Utilization of Idle Time in an <i>M</i>/<i>G</i>/1 Queueing System

TLDR

In an M/G/1 queue the server remains busy while customers are present, but when idle it takes a random vacation whose length follows a known distribution. This paper investigates how the server’s idle time can be used for additional work in a secondary system. Two vacation policies are analyzed: one where the server returns to the main queue after each vacation and serves any arrivals during the vacation, and another where the server immediately takes another vacation if the system is empty, with Laplace–Stieltjes transforms and generating functions derived for occupation, vacation, waiting times and queue length. The models are compared, and for certain special cases the optimal mean vacation times are identified.

Abstract

This paper studies an M/G/1 queue where the idle time of the server is utilized for additional work in a secondary system. As usual, the server is busy as long as there are units in the main system. However, as soon as the server becomes idle he leaves for a “vacation.” The duration of a vacation is a random variable with a known distribution function. Two models are considered. In the first, upon termination of a vacation the server returns to the main queue and begins to serve those units, if any, that have arrived during the vacation. If no units have arrived the server waits for the first arrival when an ordinary M/G/1 busy period is initiated. In the second model if the server finds the system empty at the end of a vacation, he immediately takes another vacation, etc. For both models Laplace-Stieltjes transforms of the occupation period, vacation period and waiting time are derived and generating functions of the number of units in the system are calculated. The two models are then compared to each other, and for some special cases the optimal mean vacation times are found.