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Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 291402, 6 pages
http://dx.doi.org/10.1155/2015/291402
Research Article

On Some Compound Random Variables Motivated by Bulk Queues

Maritime Faculty, University of Montenegro, Dobrota 36, 85330 Kotor, Montenegro

Received 3 June 2015; Revised 2 August 2015; Accepted 9 August 2015

Academic Editor: Hua Fan

Copyright © 2015 Romeo Meštrović. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the distribution of the number of customers that arrive in an arbitrary bulk arrival queue system. Under certain conditions on the distributions of the time of arrival of an arriving group () and its size () with respect to the considered bulk queue, we derive a general expression for the probability mass function of the random variable which expresses the number of customers that arrive in this bulk queue during any considered period . Notice that can be considered as a well-known compound random variable. Using this expression, without the use of generating function, we establish the expressions for probability mass function for some compound distributions concerning certain pairs of discrete random variables which play an important role in application of batch arrival queues which have a wide range of applications in different forms of transportation. In particular, we consider the cases when and/or are some of the following distributions: Poisson, shifted-Poisson, geometrical, or uniform random variable.