mobility weakens the distinction between multicast and unicast xinbing wang dept. of electronic...
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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
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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].
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[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
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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
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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
Previous Works & MotivationOur view on the impact of mobility
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]
Previous Works & MotivationOur view on the impact of mobility
The mobility weakens the distinction between multicast and unicast
12
We verify this observation in our paper
13
Outline Introduction
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
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
Outline Introduction
System model and main idea
The impact of mobility on capacity for restricted
mobility model
The capacity of unicast case
The capacity of multicast case
The multicast capacity gain
The impact of mobility on delay for restricted
mobility model
Discussion
Conclusion and future direction
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 capacity of unicast caseThe throughput upper-bound
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
The capacity of unicast caseThe throughput upper-bound
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.
The capacity of unicast caseThe optimal throughput achieving scheme
Pr(the node meets the relay of the next hop) (1)
26
Outline Introduction
System model and main idea
The impact of mobility on capacity for restricted
mobility model
The capacity of unicast case
The capacity of multicast case
The multicast capacity gain
The impact of mobility on delay for restricted
mobility model
Discussion
Conclusion and future direction
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
The capacity of multicast caseThe throughput upper-bound
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
The capacity of multicast caseThe throughput upper-bound
30
The capacity of unicast caseThe optimal throughput achieving scheme
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
Outline Introduction
System model and main idea
The impact of mobility on capacity for restricted
mobility model
The capacity of unicast case
The capacity of multicast case
The multicast capacity gain
The impact of mobility on delay for restricted
mobility model
Discussion
Conclusion and future direction
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
Outline Introduction
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
The delay of unicast case
The delay of multicast case
The multicast delay gain
Discussion
Conclusion and future direction
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.
The delay of unicast caseOptimal delay achieving scheme: the flooding scheme
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
The delay of unicast caseOptimal delay achieving scheme: the flooding scheme
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
The delay of unicast caseOptimal delay achieving scheme: the flooding scheme
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
The delay of unicast caseOptimal delay achieving scheme: the flooding scheme
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
The delay of unicast caseOptimal delay achieving scheme: the flooding scheme
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
The delay of unicast caseOptimal delay achieving scheme: the flooding scheme
43
Outline Introduction
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
The delay of unicast case
The delay of multicast case
The multicast delay gain
Discussion
Conclusion and future direction
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
Outline Introduction
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
The delay of unicast case
The delay of multicast case
The multicast delay gain
Discussion
Conclusion and future direction
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
Outline Introduction
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
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
……
Discussion: capacityFramework of the multicast capacity gain
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.
Discussion: capacityFramework of the multicast capacity gain
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
Discussion: capacityFramework of the multicast capacity gain
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
Outline Introduction
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
58
Conclusion and future direction
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!