## Journal of Applied Probability

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

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

Marvin K. Nakayama, Perwez Shahabuddin, and Karl Sigman

#### 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 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

**Source**

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

**Dates**

First available in Project Euclid: 21 April 2004

**Permanent link to this document**

http://projecteuclid.org/euclid.jap/1082552204

**Digital Object Identifier**

doi:10.1239/jap/1082552204

**Mathematical Reviews number (MathSciNet)**

MR2057579

**Zentralblatt MATH identifier**

1056.60085

**Subjects**

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

**Keywords**

Branching process busy period decoupling random walk single-server queue

#### Citation

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. http://projecteuclid.org/euclid.jap/1082552204.