On departure process in a production model with cyclic working and repair periods

A finite-buffer queueing system of the M/M/1/N type is used for modeling the operation of a single-machine production line with cyclic failure-free and repair periods. The arriving jobs enter randomly according to a Poisson process and are being processed individually with service times having the common exponential distribution. After an exponentially distributed working period a breakdown of the machine occurs, starting an exponentially distributed repair time during which the service process is stopped. At the completion epoch of the repair time a new working period begins and so on. A system of integral equations for conditional probability distributions of the number of jobs completely processed before the fixed time t (departure process) is built, using the concept of embedded Markov chain and the total probability law. Applying linear-algebraic approach the compact-form solution of the corresponding system written for double transforms of departure process is found.