A single-server queue with vacations and gated time-limited service

An M/G/1 queue with server vacations and gated time-limited service is analyzed. At each visit, the server serves the queue up to a fixed amount of time. When the time expires or after all candidate customers have been served, whichever occurs first, the server takes a vacation. The service policy is gated, since only those customers present at the beginning of a server visit (poling instant) are candidates for service during that visit; subsequent arrivals are deferred until the next visit. A functional equation which characterizes the amount of work, U_p, at a polling instant is derived. To solve the equation, a numerical technique is utilized in which the complementary cumulative function for U_p is closely approximated by a weighted sum of Laguerre functions with unknown coefficients. The equation is then transformed into a set of linear equations from which the coefficients can be computed. By the stochastic decomposition and Poisson-arrivals-see-time-averages properties, the average customer response time can be related to the average amount of work found by an arrival. Several numerical examples are included. The model studied is applicable to communication and computer systems where timers are used to allocate service to customers