Computing stationary distributions of the D-MAP/D-MSP$$^{(a,b) }/1$$ queueing system

Abstract

The present study investigates a single-server batch service queueing model with arrival process as discrete-time Markovian arrival process and service process as discrete-time Markovian service process. The server serves the customers in batches followed by the general bulk-service rule. We first determine the random epoch probabilities using the matrix-geometric method, where the rate matrix $${\mathbf{R}}$$ is determined by an efficient approach based on the eigenvalues and the corresponding eigenvectors of the associated characteristic equation. Next we obtain the explicit closed-form expressions for the pre-arrival, intermediate, outside observer’s and post-departure epoch probabilities by developing the relations among them in equilibrium state. Further, we provide an analytically simple approach to carry out the waiting-time distribution in the queue measured in slots as well as the distribution of the size of a service batch of an arriving customer. We also demonstrate a cost function to evaluate the optimum value of the minimum service batch size and the corresponding expected cost of the system. Finally, an adequate variety of numerical experiments are performed for validation purpose of our analytical results and they are conferred in the form of tables and graphs.