Exact and asymptotic analysis of infinite server batch service queues with random batch sizes

Batch service queueing systems are basically classified into two types: a time-based system in which the service facilities depart according to inter-departure times that follows a given distribution, such as the conventional bus system, and a demand-responsive system in which the vehicles start traveling provided that a certain number of customers gather at the waiting space, such as ride-sharing and on-demand bus. Motivated by the recent spreading of demand-responsive transportation, this study examines the M/M $$^{\textit{X}}$$ X / $$\infty $$ ∞ queue. In this model, whenever the number of waiting customers reaches a capacity set by a discrete random variable $$\textit{X}$$ X , customers are served by a group. We formulate the M/M $$^{\textit{X}}$$ X / $$\infty $$ ∞ queue as a three-dimensional Markov chain whose dimensions are all unbounded and depict a book-type transition diagram. The joint stationary distribution for the number of busy servers, number of waiting customers, and batch size is derived by applying the method of factorial moment generating function. The central limit theorem is proved for the case that $$\textit{X}$$ X has finite support under heavy traffic using the exact expressions of the first two moments of the number of busy servers. Moreover, we show that the M/M $$^\textit{X}$$ X / $$\infty $$ ∞ queue encompasses the time-based infinite server batch service queue (M/M $$^{G(x)}$$ G ( x ) / $$\infty $$ ∞ queue), which corresponds to the conventional bus system, under a specific heavy traffic regime. In this model, the transportation facility departs periodically according to a given distribution, $$\textit{G(x)}$$ G ( x ) , and collects all the waiting customers for a batch service for an exponentially distributed time corresponding to the traveling time on the road. We show a random variable version of Little’s law for the number of waiting customers for the M/M $$^{G(x)}$$ G ( x ) / $$\infty $$ ∞ queue. Furthermore, we present a moment approach to obtain the distribution and moments of the number of busy servers in a GI/M/ $$\infty $$ ∞ queue by utilizing the M/M $$^{G(x)}$$ G ( x ) / $$\infty $$ ∞ queue. Finally, we provide some numerical results and discuss their possible applications on transportation systems.