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A distributed CSMA Algorithm for Throughput and Utility Maximization in Wireless Networks

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### A distributed CSMA Algorithm for Throughput and Utility Maximization in Wireless Networks

Libin Jiang and Jean Walrand

Presented by

Ruogu Li

Department of Electrical and Computer Engineering

The Ohio State University

Contents

- Overview
- Model and Assumptions
- Adaptive CSMA for maximum throughput
- Joint scheduling and rate control
- Formulation and algorithm
- Maximizing utility
- Simulations
- Implementation considerations
- Conclusion

Overview

- In this paper, the authors studied the scheduling problem for a wireless network to achieve maximum throughput.
- Under some assumptions, they introduced a distributed adaptive Carrier Sensing Multiple Access (CSMA) algorithm for a general interference model, which
- Can achieve maximum throughput;
- Require local information only, no control message exchange is required;
- No synchronization of transmissions is needed.

Prior works

- It is well known that the maximum weight scheduling is throughput optimal.
- But it requires global information of the network, and even in centralized case it is NP-complete.
- Some low complexity throughput algorithms for particular interference models.
- But the extension to more general interference models usually involves extra challenges.
- Some distributed greedy protocols only achieve a fraction of the throughput region.

Ref: L. Tassiulas and A. Ephremids, “Stability properites of constrained queueing systems and scheduling policies for maximum throughput in multihop radio networks,” IEEE trans. Autom. Control, vol. 37, no.12, Dec 1992

A. Eryilmaz, A. Ozdaglar and E. Modiano, .Polynomial Complexity Algorithms for Full Utilization of Multi-hop Wireless Networks,. Proceedings of IEEE Infocom, 2007

X. Wu, R. Srikant, .Bounds on the Capacity Region of Multi-Hop Wireless Networks under Distributed Greedy Scheduling,. IEEE INFOCOM, 2006

Model and Assumptions

- General interference model represents by a link contention graph .
- Vertices in the graph represent links;
- Edges between vertices represent interference, i.e. if there is an edge between two links, then they cannot transmit at the same time.
- Independent set (IS) of links: links that can transmit together without interfering with each other.

1

1

Link (1,1)

Link (1,2)

2

2

Link (2,1)

Link (2,2)

Model and Assumptions

- links in the network.
- independent sets, denoted by a 0-1 vector

if link is in this set.

- Independent sets are also referred as transmission states.

Model and Assumptions

- The authors used an idealized model of CSMA
- If two links conflict, then either of the two links hears when the other one transmits.
- This implies that there are no hidden nodes in this setup. It is feasible if the range of carrier-sensing is large enough.
- Carrier sensing is instantaneous.
- Violated in actual systems because the finite speed of light.

Model and Assumptions

- The idealized CSMA protocol
- If the transmitter of a link senses the transmission of any conflicting link, it remains silent. Otherwise, it waits (or backs-off) for an exponentially distributed time period with mean .
- During the back-off time, if a conflicting link start to transmit, then link suspends its back-off and resume it after the conflicting transmit is over.
- Transmission time is exponential distributed with mean 1.

Link 1

Link 2

t

Model and Assumptions

- The idealized CSMA policy
- Assume sensing time is negligible.
- Given the continuous distribution of the back-off times, the probability of two conflicting links to start transmission is zero.
- Together with the no hidden nodes assumption.
- The collisions of general CSMA are ignored in this model.
- The transitions of the transmission states form a Continuous Time Markov Chain.

Model and Assumptions

- The CSMA Markov Chain
- State can transit to iff all conflicting links of link is not in the state , where is a 0-1 vector with element equals to 1.

{1,0,0,0}

{1,0,0,1}

{0,1,0,0}

{0,0,0,0}

{0,0,0,1}

Link 1

Link 2

{0,1,1,0}

{0,0,1,0}

Link 3

Link 4

Model and Assumptions

- Stationary distribution of CSMA MC
- Define the “transmission aggressiveness” , then for a given vector , the stationary distribution is given by

where

- For simplicity, denote .

x1={1,0}

x0={0,0}

Link 1

Link 2

x2={0,1}

Adaptive CSMA for maximum throughput

- Capacity region of a wireless network
- Assume i.i.d. traffic arrival at each link with a rate .
- is feasible iff there exists a probability distribution such that

is strictly feasible iff it is in the interior of the capacity region.

Link 1

Link 2

Adaptive CSMA for maximum throughput

- Idea: to make the probability of each transmission state “close” to .
- Recall the Kullback-Leibler (KL) distance between two distributions and is given by

or equivalently,

and we have iff the two distributions are the same.

Adaptive CSMA for maximum throughput

- Thus, define

Maximizing is equivalent to minimizing the KL distance.

- Proposition 1: If is attainable, then

where is the service rate on link .

- Proposition 2: If the arrival rate is strictly feasible, then

is attainable.

Adaptive CSMA for maximum throughput

- Use a simple gradient algorithm to solve this optimization problem.
- Algorithm 1: Adjust the transmission aggressiveness

This update happens every time units, i.e. increases by 1 every time units, and and are the average arrival and service rate during this time period.

- Note that , thus starting from zero state, we have

Adaptive CSMA for maximum throughput

- There is an assumption of time separation
- Either is very large that the CSMA Markov Chain converges between updates;
- Or is very small that the stationary distribution changes very slowly.
- This assumption is sufficient to show the convergence of the algorithm.
- In practice, simulations show that these assumptions are not necessary.
- This algorithm achieves throughput optimal at a link level, using only local information.

Joint Scheduling and Rate Control

- Dual problem of
- Directly use the expression of the service rate in a joint scheduling and flow control problem may result non-convexity.
- Rewrite as
- Associate a dual variable to the first constraint, and we can find the dual problem of this optimization problem as

Joint Scheduling and Rate Control

- The dual variable can be interpreted as the probability of transmission state .
- Proposition 3: Given some (finite) TA’s of the links, the stationary distribution of the CSMA Markov Chain maximize the partial Lagrangian of

over all possible distribution . And Algorithm 1 can be viewed as a subgradient algorithm to update .

Joint Scheduling and Rate Control

- Problem formulation
- There are flows in the networks with given routes.
- if flow use link .
- Each flow has a utility function associate to it.
- Denote the first link of the flow by , upstream link of link in flow by and downstream link by

.

Joint Scheduling and Rate Control

- Then the joint scheduling and rate control problem is
- Associate dual variable to the 2nd an 3rdconstraints to solve this problem.

Joint Scheduling and Rate Control

- Algorithm 2: Joint scheduling and rate control
- Link transmits the head packet from a flow with the maximum back-pressure when it gets the opportunity to transmit.
- Link let in the CSMA operation.
- For each source link of a flow, let

This maximizes over .

- The dual variables are updated by a sub-gradient algorithm:

by doing this, is proportional to the queue length

Joint Scheduling and Rate Control

- Approaching the maximum utility
- Modify the optimization slightly

subject to the same constraint.

- Proposition 4: The difference between the total utility resulting from solving the above optimization problem and the maximum total utility is bounded. The bound of the difference decreases with the increase of .
- The authors also introduced an enhanced version of Algorithm 2 in another version of the paper, and provided convergence analysis of the algorithm in a following up work.

Simulations

- Simulation on the joint algorithm

By solving the optimization problem analytically, the optional flow rate will be .

1

Flow 3

Flow 1

Flow 2

Implementation Considerations

- Packet collisions
- In the idealized CSMA model, the back-off time is continuous thus there is no collisions.
- However in practice, back-off time is discrete, and the duration of timeslot cannot be made arbitrarily small.
- Prior work shows that upper-bounding can reduce the probability of collisions to a reasonable level.

Implementation Considerations

- The authors introduced two ways to implement this in their algorithm
- The first one is directly impose an upper bound on , resulting the Algorithm 3 in the paper.
- The second one achieves the upper bounding on by carefully selecting the weighting factor . This is the Algorithm 4 in the paper.
- They also discussed case when are discrete values.

Conclusion

- In this paper, the authors proposed a novel CSMA based scheduling algorithm that uses local information to achieve link level optimal throughput.
- This result can be combined with other rate control mechanism forming a joint scheduling and rate control algorithm. Shown in the paper is a fixed route scenario, and the author also claimed that this is applicable in other scenarios such as anycast and multicast.

Future Work

- The convergence analysis of the joint algorithm.
- The authors have a follow up work in which they prove the convergence result for the enhanced Algorithm 2.
- Relaxing the time separation assumption.
- In the simulations the authors observed that the stationary distribution of the CSMA Markov Chain is not necessary.
- Collision freeness assumption.
- One possible direction is how to relax this assumption.
- On the other hand, there are already works on achieving collision freeness using some distributed algorithm in a time slotted system.

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