Invariant Subspace of Solving Ck/Cm/1

Abstract

在這一篇論文中，我們討論 Ck/Cm/1 的等候系統。 我們利用矩陣多項式的奇異點及向量造 C_k/C_m/1 的機率分配的解空間。而矩陣多項式的非零奇異點和一個由抵達間隔時間與服務時間所形成的方程式有密切的關係。我們證明了在 E_k/E_m/1 的等候系統中，方程式的所有根都是相異的。但是當方程式有重根時，我們必須解一組相當複雜的方程式才能得到構成解空間的向量。此外，我們建立了一個描述飽和機率為 Kronecker products 線性組合的演算方法。

In this thesis, we analyze the single server queueing system\nCk/Cm/1. We construct a general solution space of the vector for stationary probability and describe the solution space in terms of singularities and vectors of the fundamental matrix polynomial Q(w). There is a relation between the singularities of Q(w) and the roots of the characteristic polynomial\ninvolving the Laplace transforms of the interarrival and service\ntimes distributions. In the Ek/Em/1 queueing system, it is proved that the roots of the characteristic polynomial are\ndistinct if the arrival and service rates are real. When\nmultiple roots occur, one needs to solve a set of equations of matrix polynomials. As a result, we establish a procedure for describing those vectors used in the expression of saturated probability as linear combination of Kronecker products.