Equilibrium Analysis for Batch Service Queueing Systems with Strategic Choice of Batch Size

Abstract

Various transportation services exist, such as ride-sharing or shared taxis, in which customers receive services in a batch of flexible sizes and share fees. In this study, we conducted an equilibrium analysis of a variable batch service model in which customers who observe no waiting customers in an incomplete batch can strategically select a batch size to maximize the individual utilities. We formulated this model as a three-dimensional Markov chain and created a book-type transition diagram. To consider the joining/balking dilemma of customers for this model, we proposed an effective algorithm to construct a necessary and sufficient size of state space for the Markov chain provided that all customers adopt the threshold-type equilibrium strategy. Moreover, we proved that the best batch size is a non-decreasing function for i if the reward for the completion of batch service with size l is an increasing function of l assuming that a tagged customer observes i complete batches in the system upon arrival; in other words, the fee decreases as the batch becomes larger. We then derive several performance measures, such as throughput, social welfare, and monopolist’s revenue. Throughout the numerical experiment, a comparison between the present variable batch service model and regular batch service model in which customers were served in a constant batch, was discussed. It was demonstrated that the three performance measures can be optimized simultaneously in the variable batch service model, as long as the fee was set relatively high.