Consider a queueing system with waiting in which of the total number of M servers only r (r < M) servers are permanently on duty, while the remaining M -r servers are off duty. The off-duty servers are called in only when the queue reaches the length N. More precisely, we assume that if a new customer arrives when there are N enqueued customers in the system and at least one off-duty server is available, it is put on duty.If more than r servers are busy when the service of this customer is completed and the number of waiting customers is k (O~N--2), the number of servers on duty is decremented by i. The number of servers on duty thus varies from a minimum of r to a maximum of M. Particular cases of such queueing models with k = 0 were considered in [I, 2].Let v(t) be the number of enqueued customers, x(t) the number of busy servers at the moment t. In this system, we clearly have