Analysis of a renewal batch arrival queue with a fault-tolerant server using shift operator method

Motivated by the fault tolerance for manufacturing, we investigate a renewal input bulk arrival queue with a fault-tolerant server, in which the server can keep working with a low service rate even if the partial failure occurs. Only when there are no customers in the system, the partial failure can be removed. To explore the performance measures of the queue, a more generic and simpler algorithm based on the right shift operator method for solving difference equations is employed to obtain the queue-length distributions at different time epochs. The significant feature of this algorithm lies in that it does not require the derivation of the transition probability matrix for the corresponding embedded Markov chain. Furthermore, we can resort to the queue-length distribution at the pre-arrival epoch to quickly get the expected sojourn time for an arbitrary customer. Finally, with the help of Pad\(\acute{\mathrm{e}}\) approximation, several representative numerical examples are illustrated in tables and graphs, under which we show how to verify the correctness of our theoretical results through Little’s law.