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A discrete-time Geo/G/1 queue 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-i cycle, and mean time that the system resides in phase 0. In addition, we show that the M/G/1 queue 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

discrete dynamics in nature and society

On the Discrete-Time<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi>G</mml:mi><mml:mi>e</mml:mi><mml:mi>o</mml:mi><mml:mo>/</mml:mo><mml:mi>G</mml:mi><mml:mo>/</mml:mo><mml:mn fontstyle="italic">1</mml:mn></mml:math>Queue with Vacations in Random Environment

On the Discrete-TimeGeo/G/1Queue with Vacations in Random Environment