Computing system length distribution of a finite‐buffer bulk‐arrival bulk‐service queue with variable server capacity

In a recent paper, Chang et al. (2004) analysed finite‐buffer bulk‐arrival bulk‐service queue with variable server capacity: M^xG^y/1/N, and obtained queue length distributions at departure‐, arbitrary‐ and arrival‐epochs. They only obtained the distribution of the number of customers in the queue when server is idle/busy. From their analysis one cannot obtain the system length distribution or distribution of the number of customers in the batch undergoing service with the server when the server is busy. In this paper, we reinvestigate the model and first obtain the joint distribution of the number of customers in the queue and number with the departing batch at departure‐epoch and then using it we derive the joint distribution of the number of customers in the queue and number with the server at arbitrary‐ and arrival‐epoch. Besides obtaining system length distribution, we also obtain distribution of number of customers in the queue when the server is idle/busy and several performance measures viz. blocking probability, average number of customers in the system (queue) and average number of customers with the server.