An Approximate Method for Calculation Delays for a Family of Cyclic-Type Queues

The queuing model described in this paper resulted from a study of the marker-register dial-tone delay problem in No. 5 crossbar switching machines. A number of queues with Poisson arrivals of equal rates are served in a cyclic order by a server with constant service time. Upon arriving at a nonempty queue, the server chooses a customer from the queue at random. After one service time, the customer either leaves the system with a certain predetermined probability or rejoins his queue. In both cases, the server uses a fixed amount of time and moves to the next queue; thus, at most one customer leaves the system following each arrival of the server at a queue. Related models were treated by Cooper, 1 Cooper and Murray, 2 and Eisenberg. 3 In Refs. 1 and 2, the server either empties the queue being served or serves all those present at the queue in its arrival epoch. The case of two queues with different arrival rates is treated in Ref. 3. In these papers, the Laplace-Stieltjes transforms of the waiting time distributions were obtained. Attempts to obtain the distributions for 1733