Steady State Analysis and Heavy Traffic Limits for Regulated Markov Chains

Consider a continuous time finite state irreducible Markov chain whose jump transitions are partitioned into one group that is regulated and the other group that is not. The regulated transitions are only allowed to occur if there is a token available. We collect the tokens in a buer and allow a regulated transition to occur simultaneously with the removal of a token from the buffer. New tokens are added to the buer at a constant Poisson rate but the regulated transitions will be blocked if they occur too quickly. We will apply matrix analysis to the joint distribution for the state of the Markov chain and the number of tokens in the buffer. We will give a simple stability condition for the joint process and show that its steady state distribution will have a matrix geometric distribution. Moreover, we obtain from our analysis a heavy traffic limit for this joint steady state distribution which has a product form structure. This Markov chain model and steady state analysis generalizes the work of many earlier papers on specific queueing systems such as Konheim and Reiser or Latouche and Neuts, but most significantly the work of Kogan and Puhalskii.