Delays for Customers from Different Arrival Streams to a Queue

In a queue with several different arrival streams, in general, the expected delay for customers from one stream is not equal to the expected delay for customers from the other streams. Two approximations are presented here for the expected delay for customers from a particular arrival stream in an arrival process that is the superposition of independent renewal processes. Both approximations yield errors less than 10 percent, on average, when compared to simulation estimates. One approximation, extended from Holtzman (Holtzman, J. M. 1982. Mean delays of individual streams into a queue: the \Sigma GI_i/M/1 queue. Applied Probability-Computer Science: The Interface, I, Proc. Conf. in Boca Raton, 417--430.), yields better results for very sparse arrival streams and the other, empirically derived, is easier to calculate. These approximations are useful in comparing the expected delays for customers from different arrival processes to a single queue and for customers with different routes through a network of queues.