Uniform Cesaro Limit Theorems for Synchronous Processes with Applications to Queues

P. W. Glynn and K. Sigman

Stochastic Processes and their Applications, Vol. 40, 29-43 (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 θτ(k)X={X(t+τ(k)):t≥0} is the same and the cycle lengths Tn=τ(n+1)-τ(n) have finite first moment. Such processes (in general) do not converge to steady-state weakly (or in total variation) even when regularity conditions are placed on the cycles (such as non-lattice, spread-out, or mixing). Nonetheless, in the present paper we first show that the distributions of {θsX:s>0} are tight in the function space D(0,∞). Then we investigate conditions under which the Cesaro averaged functionals μt(f) converge uniformly (over a class of functions) to π(f), where π is the stationary distribution of X. We show that μt(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∈ L1(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.