Analysis of Unreliable Single Server Queueing System with Hot Back-Up Server

Abstract

In this paper, we analyze an unreliable queueing system consisting of an infinite buffer and two heterogeneous servers. The main server (server 1) is unreliable, while the server 2 is considered as the reserve server and is assumed to be absolutely reliable. The service times have the PH-type (Phase-type) distribution. If both servers are able to provide the service, they serve a customer independently of each other. The service of a customer is completed when his/her service by any of two servers is finished. After the service completion, both servers immediately start the service of the next customer, if he/she presents in the system. If the system is idle, the servers wait for arrival of the new customer. The input flow is described by the BMAP (Batch Markovian Arrival Process). Breakdowns arrive to the server 1 according to a MAP (Markovian Arrival Process). After breakdown occurrence, repair of the server starts. The repair time also has the PH-type (Phase-type) distribution. The customers, which meet the servers busy upon arrival, join a buffer. They will be picked up for the service according to the First-In-First-Out discipline. The customers arrived at the same batch are picked up for the service in random order. If a customer arriving from outside or from a buffer sees only server 2 ready for service while the server 1 is under repair, only server 2 starts the service of this customer. But if server 1 is repaired before service completion of this customer, server 1 immediately begins the service of this customer. For this model, we derive ergodicity condition, calculate the key performance measures of the system and derive an expression for the Laplace-Stieltjes transform of the sojourn time distribution of an arbitrary customer.