# State-Space Collapse via Drift Conditions Atilla Eryilmaz (OSU) and R. Srikant (Illinois) 4/10/20151.

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State-Space Collapse via Drift Conditions Atilla Eryilmaz (OSU) and R. Srikant (Illinois) 4/10/20151

Goal 4/10/20152

Motivation 3 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

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 4/10/20154

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? 4/10/20155

Heavy-Traffic Regime Consider the traffic regime where the arrival rate approaches the system capacity 4/10/20156

Foschini and Gans (1978) 4/10/20157

Lower-Bounding Queue 4/10/20159

The Lower Bound 4/10/201510

State-Space Collapse 4/10/201511 (1,1) q qq

A Useful Property of JSQ 4/10/201512

Drift Conditions and Moments 4/10/201513

Moments & State-Space Collapse 4/10/201514

The Upper Bound 4/10/201515

Using State-Space Collapse 4/10/201516

Handling Cross Terms

A Useful Identity 4/10/201518

Theorem 4/10/201519

Three-Step Procedure 4/10/201520

Wireless Networks 4/10/201521

Example Two links, four feasible rates: (0,2), (1,2), (3,1), (3,0) 4/10/201522 (0,2) (1,2) (3,1) (3,0) Capacity Region: Set of average service rates

MaxWeight (MW) Algorithm 4/10/201523 (0,2) (1,2) (3,1) (3,0) Capacity Region: Set of average service rates Arrival rates can be anywhere in the capacity region; MW stabilizes queues

Lower Bound 4/10/201524 (0,2) (1,2) (3,1) (3,0) Capacity Region: Set of average service rates Arrival rates can be anywhere in the capacity region; MW stabilizes queues

Heavy-Traffic Regime 4/10/201525 (0,2) (1,2) (3,1) (3,0) Capacity Region: Set of average service rates Arrival rates can be anywhere in the capacity region; MW stabilizes queues.

State-Space Collapse 4/10/201526 c q qq

Upper Bound 4/10/201527

Theorem 4/10/201528

Implications 4/10/201529 c q qq

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) 4/10/201530

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 4/10/201531

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 4/10/201532

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