State-Space Collapse via Drift Conditions Atilla Eryilmaz (OSU)
and R. Srikant (Illinois) 4/10/20151 Slide 2 Goal 4/10/20152 Slide
3 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 Slide 4
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 Slide 5 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 Slide 6 Heavy-Traffic Regime Consider the traffic regime
where the arrival rate approaches the system capacity 4/10/20156
Slide 7 Foschini and Gans (1978) 4/10/20157 Slide 8 Steady-State
Result for JSQ 4/10/20158 Slide 9 Lower-Bounding Queue 4/10/20159
Slide 10 The Lower Bound 4/10/201510 Slide 11 State-Space Collapse
4/10/201511 (1,1) q qq Slide 12 A Useful Property of JSQ
4/10/201512 Slide 13 Drift Conditions and Moments 4/10/201513 Slide
14 Moments & State-Space Collapse 4/10/201514 Slide 15 The
Upper Bound 4/10/201515 Slide 16 Using State-Space Collapse
4/10/201516 Slide 17 Handling Cross Terms Slide 18 A Useful
Identity 4/10/201518 Slide 19 Theorem 4/10/201519 Slide 20
Three-Step Procedure 4/10/201520 Slide 21 Wireless Networks
4/10/201521 Slide 22 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 Slide 23 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 Slide 24 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 Slide 25 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. Slide 26 State-Space Collapse 4/10/201526 c q qq
Slide 27 Upper Bound 4/10/201527 Slide 28 Theorem 4/10/201528 Slide
29 Implications 4/10/201529 c q qq Slide 30 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 Slide 31 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 Slide 32 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
