Analysis of a Two-server Queue with Consultation by Main Server Having Protected Phases of Service

In a queueing system where service is carried out in multiple phases and is subjected to interruptions, the idea of protecting certain critical phases from interruption becomes highly significant. This paper investigates a two-server queueing system in which a primary (main) server provides consultation to a secondary (regular) server. The main server performs dual functions: servicing customers and offering consultation to the regular server, with consultations being assigned preemptive priority over customer service. Customer service at the main server is subjected to stochastic interruptions. But interruptions are prohibited when the service process is in designated protected phases. Some constraints are imposed through upper bounds on the number of allowable interruptions experienced by a customer at the main server and the number of consultations provided to the regular server during an individual service period. There is a supervisory timing mechanism super clock which governs the admissibility of additional interruptions to an ongoing service at the main server. A threshold clock regulates the restart or resumption of service at both servers following each consultation event. Customer arrivals and consultation requests are modelled as mutually independent Poisson processes. It is assumed that service times at the main and regular servers are independent and follow phase-type distributions. The system’s stability condition is established, and key performance measures are evaluated through numerical analysis.