With continuously increasing use of switched Ethernet for supporting various applications, it is important to precisely characterize the performance of Ethernet switches to provide the required QoS (Quality of Service). In this paper we propose an exact mathematical model of an Ethernet switch based on a multi-dimensional Markov process. We discuss both continuous and discrete time variants of the proposed model. We put forward the problem of asymptotic analysis of steady-state distribution of the workload to evaluate the performance of the system. This approach is deeply related with the recent results on large deviations in Markovian systems. We provide a comprehensive study of a particular model called symmetrical geometric. For the special case of Binomial input flow, the asymptotic approach is compared with the exact recurrent formulae highlighting the numerical efficiency of the asymptotic approach.