On the Capacity of Wireless CSMA/CA Multihop Networks

On the Capacity of

Wireless CSMA/CA Multihop Networks

Rafael Laufer and Leonard Kleinrock

Bell Labs, UCLA

IEEE INFOCOM 2013

• Carrier sense multiple access with collision avoidance (CSMA/CA)

Before transmitting, the node verifies if the medium is idle via carrier sensing If idle, sample a random back-off interval and starts counting down

Whenever busy, freeze the counter and wait for ongoing transmission to finish

INTRODUCTION

Wireless CSMA/CA Multihop Networks

U ₂ (t)

2

1 3

• Considered unpredictable with unknown throughput limitations

Distributed nature of CSMA/CA: nodes should back off from each other

Buffer dynamics of unsaturated sources: time-varying subset of transmitters Dependence of downstream links on upstream traffic: coupled queue state

• Strong dependence among the state of transmitters

Physical proximity and traffic pattern induce correlation across the network

INTRODUCTION

Wireless CSMA/CA Multihop Networks

• Understand throughput limits of wireless CSMA/CA multihop networks

• Provide answers to specific questions regarding the network capacity

If the rate of f ₁ increases by 10%, how much can f ₂ still achieve?

If f ₃ starts, by how much must f ₁ and f ₂ slow down to keep the network stable?

• Determine the capacity region of arbitrary wireless networks

INTRODUCTION Goals

f ₂

• Theory to model the behavior of wireless CSMA/CA multihop networks

Handle buffer dynamics of unsaturated traffic sources and multihop flows Respect interference constraints imposed by the wireless medium

• Characterization of the capacity region of any wireless network

No restrictions on node placement: suitable for arbitrary networks

Agnostic to the distribution of network parameters: only averages are relevant Convex only when nodes are within range: nonconvex in general

• Feasibility test

INTRODUCTION

Key Contributions

• Single-path routing, with routes and bit rates assumed fixed

• Omnidirectional antenna communicating in a single channel

• CSMA/CA for medium access control

• Network state S composed of links transmitting

Knowledge of the feasible link sets in the network

• : fraction of time that all links in S are transmitting

MODEL AND ASSUMPTIONS

System Model

THROUGHPUT MODELING

SATURATED SINGLE-HOP FLOWS

All Nodes Within Carrier Sense Range

SATURATED SINGLE-HOP FLOWS All Nodes Within Carrier Sense Range

U ₁ (t)

t

1

U ₂ (t)

U ₃ (t) t

t

• By definition, the steady-state solution is

• Ratio between and

SATURATED SINGLE-HOP FLOWS

All Nodes Within Carrier Sense Range

• System of linear equations

• Steady-state solution

• Throughput of each flow

SATURATED SINGLE-HOP FLOWS

All Nodes Within Carrier Sense Range

SATURATED SINGLE-HOP FLOWS

Not All Nodes Within Carrier Sense Range

2 1

3

SATURATED SINGLE-HOP FLOWS

Not All Nodes Within Carrier Sense Range

U ₁ (t)

U ₂ (t) t

U ₃ (t) t

t

• Steady-state solution for this case

• General solution

• Throughput of each flow

SATURATED SINGLE-HOP FLOWS

Not All Nodes Within Carrier Sense Range

UNSATURATED SINGLE-HOP FLOWS Idle Time

U ₁ (t)

t

1

U ₂ (t)

U ₃ (t) t

t

• Steady-state solution

• Source behavior

Injecting too little traffic:  0 Injecting too much traffic:  1

UNSATURATED SINGLE-HOP FLOWS

• Why the solution is similar to the saturated case?

• Statistically equivalent to a saturated network

Average transmission times are the same Average backoff times are larger by 1/

UNSATURATED SINGLE-HOP FLOWS

UNSATURATED SINGLE-HOP FLOWS Primal Unsaturated Network

U ₁ (t)

t

1

U ₂ (t)

U ₃ (t) t

UNSATURATED SINGLE-HOP FLOWS Dual Saturated Network

U ₁ (t)

U ₂ (t) t

U ₃ (t) t

t

1

CAPACITY REGION CHARACTERIZATION

• Normalized throughput of transmitter i

• Express as a function of

• Find the inverse

• Limit the stability factors to the range

CAPACITY REGION

Characterization Algorithm

CAPACITY REGION

Two Transmitters Within Carrier Sense Range

CAPACITY REGION

Two Transmitters Within Carrier Sense Range

1 y ₁

y ₂

1 1

1 





2 2

1 



 1

2 1

2

1  







2 1

1

1  







 1 

2

2 2 1

1 y

y 

 





 ₂ 

1 1

1 1

1 y

y 

 





1  0



1  1



2  0

  ₂  1

CAPACITY REGION

Three Transmitters Within Carrier Sense Range

CAPACITY REGION

Three Transmitters Within Carrier Sense Range

1 y ₁

y ₂ 1

y ₃ 1

CAPACITY REGION

Three Transmitters Not Within Carrier Sense Range

2 1

3

CAPACITY REGION

Three Transmitters Not Within Carrier Sense Range

1 y ₁

y ₂

1 1

1 





2 2

1 



 1

1 1 1 2 1

2

1     











1 1 1 2 1

1 1 1

1     

















 

 1  ¹

1 1

2 2 1

2 2

2

 



 



 





y y

y y



 ₂ 

1 1

1 1

1 y

y 

 





Capacity lost due to the lack of

synchronization between nodes

CAPACITY REGION

Three Transmitters Not Within Carrier Sense Range

1 y ₁

y ₂

1

FEASIBILITY TEST

• Does the network support a given rate vector ?

• Normalized throughput depends only on average values

approximates the total transmission time as approximates the total time as

• Plug into the expression and check if

FEASIBILITY TEST

Feasibility of Input Rates

SIMULATION RESULTS

SIMULATION SCENARIO

MIT Roofnet Network: Single-Hop Flows

SIMULATION RESULTS

Single-Hop Flows (ρ = 1.00)

SIMULATION RESULTS

Single-Hop Flows (ρ = 0.50)

SIMULATION RESULTS

Single-Hop Flows (ρ = 0.25)

SIMULATION RESULTS

Single-Hop Flows (ρ = 0.01)

• Capacity of wireless CSMA/CA multihop networks poorly understood

• Theory able to model the network behavior

Buffer dynamics of unsaturated sources and multihop flows

Wireless CSMA/CA multihop networks are not erratic, but predictable

• System of nonlinear equations characterizes the network capacity

Agnostic to the distribution of network parameters, only averages relevant

• Knowledge of the underlying process governing CSMA/CA networks

Opens up new areas of research

Routing optimization and network provisioning

CONCLUSIONS

On the Capacity of

Wireless CSMA/CA Multihop Networks