Markovian couplings staying in arbitrary subsets of the state space

Abstract

Let (X t ) and (Y t ) be continuous-time Markov chains with countable state spaces E and F and let K be an arbitrary subset of E x F. We give necessary and sufficient conditions on the transition rates of (X t ) and (Y t ) for the existence of a coupling which stays in K. We also show that when such a coupling exists, it can be chosen to be Markovian and give a way to construct it. In the case E=F and K ⊆ E x E, we see how the problem of construction of the coupling can be simplified. We give some examples of use and application of our results, including a new concept of lumpability in Markov chains.

Monotonicity Properties for Multi-Class Queueing Systems

“…We give sufficient conditions on two allocation policies in order to compare sample-path wise the workload and the number of users of the various classes. We obtain weaker sufficient conditions on the transition rates than [25,24]. Since our result is a pure sample-path comparison, it holds for arbitrary arrival processes, service time processes and rate region variations.…”

“…In particular, stochastic comparison is often used. In the seminal paper [25] (see also [24]) necessary and sufficient conditions on the transition rates are given for the existence of a stochastic ordering between two Markov processes defined on ordered state spaces, starting from any two ordered initial states. It turns out that these conditions are often too strong in a queueing context.…”

Comparison of bandwidth-sharing policies in a linear network

“…The main goal of this paper is to give sufficient conditions on two policies in order to compare sample-path wise the workload and the number of users of the various classes. We obtain weaker sufficient conditions on the transition rates than [16,15], which can be explained from the fact that we only compare the two processes starting in the same initial state, as opposed to starting from any two ordered initial states as in [16,15]. From the performance point of view, starting from the same initial state does not diminish the applicability of the results, since the steady-state behavior does not depend on the initial state.…”

“…. , μL, for which these conditions are satisfied, and a stochastic ordering relation on the total number of users as in the framework of [15,16] does not hold.…”

“…In particular, stochastic comparison is often used. In the seminal paper [16] (see also [15]) necessary and sufficient conditions on the transition rates are given for the existence of a stochastic ordering between two Markov processes defined on ordered state spaces, starting from any two ordered initial states. It turns out that these conditions are often too strong in a queueing context.…”