Queues with a variable number of servers

In modern communication networks such as the broadband integrated services digital network, it is possible to re-negotiate the bandwidth of a virtual channel (in queueing terminology the service rate of a server or the number of servers) during the course of a communication session. Many communication protocols have been proposed to take advantage of this flexibility. The basic principle of these protocols is to request more bandwidth when needed and release extra bandwidth when not used. Motivated by one such protocol, we study the steady-state behavior of an M/M/s queueing system in which the number of servers varies between a lower limit and an upper limit. New servers are hired when there are waiting customers in the queue and idle servers are released when the queue has been empty for some time. Using the matrix-analytical technique, we derive formulas in matrix-geometric form for the steady-state probabilities, and formulas for standard performance measures. One interesting feature of our solution is that the rate matrix R can be obtained explicitly without the need for iteration as in many other cases. We also study the optimization problem in which the server utilization is to be maximized subject to the performance constraint that the mean queueing delay does not exceed a pre-determined target. Finally, we compare the optimal utilization with that of the classical M/M/s system in which the number of servers is fixed.