Optimal Throughput Allocation in General Random Access Networks P. Gupta, A. Stolyar Bell Labs, Murray Hill, NJ March 24, 2006

2 u Slotted Aloha is a classical random access model. Applies to the situations when all transmitters interfere with each other (“shared transmission medium”) u Slotted Aloha is relatively well studied, allows efficient control u 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) u Focus of this work is on –characterization of efficient - Pareto optimal - throughput allocations –dynamic distributed controls producing optimal throughputs, given a specific objective Motivation

3 Slotted Aloha Throughputs: 1N32 p1p1 p2p2 p3p3 pNpN 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

4 General (“node-centric”) random access model Transmission and interference graph: Throughput region: What is the Pareto boundary M* ? p1p1 p 12 p 21 p 23 p 32 p 43 p 34 p3p3 p2p2 p4p4 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

5 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 w nm =1

6 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*

7 Simple interpretation of Slotted Aloha throughput region Throughputs: 1N32 p1p1 p2p2 p3p3 pNpN From Th1: for any  n w n = 1, p=w solves max   w n log  n (p) Theorem (Massey-Mathys’85): Pareto (“north-east”) boundary M* of region M is given by

8 Dynamic throughput allocation: basic procedure The characterization of Pareto boundary suggests the following basic procedure: –Each node n »maintains a dynamic weight w nm for its outgoing links (nm) »maintains and periodically broadcasts its “incoming weight” W n in »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

9 Weighted proportional fairness s.t. minimum throughputs Problem: for some fixed positive weights Algorithm:

10 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 r nm in place of s nm (t) and r nm resp. (the alg. becomes a GPD alg., S’05) –For the stability of the queues (Th3), non-convexity of M is not a problem

11 Example 1 All  nm =1, so that we maximize  log  nm Two cases: –All r nm =0: no min rate constraints –r 5,9 =0.1 and the rest are 0 Parameter  =0.001

12 Example 1: steady-state throughputs

13 Example 1: link weight and access probability dynamics

14 Example 2 All  nm =1, so that we maximize  log  nm Two cases: –All r nm =0: no min rate constraints –R 2,1 =1/7 and the rest are 0 Parameter  =0.001

15 Example 2: steady-state throughputs

16 u 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 u This characterization can be exploited for efficient “greedy” dynamic throughput controls u Need more work on convergence properties of dynamic controls Conclusions