Research On A Two-queue Cyclic Polling System With Batch Markovian Arrival Process And Priority Service

With the enormous development in the field of information technology,the data diversity is becoming obvious more and more.Different data may have different service priority levels,and the Quality of Service required by the data with different service priority levels are different.So,it is necessary that the appropriate data service scheme be configured in the information systems according to the specific situation.As one theoretical tool,the polling system can be used to design or analyze the data service scheme,and further optimize the scheme.Therefore,it is worth to study the polling systems with priority service.In this thesis,a two-queue cyclic polling system with batch Markovian arrival processes and priority service is analyzed.This system consists of one server and two queues,these two queues are called queue 1 and queue 2 respectively,and the buffers of the two queues are both finite.Customers arrive at the two queues according to two independent batch Markovian arrival processes(BMAPs),and the service priority levels of the customers in different queues satisfy:queue 1 > queue 2.When a batch of customers arrive at the queue 1 or queue 2,if there are not enough capacities in the buffers of the queue 1 or queue 2,then a part of the customers can enter into the corresponding queue;if there are no capacities,then all of the arriving customers are rejected.The server attends the two queues in the cyclic manner,and the switchover times of the server transferring from a queue to the other one are considered.The queue 1 is served according to the gated service discipline,and the queue 2 is served according to a state-dependent timelimited service discipline.The service discipline attached to the queue 2 satisfies the preemptive repeat-different property.For each queue,the service order is FCFS(First Come First Served).For the two-queue cyclic polling system with batch Markovian arrival processes and priority service,the main contributions of this thesis are as follows:(1)Introduce and construct the joint arrival process of multiple independent batch Markovian arrival processes. Before analyzing the polling system,the joint arrival process of multiple independent BMAPs is introduced,and its parameter matrices are constructed.Moreover,it is proved that these parameter matrices satisfy the same properties as the ones corresponding to the BMAP;subsequently,these properties are used in analyzing the polling system with batch Markovian arrival process.(2)Analyze and obtain the joint queue length stationary distribution at arbitrary time,and the expressions for the mean queue lengths of different queues at arbitrary time.An embedded Markov chain is constructed,and it displays the state transitions of the polling system at the service complication and switchover termination epochs.A semi- regenerative process is constructed,then based on the previous embedded Markov chain and the limit distribution theorem of the semi-regenerative process,the joint queue length stationary distribution at arbitrary time is obtained.According to this stationary distribu-tion,the expressions for the mean queue lengths of different queues at arbitrary time are obtained.(3)Analyze and obtain the Laplace-Stieltjes transforms of the waiting time distributions of the virtual customers with different service priority levels,the expressions for the mean waiting times of the virtual customers with different service priority levels,and the ex- pressions for the mean waiting times of the actual customers with different service priority levels.An additional event and its counting process are introduced.An embedded Markov chain accompanied by the additional event is constructed,and it displays the state tran-sitions of the polling system accompanied by the additional event,at the service compli-cation and switchover termination epochs.A semi-regenerative process is constructed, then based on the embedded Markov chains in(2)as well as(3)and the limit distribution theorem of the semi-regenerative process,the Laplace-Stieltjes transforms of the wait- ing time distributions of the virtual customers with different service priority levels are obtained.According to these Laplace-Stieltjes transforms,the expressions for the mean waiting times of the virtual customers with different service priority levels are obtained. In addition,based on the mean queue lengths,and according to the Little's Law,the ex- pressions for the mean waiting times of the actual customers with different service priority levels are obtained.(4)Analyze and obtain the stability condition of the polling system with infinite buffers.When the buffers of the polling system are infinite,two embedded Markov chains are constructed,in which the embedded epochs are the instants when the server attends the queue 1.One embedded Markov chain displays the state transitions of the queue 1 at the embedded epochs,provided that the number of customers in the queue 2 is infinite;the other displays the state transitions of the whole system at the embedded epochs.Subse-quently,based on these two embedded Markov chains,the local stability condition and the global stability condition of the polling system are obtained respectively.Moreover,the calculations of the variables related to the stability conditions are given.(5)Propose a packet transmission protocol with priority and fairness,and give the optimization method of the protocol.A packet transmission protocol with priority and fairness is proposed,and it corresponds to the polling system analyzed in this thesis.Then,according to the analysis results of the polling system,an optimization scheme for the packet transmission protocol is given based on a cost function.Finally,some numerical examples are carried out to illustrate the optimization process of the protocol and the variation of the(virtual and actual)packet mean waiting times against the packet arrival rates.