When traffic sources are statistically multiplexed over a common link, the sum of the peak rates of the sources exceeds the throughput of the link. The excess may be stored in a buffer, but when this overflows, information is lost. When a source is bursty, the peak rate is attained only for very short periods of time, whereas between bursts the source is idle. Because the sources are independent, the chance that many bursts arrive simultaneously is small, but these rare events do occur and the mean time until overload is a key design parameter. Here we model the multiplexor as a multidimensional Markov process with a set of forbidden states that represent the exceedance of the link capacity. We use the theory of induced Dirichlet forms to estimate the Laplace transform of the hitting time of this forbidden set. We obtain an upper bound on the probability that the link capacity is exceeded during a fixed time interval along with a lower bound for the mean time until the link capacity is exceeded. This provides the network designer with a degree of assurance about the probability and frequency of overloads.