Sojourn times in the M/G/1 queue with deterministic feedback

In this paper we consider an M/G/1 queueing model, in which each customer is fed back a fixed number of times. For the case of negative exponentially distributed service times at each visit, we determine the Laplace-Stieltjes transform of the joint distribution of the sojourn times of the consecutive visits. As a by-result, we obtain the (transform of the) total sojourn time distribution; it can be related to the sojourn time distribution in the M/D/l queue with processor sharing. For the case of generally distributed service times at each visit, a set of linear equations is derived, from which the mean sojourn times per visit can be calculated.