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On finite exponential moments for branching processes and busy periods for queues

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Marvin K. Nakayama ,
Perwez Shahabuddin  and
Karl Sigman
Marvin K. Nakayama
Affiliation:
Department of Computer Science, New Jersey Institute of Technology, Newark, NJ 07102, USA. Email address: marvin@njit.edu
Perwez Shahabuddin
Affiliation:
Department of Industrial Engineering and Operations Research, Columbia University, 500 West 120th Street, New York, NY 10027-6699, USA. Email address: psl47@columbia.edu
Karl Sigman
Affiliation:
Department of Industrial Engineering and Operations Research, Columbia University, 500 West 120th Street, New York, NY 10027-6699, USA. Email address: ks20@columbia.edu
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Abstract

Using a known fact that a Galton–Watson branching process can be represented as an embedded random walk, together with a result of Heyde (1964), we first derive finite exponential moment results for the total number of descendants of an individual. We use this basic and simple result to prove analogous results for the population size at time t and the total number of descendants by time t in an age-dependent branching process. This has applications in justifying the interchange of expectation and derivative operators in simulation-based derivative estimation for generalized semi-Markov processes. Next, using the result of Heyde (1964), we show that, in a stable GI/GI/1 queue, the length of a busy period and the number of customers served in a busy period have finite exponential moments if and only if the service time does.

Keywords

Branching processbusy perioddecouplingrandom walksingle-server queue

MSC classification

Primary: 60G10: Stationary processes 60G55: Point processes 60K30: Applications (congestion, allocation, storage, traffic, etc.) 60K25: Queueing theory
Type
Part 5. Properties of random variables
Information
Journal of Applied Probability , Volume 41 , Issue A: Stochastic Methods and their Applications , 2004 , pp. 273 - 280
Copyright
Copyright © Applied Probability Trust 2004 

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