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. The paper shows that a continuous time Harris recurrent Markov process (HRMP) is of this type and gives a different proof of the ergodic properties of HRMP’s based on this fact. It next introduces the new notion of a market point process (mpp) governed by a HRMP and by doing so obtains 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. The paper also presents a general framework for representing queues in continuous time as HRMP’s when the input is a mpp governed by a HRMP and thus obtains a continuous time analogue of [20]. In this context the paper considers a single server queue with feedback and a tandem queue. Finally, due to the potential for regenerative type simulation, it states a CLT for one-dependent regenerative processes.