Abstract
A discrete-timeGeo/G/1queue with vacations in random environment is analyzed. Using the method of supplementary variable, we give the probability generating function (PGF) of the stationary queue length distribution at arbitrary epoch. The PGF of the stationary sojourn time distribution is also derived. And we present the various performance measures such as mean number of customers in the system, mean length of the type-icycle, and mean time that the system resides in phase0. In addition, we show that theM/G/1queue with vacations in random environment can be approximated by its discrete-time counterpart. Finally, we present some special cases of the model and numerical examples.
Highlights
During the last three decades queueing systems with vacations have been largely studied; see, for example, the surveys (Doshi [1]) and the monographs (Takagi [2] and Tian and Zhang [3])
We consider a discrete-time vacation queue operating in random environment where the time axis is segmented into slots of equal length
When the system operates in phase i, i = 1, 2, . . . , n, customers arrive according to geometrical process with rate λi, where λi is the probability that a customer arrives in a slot, and the service times are independent and identically distributed according to a general distribution {si,k}∞ k=1 with probability generating function Si(z) = ∑∞ k=1 si,kzk and mean 1/μi
Summary
During the last three decades queueing systems with vacations have been largely studied; see, for example, the surveys (Doshi [1]) and the monographs (Takagi [2] and Tian and Zhang [3]). Queueing systems with vacations are characterized by the feature that each time a busy period ends and the system becomes empty, the server starts a vacation of random length of time. Atencia and Moreno [11] studied a discrete-time Geo/G/1 retrial queue, where the server is subject to starting failures. To the best of our knowledge, there has been no research on the discrete-time queueing system with vacations in random environment. The new service rate may change with the changes of the customers’ arrival rate, environmental conditions, and operator experience This motivated us to study the Geo/G/1 queue with vacations in random environment in this paper.
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