Mathematical Modelling of a Multi-Stage Feedback Queue with Discretionary Server Downtime

Abstract

In this section, we take into account an M/G/1x queue in the presence of multiple service tiers, customers who provide feedback at random, and a server who is on vacation. The customer is given the option to progress through a series of k progressively more complex steps in the system. A customer is free to abandon the system after receiving service in the first of  k stages (k > 1). There is a specific order to the stages, starting with the first stage service and continuing on to the second stage service and so on. The customer has multiple opportunities for subsequent service after each phase. However, the re service can only be used from the very beginning, so it doesn't matter which stage he left. Therefore, the customer who requires a different service should go to the very end of the line and wait there. When a server finishes a service for a customer, they may take a short break or move on to the next customer immediately, depending on their schedule. In this article, we extend this model to accommodate a batch arrival queue that allows for a server to take a break.