Celebratio Mathematica — Varadhan

Works authored by Varadhan

Works connected to B. Gopinath

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J. A. Gubner, B. Gopinath, and S. R. S. Varadhan: “Bounding functions of Markov processes and the shortest queue problem,” Adv. in Appl. Probab. 21 : 4 (1989), pp. 842–860. MR 1039631 Zbl 0707.60086 article

We prove a theorem which can be used to show 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 ‘joinging 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.

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