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Convexity in queues with general inputs

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2 Author(s)
M. J. Neely ; Dept. of Electr. Eng., Univ. of Southern California, Los Angeles, CA, USA ; E. Modiano

In this correspondence, we develop fundamental convexity properties of unfinished work and packet waiting time in a queue serving general stochastic traffic. The queue input consists of an uncontrollable background process and a rate-controllable input stream. We show that any moment of unfinished work is a convex function of the controllable input rate. The convexity properties are then extended to address the problem of optimally routing arbitrary input streams over a collection of K queues in parallel with different (possibly time-varying) server rates (μ1(t),...,μK(t)). Our convexity results hold for stream-based routing (where individual packet streams must be routed to the same queue) as well as for packet-based routing where each packet is routed to a queue by probabilistic splitting. Our analysis uses a novel technique that combines sample path observations with stochastic equivalence relationships.

Published in:

IEEE Transactions on Information Theory  (Volume:51 ,  Issue: 2 )