Optimal Operating Policies for M/G/1 Queuing Systems

Abstract

We consider the economic behavior of a M/G/1 queuing system operating with the following cost structure: a server start-up cost, a server shut-down cost, a cost per unit time when the server is turned on, and a holding cost per unit time spent in the system for each customer. We prove that for the single server queue there is a stationary optimal policy of the form: Turn the server on when n customers are present, and turn it off when the system is empty. For the undiscounted, infinite horizon problem, an exact expression for the cost rate as a function of n and a closed form expression for the optimal value of n are derived. When future costs are discounted, we obtain an equation for the expected discounted cost as a function of n and the interest rate, and prove that for small interest rates the optimal discounted policy is approximately the optimal undiscounted policy. We conclude by establishing the recursion relation to find the optimal (nonstationary) policy for finite horizon problems.

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

“…For example, the second model has been partially used by Cooper [2] to analyze a system of queues served in cyclic order. In that study, for any given queue, say i, the "vacation time" is the length of time the server spends idle or working [3] analyzed an M/G/1 system where the service facility is turned off when no customers are present and is turned on only when the rth unit has arrived. Yadin and Naor's model differs from ours in that the server returns to the main queue immediately when the r th unit arrives, while in our model the server reutrns to the main queue only upon termination of its vacation-independent of the number of customers present there.…”

MULTI Phase M/G/1 Queue with Bernoulli Feedback and Multiple Server Vacation

“…[7,[9][10][11][12][13][14][15].Much research [8, 16 -20]has studied queueing models under vacations. For complete reference on vacation models, one may refer to Doshi [17]and Tagagi [3].M/G/1 vacation models under the various service disciplines have been investigated [8,18,[21][22][23][24].…”

Reservicing some customers in M/G/1 queues under three disciplines

“…Earlier authors include Yadin and Naor [12], Heyman [4], Sobel [9], and Bell [1]. Yadin and Naor [12] proposed shutdown control for an M/M/1 queue in order to increase the length of individual idle periods.…”

“…Yadin and Naor [12] proposed shutdown control for an M/M/1 queue in order to increase the length of individual idle periods. Soon after, Heyman [4] proved that (0,N) control (i.e., the server turns off if the system size is zero and turns on when the system size is N) is the optimal policy in an M/G/1 queue by considering the start-up and shutdown cost of a server, a cost per unit time when a server is running and a customer is waiting cost. This model was modified by Heyman and Marshal [5] by allowing for interarrival distributions that have increasing rates.…”