The Geom^X/G/1 queue with Bernoulli feedback and server setup/closedown times

In this paper ,we discuss the Geom^X/G/1 queue with Bernoulli feedback and server setup/closedown times.We derive the probability generating function (PGF) of the steady-state queue length immediately after a service completion by the embedded Markov chain and the PGF of the queue length immediately after an arbitrary slot boundary .We also get the PGF of the stationary waiting times under first come first service (FCFS) service discipline .From the results ,we obtain the conclusion the steady-state queue length and waiting times have the property of stochastic decomposition .Finally,we get the generating function of busy period and give several special cases to verify the effect of our model.