Uniform Cesaro Limit Theorems for Synchronous Processes with Applications to QueuesP. W. Glynn and K. Sigman Stochastic Processes and their Applications, Vol. 40, 2943 (1992) 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)}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. Such processes (in general) do not converge to steadystate weakly (or in total variation) even when regularity conditions are placed on the cycles (such as nonlattice, spreadout, or mixing). Nonetheless, in the present paper we first show that the distributions of {θ_{s}X:s>0} are tight in the function space D(0,∞). Then we investigate conditions under which the Cesaro averaged functionals μ_{t}(f) converge uniformly (over a class of functions) to π(f), where π is the stationary distribution of X. We show that μ_{t}(f) converges to π(f), uniformly over f satisfying f_{∞}≤1 (total variation convergence). We also show that to obtain uniform convergence over all f satisfying f≤g (g∈ L_{1}(π) fixed) requires placing further conditions on the PRS. This is in sharp contrast to both classical regenerative processes and discrete time Harris recurrent Markov chains (where renewal theory can be applied) where such uniform convergence holds without any further conditions. For continuous time positive Harris recurrent Markov processes (where renewal theory cannot be applied) we show that these further conditions are in fact automatically satisfied. In this context, applications to queueing models are given.
