Journal of Applied Probability

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

Marvin K. Nakayama, Perwez Shahabuddin, and Karl Sigman

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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 descendents 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 descendents 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.

Article information

J. Appl. Probab. Volume 41A, Issue (2004), 273-280.

First available in Project Euclid: 21 April 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G10: Stationary processes 60G55: Point processes 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx] 60K25: Queueing theory [See also 68M20, 90B22]

Branching process busy period decoupling random walk single-server queue


Nakayama, Marvin K.; Shahabuddin, Perwez; Sigman, Karl. On finite exponential moments for branching processes and busy periods for queues. J. Appl. Probab. 41A (2004), 273--280. doi:10.1239/jap/1082552204.

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