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