Dynamic service management and priority handling in dual server queuing systems with breakdowns

This paper examines a dual-server system comprising server 1 and server 2, each with its dedicated queue (queue 1 and queue 2). Queue 1, with lower priority, can handle an unlimited number of tasks, while queue 2, with higher priority, has a maximum capacity of N tasks. Task arrivals follow a Markovian process, and their service times adhere to a phase-type distribution. In the event of server 2's breakdown, server 1 seamlessly takes over, minimising service disruptions. To address customer dissatisfaction, server 1 provides prompt feedback during these transitions. The study employs the quasi-birth-death (QBD) process, utilising matrix analytic methods with finite-dimensional block matrices to analyse steady-state behaviour. Performance metrics like busy periods, costs, and other relevant measures are assessed, yielding insights into system efficiency. Numerical computations and graphical representations showcase the system's performance across various scenarios, providing a comprehensive understanding of its operation.