Research On The Stability Of Large Queuing Networks Based On Mean Field Theory

With the advancement of science and technology,queuing systems consisting of multiple queues have increasingly appeared in computer communication technology,transportation and telecommunication networks.For such queuing systems,a central research question is which server to assign incoming jobs to achieves the best performance.The load balancing mechanism is one of the main measures to optimize the performance of the queuing network,which can effectively improve the service rate of the queuing system and shorten the length of the queue.Joining the shortest queuing rule is one of the commonly used load balancing mechanisms.First,in previous researches,the shortest queuing rule is mostly used in queuing networks with a small number of queues.When the number of queues N is large,it will become a high-dimensional interactive system after introducing a balance mechanism.It is complex and difficult to study.For such large queuing systems with interactions,the main research method is the mean-field model,which has the advantage that the limit properties of queuing systems can be studied by finding the solutions of deterministic systems.This paper focuses on large queuing networks The shortest queuing rule is introduced,and the mean field theory is used to study its stationarity and analyze its performance.First consider the queuing network consist of N parallel queues(N is large but finite).Each queue has a Poisson arrival with an arrival rate ofl,and,in addition,there is an smart arrival with an arrival rate ofNl_s,from which customers join the shortest queue in the queuing network,and the service rate is μ,i=1,2,3,...(i represents the number of customers).For this queuing system,first,by establishing a mean field interaction model to study the limit behavior of the queuing system,it is obtained that the empirical measure of the queue length in the queuing system converges to the solution of the nonlinear master equation.Then,the stationary distribution of the queuing system is solved,and finally it is verified that the performance of the system has been improved.In an open Jackson network with J nodes(J is large but finite),a smart arrival flow with an arrival rate of Jλ_s is introduced,and customers who arrive intelligently join the shortest queue in the Jackson network.For this queuing system,first assume that the total arrival rate of each node is a constant.By establishing a mean field interaction model,it is obtained that the empirical measure of queue length in the Jackson network converges to the solution of the nonlinear master equation.Finally,it is verified that the Jackson network under the shortest queuing rule the performance has been improved.