Networks of Queues with Delayed Feedback.

摘要

The problem of queueing systems with feedback has received little attention in the queueing literature. The queue with instantaneous feedback has been more studied, recently, than the queue with randomly delayed feedback. In this paper the author presents some results for M/M/1 queues with finite capacities with delayed feedback. The delay in the feedback to an M/M/1 queue is modelled as another M-server queue with a finite capacity. Then the output from server 1 progresses to server 2 with probability p and leaves the system with probability q. Arrivals at server 2 join a waiting line, if necessary, are served and return to server 1 with probability 1. In this way server 2 acts as a delay mechanism in the feedback stream. All new arrivals are to server 1. The steady state probabilities for the two dimensional Markov process (N(t), M(t)) are solved when N(t) = queue length at server 1 at t and M(t) = queue length at server 2 at t. A sample problem is run to show that the solution methods can be programmed and meaningful results obtained numerically.