Research On Queueing System With Negative Customers And Server Breakdown

With the wide application of random service systems,more scholars have devoted themselves to the study of queueing models with negative customers and server breakdown.Combined with embedded Markov chain method,the steady-state conditions and performance indexes of queueing system are analyzed,which is beneficial to the optimization and management of actual service system.Based on this,considering different service systems in real life,this thesis constructs the queueing models with different strategies and analyzes queueing models with negative customers and server breakdown.Firstly,the research background and significance of the queueing model with negative customers and server breakdown are introduced.The basic knowledge of queueing theory and the analysis methods involved in the study are briefly described.And the research status of queueing model with different strategies such as negative customers,vacation and server breakdown is specifically expounded.And this provides theoretical basis for the study.Secondly,this thesis discusses an M/G/1 G-queue with server breakdown,working vacations and Bernoulli vacation interruption.In this model,the main influence of negative customers is to take away a positive customer who is receiving service during normal busy period and cause server failure.In order to improve the service efficiency of equipment,upon the completion of a service in the vacation period,if the system is not empty,the vacation can be interrupted with probability p or continues with probability1-p.The expressions of the probability generating functions of the server state and the queue length under steady state condition are obtained.The effect of the parameters on the queue length are explored by numerical examples.The cost analysis provides optimization basis for actual production system.Thirdly,combined with the practical application of computer network system,the M/G/1 G-queue with working breakdown and single vacation is analyzed.In this system,the server working at a lower service rate instead of stopping services during the working breakdown period.The various queueing indexes are given.In addition,the reliability index is obtained.The effect of some parameters on the mean queue is analyzed and demonstrated by computer.Finally,based on the above analysis,the M/G/1 queueing model is extended to the M~X/G/1 queueing model.And an M~X/G/1 G-queue with single vacation,setup times and working breakdown is studied.The arrival process of positive customers in batches is a compound Possion process.The server turns off if the system becomes idle at a vacation completion instant.And the service of the first positive customer must take a setup time from the turned-off server.The embedded Markov chain is constructed and the steady-state indexes of the system are discussed.MATLAB is used to simulate numerical examples and draw diagrams to explain the effect of system parameters on mean queue length.The cost function is constructed.Then the minimum value and approximate optimal solution of the cost function are obtained based on parabolic method,which makes the model have more practical guiding value.