Stationary Analysis on the Discrete Time Queue with Working Vacation and Setup-closed Time
|Course||Probability Theory and Mathematical Statistics|
|Keywords||GI/Geom/1queue Geom/G/1queue Single(Multiple) working vacation Matrix geometric solution Stochastic decomposition|
Since Servi and Finn firstly introduced working vacation policy into a queue,working vacation model has been widely studied.(The working vacation policy meansthat the server can provide service at a lower rate rather than completely stop serviceduring a vacation). The working vacation model has been a powerful tool for modelingand analysis in some fields, such as, optical fiber communication, random network.Close-down period (or set-up period) in a queue refers to a period for the equipmentclosed (or set up) when there are no customer in the system(or there are customers arrivethe system). On the one hand, setting a period of close-down time (or set-up time) canavoid damage to the system caused by instantly turning off (or turning on) the equipment;On the other hand, the close-down period can also serve customer at normal rate andreduce the customer’s waiting time as much as possible, ensure the system flow. In thispaper, discrete-time queue models with working vacation and close-set-up period areanalized, the effect of close, setup period and working vacation on the performance indexare demonstrated.Firstly, the thesis considers a GI/Geom/1queue with single working vacation andset-close time. The instants before the customers arrived the system are regarded asembedded points, an embedded Markov chain is obtained. Using the matrix-geometricsolution, the steady state distributions for the queue length and the probability generatingfunction of waiting time, the stochastic decomposition results, mean queue length andmean waiting time are derived.Secondly, the paper studies a Geom/G/1queue with multiple vacation and set-uptime. The instants after the customers left the system are regarded as embedded points, anembedded Markov chain is obtained. The transition probability matrix of Markov chainobtained, the generating function of queue length and LST of waiting time and their meanvalue are gained.Finally, the numerical results are presented using Matlab software, which demon-strats the effect of different system parameters on the performance indices, such as, the steady-state queue length and waiting time.