Minimization of the blocking time of the unreliable Geo / G D / 1 queueing system ∗

In this paper we study the blocking time of an unreliable single-server queueing system Geo/GD/1. The service can be interrupted upon explicit or implicit breakdowns. For the successful finish of the service we use a special service discipline dividing the pure service time X (assumed to be a random variable with known distribution) in subintervals with deterministically selected time-points 0 = t0 < t1 < . . . < tk < tk+1; tk < X ≤ tk+1, and making a copy at the end of each subinterval (if no breakdowns occur during it) we derive the probability generating function of the blocking time of the server by a customer. As an application, we consider an unreliable system Geo/D/1 and the results is that the expected blocking time is minimized when the timepoints t0, t1, . . . are equidistant. We determine the optimal number of copies and the length of the corresponding interval between two consecutive copies.