Subexponential asymptotics for steady state tail probabilities in a single-server queue with regenerative input flow

The present paper is devoted to queueing systems with regenerative input flow in the presence of heavy tails. Our goal is to develop an asymptotics for the probability of the waiting time process in a stationary regime to exceed a high level. In this paper, we consider the total service time of customers arriving during the time-interval $[0,t]$ as an input flow $X(t)$. This allows us to consider a case when the service times $\{\eta_n\}_{n=1}^\infty$ are dependent random variables that, besides, may be dependent on a number of customers arriving in $[0,t]$. We obtain conditions for the virtual waiting time process in steady state to have a subexponential distribution function. We apply this result to a system with a Markov modulated semi-Markov input flow. We also consider a queue with a doubly stochastic Poisson flow in the case when the random intensity is a regenerative process. We show that these results could be transferred to corresponding systems with an unreliable server.