Queueing Models With Very Heavy Tails In Heavy Traffic

In1917, A.K.Erlang put forward the congestion theory on communicationbusiness, and analyzed the communication traffic with the statistical method, whichformed a new branch of probability theory. When people want to use phone, users willhave to wait if telephone exchanges in all the bolt has been occupied. This is a kind ofinvisible queueing phenomenon. And as in storage-forwarding data transmissionnetwork, when information get to network node and wait for processing andtransmission, it would form a line up which is visible although we are not easy to see.The queueing phenomenon is due to the randomness of the customer demand and thelimitation of service facilities.In this paper, we introduces a special queueing model which is the on-off modelsand M/P_α/1（0＜α＜1） with very heavy tails in heavy traffic.It is different fromthe traditional model in the following three aspects. Firstly, this paper discusses theservices in the busy period, queue length and waiting time of the queueing models inthe SkorohodJ₁topology on the function space D, which has very heavy tailsdistribution. That means the variable has infinitely variance. Furthermore the modelutilizes continuous mapping theorem getting the limit theorem of every process inheavy traffic, and reflects it on non-gaussian stable processes. Finally, in thesimulation part, this paper analyzes five indicators about arriving time, service time,waiting time, departure time and queue length. And it makes a comparison betweenthe five process diagrams of M/P_0.5/1and the basic information chart of theM/M/1queueing model, and then compares it to the dual model P_0.5/M/1.Werandomly selected10customers’ information in order to make us have a betterunderstanding of the differences of the five index when service and arriving with veryheavy tails distribution respectively. This kind of queueing model is mainly applied inthe communication system.