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Regenerative derivatives of regenerative sequences

Part of: Special processes Probabilistic methods, simulation and stochastic differential equations Markov processes

Published online by Cambridge University Press:  01 July 2016

Paul Glasserman
Paul Glasserman*
Affiliation:
Columbia University
*
∗Postal address: 403 Uris Hall, Columbia Business School, New York, NY 10027, USA. E-mail address: pglasser@research.gsb.columbia.edu.
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Abstract

Given a parametric family of regenerative processes on a common probability space, we investigate when the derivatives (with respect to the parameter) are regenerative. We primarily consider sequences satisfying explicit, Lipschitz recursions, such as the waiting times in many queueing systems, and show that derivatives regenerate together with the original sequence under reasonable monotonicity or continuity assumptions. The inputs to our recursions are i.i.d. or, more generally, governed by a Harris-ergodic Markov chain. For i.i.d. input we identify explicit regeneration points; otherwise, we use coupling arguments. We give conditions for the expected steady-state derivative to be the derivative of the steady-state mean of the original sequence. Under these conditions, the derivative of the steady-state mean has a cycle-formula representation.

Keywords

REGENERATIVE PROCESSDERIVATIVE ESTIMATIONPERTURBATION ANALYSISQUEUESHARRIS CHAINSCOUPLING

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60J05: Discrete-time Markov processes on general state spaces 65C99: None of the above, but in this section
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 1 , March 1993 , pp. 116 - 139
Copyright
Copyright © Applied Probability Trust 1993 

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