A single server tandem queue

Avi-Itzhak, Maxwell and Miller (1965) studied a queueing model with a single server serving two service units with alternating priority. Their model explored the possibility of having the alternating priority model treated in this paper with a single server serving alternately between two service units in tandem. Here we study the distribution of busy period, virtual waiting time and queue length and their limiting behavior. 1. Concepts and definitions We consider a queueing process with two service units, unit 1 and unit 2, and a single server. The server attends to the two units alternately according to some switching rule. A switching rule (Neuts and Yadin (1968)) is a rule describing how the server changes from one unit to the other. The server may change from one unit to the other either by a non-zero switching rule or by a zero switching rule. By a non-zero switching rule the server continues to serve in a unit until some upper number of consecutive services has been completed and then he switches to the other unit. By a zero switching rule the server stays in a unit until the queue in it becomes empty and then he switches to the other unit. In this paper we discuss a non-zero switching rule for unit 1 and zero switching rule for unit 2. Since the analysis of the non-zero switching rules for both units needs heavy notations, we omit the discussion of it here. The zero switching rules for both units is dealt with separately.