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Bounding functions of Markov processes and the shortest queue problem

Published online by Cambridge University Press:  01 July 2016

John A. Gubner*
University of Maryland
B. Gopinath*
Bell Communications Research
S. R. S. Varadhan*
Courant Institute of Mathematical Sciences
Present address: Department of Electrical and Computer Engineering, University of Wisconsin, Madison, WI 53706, USA.
∗∗Present address: Department of Electrical Engineering, Rutgers University, New Brunswick, NJ 08903, USA.
∗∗∗Postal address: Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA.


We prove a theorem which can be used to show that the expectation of a non-negative function of the state of a time-homogeneous Markov process is uniformly bounded in time. This is reminiscent of the classical theory of non-negative supermartingales, except that our analog of the supermartingale inequality need not hold almost surely. Consequently, the theorem is suitable for establishing the stability of systems that evolve in a stabilizing mode in most states, though from certain states they may jump to a less stable state. We use this theorem to show that ‘joining the shortest queue' can bound the expected sum of the squares of the differences between all pairs among N queues, even under arbitrarily heavy traffic.

Research Article
Copyright © Applied Probability Trust 1989 

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This work was supported by Bell Communications Research. The research of the first author was also supported in part by the University of Maryland Systems Research Center under NSF Grant OIR-85-00108.


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