On Flow-Aware CSMA in Multi-Channel Wireless Networks Mathieu - - PowerPoint PPT Presentation

On Flow-Aware CSMA

in Multi-Channel Wireless Networks

Mathieu Feuillet Joint work with Thomas Bonald CISS 2011

Outline

Model Background Standard CSMA Flow-aware CSMA Conclusion

Outline

Model Background Standard CSMA Flow-aware CSMA Conclusion

Conflict graphs

The network is represented by a set of conflict graphs, one per channel. 1 2 3 1 2 3 Channel 1 Channel 2

Conflict graphs

The network is represented by a set of conflict graphs, one per channel. 1 2 3 1 2 3 Channel 1 Channel 2 Conflict graphs on different channels can be different.

Conflict graphs

1 2 3 1 2 3 Channel 1 Channel 2 Schedules:

◮ ∅, {(1, 1)}, {(2, 1)}, {(3, 1)},{(1, 1), (3, 1)}.

Conflict graphs

1 2 3 1 2 3 Channel 1 Channel 2 Schedules:

◮ ∅, {(1, 1)}, {(2, 1)}, {(3, 1)},{(1, 1), (3, 1)}.

Conflict graphs

1 2 3 1 2 3 Channel 1 Channel 2 1 Schedules:

◮ ∅, {(1, 1)}, {(2, 1)}, {(3, 1)},{(1, 1), (3, 1)}.

Conflict graphs

1 2 3 1 2 3 Channel 1 Channel 2 2 Schedules:

◮ ∅, {(1, 1)}, {(2, 1)}, {(3, 1)},{(1, 1), (3, 1)}.

Conflict graphs

1 2 3 1 2 3 Channel 1 Channel 2 3 Schedules:

◮ ∅, {(1, 1)}, {(2, 1)}, {(3, 1)},{(1, 1), (3, 1)}.

Conflict graphs

1 2 3 1 2 3 Channel 1 Channel 2 1 3 Schedules:

◮ ∅, {(1, 1)}, {(2, 1)}, {(3, 1)},{(1, 1), (3, 1)}.

Conflict graphs

1 2 3 1 2 3 Channel 1 Channel 2 Schedules:

◮ ∅, {(1, 1)}, {(2, 1)}, {(3, 1)},{(1, 1), (3, 1)}. ◮ {(1, 2)}, {(2, 2)}, {(3, 2)},{(1, 2), (3, 2)}.

Conflict graphs

1 2 3 1 2 3 Channel 1 Channel 2 Schedules:

◮ ∅, {(1, 1)}, {(2, 1)}, {(3, 1)},{(1, 1), (3, 1)}. ◮ {(1, 2)}, {(2, 2)}, {(3, 2)},{(1, 2), (3, 2)}. ◮ {(1, 1), (2, 2)}, {(1, 2), (2, 1), (3, 1)},. . .

Conflict graphs

1 2 3 1 2 3 Channel 1 Channel 2 1 2 Schedules:

◮ ∅, {(1, 1)}, {(2, 1)}, {(3, 1)},{(1, 1), (3, 1)}. ◮ {(1, 2)}, {(2, 2)}, {(3, 2)},{(1, 2), (3, 2)}. ◮ {(1, 1), (2, 2)}, {(1, 2), (2, 1), (3, 1)},. . .

Conflict graphs

1 2 3 1 2 3 Channel 1 Channel 2 2 1 3 Schedules:

◮ ∅, {(1, 1)}, {(2, 1)}, {(3, 1)},{(1, 1), (3, 1)}. ◮ {(1, 2)}, {(2, 2)}, {(3, 2)},{(1, 2), (3, 2)}. ◮ {(1, 1), (2, 2)}, {(1, 2), (2, 1), (3, 1)},. . .

Time-scale separation

We assume time-scale separation between flow-level and packet-level dynamics.

◮ At the packet level, the number of flows is fixed. ◮ At the flow level, the packet-level dynamics are at

equilibrium: the throughput of each node is the fraction

• f time it is active.

Time-scale separation

We assume time-scale separation between flow-level and packet-level dynamics.

◮ At the packet level, the number of flows is fixed. ◮ At the flow level, the packet-level dynamics are at

equilibrium: the throughput of each node is the fraction

• f time it is active.

Example: 1 2 3 ϕ1 = p{1} + p{1,3}, ϕ2 = p{2}, ϕ3 = p{3} + p{1,3}.

Capacity Region

Defined as the set of all feasible link throughputs ϕk =

• i:k∈Si pi

Schedule i Probability of schedule i Throughput of node k

Capacity Region

Defined as the set of all feasible link throughputs ϕk =

• i:k∈Si pi

Schedule i Probability of schedule i Throughput of node k

Example: 1 2 3 Schedules: ∅, {1}, {2}, {3}, {1, 3}. Capacity region: {ϕ1 + ϕ2 ≤ 1, ϕ2 + ϕ3 ≤ 1}.

Stability region

Defined as the set of traffic intensities such that the network is stable. ρk = λk × σk Example: 1 2 3 ρ1 ρ2 ρ3 The stability region depends on the algorithm.

Stability region

Defined as the set of traffic intensities such that the network is stable. ρk = λk × σk

Flows arrival rate

Stability region

Defined as the set of traffic intensities such that the network is stable. ρk = λk × σk

Flows arrival rate Mean flow size

Stability region

Defined as the set of traffic intensities such that the network is stable. ρk = λk × σk

Traffic intensity Flows arrival rate Mean flow size

Optimal stability region

The stability region of any algorithm is included in the capacity region. The interior of the capacity region is called the optimal stability region.

Optimal stability region

The stability region of any algorithm is included in the capacity region. The interior of the capacity region is called the optimal stability region. Example: 1 2 3 Optimal stability region: {ρ1 + ρ2 < 1, ρ2 + ρ3 < 1}.

Outline

Model Background Standard CSMA Flow-aware CSMA Conclusion

Maximal Weight scheduling

T assiulas & Ephremides 92 maxwi(x) =

• k∈Si

xk

Maximal Weight scheduling

T assiulas & Ephremides 92 maxwi(x) =

• k∈Si

xk Example: 1 2 3 x1 + x3 > x2 ⇒ schedule {1, 3} x1 + x3 < x2 ⇒ schedule {2}

Suboptimal algorithms

Maximal queue scheduling

Mc Keown 95 maxxk Dimakis & Walrand 06 Efficiency 8

9 ≈ 0.89

Suboptimal algorithms

Maximal queue scheduling

Mc Keown 95 maxxk Dimakis & Walrand 06 Efficiency 8

9 ≈ 0.89

Maximal size scheduling

Charporkar, Kar & Sarkar 95 max |Si| Bonald & Massoulié 01 Efficiency ≈ 0.76

Optimal Algorithms

Adaptative rate-based CSMA Jiang & Walrand 08

◮ Measure the packet input and output rates and adapt the

back-off accordingly.

◮ Learning algorithm.

Optimal Algorithms

Adaptative rate-based CSMA Jiang & Walrand 08

◮ Measure the packet input and output rates and adapt the

back-off accordingly.

◮ Learning algorithm.

Adaptative queue based CSMA Rajagopolan, Shah & Shin 09

◮ Adapt the back-off according to the loglog of the queue

length.

◮ Some technical assumptions.

Outline

Model Background Standard CSMA Flow-aware CSMA Conclusion

Standard CSMA

Probe a channel at random, after some random backoff time. 1 1 1 α/2 α/2 1 1 ∼ exp(α) Back-off ∼ exp(1) Transmission ∼ exp(α) Back-off ϕ1(x) = α 1 + α

Standard CSMA

Probe a channel at random, after some random backoff time. 1 1 1 α/2 α/2 1 1 ∼ exp(α) Back-off ∼ exp(1) Transmission ∼ exp(α) Back-off ϕ1(x) = α 1 + α Stability region:

• ρ1 <

α 1+α

α→∞ − − − → {ρ1 < 1}

Standard CSMA

wi(x) =

• (k,j)∈Si αk,j✶{xk>0}

Weight of schedule i Ratio of transmission time to virtual backoff time at link k

• n channel j

Example: Assume αk,j = α for k = 1, 2, 3 and j = 1, 2. 1 2 3 1 2 3 1 2 3 α α α 1 1 1 α α α 1 1 1

Standard CSMA

wi(x) =

• (k,j)∈Si αk,j✶{xk>0}

Weight of schedule i Ratio of transmission time to virtual backoff time at link k

• n channel j

Example: Assume αk,j = α for k = 1, 2, 3 and j = 1, 2. 1 2 3 1 2 3 1 2 3 α α α 1 1 1 α α α 1 1 1 ∅ : wi(x) = 1

Standard CSMA

wi(x) =

• (k,j)∈Si αk,j✶{xk>0}

Weight of schedule i Ratio of transmission time to virtual backoff time at link k

• n channel j

Example: Assume αk,j = α for k = 1, 2, 3 and j = 1, 2. 1 2 3 1 2 3 1 2 3 α α α 1 1 1 α α α 1 1 1 1 {(1, 1)} : wi(x) = α

Standard CSMA

wi(x) =

• (k,j)∈Si αk,j✶{xk>0}

Weight of schedule i Ratio of transmission time to virtual backoff time at link k

• n channel j

Example: Assume αk,j = α for k = 1, 2, 3 and j = 1, 2. 1 2 3 1 2 3 1 2 3 α α α 1 1 1 α α α 1 1 1 1 2 {(1, 1), (2, 2)} : wi(x) = α2

Standard CSMA

wi(x) =

• (k,j)∈Si αk,j✶{xk>0}

Weight of schedule i Ratio of transmission time to virtual backoff time at link k

• n channel j

Example: Assume αk,j = α for k = 1, 2, 3 and j = 1, 2. 1 2 3 1 2 3 1 2 3 α α α 1 1 1 α α α 1 1 1 1 2 3 {(1, 1), (2, 2), (3, 1)} : wi(x) = α3

Standard CSMA

wi(x) =

• (k,j)∈Si αk,j✶{xk>0}

Weight of schedule i Ratio of transmission time to virtual backoff time at link k

• n channel j

Example: Assume αk,j = α for k = 1, 2, 3 and j = 1, 2. 1 2 3 1 2 3 1 2 3 α α α 1 1 1 α α α 1 1 1 wi(x) = αn for n active links

Standard CSMA

ϕk(x) =

• i:k∈Si pi(x)

Throughput of link k Probability of schedule i

Example: Assume αk,j = α for k = 1, 2, 3 and j = 1, 2. 1 2 3 1 2 3 1 2 3 α α α 1 1 1 α α α 1 1 1 ϕ1(x) = ϕ3(x) = 2α + 6α2 + 2α3 1 + 6α + 8α2 + 2α3 , ϕ2(x) = 2α + 4α2 + 2α3 1 + 6α + 8α2 + 2α3

Suboptimality of standard CSMA

1 channel

α ≫ 1

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

ρ1 = ρ3 ρ2

Optimal Actual

1 2 3 Homogeneous case ρ1 = ρ2 = ρ3 Efficiency ≈ 0.84.

Suboptimality of standard CSMA

2 channels

α ≫ 1

0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

ρ1 = . . . = ρ4 ρ5

Optimal Actual

1 5 2 3 4 Homogeneous case ρ1 = . . . = ρ5 Efficiency ≈ 0.85

Outline

Model Background Standard CSMA Flow-aware CSMA Conclusion

Flow-aware CSMA

Proposed modification of CSMA: Exponential backoff time for each flow 1 1 1 αx1/2 αx1/2 1 1 ∼ exp(αx1) Back-off ∼ exp(1) Transmission ∼ exp(αx1) Back-off ϕ1(x) = αx1 1 + αx1

Flow-aware CSMA

Proposed modification of CSMA: Exponential backoff time for each flow 1 1 1 αx1/2 αx1/2 1 1 ∼ exp(αx1) Back-off ∼ exp(1) Transmission ∼ exp(αx1) Back-off ϕ1(x) = αx1 1 + αx1 Stability region: {ρ1 < 1}

Flow-aware CSMA

wi(x) =

• k∈Si αk,jxk

Weight of schedule i Ratio of transmission time to virtual back-

• ff time at link k on

channel j

Example: Assume αk,j = α for k = 1, 2, 3 and j = 1, 2. 1 2 3 1 2 3 1 2 3 αx1 αx2 αx3 1 1 1 αx1 αx2 αx3 1 1 1

Flow-aware CSMA

wi(x) =

• k∈Si αk,jxk

Weight of schedule i Ratio of transmission time to virtual back-

• ff time at link k on

channel j

Example: Assume αk,j = α for k = 1, 2, 3 and j = 1, 2. 1 2 3 1 2 3 1 2 3 αx1 αx2 αx3 1 1 1 αx1 αx2 αx3 1 1 1 ∅ : wi(x) = 1

Flow-aware CSMA

wi(x) =

• k∈Si αk,jxk

Weight of schedule i Ratio of transmission time to virtual back-

• ff time at link k on

channel j

Example: Assume αk,j = α for k = 1, 2, 3 and j = 1, 2. 1 2 3 1 2 3 1 2 3 αx1 αx2 αx3 1 1 1 αx1 αx2 αx3 1 1 1 1 {(1, 1)} : wi(x) = αx1

Flow-aware CSMA

wi(x) =

• k∈Si αk,jxk

Weight of schedule i Ratio of transmission time to virtual back-

• ff time at link k on

channel j

Example: Assume αk,j = α for k = 1, 2, 3 and j = 1, 2. 1 2 3 1 2 3 1 2 3 αx1 αx2 αx3 1 1 1 αx1 αx2 αx3 1 1 1 1 2 {(1, 1), (2, 2)} : wi(x) = α2x1x2

Flow-aware CSMA

wi(x) =

• k∈Si αk,jxk

Weight of schedule i Ratio of transmission time to virtual back-

• ff time at link k on

channel j

Example: Assume αk,j = α for k = 1, 2, 3 and j = 1, 2. 1 2 3 1 2 3 1 2 3 αx1 αx2 αx3 1 1 1 αx1 αx2 αx3 1 1 1 1 2 3 {(1, 1), (2, 2), (3, 1)} : wi(x) = α3x1x2x3

Flow-aware CSMA

ϕk(x) =

• i:k∈Si pi(x)

Throughput of link k Probability of schedule i

Example: Assume αk,j = α for k = 1, 2, 3 and j = 1, 2. 1 2 3 1 2 3 1 2 3 αx1 αx2 αx3 1 1 1 αx1 αx2 αx3 1 1 1 ϕ2(x) ∝ 2α3x1x2x3 + 2α2x2(x1 + x3) + 2αx2

Optimality of flow-aware CSMA

Theorem

The flow-aware CSMA algorithm is optimal: for any number

• f channels and any sets of conflicts graphs, it stabilizes the

network for all traffic intensities inside the capacity region. Sketch of proof:

◮ When the number of flows is high, the algorithm behaves

as max-weight: the algorithm selects schedules of high weight with high probability.

◮ Proof based on Foster’s criterion.

Throughput performance

. . . . . . Homogeneous load α = 1

0.2 0.4 0.6 0.8 1 0.2 0.4

Load Flow throughput Standard CSMA

1 channel 2 channels 3 channels 0.2 0.4 0.6 0.8 1 0.2 0.4

Load Flow throughput Flow-aware CSMA

1 channel 2 channels 3 channels

Time-scale separation

1 5 2 3 4 α = 1 1 packet per flow

0.2 0.4 0.6 0.2 0.4

Load Flow throughput Center link

With Without 0.2 0.4 0.6 0.2 0.4

Load Flow throughput Edge link

With Without

Outline

Model Background Standard CSMA Flow-aware CSMA Conclusion

Conclusion

A simple, distributed, asynchronous access scheme that is provably optimal. A theoretical issue:

◮ Time-scale separation

Extensions of the model:

◮ Collisions ◮ Throughput performance