Analysis of a Multi-Server Queue with Group Service and Service Time Dependent on the Size of a Group as a Model of a Delivery System

Abstract

In this paper, we consider a multi-server queue with a finite buffer. Request arrivals are defined by the Markov arrival process. Service is provided to groups of requests. The minimal and maximal group sizes are fixed. The service time of a group has a phase-type distribution with an irreducible representation depending on the size of the group. The requests are impatient. The patience time for an arbitrary request has an exponential distribution. After this time expires, the request is lost if all servers are busy or, if some server is idle, with a certain probability, all requests staying in the buffer start their service even if their number is below the required minimum. The behavior of the system is described by a multi-dimensional continuous-time Markov chain that does not belong to the class of level-independent quasi-birth-and-death processes. The algorithm for the computation of the stationary distribution of this chain is presented, and expressions for the computation of the queuing system’s performance characteristics are derived. The description of a delivery system operation in terms of the analyzed queuing model is given, and the problem of the optimization of its operation is numerically solved. Multi-server queues with a phase-type distribution for the group service time that are dependent on the size of the group, the account of request impatience, and the correlated arrival process have not previously been analyzed in the existing literature. However, they represent a precise model of many real-world objects, including delivery systems.