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Pair formation in a Markovian arrival process with two event labels

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

Published online by Cambridge University Press:  14 July 2016

Marcel F. Neuts  and
Attahiru Sule Alfa
Marcel F. Neuts*
Affiliation:
University of Arizona
Attahiru Sule Alfa*
Affiliation:
University of Manitoba
*
∗ Postal address: Department of Systems and Industrial Engineering, University of Arizona, Tucson, AZ 85721, USA. Email address: marcel@mindspring.com
∗∗ Postal address: Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, Manitoba R3T 5V6, Canada. Email address: alfa@ee.umanitoba.ca
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Abstract

The stochastic process resulting when pairs of events are formed from two point processes is a rich source of questions. When the two point processes have different rates, the resulting stochastic process has a mean drift towards either -∞ or +∞. However, when the two processes have equal rates, we end up with a null-recurrent Markov chain and this has interesting behavior. We study this process for both discrete and continuous times and consider special cases with applications in communications networks. One interesting result for applications is the waiting time of a packet waiting for a token, a special case of this pair-formation process. Pair formation by two independent Poisson processes of equal rates results in a point process that is asymptotically a Poisson process of the same rate.

Keywords

Point processpair formationMAPalgorithmic probabilitynull recurrence

MSC classification

Primary: 60J99: None of the above, but in this section
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J27: Continuous-time Markov processes on discrete state spaces 65C40: Computational Markov chains
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 4 , December 2004 , pp. 1124 - 1137
Copyright
Copyright © Applied Probability Trust 2004 

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