On a many-to-many matched queueing system with flexible matching mechanism and impatient customers

Abstract

In this paper, we focus on a kind of many-to-many matching system which is applicable to many real-life matching systems. We use customers and servers to represent the inputs at both ends. To portray the inputs’ behavioral features in many actual matching service systems, we introduce the customer impatience and flexible matching mechanism. Considering the possibility of customer reneging from the system, an infinite-dimensional non-homogeneous QBD (quasi-birth-and-death) process is constructed to depict the process of queue lengths at both ends. Methodologically, we capture the joint stationary distribution of the queue lengths at both sides by truncating the Markov chain. In the above procedure, the matrix G at any truncation point is obtained, and two detailed algorithms are provided to determine the truncation point and the approximate stationary distribution. For the system without customers’ impatience, the stationary distribution of the queue lengths is also analyzed by using matrix analytic method. Subsequently, several important performance measures (e.g., the loss probability of customers) are obtained. In addition, by reconstructing the three-dimensional and two-dimensional absorbing Markov chains, we investigate the distributions of customer’s waiting time for matching and conditional clearing time of the system, and apply RG-factorizations to calculate the specific results. Finally, numerical examples are used to reveal the effects of different parameters on the performance measures.