Research On Vacation Queueing System In Multi-phase Operational Environment

Due to the important applications in communication networks,inventory management,production management,flexible manufacturing systems,etc,queueing system with vacations has received wide attention of researchers since its birth.In most of the existing models on vacation queue,the arrival and service rates are homogeneous,but in actual problem,they may change with changes of the environment of operation.This dissertation is devoted to studying a class of queueing system with vacations in multi-phase operational environment,which can be understood as a class of nonhomogeneous vacation queueing system.The main works are as follows:Firstly,we consider an M/M/1 vacation queue in multi-phases operational environment.By using the stability condition of classical M/M/1 vacation queue,we obtain the necessary and sufficient condition for the stability of the system that we consider.By using the method of probability generating function and solving the equilibrium equations,we derive the probability generating function of stationary queue length at arbitrary epochs.We also derive some important performance measures of the system.Then,we investigate some special cases of our model.Finally,we give some numerical examples to demonstrate the impact of parameters on some crucial performance measures of the system.Secondly,we discuss an M/G/1 vacation queue in multi-phases operational environment,which extends the above M/M/1 queueing system.Because the service times follow a general distribution,the queueing system we discuss is a non-Markovian system.So,using the method of supplementary variables,we represent the state of the system by constructing a Markov process.Then,we can obtain the stationary equations of the system.By solving these stationary equations,we obtain the probability generating function of stationary queue length at arbitrary epochs.The stationary sojourn time distribution of an arbitrary customer is also derived.In addition,the stochastic decomposition property is investigated.Finally,we present some special cases of the model and some numerical results,Thirdly,we deal with an M/G/1 queue with Min(N,V)vacation policy in multi-phases operational environment.Under Min(N,V)vacation policy,a better balance between quick response time and efficient use of resources can be most likely achieved,but the boundary conditions of the system become more complex.By solving the Kolmogorov equations and boundary conditions equations,the distribution for the stationary queue length at arbitrary epochs is derived along with some important performance measures of the system.Then,we discuss some special cases of the model,and provide some numerical examples to illustrate the effect of several parameters on some crucial performance measures of the system,including the effect of N on the mean queue length and the probability that the system is empty.Fourthly,we study a GI/M/1 vacation queue in multi-phases operational environment,where the inter-arrival times follow a general distribution,causing the analysis method to be different from the ones for the above several systems.By using the embedded Markov chain,we obtain the distribution of the stationary queue length at arrival epochs;By using the semi-Markov process,we obtain the distribution of the stationary queue length at arbitrary epochs.Furthermore,we also obtain some performance measures such as the stationary waiting time distribution.Finally,we demonstrate the influence of the model parameters on several performance characteristics through some numerical experiments,especially the inter-arrival time distribution and multiple phases of operation on several performance characteristics.Fifthly,we investigate a Geo/G/1 discrete-time vacation queue in multi-phases operational environment.Discrete-time queue and continuous-time queue have big differences on system description,analysis method,expression of the results,etc.Choosing the remaining service time of the customer being served as supplementary variable,we get the equilibrium equations of the system.Then we obtain the probability generating function of stationary queue length at arbitrary epochs and some other important performance measures of the system.In addition,we analyze the relationship between the discrete-time system and its continuous-time counterpart,and show that the continuous-time system can be approximated by the discrete-time system.Finally,some special cases of the system we consider are analyzed,and some numerical examples are presented.