Modeling And Optimizing For M/M/1 Queuing System Based On Double-threshold Control Policy

In recent years,with continuous development of queuing theory in both breadth and depth,it has been widely studied and applied to a variety of service systems,such as production management,traffic system,call center and communication transmission system and so on,and has achieved fruitful results of theory and practice.For the commonly congestion phenomena in the queuing system,the scholars have put forward a variety of strategies for modeling and optimization of the queuing system in order to reduce the system's average queue length and customers' waiting time,and further improve the operation efficiency of the service counter.Among them,the research of the queuing system with threshold control policy is gradually becoming a new,important and hot content in queuing theory.It's pretty common of the queuing phenomena with threshold control policy in our life.For example,the free release policy in highway toll is successively developed in various provinces and cities in recent years,the common method of producing the incoming material is seen by using two rates with a high gear and a low gear in random production system,and the policy of switching different broadband is applied to control bandwidth costs in Communication network.According to these actually queue phenomena,this thesis proposes a control policy with double thresholds(m,N)of queue system to coordinate service system operating costs and customer waiting time.Which is to say,the service desk has two service rates with a fast one b? and a slow one i?,and switches its service rates at two pre-set thresholds for a large one and a small one based on queue length.When the queue length gradually increases from zero to N,the running rate of the service desk changes to the fast one b? from the slow one i?,until the queue length gradually drops to m,the operating rate of service desk switches to the slow one i? from the fast one b?.First of all,the thesis studies the M/ M/1 queuing model with a single threshold control policy(which is to say the N-policy)of queuing model based on the traditional M/ M/1 queuing model.Then the steady-state probability formula and the key performance indicators of the model are given out.And comparison of the queuing model under single threshold with the traditional queuing model are conducted by means of numerical experiments,which proves that the single threshold control policy has more superiority in easing congestion and improving the operational efficiency of the service system.Furthermore,the Arena software is applied to simulate this model,and the simulation results verify the accuracy of those formulas developed by mathematical model and some general rules.Secondly,the thesis studies the M/ M/1 queuing model with double threshold control policy.The overlap region with a same queue length and a different service rate is represented by the state code in the process of system state transition,and the state transfer matrix of the model and many key performance indicators such as the steady-state probability,average captain,average waiting time and the state transition probability and so on are obtained by the method of Markov processes and birth and death process.And the further numerical simulations discuss the influences of the two thresholds on those key indicators,some common rules proof the robustness of indicators through these analysis.Finally,based on the results of traditional M/ M/ 1 queue model and the queuing model with a single threshold control policy,this article constructs a relative net income function based on cost structure for the control policy with double thresholds comparing with no control policy queuing model,and gives the optimization model of net revenue function.Then,the net search algorithm is designed to obtain the optimal results of the biggest profit and the corresponding thresholds.Furthermore,the numerical experiments are also operated to study the influences of the changes of double thresholds and system parameters on the system net income and average queue length.Some varying regulations of the net income function and key performance indicators on system parameters are developed,which demonstrates that the dual thresholds control policy is more superiority than single threshold.Based on the common phenomena in our life,which is to solve the congestion problems between congestion and service cost by controlling the serve rate with queue length,the queuing model with threshold control policy is studied in this thesis.On the one hand,the results could enrich the achievements of the application of queue theory in the field of the threshold control policy emerged in recent years;on the other hand,the results might give some reality guiding advices to optimize the management of service queuing system,improve service levels,and promote the system efficiency for managers of the service agency.It is hoped that our research could provide some references to the management decisions in transportation,production and services system and so on.