Two different kinds of heavy-traffic limit theorems have been proved for s-server queues. The first kind involves a sequence of queueing systems having a fixed number of servers with an associated sequence of traffic intensities that converges to the critical value of one from below. The second kind, which is often not thought of as heavy traffic, involves a sequence of queueing systems in which the associated sequences of arrival rates and numbers of servers go to infinity while the service time distributions and the traffic intensities remain fixed, with the traffic intensities being less than the critical value of one. In each case the sequence of random variables depicting the steady-state number of customers waiting or being served diverges to infinity but converges to a nondegenerate limit after appropriate normalization. However, in an important respect neither procedure adequately represents a typical queueing system in practice because in the (heavy-traffic) limit an arriving customer is either almost certain to be delayed (first procedure) or almost certain not to be delayed (second procedure). Hence, we consider a sequence of (GI/M/S) systems in which the traffic intensities converge to one from below, the arrival rates and the numbers of servers go to infinity, but the steady-state probabilities that all servers are busy are held fixed. The limits in this case are hybrids of the limits in the other two cases. Numerical comparisons indicate that the resulting approximation is better than the earlier ones for many-server systems operating at typically encountered loads.

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In this paper a queueing model is developed for a system serving two types of customers. Primary customers with a negative exponential service distribution arrive randomly to a group of servers. When all servers are busy with other primary customers new arrivals are assumed to leave without service. Secondary customers with a general distribution of service times can be served only by servers which are not occupied by a primary customer and join a queue if all servers are busy. Primary customers have preemptive priority and are always served ahead of secondary customers. The buffer containing the service required by the delayed secondary customers empties at a rate proportional to the number of servers not occupied by primary customers. Recursive formulas are derived for the distribution and the moments of the content of the buffer in statistical equilibrium. Numerical examples are presented.

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