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Limiting conditional and conditional invariant distributions for the Poisson process with negative drift

Part of: Special processes Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  14 July 2016

Raúl Fierro ,
Servet Martínez  and
Jaime San Martín
Raúl Fierro*
Affiliation:
Universidad Católica de Valparaíso
Servet Martínez*
Affiliation:
Universidad de Chile
Jaime San Martín*
Affiliation:
Universidad de Chile
*
∗Postal address: Departamento de Matemáticas, Universidad Católica de Valparaíso, Casilla 4059, Valparaíso, Chile.
∗∗Postal address: Departamento de Ingenierí a Matemática, Universidad de Chile, Casilla 170/3-Correo 3, Santiago, Chile.
∗∗Postal address: Departamento de Ingenierí a Matemática, Universidad de Chile, Casilla 170/3-Correo 3, Santiago, Chile.
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Abstract

In this paper we study the conditional limiting behaviour for the virtual waiting time process for the queue M/D/1. We describe the family of conditional invariant distributions which are continuous and parametrized by the eigenvalues λ ∊ (0, λc], as it happens for diffusions. In this case, there is a periodic dependence of the limiting conditional distributions on the initial point and the minimal conditional invariant distribution is a mixture, according to an exponential law, of the limiting conditional distributions.

Keywords

Yaglom limitquasi-stationary distributionsvirtual waiting time process

MSC classification

Primary: 60B10: Convergence of probability measures 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 4 , December 1999 , pp. 1194 - 1209
Copyright
Copyright © Applied Probability Trust 1999 

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