Performance Analysis And Optimal Control Policy Of M/G/1 Queue With Single Vacation Under The (p,N)-Policy Control

This dissertation considers an M/G/1 queueing model with uninterrupted single vacation under the control of randomized setup(p,N)-policy.With a direct probability decomposition technique,that is,employing the renewal process theory,total probability decomposition technique and Laplace transform tool,we investigate the transient and steady-state distributions of the queue length at any time t in detail.The sensitivity analysis of the idle rate and additional expected queue length of the system to some main parameters of the system is carried out with numerical examples.Furthermore,on the basis of establishing the cost model,the optimal control policy to minimize the system cost is discussed by using the renewal reward theorem and Matlab Software tool.This dissertation is divided into three Chapters as follows.1)In the Chapter 1 of this dissertation,by combining the randomized setup(p,N)-policy with the server’s single vacation,we propose a new queueing model—an M/G/1 queuing model with uninterrupted single vacation under the control of randomized setup(p,N)-policy,in which whenever the system becomes empty,the sever immediately takes an uninterrupted vacation.The randomized setup(p,N)-policy means that when the vacation ends and the server returns to the system,if the number of customers in the system is greater than or equal to a given control threshold value N(N≥1),the server starts its service immediately.If there are less than N customers but at least one customer in the system,the server begins its service with probability p(0 ≤p ≤1)or stays idle with probability 1-p until there are N customers in the system and starts its service at once.If no customer arrives in the system during the vacation,the server stays idle(but on duty)until there is one customer arrives in the system and starts its service immediately.Then,we use the renewal process theory,total probability decomposition technique and Laplace transform tool to discuss in detail the transient probability distribution of the queue length at any time t that the system starts from any initial state.The expressions of the Laplace transform of the transient queue size distribution concerning time t are obtained.2)In the Chapter 2 of this dissertation,on the basis of transient analysis in the Chapter 1,we further discuss the steady-state probability distribution of the queue length at any time t and the embedded Markov chain of the queue length of the system.First of all,using L’Hospital’s rule and some algebraic manipulations,we obtain the recursive expressions of the steady-state queue length distribution,the probability generating function of the steady-state queue length distribution and the expression of the average steady-state queue length as well as the random decomposition structure of the steady-state queue length.Then,in the steady-state,the embedded Markov chain of the queue length is also studied,and the probability generating function expression of the steady-state distribution for the embedded Markov chain.Secondly,some other important queueing performance indexes of the system and some special cases are derived.Finally,we employ numerical examples to discuss the sensitivity of the idle rate and additional average queue length to some main parameters of the system and the application of the steady-state probability distribution in the optimal design of the system capacity.3)In the Chapter 3 of this dissertation,based on the established cost model,employing the renewal reward theorem,the expression of the cost objective function of the system in a long-term unit time is derived.When the probability p is determined,computational experiments are implemented to investigate the one-dimensional optional control strategy N*and the two-dimensional optional control strategy(N*,T*)for economizing the system cost by using Matlab Software.