Research On The Queue Length Distribution In Delayed Arrival Queuing Model

In the traditional queuing network,the transfer of customers between nodes is usually assumed to be instantaneous,which often does not conform to the actual situation.This thesis considers a single-node queuing model with delayed arrival of customers,and analyzes three possible situations in the process of the transfer of customers between nodes.In view of the above problems,this thesis mainly completes the following three tasks:（1）Firstly,the problem of delayed arrival queuing problem is discussed.Adding a delayed arrival process to the queue before customers enter the service system and model it as a delayed arrival queuing model.The added delayed arrival process can be modeled as a virtual node.Based on the order in which customers enter the delayed node and leave the delayed node without changes,this chapter establishes a single-node order preserving type delayed arrival system,namely M/M/（K）→·^（X）/M/1 queuing model.The transition of customers from delayed node to formal node is described by probability method,the steady probability distribution of the length of the model system is solved by constructing M/G/1 type markov chain and using matrix geometry method.Numerical examples are given at the end of this chapter.（2）Then,based on the situation in chapter three where customers do not need to preserve the order in the delayed node,establishing a without order preserving type delayed arrival system,namely M/M/（K）→·/M/1 queuing model.By constructing quasi-birth-death-process and using the matrix geometry method,the algorithm of solving system steady-state probability is presented.Numerical examples are given at the end of this chapter.（3）Finally,the queue-change phenomenon of customers in the delay node is studied.Considering that there are two parallel queues in the delay node,the queue-change model is established to solve the problem that customers choose the wrong queue with a certain probability when entering the delay node.By constructing GI/M/1 type markov chain and using matrix geometry method,an algorithm is given to solve the steady-state probability distribution of queue length in the model.Through numerical examples,it is found that the probability of customer change queues has obvious influence on the distribution of queue length of two service stations in the delay node.