Performance Analysis Of Queueing Systems With Bi-Level Randomized（p,N₁,N₂）-Policy

In order to reduce the cost caused by the frequent switching,the investigation concerning controllable queueing models with control policy have received great attention from scholars.This thesis proposes a bi-level randomized（p,N₁,N₂）-policy.That is,whenever the system becomes empty,the server is closed down immediately.When the number of customers arriving in the system reaches a given low threshold value N₁（≥1）,the server is activated for work with probability p（0≤p≤1）or still left off with probability（1-p）until the number of customers arriving in the system reaches a given high threshold value N₂（≥ N₁）.This thesis studies four queueing models with the bi-level randomized（p,N₁,N₂）-policy.It is divided into the following five chapters:In the first chapter of this thesis,an introduction is presented.It briefly introduces the origin and research background of queuing theory,and outlines the research status of queueing systems with vacation and control policy.In the second chapter of this thesis,we establish an M/G/1 queueing model with bi-level randomized（p,N₁,N₂）-policy.First,we study the transient queue length distribution at any time by using the total probability decomposition technique and Laplace transform.Then,employing L’Hospital’s rule and some algebraic operations,the recursive formulas of the steadystate queue-length distribution are presented.Meanwhile,the expressions of its probability generating function of the steady-state queue-length distribution,the expected queue size and other queuing performance indicators are obtained.Moreover,the influence of the system parameters on the steady-state queue-length distribution is discussed through numerical examples as well as the role of the steady-state queue-length distribution in the system capacity optimization design.Finally,on the basis of the cost model established in this dissertation,we find the optimal bi-level threshold values（N₁^*,N₂^*）and its corresponding minimum expected cost that satisfy the average waiting time constraints by numerical calculation.In the third chapter of this thesis,we propose an M/G/1 queue model with bi-level randomized（p,N₁,N₂）-policy and single vacation by combining the server’s vacation rule.In this queueing system,the single vacation of the server is interruptible,that is,whenever the system becomes empty,the server takes a vacation immediately.During the vacation process,once the system accumulates N₁（≥1）customers,the server ends the vacation with probability p（0≤p≤1）and immediately starts the system to serve customers,or the server continues the vacation with probability（1-p）until the number of customers in the system reaches N₂（≥N₁）.When the number of customers in the system reaches N₂,the server immediately ends the vacation to serve customers or only after the end of this vacation to serve customers.Applying the analysis method and analysis route in chapter 2,a series of queueing performance indicators,such as the transient queue length distribution,the steady-state queue length distribution and so on,are discussed.Meanwhile,the departure process of the system is studied.And through numerical examples,the influence of the system parameters on the additional average queue length and the optimal design of the system capacity are also investigated.Finally,based on the established cost model,the optimal vacation time of the server and the optimal bi-level threshold values（N₁^*,N₂^*）of the system and its corresponding minimum expected cost are studied under the constraint of the average waiting time of the customer.In the fourth chapter of this thesis,considering the difference between discrete-time queueing systems and continuous-time queueing systems,and the fact that discrete-time queueing models have wider applications in mobile communication and other fields.Therefore,this dissertation extends the model studied in chapter 2 to the corresponding discrete-time queueing system,and propose a Geo/G/1 discrete-time queueing model with bi-level randomized（p,N₁,N₂）-policy.Then,a series of queueing performance indexes,such as transient and steady-state queue length distribution at epochs n^-,n,n⁺ and outside observer’s observation epoch,are discussed by using the renewal process theory,total probability decomposition technique and z-transform.Furthermore,through numerical examples,the effects of the system parameters on the probability that the system is empty and the mean queue size are discussed as well as the optimal design of the system capacity.Finally,employing the renewal reward theorem,we establish a cost function to investigate the cost optimization problem under the constraint of the average waiting time,and obtain optimum threshold values（N₁^*,N₂^*）and the corresponding minimum expected cost.In the fifth chapter of this thesis,this thesis considers a discrete-time Geo/G/1 queueing system with bi-level randomized（p,N₁,N₂）-policy and uninterrupted single vacation.In this discrete-time queueing system,the single vacation of the server is uninterrupted,i.e.,whenever the system becomes empty,the server shuts down the system and takes a full vacation.When the server returns from vacation,if the number of customers in the system is greater than or equal to N₂,the server immediately starts the system to serve customers.If the number of customers in the system is greater than or equal to N₁（≥1）but less than N₂（≥N₁）,the server starts the system to serve customers with probability p（0≤p≤1）or stays in the system with probability（1-p）until the number of customers in the system is N₂.If the accumulated number of customers in the system is less than N₁（including there is no customer in the system）,the server stays in the system and begins to serve the customers only when the number of customers in the system equals N₁.Using the analysis techniques and lines of analysis in chapter 4,a series of queueing performance indexes such as transient and steady-state queue length distribution at epochs n^-,n,n⁺ and outside observer’s observation epoch are discussed.Meanwhile,the departure process of the system is analyzed.Furthermore,numerical examples are presented to investigate the effects of system parameters on the probability that the system is empty and the mean queue size.At last,the cost optimization problem of the system is discussed under the constraint of the average waiting time.And the optimum threshold values（N₁^*,N₂^*）and the corresponding minimum expected cost are obtained.