Tightness of Synchronous Processes

P. W. Glynn and K. Sigman

Technical Report, Department of Operations Research, Stanford University (1989)

Let X = (X(t) : t≥0) be a positive recurrent synchronous process (PRS), that is, a process for which there exists an increasing sequence of random times τ=(τ(k)) such that for each k the distribution of θ_τ(k) o X = (X(t+τ(k)):t≥0) is the same and the cycle lengths (T_n = τ(n+1)-τ(n) have finite first moment. Whereas the ergodic properties of such processes are well known in the literature, the same is not so for the distributional properties of either the marginals X(t) or more generally the shifted processes θ_s o X = (X(s+t) : t≥0) in function or space. The present paper shows that these distributions are in fact tight. In contrast to classical regenerative processes the standard types of regularity assumptions (non-lattice cycle length distribution, mixing) do not ensure weak convergence to steady-state for a PRS. Applications are given in the context of one-dependent regenerative (od-R) processes. These arise in the queueing models that motivated this paper.