Joint Analysis of Served and Lost Customers in Busy Periods of MX/M/1/n Systems with Balking

We derive the joint probability-generating function of the number of customers served and the number of customers lost in busy periods of MX/M/1/n queueing systems with balking. Capitalizing on the Markov-regenerative nature of these systems, we propose an efficient recursive procedure to compute this joint distribution. The model incorporates customer impatience, where the admission of customers is modulated by the number of customers in the system at arrival instants. Specifically, customers arriving in batches that find a place in the system may opt to join or balk, depending on the number of customers already in the system. The performance analysis of such systems is an important issue for business management, as customer impatience may result in the loss of potential customers, thus reducing the number of served customers. In addition, we perform a sensitivity analysis of the corresponding joint probability function of the numbers of customers served and lost in busy periods and other performance measures, considering both partial and total customer batch blocking policies, across various batch size distributions and balking probability profiles.