Perturbation Analysis Of Transient Behavior Of M/M/m/m Queue

The Erlang loss model（M/M/m/m queue）is one of the basic models in queueing theory.It is of great significance in the analysis of blocking in teletraffic.Assuming that in this model,the maximum server capacity of the server is m.When the number of occupied servers N（t）<m,the arrival rate is λ₀,and when N（t）≥m,the arrival rate is 0.The service rate of each server is μ.When 0 ≤n ≤m,The probability of the number of servers being occupied N（t）=n at time t isp_n（t）.And the balance equations forp_n（t） is where δ is the Kronecker delta.It is often involved to obtain exact solutions for this type of differential difference equations,so it is necessary to obtain an asymptotic result that can be expressed by elementary function or fully studied special functions.The WKB method can be directly applied to a given equation when the exact solution is not yet clear,and is a commonly used analytical solution.In many practical situations,the number of servers m is quite large,so obtaining the asymptotic results for limit m >>1 is very useful.This paper discusses the sign problem of the eigenvalues of a class of tridiagonal matrix in the M/M/m/m queue transient equilibrium model,and gives the bounds of the maximum eigenvalues according to the module by the power method,that is,when m → +∞,the maximum eigenvalues according to the module|λ|_max =O m.Proved the following conclusionTheorem: If the tridiagonal matrix in the form of matrix B has only real eigenvalues,then for the determined m,the eigenvalue is not greater than zero.Where w_i =-mα-i-β,α and β are nonnegative arbitrary constants.In Chapter 3,we assume n₀ =0,based on the asymptotic approximation ofp_n（t） proposed by Knessl using the WKB method at m>>1,λ₀=O（m）.Then we provide an approximate solution forp_n（t）.In Chapter 4,we compared the numerical and approximate solutions obtained by the classical fourth order Runge Kutta method,and discussed the relative error between the expected exact solution and the expected approximate solution of the probability distribution,verifying the effectiveness of the approximate solution.