State-Space Collapse via Drift Conditions Atilla Eryilmaz (OSU) and R. Srikant (Illinois) 4/8/2015 1 Goal 4/8/2015 2 Motivation • Parallel servers • Jobs are buffered at a single queue • When a server becomes idle, it grabs the first job from the queue to serve • All servers are fully utilized whenever possible 4/8/2015 3 Multiple queues • Jobs arrive and choose to join the shortest queue upon arrival • Total queue length is the same as in the case of a single queue if jobs “defect” to a different queue whenever one becomes empty 4/8/2015 4 Multi-Path Routing • Choice of paths from source to destination: route each packet on currently least-congested path • JSQ is an abstraction of such routing scheme. It is not possible for packets to defect from one path to another. • Is JSQ still optimal in the sense of minimizing queue lengths? 4/8/2015 5 Heavy-Traffic Regime • Consider the traffic regime where the arrival rate approaches the system capacity 4/8/2015 6 Foschini and Gans (1978) 4/8/2015 7 Steady-State Result for JSQ 4/8/2015 8 Lower-Bounding Queue 4/8/2015 9 The Lower Bound 4/8/2015 10 State-Space Collapse (1,1) q q 4/8/2015 11 A Useful Property of JSQ 4/8/2015 12 Drift Conditions and Moments 4/8/2015 13 Moments & State-Space Collapse 4/8/2015 14 The Upper Bound 4/8/2015 15 Using State-Space Collapse 4/8/2015 16 Handling Cross Terms Theorem 4/8/2015 19 Three-Step Procedure 4/8/2015 20 Wireless Networks 4/8/2015 21 Example • Two links, four feasible rates: (0,2), (1,2), (3,1), (3,0) Capacity Region: Set of average service rates (0,2) (1,2) (3,1) (3,0) 4/8/2015 22 MaxWeight (MW) Algorithm Capacity Region: Set of average service rates (0,2) (1,2) (3,1) Arrival rates can be anywhere in the capacity region; MW stabilizes queues (3,0) 4/8/2015 23 Lower Bound Capacity Region: Set of average service rates (0,2) (1,2) (3,1) Arrival rates can be anywhere in the capacity region; MW stabilizes queues (3,0) 4/8/2015 24 Heavy-Traffic Regime Capacity Region: Set of average service rates (0,2) (1,2) . (3,1) Arrival rates can be anywhere in the capacity region; MW stabilizes queues (3,0) 4/8/2015 25 State-Space Collapse c q q 4/8/2015 26 Upper Bound 4/8/2015 27 Theorem 4/8/2015 28 Implications c q q 4/8/2015 29 Use Beyond Heavy-Traffic Regime • Each face of the capacity region provides an upper and lower bound • Treat these as constraints • From this the best upper and lower bounds can be obtained o Similar to Bertsimas, Paschalidis and Tsitsiklis (1995), Kumar and Kumar (1995), Shah and Wischik (2008) 4/8/2015 30 Stability and Performance • Stability of control policies can be shown by considering the drift of a Lyapunov function • Setting this drift equal to zero gives bounds on queue lengths in steady-state • But these are not tight in heavy-traffic • The moment-based interpretation of state-space collapse and the upper bounding ideas to use this information provide tight upper bounds 4/8/2015 31 Conclusions • An approach to state-space collapse using exponential bounds based on drift conditions • A technique to use to these bounds in obtaining tight upper bounds • Demonstrated for o JSQ o MaxWeight o MaxWeight with fading is an easy extension 4/8/2015 32