A Short Note on the System-Length Distribution in a Finite-Buffer $$GI^X/C$$-MSP/1/N Queue Using Roots

Abstract

This paper deals with a renewal input finite-buffer singleserver queue, where the arrivals occur in batches and the server serves the customers singly. It is assumed that the inter-batch arrival times are generally distributed and the successive service times are correlated. The correlated single-service process is exhibited by a continuous-time Markovian service process (C-MSP ). As the buffer capacity N (including the one in service) is finite, the partial-batch rejection policy is considered here. Steady-state distributions at different epochs, namely pre-arrival and arbitrary epochs are obtained. These distributions are used to obtain some important performance measures, e.g. the blocking probability of the first, an arbitrary, and the last customer of a batch, the average number of customers in the system and the mean waiting time in the system. The proposed analysis is based on the roots of a characteristic equation which is derived from the balance equations of an embedded Markov chain at pre-arrival epochs of a batch. For this non-renewal service finite-buffer queueing model, we implement a novel as well as simple procedure for deriving the characteristic equation and then finding the stationary probability vectors in terms of the roots of the characteristic equation. Finally, some numerical results are presented in the form of tables for the case of a phase-type inter-batch arrival distribution.