The age of the arrival process in the G/M/1 and M/G/1 queues

This paper shows that in the G/M/1 queueing model, conditioning on a busy server, the age of the inter-arrival time and the number of customers in the queue are independent. The same is the case when the age is replaced by the residual inter-arrival time or by its total value. Explicit expressions for the conditional density functions, as well as some stochastic orders, in all three cases are given. Moreover, we show that this independence property, which we prove by elementary arguments, also leads to an alternative proof for the fact that given a busy server, the number of customers in the queue follows a geometric distribution. We conclude with a derivation for the Laplace Stieltjes Transform (LST) of the age of the inter-arrival time in the M/G/1 queue.