Abstract
The transient behavior of a single-server queuing system, e.g., a computer, having two classes of customers, of high and low priority, is difficult. When the customer service times are Poisson and exponentially distributed, one is coping with a random walk in two dimensions of known intractability. The server busy period distribution for the M/G/1 queue is of related interest. Even though this distribution is known explicitly in real time for arbitrary service-time distribution, the form of the answer as an infinite series involving convolutions provides limited insight. For the two Poisson streams with exponential service times, the effective service-time distribution of an equivalent single stream is completely monotone. It will be shown that the busy-period density of the server is itself completely monotone, and that this and related results provided improved understanding of the system behavior. Some interest is focused on the spectral distribution function of the completely monotone density as a working tool.
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