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Connecting Renewal Age Processes with M/D/1 and M/D/∞ Queues Through Stick Breaking

We consider the length of a busy period in the M/D/∞ queue and show that it coincides with the sojourn time of the first customer in an M/D/1 processor-sharing queue. We further show that the busy period is intimately related with the stationary waiting time in the M/D/1 first-come-first-served queue. We present three characterizations for the distribution function of the busy period and an asymptotic expression for its tail distribution. The latter involves complex-valued branches of the Lambert W function.