626 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 16, NO. 5, JUNE 1998 Matrix-Geometric Solutions of M/G/1-Type Markov Chains: A Unifying Generalized State-Space Approach Nail Akar, Member, IEEE, Nihat Cem O˘ guz, and Khosrow Sohraby, Senior Member, IEEE Abstract—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 networ ks. The approach unifies finite- and infinite-level Markov chains of this type through a generalized state-space representation for the probability gener- ating function of the stationary solution. When the underlying probability generating matrices are rational, the solution vector for level , , is shown to be in the matrix-geometric form for the infinite-level case, whereas it takes the modified form , for the finite-level case. The matrix parameters in the above two expressions can be obtained by decomposing the generalized system into forward and backward subsystems, or, equivalently, by finding bases for certain generalized invariant subspaces of a regular pencil We note that the computation of such bases can efficiently be carried out using advanced numerical linear algebra techniques including matrix-sign function iterations with quadratic convergence rates or ordered generalized Schur decomposition. The simplicity of the matrix-geometric form of the solution allows one to obtain various performance measures of interest easily, e.g., overflow probabilities and the moments of the level distribution, which is a significant advantage over conventional recursive methods. Index Terms— ATM multiplexer analysis, generalized difference equations, generalized invariant subspaces, generalized Schur decomposition, matrix-sign function, M/G/1-type Markov chains, polynomial matrix fractional descriptions. I. INTRODUCTION I N this paper, we study Markov chains of M/G/1 type with finite or infinite number of levels. The state space of an infinite-level (or simply infinite) M/G/1-type Markov chain consists of integer pairs where the level of the chain, takes on an infinite set of values and the phase of the chain, takes on a finite set of values The transition probability matrix of this chain has the block- Manuscript received December 1997; revised February 1998. This work was supported in part by DARPA/ITO under Grant A0-F316 and by NSF under Grant NCR-950814. N. Akar is with Technology Planning & Integration, Sprint, Overland Park, KS 66212 USA. N. C. O˘ guz and K. Sohraby are with the Department of Computer Science Telecommunications, University of Missouri-Kansas City, Kansas City, MO 64110 USA. Publisher Item Identifier S 0733-8716(98)04100-6. partitioned form [29] . . . . . . . . . (1) where and are matrices. Assuming that is irreducible and positive recurrent, we find the stationary probability vector which satisfies (2) where is and is an infinite column vector of ones. When the number of levels is finite, say the transition probability matrix takes the block upper-Hessenberg form . . . . . . . . . . . . (3) where and are and constitute the boundary at level We then study the solution vector which satisfies (2), with this time being a column vector of ones of length Throughout the paper, will denote a column vector of ones of suitable size. Both infinite and finite M/G/1-type Markov chains arise frequently in the performance analysis of ATM (asynchronous transfer mode) networks. In an ATM network, the basic unit of information is a fixed-length cell and the sharing of common network resources (bandwidth, buffers, etc.) among virtual connections is made on a statistical multiplexing basis. Statistical quality of service guarantees are integral to an ATM network, necessitating accurate traffic and performance analysis tools to determine the cell loss rate, cell delay, and cell delay variation in an ATM node (switch, multiplexer, etc.). This is, in general, difficult due to multiplexing of typically a large number of connections and burstiness of individual cell streams at possibly different time scales. One popular approach is to approximate such complex nonrenewal input processes by analytically tractable Markovian models either at 0733–8716/98$10.00  1998 IEEE