New results on bulk service queue with finite-buffer: M/G (a,b)/1/N

Abstract

We consider a single-server finite-buffer queue with general bulk service rule where customers arrive according to a Poisson process and service times of the batches are arbitrarily distributed. Using the embedded Markov chain technique we first obtain the steady-state joint distribution of the number of customers in the queue and number with the departing batch at departure-epoch. Further in order to obtain the joint distribution of the number of customers in the queue and number with the server at arbitrary-epoch, we develop relations between arbitrary- and departure-epoch probabilities. Finally, we also obtain the distributions of number of customers in the queue (system), and number with the server. Various performance measures of interest such as average number of customers in the queue (system) and average number of customers with the server etc. are obtained. The model was earlier discussed by Gold and Tran-Gia (Queueing Syst 14:413–426, 1993) and Chaudhry and Gupta (Queueing Syst 31:95–100, 1999) wherein they first obtained the distribution of the number of customers in the queue at departure-epoch without taking into the consideration of the size of the departing batch. Then they obtained the distribution of the number of customers in the queue at arbitrary-epoch when server is idle/busy.