Analysis Of M/G/1 Queueing System With Bi-level Threshold(m,N)-policy And Uninterrupted Single Vacation

In this dissertation,we study two M/G/1 queueing models with bi-level threshold(m,N)-policy and single server vacation without interruption.It is divided into two chapters as follows.In the first chapter of this dissertation,based on the actual situation,the vacation rule of“uninterrupted single vacation”is introduced into the M/G/1 queueing system with s-tartup time?bi-level threshold(m,N)—policy.In this queueing system,both the starting threshold m(m?1)of the system and the threshold N(N?m)for the server to start serv-ing customers are positive integer and are given in advance.Once the system becomes empty,the server takes a vacation,and the length of the vacation is a random variable.When the server is transferred on vacation,if the number of waiting customers in the system is no less than m,the server immediately starts the system,which will take a random length of time.When the system startup is completed,if the number of waiting customers in the system is no less than N,the server begins service immediately until the system becomes empty again.The server's vacation time and the startup time of the system are assumed to follow general distributions.Firstly,using the renewal process theory?the Laplace transform tool and the to-tal probability decomposition method in probability theory,the probability distribution of the system queue-length at any time t,i.e.the transient solution of the system queue-length dis-tribution is discussed,and the steady-state probability distribution of the system queue-length,i.e.the steady-state solution of the system queue-length distribution is successively discussed.We obtain the expressions of the Laplace transform of the transient solution with respect to time t.And then through some algebraic operations,the recursive formulas of the steady-state solution of the system queue-length distribution with important application value are presented.Second,based on the discussion above,the stochastic decomposition structure of the steady-state queue-length,the explicit expression of the average queue size are given.In addition,the relevant results of the system are also obtained under some special circumstances.Finally,in Section 1.6 of this chapter,the cost problem of the system is discussed by establishing the cost model and cost objective function.And with the help of MATLAB software,the optimal value(m*,N*)of two-dimensional decision variables that minimizes the expected cost per unit time of the system's long-term operation is determined by a numerical example.In the second chapter of this dissertation,by introducing the"delayed vacation' into the M/G/1 queueing system studied in the first chapter,we further constructs the model of M/G/1 queueing system with bi-level threshold(m,N)-policy and delayed single vaca-tion without interruption,which is more complex and difficult to study.Applying the same arguments used in the first chapter,the corresponding queuing indicators of this model are dis-cussed.Some corresponding results,such as the expressions of the Laplace transform of the transient queue-length distribution with respect to time t,the recursive formulas of the steady-state solution of the system queue-length distribution,the stochastic decomposition structure of the steady-state queue-length and the explicit expression of the average queue size,are ob-tained.Meanwhile,the relevant results of the system in some special cases are given.Finally,in Section 2.6 of this chapter,by establishing the cost model and cost objective function,we discuss the expected cost per unit time of the system's long-term operation.With the help of MATLAB software,the optimal value(m*,N*)of two-dimensional decision variables that minimizes the expected cost is numerically determined.