Department of Applied Mathematics & Physics, Kyoto Univiversity

Technical Report #98011 (July, 1998)

Queue length distribution in a FIFO single-server queue with multiple arrival streams having different service time distributions
by Tetsuya Takine

This paper considers the queue length distribution in a class of FIFO
single-server queues with (possibly correlated) multiple arrival
streams, where the service time distribution of customers from each
arrival stream may differ from one another among streams. It is widely
recognized that the queue length distribution in a FIFO queue with
multiple non-Poissonian arrival streams having different service time
distributions is very hard to analyze, since we have to keep track of
the complete order of customers in the queue to describe the queue
length dynamics. In this paper, we provide an alternative way to solve
the problem for a class of such queues. We characterize the stationary
joint queue length distribution in terms of the joint probability
generating function, by considering the joint distribution of the
number of customers arriving from each stream during the stationary
attained waiting time. Further we provide recursion formulas to
compute the stationary joint queue length distribution, the stationary
distribution of the total number of customers and the marginal
distribution of the number of customers in each stream.