Research And Analysis Of A Modified Queuing Theory Model

In the traditional queuing theory,customers arrive at the system with the same distribution,receive services with the same distribution,and then leave the system and the models have M/M/1,M/G/1,and G/G/1.Different from the traditional queuing theory,this paper studies the modification of the queuing theory.The service time of customers who arrive in the system for the first,time when the system is empty is not the same as the service time of customers who arrive in the system when the system is not empty.In the modified queuing theory,the distribution of customers arriving at the system is still subject to a Poisson distribution with a rate of A.The main contents of this paper are as follows:First,introduce the background of the problem of modified queuing theory,and then study the length of the queue,customer waiting time in the system,and the proportion of system idle time to compare the relationship between modified queuing theory and traditional queuing theory models.And are not homogeneous,so two discrete-time Markov chains are introduced;the relationship between the modified queuing theory and two discrete-time Markov chains is further analyzed.Secondly,by using the knowledge of queuing theory,it is proved that the queue length,the waiting time of customers in the system and the proportion of idle time of the system are between M/G/1 model and M/D/1 model.The matrix of transition probability of two discrete time markov chains is found,and it is proved that the modified queuing theory is a positive recurrent with a stationary distribution,and the limit average queue length and the limit idle time ratio of the modified queuing theory are between the two discrete time markov chains.And give a few guesses.Finally,through numerical simulation,verify that the numerical solution of the queuing theory is consistent with the theoretical solution;verify that the modification of the queuing theory is the positive recurrent;verify that the queue length of the modified queuing theory,the system customer waiting time,and the system idle time ratio are between MG1 and In M/D/1,verify that when ??0,modify the queue length,system customer waiting time,and system idle time ratio of the queuing theory to the M/G/1 distribution queue length,system customer waiting time,and system idle time ratio;When??1,modify the queue length of the queuing theory,the system customer waiting time,whether the system idle time ratio tends to the M/D/1 distribution queue length,the system customer wait time,and the system idle time ratio.Then the numerical solution of the discrete time markov chain R(n)is obtained.