This paper focuses on the situation of a large number n of independent, identically distributed Markov chains. The fractions of these n Markov chains being in any specific state are called state frequencies (or: ‘empirical distribution’). We provide a large deviations analysis of the probability of these state frequencies reaching a distribution β from a given initial distribution α in a fixed amount of time t. This probability is characterized asymptotically in the number of Markov chains. Apart from these asymptotics, it is also shown how the rare event occurs: the ‘most likely path’ from α to β in t units of time is explicitly calculated. It is argued that the analysis provides useful insight into the likelihood of extreme fluctuations of the input rate of an asynchronous transfer mode switching element of an integer programming router. A number of numerical examples are included.
