Waiting time analysis for a queueing system with time-limited service for exponential timer

We consider a single-queue with exhaustive or gated time-limited services and server vacations, in which the length of each service period at the queue is controlled by a timer, i.e., the server serves customers until the timer expires or the queue becomes empty, whichever occurs first, and then takes vacations. The customer whose service is interrupted due to the timer expiration may be attended according to nonpreemptive or preemptive service disciplines. For the M/G/1 exhaustive/gated time-limited service queueing system with an exponential timer and four typical preemptive/nonpreemptive service disciplines, we derive the Laplace–Stieltjes transforms and the moment formulas for waiting times and sojourn times through a unified approach, and provide some new results for these time-limited service disciplines.