Two stage \(\mathrm{M}^{[x]}/\mathrm{G}/1\) heterogeneous service, Bernoulli feed back, Bernoulli schedule server vacation random break down, setup time and restricted admissibility (Q2928717)

An unreliable single server is investigated with Batch-Poisson arrivals, 2-stage services with general service time, Bernoulli feedback, and Bernoulli vacations after a customer's second stage of service is finished. The server's up-times and the repair times after breakdown are exponential. Admission of batches is regulated by a Bernoulli decision where the admission probability depends on whether the server is at vacation or not. A Markovian state description is constructed and, starting with an empty system, the transient state probabilities are derived in a complicated transform version. Assuming steady state conditions, transform expressions are given for the stationary distribution and some performance metrics are derived.