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Matrixgeometric Solutions of M/G/1 Type Markov Chains: A Unifying Generalized Statespace Approach

Summary: Matrix­geometric Solutions of M/G/1 Type Markov
Chains: A Unifying Generalized State­space Approach
Nail Akar Nihat Cem O–guz and Khosrow Sohraby \Lambda
Technology Planning & Integration Computer Science Telecommunications
Sprint University of Missouri­Kansas City
9300 Metcalf Avenue 5100 Rockhill Road
Overland Park, KS 66212 Kansas City, MO 64110
akar@sprintcorp.com fncoguz,sohrabyg@cstp.umkc.edu
In this paper, we present an algorithmic approach to find the stationary probability distribution
of M/G/1 type Markov chains which arise frequently in performance analysis of computer and
communication networks. The approach unifies finite­ and infinite­level Markov chains of this
type through a generalized state­space representation for the probability generating function of
the stationary solution. When the underlying probability generating matrices are rational, the
solution vector for level k, x k , is shown to be in the matrix­geometric form x k+1 = gF k H, k – 0,
for the infinite­level case whereas it takes the modified form x k+1 = g 1 F k
1 H 1 + g 2 F K \Gammak\Gamma1
2 H 2 ,
0 Ÿ k ! K, for the finite­level case. The matrix parameters in the above two expressions can be
obtained by decomposing the generalized system into forward and backward subsystems , or equiv­


Source: Akar, Nail - Department of Electrical and Electronics Engineering, Bilkent University


Collections: Engineering