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. Explicit expressions for the density functions of this age conditioning on a busy server and conditioning on an idle server 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. Also, we show that the residual inter-arrival time and the number of customers in the system given that the server is busy are independent. Moreover, we develops an explicit form for the density function of the conditional residual inter-arrival time given a busy and given an idle server. This is also repeated for the inter-arrival time itself. 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.