Journal of Engineering Mathematics, Vol. 7, No. 2, April 1973 Noordhoff International Publishing-Leyden Printed in The Netherlands Queue-dependent servers V. P. SINGH IBM Components Division, East Fishkill, New York 12533, USA (Received April 4, 1972) 123 SUMMARY A Markovian queue with number of servers depending upon queue length is discflssed. Whenever the queue in front of the first server reaches a certain length, the system starts another server. There are costs associated with the opening of a new server and the waiting of the customers. A relationship among the costs, traffic intensity and the queue size is obtained. 1. Introduction In many situations when there are too many people waiting to be served in front of a service facility, the system opens another service facility to reduce congestion. For example, this happens in the banks and at the checkout counters in the department and grocery stores over the weekends. In this paper we study a Markovian queue in such a way that a new service facility is provided by the system whenever the queue in front of a server reaches a certain length. The different servers may either have the same or different service rates. A new service facility is started at some cost to the system. There is also a cost associated with the difference in the average number of customers in a single server system and the new system. For the case of two homogeneous servers, a relationship is developed among the costs, the traffic intensity p and the maximum allowable queue size N in front of the first server. For different values of N and p, the ratio of the costs is given in a table. This situation is then discussed for the case of three homogeneous servers. Finally the case of two heterogeneous servers is discussed. 2. The queue M]M[2 with number of servers depending on queue length Customers arrive at a single service counter following an orderly, stationary Poisson stream without after effects with parameter 2. Whenever there are N customers In the queue, the service system starts another server to reduce congestion. It costs c2 dollars to the system to provide the second server. The service time distribution for each server is negative exponential with parameter/~. Let P, denote the steady-state probability that there are n customers in the system at any time. Then the balance equations for the above system take the following form: 2Po =/~P1 , (2+g)P. =2P,_l+#P.+I, l<=n<N, (2+p)PN=J.PN_I+2pPN+a, n = N , ()~+2#)P. = 2 P . 1+2/~P.+1, n>N. The solution of the above system of equations is Po- (1-p)(2-p) where p 2 2_p_pN+l , # Journal of Engineering Math., Vol. 7 (1973) 123-126