m

According to the definition of multicast delay gain, the multicast delay upper-bound gain of restricted mobility model can be expressed as

48




mobility model


mobility model

Discussion


49

Discussion: capacityThe upper-bound of the multicast capacity gain

The multicast session can be treated as multiple unicast sessions.

If the multicast is finished during one unicast session, the multicast capacity gain is maximized.

( ) ( )n m

50

Discussion: capacityThe lower-bound of the multicast capacity gain

The multicast session can be treated as multiple unicast sessions.

The random i.i.d. mobility model can achieve the lower-bound of multicast capacity gain

( ) (1)n

51

Discussion: capacityFramework of the multicast capacity gain

Mobility weakens the distinction between multicast and unicast, which is the cost of the capacity enhancement. Restricted mobility

m

( )n

(1)

0

0 2

2 3

3

Upper-bound

Lower-bound

m

2( )n l n

Multicast capacity gain range of restricted mobility

Random i.i.d. mobility

Random walk (step length l)

One-dimensional static m

( )n

Brief explanation: the unpredictability of mobility decreases the opportunity of flow aggregation. (Please see page 7)

52

Other affecting factors

The distribution of nodes

The number of destinations

……


53

The distribution of nodes also impacts the multicast gain

The distribution of nodes determines the number of nodes covered by each hop.

For a given transmission range, the opportunity of flow aggregation increases with the number of nodes covered by each hop. Therefore, we have following conjecture:

SourceDestination

Transmission range

Conjecture: The multicast gain is high if the nodes are distributed (or probabilistically distributed) close to a line (or a curve). An example can be found in page 46.


54

The multicast gain is also related with the number of destinations

Brief explanation: the additional destinations may help increase the opportunity of flow aggregation.

Sum of reduced flows 2 8

Source

Destination

3

1 1

6

3

1 1

2

1

Aggregated flows


55

Discussion: delayFramework of multicast delay gain

In multicast case, assuming the unicast delay from source i to one of its destinations j is , the multicast delay gain can be expressed as

Since in flooding scheme, the upper-bound of multicast delay gain is and the lower-bound is .

_ , ( )m uni jD n

_ ,

( )( )

( )multi

m uni jj

D nn

D n

_ ,( ) max ( )multi m uni j

jD n D n

( )m (1)

56

Discussion: delayThe lower-bound achieving scheme

Random i.i.d. mobility

n-3n/m nodes

3n/m nodes( ( ))r n

3p n

0O1O

With probability n-3, one node in one part can move to another part in one time slot, and then it moves back to its initial part in the next time slot. Therefore,

where poly-logarithmic factors are ignored, i.e., the multicast delay gain is .

3_ ,max ( ) ( )m uni j

jD n n 3

_ , ( ) ( )m uni jj

D n n

(1)

57




mobility model


mobility model

Discussion


58


This paper studies the essential roles of multicast scheduling and mobility in one-to-many transmission networks. Based on the restricted mobility model, the theoretical analysis indicates that the mobility weakens the distinction between multicast and unicast, i.e., the probability of flow aggregation.

We further propose another two affecting factors of the multicast capacity gain, i.e., the distribution of the nodes and the number of destinations.

Therefore, three interesting future directions arise: What is the optimal delay/throughput tradeoff for restricted mobility

model? And what is the corresponding scheme? What is the impact of mobility on multicast in a general mobility model? Is there any other affecting factors of the multicast capacity gain?

59

Comments ?

Questions ?

Thank you!

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Mobility Weakens the Distinction between Multicast and Unicast Xinbing Wang Dept. of Electronic...

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