Reversible Markov Processes on General Spaces: Spatial Birth-Death and Queueing Processes

with respect to π, if πis a measure on IE such thatπ(dx)q(xdy) = π(dy)q(ydx).Reversibility was introduced by Kolmogorov; see the review [1] and its applications to queueing in[3,6,8], which are for processes on countable state spaces. We present a canonical representation of thestationary distribution of Xon a general state space. This involves representing two-way communicationby certain Radon-Nikodym derivatives for measures on product spaces, using a result from [7]. This is notneeded for classical processes on discrete spaces or for kernels with with density functions (e.g., q(xdy) =r(x,y)µ(dx)). Included is a Kolmogorov criterion that establishes the reversibility of ψ-irreducible Markovjump processes [5].Thesecondpartofthestudyderivesstationarydistributionsfortwoclassesofreversiblemeasure-valuedMarkov processes:(1) Spatial birth-death processes with single and multiple births and deaths (the total population is neverinfinite, which is different from infinite-population systems [2,4]).(2) Spatial queueing systems in which customers move in a space where they receive services, analogous toservices in queueing networks [3,6,8].Sufficient conditions for ergodicity of spatial queues are also presented.The following is the main result for a