The Stationary Solution of a Multiserver Queuing Model with Discouragement

Several authors, including Morse, Ancker and Gafarian, and Cox and Smith have studied single-server queuing models in which the arrival rate is a decreasing function of queue length. This paper analyzes a multi-server generalization of the model given by Cox and Smith. A generating function is derived for the equilibrium distribution of queue length, and from it are derived expressions for the probability of finding all servers busy on arrival, and for the mean number of customers in the system and waiting for service to begin. Also, an expression is obtained for the proportional loss in custom caused by discouragement. Finally, the distribution of waiting time is obtained using characteristic functions and the mean waiting time is calculated. The variations of these quantities with traffic intensity and number of servers are illustrated graphically.