A Sequence of Servers with Arbitrary Input and Regular Service Times Revisited

The March 1965 issue of Management Science saw a sequence of two papers by Avi-Itzhak and Yadin, the first presenting analysis of a two-server tandem system, with manufacturing blocking and no intermediate queue, leading to the second paper, which analyzes an arbitrary number of tandem servers with manufacturing blocking.

In this work we present generalized results for the same systems under k-stage blocking. One-stage blocking is the so-called manufacturing blocking in which a job entering the jth service position in the sequence requires attendance by the jth server only. Upon completion of service the job will move to position (j + 1) if that position is not occupied. Otherwise, it will continue to stay in position j, blocking it, until the next position is vacated. Two-stage blocking is the so-called communication blocking in which a job entering the jth position needs simultaneous attendance by the jth and the (j + 1)-th servers. Thus, start of service may be delayed until the (j + 1)-th server is freed. In k-stage blocking, the job requires at each position in the sequence the joint attendance of the server of that position together with the servers of the next k − 1 positions.

One interesting result is that for k > 1 the waiting times are not order-insensitive while the G/D/1 equivalence is maintained.