State-Space Collapse via Drift
Conditions
Atilla Eryilmaz (OSU) and R. Srikant (Illinois)
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Goal
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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
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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
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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?
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Heavy-Traffic Regime
• Consider the traffic regime where the arrival rate
approaches the system capacity
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Foschini and Gans (1978)
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Steady-State Result for JSQ
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Lower-Bounding Queue
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The Lower Bound
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State-Space Collapse
(1,1)
q
q
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A Useful Property of JSQ
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Drift Conditions and Moments
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Moments & State-Space Collapse
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The Upper Bound
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Using State-Space Collapse
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Handling Cross Terms
Theorem
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Three-Step Procedure
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Wireless Networks
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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)
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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)
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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)
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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)
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State-Space Collapse
c
q
q
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Upper Bound
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Theorem
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Implications
c
q
q
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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)
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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
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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
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