1 Type Structured Chains: A

IEEE INFOCOM 2001

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Combined M/G/1-G/M/1 Type Structured Chains: A Simple Algorithmic Solution and Applications Reza Jafari and Khosrow Sohraby Computer Science Telecommunications University of Missouri - Kansas City 5100 Rockhill Road Kansas City, MO 64110, USA frjafari; sohraby g@cstp:umkc:edu

Abstract | In this paper, we consider combined M/G/1G/M/1 type Markov chains with block-structured transition. It is assumed that all the blocks are generated with rational generating matrices. We provide an algorithmic approach to nd the stationary probability distribution based on well-known concepts in linear system theory. These chains arise in the (correlated) G/G/1 queueing systems and known structured Markov chains such as canonical and non-canonical M/G/1 and G/M/1 types which frequently arising in teletrac analysis of computer and communications networks are special cases. We provide a truncation-free algorithmic solution in simple geometric form of the the stationary probability vector of the chain taking full advantage of the rational generating matrices. Keywords | Discrete-time Queueing Models, In nite-state Markov Chain, Rational Generating Matrix, Matrix Geometric Solution, Correlated G/G/1 Queue

I. INTRODUCTION

lowing

B0 em0 + Ck em0 +

kX ?1 i=1

1 X i=1

Ri em +

Bi em = em0 ;

1 X i=0

Ai em = em; k  1

where em is an m  1 vector of ones. The state space of P consists of integer pairs (i; j ) , where i is called the level of the chain and takes the values f 0; 1; 2;


g and j is called the phase of the chain and takes on the nite set of values 0  j  m ? 1, for levels f1; 2;


g and 0  j  m0 ? 1, for level 0. The transition from level i; i  1, to level k + i is governed by the matrix Ak ; k  0, and from level k + i to level i is governed by Rk ; k  1. Transition from level zero to level k, is given by Bk and from level k, k > 0, to level zero by Ck . Our goal is to nd the unique steady-state probability vector    = 0 1


that satis es

N this paper, we study an irreducible discrete time comIbined M/G/1-G/M/1 type Markov chains. A large class of teletrac models which has been studied in the literatures such as G/M/1 [1] and M/G/1 [2] are special cases. This work is motivated by the increasing interest in recent 1 X years in discrete-time analysis of queueing systems arising (2)  = P; 0 em0 + i em = 1 in modern high-speed networks such as ATM. i=1 Consider an irreducible and positive recurrent discretetime Markov chain with the in nite state block-partitioned where 0 is 1  m0 and i ; i  1, is a 1  m vector. The rst row and rst column of blocks of P represent stochastic probability transition matrix P as the boundaries of the chain. The rest of P is a block2 3 structured Toeplitz matrix. When matrix P is skip-free to B0 B1 B2 B3 B4


6 C 7 the left in blocks, we have the probability transition matrix 6 1 A0 A1 A2 A3


7 6 7 of an M=G=1 type chain ([2], [3]). Also, if P is skip-free R1 A0 A1 A2


77 P = 666 CC2 R (1) to the right in blocks the chain is called an G=M=1 type 3 2 R1 A0 A1


7 6 7 chain ([1]). 4 C4 R3 R2 R1 A0


5 In [4] a state reduction method has been proposed to














obtain the numerical solution of chain P . This method where the elements Ai ; i  0; Bi ; i  1; B0 ; Ci ; i  1; uses truncation of sequences fAi g and fRi g in number and Ri ; i  1; are nite , non-negative matrices of di- of its algorithmic steps which is not appropriate for nonmensions m  m, m0  m, m0  m0 , m  m0 , and m  m polynomial generating functions. In this paper, a simple respectively. Since P is stochastic it should satisfy the fol- geometric solution is provided. Our solution methodology,

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IEEE INFOCOM 2001

takes full advantage of rational generating functions forming the blocks of the chain. We note that if the matrix sequences Ai or Ri are nite (i.e., their generators are polynomials), as in [3], our algorithmic solution results in (geometric) solution consisting of matrices with high dimensions, especially of the degrees of the polynomials are not small. In such cases, we do not recommend this approach and we expect that the solution provided in [4] be more ecient (possibly both in CPU time and memory). Our approach is best suited when the underlying generator matrices are of rational form resulting in in nite sequences Ai and/or Ri . As stated in [3], this appears frequently in modern teletrac models. Before we proceed, we discuss about condition of stability of this chain heuristically. The proof of the stability ,in general, is a dicult problem. Let de ne

A=(

1 X i=0

Ai +

1 X i=1

Ri )

(3)

to be less than the average number of departure, or the expected value of random variable un be less than zero, i.e.,

E [un ] < 0

(5)

The above condition for stability can be written as 1 X

i=?1

i u~i < 0 !

1 X i=1

iu~i