Performance Forecasting for Multi-Server Retrial Queue with Possibility of Processing Repetition and Server Reservation for Repeating Users

This study focuses on forecasting and optimizing the performance of a real-world object modelled by a multi-server queueing system that processes two types of users: primary (new) users and repeating users. The repeating users are those who succeeded in entering processing upon arrival and then decided to repeat it. These users have privilege and can enter processing when they wish once at least one device is idle. The primary user is admitted to the system only if the number of occupied devices is less than some threshold value and the quantity of repeating users residing in the system does not exceed certain thresholds. Repeating users are impatient and non-persistent. Arrivals of primary users are described by the Markovian arrival process. Processing times of primary and repeating users have distinct phase-type distributions. Utilizing the concept of the generalized phase–time distributions, the dynamics of this queueing system are formally characterized by the multidimensional Markov chain, which is examined in this paper. The ergodicity condition is derived. The relation of the key performance characteristics of the system and the thresholds defining the policy of the primary user’s admission is numerically highlighted. Optimal threshold selection is demonstrated numerically.