One-Dependent Regenerative Processes and Queues in Continuous Time

Motivated by the study of queues in continuous time, a “one-dependent” regenerative process is defined and its ergodic properties are given. We show that a continuous time Harris recurrent Markov process (HRMP) is of this type and give a different proof of the ergodic properties of HRMP's based on this fact. We next introduce the new notion of a marked point process (mpp) governed by a HRMP and by doing so obtain a class of mpp's that are one-dependent regenerative. This class is shown to include the superposition of independent renewal processes, the departure process from a FIFO GI/GI/c queue, Markov modulated arrivals and the loss stream (overflow) from a GI/GI/1/0 queue. We also present a general framework for representing queues in continuous time as HRMP's when the input is a mpp governed by a HRMP and thus obtain a continuous time analogue of [20]. In this context we consider a single server queue with feedback and a tandem queue. Finally, due to the potential for regenerative type simulation, we state a CLT for one-dependent regenerative processes.