Randomly timed gated queueing systems

Timers are used as a control mechanism to constrain the service periods in various telecommunication networks. In this paper we introduce and analyze a general family of timer-type models, called Randomly Timed Gated (RTG) queueing systems, which, in addition, extends and generalizes the family of M/G/1-type queues with server intermissions. The control mechanism is such that, whenever the server reenters the system, an exponential Timer T is activated. If the server empties the queue before the Timer's expiration, it immediately leaves for another intermission. Otherwise (if the Timer expires when there is still work in the system), the server obeys one of the following rules, each leading to a different model: (1) The server completes all the work accumulated up to time T, and leaves. (2) The server completes only the service of the job currently being served, and leaves. (3) The server leaves immediately. Consequently, the RTG queue forms a unified structure for a vast collection of timer-type models: Under rule (1) it spans an entire spectrum of regimes lying between the Gated and the Exhaustive. Under rule (2) it spans an entire spectrum between the 1-Limited and the Exhaustive disciplines. The analysis is achieved through a general solution of an infinite set of linear equations where the unknowns are the state-dependent joint transforms of two key characteristics: the length of a busy period starting with r jobs (r = 0, 1, 2, ...), and the number of jobs left behind at the end of such a busy period. We show that, in steady state, a complete description of the RTG system can be obtained explicitly in terms of the probability generating function (PGF) of the queue size at a start of a busy period. A fixed point equation for this PGF can be constructed when the structure of the intermission interval is specified. We derive various performance measures for the three models in steady state.