A note on an M/G/1 queue with a waiting server, timer and vacations

We analyze a generalized protocol of an M/G/1 queue with server vacations where after returning from a vacation to an empty system, the server (as in many real-life situations and common also to human behaviour) activates a Timer and waits dormant. If an arrival occurs before the Timer expires, a busy period starts immediately. If the Timer is shorter than the inter-arrival time, the server does not wait any more and leaves for a new vacation, etc. We derive transforms and performance measures of the system's key variables and show how the general results reduce to their two extreme cases: (i) zero Timer yields the multiple vacation model and (ii) infinite Timer yields the single vacation case.