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/$\infty$ 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. Keywords: Lambert W function; Processor sharing; Queues; Renewal age process; Stick breaking. Mathematics Subject Classification: 60K25; 97I80.