A simple method to obtain the stochastic decomposition structure of the busy period in Geo/Geo/1/N vacation queue

Abstract

In this paper, some tips and tricks for algebraic manipulations are utilized to explicitly get the mean and variance of the duration of the busy period in a discretetime finite-buffer vacation queue. Applying the law of total expectation, the closedform expressions for the first two moments of the busy period initiated with an arbitrary number of customers are firstly derived. Then, by employing the queue length distri-bution at vacation termination and the quantities that mentioned above, we give the stochastic decomposition structure of the busy period. Finally, in order to ensure the reliability of the analytical approach, an effective way to validate the correctness of our results along with a numerical example is also provided. We may find that these simple tips and tricks can greatly reduce the difficulty of problem solving.