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We develop an importance sampling technique that can be used to speed up the simulation of a model of a buffered communication multiplexer fed by a large number of independent sources. The sources generate traffic according to a periodic function with a random phase. This traffic model accommodates a wide range of situations of practical interest, including ON-OFF periodic traffic models and sequences of bit rates generated by actual variable bit rate sources, such as MPEG video compressors. The simulation seeks to obtain estimates for the buffer overflow probability that in most cases of interest is very small. We use a large deviations result to devise the change of measure used in the importance sampling technique and demonstrate through numerical results that this change of measure leads to a dramatic reduction in the required simulation time over direct Monte Carlo simulation. Possible practical applications include short-term network resource planning and even real-time call admission control.