Nearly Deterministic Queue Models In Heavy Traffic

Nearly deterministic queues are extensively applied in different fields.Therefore,it is of great theoretical significance and practical value to study nearly deterministic queue models.This thesis mainly studies the performance indicators for nearly deterministic queue models,such as waiting time and queue length.The thesis firstly establishes a standard sequence framework(1)and(2),so as to get the heavy-traffic limit for the waiting time in G/G/s model under the two frameworks respectively.Then it refines arrival process and service process of G/G/s model for a period of n and introduces two heavy-traffic conditions:(1)(1)(1-?n)n(?)? ?,0 ??<?(n ? ?);(2)(1-?n)n ? ?,0 ? ?<?(n??),so as to obtain the heavy-traffic limit for waiting time in Gn/Gn/s model.And using first-passage function to study waiting time,the thesis finds out the expression and heavy-traffic limit for the first-passage function of the waiting time in Gn/Gn/s model.Secondly,based on the heavy-traffic limit for waiting time in Gn/Gn/s model,in combination with the cyclic thinning nature of the steady-state point process,the thesis gets the heavy-traffic limit for the steady-state waiting time in the model.If both the arrival process and the service process are independently and identically distributed in Gn/Gn/s model,GIn/GIn/s model is thereby attained.The thesis mainly studies the steady-state waiting time and queue length in GIn/GIn/s model to obtain the model's nature in terms of stationary delay probabilities for the steady-state waiting time,mean steady-state waiting time,the heavy-traffic limit for steady-state waiting time and steady-state queue length.Furthermore,it extends the general arrival process to a special arrival process whose arrival rates varies with time,taking the Mt/GI/? model with cosine arrival rates for example,the steady-state probability quality function and accumulated distribution function in Mt/GI/? model are derived.In the last part,taking single-server queue for example,the thesis conducts an simulation with Matlab as to the waiting time and queue length process in Gn/Gn/1 model,and carries out an approximate evaluation for the steady-state waiting time in GIn/GIn/1 model.