Queues with Mixed Renewal and Poisson Inputs

The concept of a piecewise Markov process 1 is used to analyze two queueing systems the inputs of which are composed of two independent streams. One of the streams is Poissonian with intensity X and the other (called a GI stream because of its General Independent Distribution of intervals between arrivals).is assumed to be a renewal process with intensity v. We assume the service times of all the customers are independent and identically distributed according to an exponential distribution with mean n~ Such models are denoted by GI + M / M / c in Kendall's notation, where the " M " refers to the Markovian character 1305 1306 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1972 of Poissonian arrivals and of exponential service and c refers to the number of servers. The state of the system (number of customers waiting and in service) seen by an arriving Poissonian customer will generally differ from t h a t of the GI customer. Consequently, in a system where delayed customers wait for service such as in the GI + M / M / l queue, the service received by the two types of customers will differ. Whether the GI customers receive better or worse service than the Poissonian customers depends on the variability of the interarrival times of the GI stream. In Section II, we analyze the GI + M / M / l queue. The delay distribution with order-of-arrival service for the two types of customers is derived for GI streams the interarrival time distributions of which have rational Laplace-Stieltjes transforms.