Reliability modelling using G-queues

A new technique for modelling unreliable queueing nodes is presented, based on the notion of queues with negative customers – called G-queues. A breakdown at a server is represented by the arrival of a negative customer which causes some customers to be lost. We first model an M/M/1 queue with breakdowns and instantaneous repairs which is relevant in systems such as packet-switched telephone networks where a ‘failure’ is interpreted as the loss of a call. The analysis is then extended to the case of exponential repair times which provides a more conventional model of unreliable systems and a variation on the contemporary notion of G-queues. Specifically, we derive expressions for the Laplace transform of the sojourn time density in a single server queue with exponential service times, independent Poisson arrival streams of positive customers and negative customers with batch killing, and both instantaneous and exponential repair times. We apply our model to approximate the performance of a bank of parallel, unreliable servers by modifying the arrival process to a Markov modulated Poisson process and considering an approximate decomposition of the underlying Markov chain under the assumption that arrivals and service occur much faster than breakdowns and repairs. Numerical validation with respect to simulation suggests that the approximation is accurate, especially for non-heavily utilised servers. Finally we indicate how to extend our approach to arbitrarily connected Markovian queueing networks with breakdowns.