Optimal Throughput Allocation in General Random Access Networks

P. Gupta, A. Stolyar Bell Labs, Murray Hill, NJ

March 24, 2006

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Slotted Aloha is a classical random access model. Applies to the situations when all transmitters interfere with each other (“shared transmission medium”)

Slotted Aloha is relatively well studied, allows efficient control

We want to study more general models, where

– not all transmitters interfere with each other (ad-hoc nets, etc.)

– or cause different levels of interference (not in this talk)

Focus of this work is on

– characterization of efficient - Pareto optimal - throughput allocations

– dynamic distributed controls producing optimal throughputs, given a specific objective

Motivation

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Slotted Aloha

Throughputs:

1 N32

p1 p2 p3 pN

Access probabilities in a slot

Throughput region:

Theorem (Massey-Mathys’85):Pareto (“north-east”) boundary M* of region M is

given by

- Example: the result provides guiding principle for

WLAN RC-MAC in Gupta-Sankarasubramaniam-S’05

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General (“node-centric”) random access model

Transmission and interference graph:

Throughput region:

What is the Pareto boundary M* ?

p1

p12

p21

p23

p32

p43

p34

p3

p2

p4

Throughputs:

- this model is a generalization of that in Kar-Sarkar-Tassiulas’04

- a different - “link-centric” - model with “random interference” is in Gupta-S’05

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Auxiliary problem: max weighted proportional fairness objective

Relatively easily solvable, because

Problem: for some fixed positive weights

Unique optimizer

total weight of all “incoming links” to node mnode n interferes with these nodes

- generalization of Kar-Sarkar-Tassiulas’04 where wnm=1

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Pareto boundary characterization

Question: If we vary weights w, do vectors (p(w)) “fill” the entire set M* ?

For any set of weights w and the corresponding optimizer p(w), throughput vector (p(w)) is on the Pareto boundary M*

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Simple interpretation of Slotted Aloha throughput region

Throughputs:

1 N32

p1 p2 p3 pN

From Th1: for any n wn = 1, p=w solves

max wn log n (p)

Theorem (Massey-Mathys’85):Pareto (“north-east”) boundary M* of region M is

given by

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Dynamic throughput allocation: basic procedure

The characterization of Pareto boundary suggests the following basic procedure:

– Each node n

» maintains a dynamic weight wnm for its outgoing links (nm)

» maintains and periodically broadcasts its “incoming weight” Wnin

» calculates (or estimates) the sum of incoming weights of the nodes it interferes with

» sets its access probabilities according to the above formula

– Node n dynamically adjusts weights of its links, based on their “satisfaction” with the current throughput

– As different nodes vary their link weights, the throughputs vary, but stay on the Pareto boundary

total weight of all incoming links to node mnode n interferes with these nodes

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Weighted proportional fairness s.t. minimum throughputs

Problem: for some fixed positive weights

Algorithm:

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Fluid limit dynamics

– To prove Th2, convexity of log M is good enough

– For a proof of convergence, non-convexity of M is a problem

– If weights are updated on slower time scale, convergence is provable for the algorithm using log nm(t) and log rnm in place of snm(t) and rnm resp. (the alg. becomes a GPD alg., S’05)

– For the stability of the queues (Th3), non-convexity of M is not a problem

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Example 1

All nm=1, so that we maximize log nm

Two cases:

– All rnm=0: no min rate constraints

– r5,9=0.1 and the rest are 0

Parameter =0.001

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Example 1: steady-state throughputs

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Example 1: link weight and access probability dynamics

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Example 2

All nm=1, so that we maximize log nm

Two cases:

– All rnm=0: no min rate constraints

– R2,1=1/7 and the rest are 0

Parameter =0.001

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Example 2: steady-state throughputs

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For generalized (“node-centric”) Slotted Aloha model, we characterized Pareto boundary of the throughput region as a set of solutions to weighted prop. fairness problem

This characterization can be exploited for efficient “greedy” dynamic throughput controls

Need more work on convergence properties of dynamic controls

Conclusions