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

摘要

In this paper, some tips and tricks for algebraic manipulations are utilized toexplicitly get the mean and variance of the duration of the busy period in a discretetimefinite-buffer vacation queue. Applying the law of total expectation, the closedformexpressions for the first two moments of the busy period initiated with anarbitrary number of customers are firstly derived. Then, by employing the queuelength distri-bution at vacation termination and the quantities that mentioned above,we give the stochastic decomposition structure of the busy period. Finally, in order toensure the reliability of the analytical approach, an effective way to validate thecorrectness of our results along with a numerical example is also provided. We mayfind that these simple tips and tricks can greatly reduce the difficulty of problemsolving.