Optional Vacation And Dual Thresholds Controled Vacation Strategies Of M/M/1 Queue

Nowadays,with the rapid development of technology,the transportation of network communication is more and more complex,and the requirement of system performance and service quality will be higher and higher.Queuing theory,as an effective tool to solve the application research,needs to introduce a new vacation strategy.Under this background,we introduce second optional vacation and dual thresholds controlled strategies based on working vacation,respectively to establish the corresponding mathematical model and analysis,and obtain some important performance index.The main contents are as follows:According to the classic vacation continuous time M/M/1 queueing system,we add the second optional vacation strategy.Based on the model assumes we get the two-dimensional Markov chain,and by analyzing the status transfer rule,we get its infinitesimal member.By using matrix geometric method and QBD,we get a steady state distribution of the system,average waiting time and average captain.Finally,we propose numerical analysis and use Matlab to analyze the trends of system performance index with the parameters change.Based on the M/M/1 queueing system with working vacation,we introduce the double thresholds controlled strategy.By using matrix geometric method and QBD,we get a steady state distribution of the system.On the basis,we present the distributions and the conditional stochastic decomposition structures for the number of customers and the waiting time.We also give the additional queue length probability generating function(PGF)as well as a Laplace-Stieltjes transform(LST)of the additional delay.In a word,this paper discusses two kinds of special vacation queue models,which not only enrich the theory of continuous time queue systems with vacation strategies,but also provide a more complex model to approach life.According to the numerical analysis,we discuss the trend of the performance index.It has a certain practical significance.