On Some Compound Random Variables Motivated by Bulk Queues

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.