Synchronized Abandonment in Discrete-Time Renewal Input Queues with Vacations

We consider a discrete-time infinite buffer renewal input queue with multiple vacations and synchronized abandonment. Waiting customers get impatient during the server's vacation and decide whether to take service or abandon simultaneously at the vacation completion instants. Using the supplementary variable technique and difference operator method, we obtain an explicit expression to find the steady-state system-length distributions at pre-arrival, random, and outside observer's observation epochs. We provide the stochastic decomposition structure for the number of customers and discuss the various performance measures. With the help of numerical experiments, we show that the method formulated in this work is analytically elegant and computationally tractable. The results are appropriate for light-tailed inter-arrival distributions and can also be leveraged to find heavy-tailed inter-arrival distributions.