Abstract
We consider an M/M/cqueueing system with impatient customers and a synchronous vacation policy, where customer impatience is due to the servers’ vacation. Whenever a system becomes empty, all the servers take a vacation. If the system is still empty, when the vacation ends, all the servers take another vacation; otherwise, they return to serve the queue. We develop the balance equations for the steady-state probabilities and solve the equations by using the probability generating function method. We obtain explicit expressions of some important performance measures by means of the two indexes. Based on these, we obtain some results about limiting behavior for some performance measures. We derive closed-form expressions of some important performance measures for two special cases. Finally, some numerical results are also presented.
Highlights
Queueing systems with vacations have been developed for wide range of applications in flexible manufacturing and computer communication systems over more than two decades
The service is provided by c servers, who serve the customers on a firstcome first-served (FCFS) basis
For M/M/1 queueing model with impatient customers and multiple vacation policy, Altman and Yechiali [12] show that the probability P.0[P.1] is an increasing concave function of ξ, having its limits at 1 − (λ/μ) [at (λ/μ)]
Summary
Queueing systems with vacations have been developed for wide range of applications in flexible manufacturing and computer communication systems over more than two decades. Altman and Yechiali [12] presented a comprehensive analysis for M/M/c queueing models with server vacations and customer impatience, where customers became impatient only when the servers were on vacation They considered asynchronous vacation policy for both the single and the multiple vacation cases by using the probability generating function method. They did not obtain the detailed results for the stationary probabilities and some expected performance measures such as the mean queue length and expected waiting time. Customers whose orders are back-ordered may become impatient and decide to cancel their orders if the customers’ waiting time exceeds a customer’s level of patience This is especially likely when the facility performs optional jobs.
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