Discrete Time Working Vacation Queue Under Different Vacation Interruption Policies

Working vacation queue is an extension and development of classical vacationqueuing theory, initially was promoted by modeling and analysis of Gateway Router inlight communication in which Servi and Finn introduced a semi-vacation policy in M/M/1multiple vacations queue system. It has been widely applied to computer communicationnetworks, flexibility manufacture system, asynchronously transfer mode and so on. Theworking vacation with interruption was firstly led into the Geom/Geom/1single vacationsystem by Li and Tian, and the detailed analysis of the GI/Geom/1queue with workingvacation with interruption was given. It's no difficult to imagine that if selecting when theworking vacation can interrupt opportune or taking conversion in different stages, thestudying of working vacation with interruption would provide theoretical basis andanalysis methods to optimized design when the system is running in low service rate andwould be more theoretical significance and practical value. So based on the above idea theresearching on Geom/Geom/1queue with working vacation interruption under N-policyand different converting policies is started, and the result is spread to Geom/Geom/cqueue.Firstly, the current researching situation and practical value of discrete time vacationqueue are introduced. The definition of working vacation and vacation interruption policyare also elaborated. And the entry protocol and matrix geometric solution are briefly givento pave the way for back model's analysis.Secondly, the Geom/Geom/1queue model with working vacation interruption underN-policy is given. By means of imbedding Markov chain solution, one step transitionprobability matrix of the system is obtained. And using matrix geometric solution thededucing process of steady-state queue length and other corresponding steady state indexis given. The special numerical examples are given to verify the model's results, and thevariation of the index with several parameters is analyzed by figures. Spreading abovemodels, the Geom/Geom/1queue with reciprocal transformation between N and Q policyduring working vacation period, and the influence which some parameters' variation brings to average queue length and customers' average residence time is obtained.Finally, as the spreading of the queue with working vacation interruption, a multipleservices queue with synchronic vacation and interruption is given. The positive recurrentcondition which process fulfills and the proof of existence of the rate matrix are given indetail, thus the correctness of the system is further proved by numerical examples.