An algorithmic approach to analysing the reliability of a controllable unreliable queue with two heterogeneous servers

We consider a Markovian queueing system with two unreliable heterogeneous servers and one common queue. The servers serve customers without preemption and fail only if they are busy. Customers are allocated to one or the other server via a threshold control policy which prescribes using the faster server whenever it is free and the slower server only when the number of waiting customers exceeds a specified threshold level that depends on the state of the faster server. This paper focuses on the reliability analysis of a system with unreliable heterogeneous servers. First, we obtain the stationary state distribution using a matrix-geometric solution method. Second, we analyse the lifetimes of the servers and of the system. We provide algorithms for calculating the stationary reliability characteristics, reliability functions in terms of the Laplace transform and the mean times to the first failure. A new reliability measure is introduced in the form of the discrete distribution function of the number of failures during a specified life time that is derived from a probability generating function. The effects of various parameters on these reliability characteristics are analysed numerically.