Burst arrival queues with server vacations and random timers

We derive performance measures for burst arrival (e.g. messages of variable length packets) M / G / 1 queues with server vacations, controlled by the, so called, Randomly Timed Gated (RTG) protocol, operating as follows : Whenever the server returns from a ( general-type ) vacation and initiates a busy period, a Timer with random duration T is activated. If the server empties the queue before time T, he leaves for another vacation. Otherwise ( i.e. if there are still customers (packets) in the system when the timer expires ), two versions of terminating the busy period, each leading to a different model, are studied : (i). The server completes service (e.g. transmission) only to the customer being served at time T and leaves. (ii). The server leaves immediately. We derive both state-dependent and steady-state performance measures as a function of the (vacation-type dependent variable) number of customers present at the initiation of a busy period. When the vacation policy is specified ( i.e. Multiple or Single ), we obtain explicit formulas for the various performance measures, derive the distribution and mean of the waiting and sojourn times of a customer, and compare between the two versions. Analysis of the conditions for stability concludes the paper.