Mean Field for Performance Models with Deterministically-Timed Transitions

We extend the mean-field (a.k.a. fluid-analysis) approach for massively-parallel continuous-time Markov chains (CTMCs) to models with both Markovian and deterministically-timed transitions. We introduce a new low-level formalism for specifying massively-parallel models with generally-timed transitions, the population generalised semi-Markov process (PGSMP). We then show how systems of coupled delay differential equations (DDEs) which approximate transient component counts may be mechanistically derived from such models. This is possible not only in the case of non-competing deterministic delays, but also where the deterministic transitions may race locally with simultaneously enabled Markovian ones. For a large class of PGSMP models, we are able to prove mean-field convergence formally and to construct a second-order limit process. In the steady state, we show that it is not always straightforward to apply the fixed point approach to approximate the stationary behaviour of a PGSMP. To address this, we provide a new algorithm for computing steady-state mean-field approximations for PGSMP models leveraging the theory of Markov regenerative processes. Throughout, our techniques are illustrated on a number of simple worked examples drawn from the areas of peer-to-peer networks and wireless sensor networks.