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

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.

A Push–Pull Network with Infinite Supply of Work

“…The queue of jobs of stream 1 between the two machines will in that case be positive recurrent. In fact machine 2 will behave like a server with vacations (see [23]). The corresponding queueing network (see Fig.…”

Optimal two-threshold policies in an M/G/1 queue with two vacation types

“…When both types of vacations are exponentially distributed, we can compute il, ill and R (2) explicitly for use in (1), (2), (4) and (7)- (9). Note that R can be either the FRT of a long vacation or the FRT of a short vacation.…”

“…Skinner[ll] and Cooper [1] investigated the M/G/1 model with multiple vacations in which the server continues to take vacations until he finds at least one customer in the system. Levy and Yechiali [9] studied systems with multiple vacations and also introduced the single vacation model, where the server takes just one vacation whenever he empties the system. The literature on vacation models is extensive and growing.…”

M/G/1 Vacation Queueing Systems with Server Timeout

“…Queueing systems with server vacations have attracted the attention of many researchers since the idea was first discussed in the paper of Levy and Yechiali [1]. Several excellent surveys on these vacation models have been done by Doshi [2] [3], and the books by Takagi [4] and Tian and Zhang [5] are devoted to the subject.…”