This article considers the performance of random block codes over the Gilbert-Elliott channel and characterizes the queueing performance under maximum-likelihood decoding. The probability of decoding failure is upper bounded using an approximation that works well in the rare-transition regime and the bound is used to perform a queueing analysis. A Poisson arrival process is chosen to allow fair comparisons between different block lengths and code rates. A Markov chain, based on the queue length and channel state, is constructed and used to analyze the tail probability of the queue. Our methods are used to evaluate both the probability of decoding failure, under a constraint of the probability of undetected error, and the queueing performance. The main result is that, for random coding on the Gilbert-Elliott channel, the performance analysis based on upper bounds provides a very good estimate of both the system performance and the optimum code parameters.