ABSTRACT

Invariant measures of Markov chains in discrete or continuous time with a countable set of states are characterized by its steady state recurrence relations. Exemplarily, we consider transition matrices and Q-matrices with upper bandwidth n and lower bandwidth 1 where the invariant measures satisfy an (n + 1)-order linear difference equation. Markov chains of this type arise from applications to queueing problems and population dynamics. It is the purpose of this paper to point out that the forward use of this difference equation is subject to some hitherto unobserved aspects. By means of the concept of generalized continued fractions (GCFs), we prove that each invariant measure is a dominated solution of the difference equation such that forward computation becomes numerically unstable. Furthermore, the GCF-based approach provides a decoupled recursion in which the phenomenon of numerical instability does not appear. The procedure results in an iteration scheme for successively computing approximants of the desired invariant measure depending on some truncation level N. Increasing N leads to the desired solution. A comparison study of forward computation and the GCF-based approach is given for Q-matrices with upper bandwidth 1 and 2.

Keywords:

Invariant Measures of Markov Chains, Inherent Numerical Instability of Linear Difference Equations, Generalized Continued Fractions, Convergence Criteria for Generalized Continued Fractions, Truncation Procedures for Infinite Matrices

1. Introduction

This paper is dedicated to discrete-time Markov chains $X={\left(X_k\right)}_{k\ge 0}$ with a countably infinite set of states (labelled as $\mathrm{0,1,2,}\cdots$ ) and a stationary one-step transition matrix $P={\left(P_{ik}\right)}_{0\le i,k<\infty }$ of the form

$P=\left[\begin{array}{ccccccccc}{P}_{00}& P_{01}& P_{02}& \dots & P_{0n}& O& O& \dots & \dots \\ P_{10}& P_{11}& P_{12}& \dots & P_{1n}& P_{1,n+1}& O& \dots & \dots \\ O& P_{21}& P_{22}& \dots & P_{2n}& P_{2,n+1}& P_{2,n+2}& O& \dots \\ O& O& P_{32}& \dots & P_{3n}& P_{3,n+1}& P_{3,n+2}& P_{3,n+3}& \\ \ddots & \ddots & \ddots & & \ddots & \ddots & \ddots & \ddots & \ddots \end{array}\right]$ (1)

with $n\ge 1$ and $P_{k\mathrm{,}k+n}\ne 0$ for at least one $k\in {\mathbb{N}}_0=\left\{\mathrm{0,1,2,}\cdots \right\}$ . In addition we assume that X is irreducible and recurrent. Notice that the irreducibility of X implies

$P_{k,k-1}\ne 0\text{for}k=1,2,\cdots$ (2)

For a recurrent irreducible Markov chain, the system

$\{\begin{array}{l}{x}_k={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}}{x}_i{P}_{ik}\left(k=0,1,2,\cdots \right)\\ x_0=1,x_k\ge 0\left(k=1,2,\cdots \right)\end{array}$ (3)

has a unique solution which is strictly positive, see Karlin and Taylor ( [1] , chapter 11). Every constant positive multiple of $x=\left(x_0,x_1,\cdots \right)$ is called an invariant measure of X.

Substituting (1) into (3) and rearranging yields the (n + 1)th-order homogeneous linear recurrence relation

$P_{k+n+1,k+n}{x}_{k+n+1}+\left(P_{k+n,k+n}-1\right)x_{k+n}+\cdots +P_{k,k+n}{x}_k=0\left(k=0,1,2,\cdots \right)$ (4)

coupled to the side conditions

$\{\begin{array}{l}{P}_{00}{x}_0+P_{10}{x}_1=x_0\\ P_{01}{x}_0+P_{11}{x}_1+P_{21}{x}_2=x_1\\ ⋮\\ P_{0,n-1}{x}_0+P_{1,n-1}{x}_1+\cdots +P_{n,n-1}{x}_n=x_{n-1}\end{array}$ (5)

From the theory of linear difference equations (see Miller [2] ) it is known that there exist $n+1$ linearly independent functions $x^{\left(0\right)}\mathrm{,}{x}^{\left(1\right)}\mathrm{,}\cdots \mathrm{,}{x}^{\left(n\right)}$ defined on ${\mathbb{N}}_0$ which take on prescribed values at $k=0,1,\cdots ,n$ and satisfy the homogeneous recurrence relation (4) for all $k\ge 0$ .

Since $P_{k+n+\mathrm{1,}k+n}\ne 0$ for all k the desired solution ${\left(x_k\right)}_{k=0}^{\infty }$ is uniquely determined by its initial values $x_0\mathrm{,}{x}_1\mathrm{,}\cdots \mathrm{,}{x}_n$ and (5). An application of the recurrence relation (4) then should directly lead to the desired solution x. But unfortunately forward computation of x by means of the homogeneous equation (4) is not a meaningful procedure. In the sequel it is shown that the solution space S of the recurrence relation (4) is the direct sum

$S=S_1\oplus S_2$

of two subspaces $S_1$ and $S_2$ with the property, that every solution $y\in S_2$ dominates over every solution $x\in S_1$ , i.e.

$\underset{k\to \infty }{\lim }\frac{x_k}{y_k}=0.$

In addition, we have $\dim S_1=n,\dim S_2=1$ and $x\in S_1$ .

As pointed out by Cash [3] , Gautschi [4] [5] , Lozier [6] and Mattheij [7] , forward computation of a dominated solution becomes numerically unstable. To avoid numerical instability, it is therefore necessary to construct a decoupled recursion in which the dominant solution $y\in S_2$ does not appear. In the sequel, it is shown that with each transition matrix P of the form (1) one can associate a set of so-called generalized continued fractions (GCFs) being convergent if the underlying Markov chain is irreducible and recurrent. It turns out that the limits of these fractions coincide with the coefficients of the desired decoupled recursion.

The phenomenon of inherent instability has been first recognized in connection with the computation of higher transcendental functions such as Bessel and associated Legendre functions (see Gautschi [4] and Wimp [8] ). Our experience seems to indicate that also most of the recurrence relations arising from stochastic modelling are subject to numerical instability and require special attention. Transition matrices and Q-matrices with lower bandwidth n and upper bandwidth 1 have been discussed in [9] .

Remark. Most of our considerations concerning the computation of invariant measures of Markov chains may be extended to more general infinite systems of equations. In Section 3, we will point out the properties of invariant measures which we explicitly used.

2. Generalized Continued Fractions and Linear Difference Equations

This chapter deals with the computation of dominated solutions of homogeneous linear difference equations by means of generalized continued fractions (GCFs). The theory of GCFs was initiated by Jacobi [10] and Perron [11] [12] and played a leading role in different areas of mathematics, especially in number theory, ergodic theory, linear difference equations and Padé approximations.

Defintion 1. A generalized continued fraction (GCF) of dimension $n$ is an $\left(n+1\right)$ -tuple $\left(a^{\left(1\right)}\mathrm{,}\cdots \mathrm{,}{a}^{\left(n\right)}\mathrm{,}b\right)$ of real-valued sequences and a convergence structure as follows. Let $A^{\left(1\right)}\mathrm{,}\cdots \mathrm{,}{A}^{\left(n\right)}\mathrm{,}B$ be the principal solutions of the corresponding homogeneous linear difference equation

$x_{k+n+1}=b_k{x}_{k+n}+a_k^{\left(n\right)}{x}_{k+n-1}+\cdots +a_k^{\left(1\right)}{x}_k\left(k=0,1,2,\cdots \right)$ (6)

satisfying

$\{\begin{array}{l}{A}_j^{\left(i\right)}={\delta }_{i,j+1}=\{\begin{array}{l}1\text{for}i=j+1\\ 0\text{for}i\ne j+1\end{array}\left(i=1,\cdots ,n;j=0,\cdots ,n\right)\\ B_j=0\text{for}j=0,\cdots ,n-1\text{and}{B}_n=1.\end{array}$ (7)

The GCF is said to converge iff all the limits

${\xi }^{\left(i\right)}=\underset{k\to \infty }{\lim }{A}_k^{\left(i\right)}/B_k\left(i=1,\cdots ,n\right)$ (8)

do exist.

Convergence of a GCF is indicated by the notation

$\left[\begin{array}{c}{\xi }^{\left(1\right)}\\ ⋮\\ {\xi }^{\left(n\right)}\end{array}\right]\stackrel{k}{=}\left[\begin{array}{cccc}{a}_0^{\left(1\right)}& a_1^{\left(1\right)}& a_2^{\left(1\right)}& \cdots \\ ⋮& ⋮& ⋮& \\ a_0^{\left(n\right)}& a_1^{\left(n\right)}& a_2^{\left(n\right)}& \cdots \\ b_0& b_1& b_2& \cdots \end{array}\right].$

Terminating a GCF after the N-th column we get the so-called N-th approximants

$\left[\begin{array}{c}\frac{A_{N+n+1}^{\left(1\right)}}{B_{N+n+1}}\\ ⋮\\ \frac{A_{N+n+1}^{\left(n\right)}}{B_{N+n+1}}\end{array}\right]=\left[\begin{array}{cccc}{a}_0^{\left(1\right)}& a_1^{\left(1\right)}& \cdots & a_N^{\left(1\right)}\\ ⋮& ⋮& \ddots & ⋮\\ a_0^{\left(n\right)}& a_1^{\left(n\right)}& \cdots & a_N^{\left(n\right)}\\ b_0& b_1& \cdots & b_N\end{array}\right]\mathrm{.}$

The term “GCF” becomes obvious if forward computation of the N-th approximant of a GCF is replaced by the equivalent backward algorithm. It is known that the recurrence relation (6) can be converted into a first order vector recursion of the form

$u_{k+1}=W_k{u}_k\left(k=0,1,2,\cdots \right)$ (9)

by putting

$u_k=\left[\begin{array}{c}{x}_k\\ x_{k+1}\\ ⋮\\ x_{k+n-1}\\ x_{k+n}\end{array}\right]\text{and}{W}_k=\left[\begin{array}{cccccc}0& 1& 0& \cdots & 0& 0\\ 0& 0& 1& \cdots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \cdots & \cdots & 0& 1\\ a_k^{\left(1\right)}& a_k^{\left(2\right)}& \cdots & \cdots & a_k^{\left(n\right)}& b_k\end{array}\right]\left(k=0,1,2,\cdots \right).$

Comparing (6) and (9) it is seen, that the numerators $A_{N+n+1}^{\left(1\right)}\mathrm{,}\cdots \mathrm{,}{A}_{N+n+1}^{\left(n\right)}$ and the denominators $B_{N+n+1}$ of the GCF appear in the last row of the matrix

$W_N{W}_{N-1}\cdots W_0.$

Consider now the backward recurrence scheme

${}_{\left(N\right)}v{}_k={}_{\left(N\right)}v{}_{k+1}{W}_k\left(k=N,N-1,\cdots ,0\right)$ (10)

with initial vector ${}_{\left(N\right)}v{}_{N+1}=\left(\mathrm{0,}\cdots \mathrm{,0,1}\right)$ . Then

${}_{\left(N\right)}v{}_0={}_{\left(N\right)}v{}_{N+1}{W}_N{W}_{N-1}\cdots W_0=\left(A_{N+n+1}^{\left(1\right)},\cdots ,A_{N+n+1}^{\left(n\right)},B_{N+n+1}\right).$

Writing (10) in expanded form and inserting the structure of $W_k$ , we get

${}_{\left(N\right)}v{}_k^{\left(1\right)}={}_{\left(N\right)}v{}_{k+1}^{\left(n+1\right)}{a}_k^{\left(1\right)}\left(k=N,N-1,\cdots ,0\right),$

${}_{\left(N\right)}v{}_k^{\left(i\right)}={}_{\left(N\right)}v{}_{k+1}^{\left(i-1\right)}+{}_{\left(N\right)}v{}_{k+1}^{\left(n+1\right)}{a}_k^{\left(i\right)}\left(i=2,\cdots ,n,k=N,N-1,\cdots ,0\right),$

${}_{\left(N\right)}v{}_k^{\left(n+1\right)}={}_{\left(N\right)}v{}_{k+1}^{\left(n\right)}+{}_{\left(N\right)}v{}_{k+1}^{\left(n+1\right)}{b}_k\left(k=N,N-1,\cdots ,0\right)$

or, equivalently

$\{\begin{array}{l}{}_{\left(N\right)}r{}_k^{\left(1\right)}:=\frac{{}_{\left(N\right)}v{}_k^{\left(1\right)}}{{}_{\left(N\right)}v{}_k^{\left(n+1\right)}}=\frac{a_k^{\left(1\right)}}{b_k+{}_{\left(N\right)}r{}_{k+1}^{\left(n\right)}}\left(k=N,N-1,\cdots ,0\right)\\ {}_{\left(N\right)}r{}_k^{\left(i\right)}:=\frac{{}_{\left(N\right)}v{}_k^{\left(i\right)}}{{}_{\left(N\right)}v{}_k^{\left(n+1\right)}}=\frac{a_k^{\left(i\right)}+{}_{\left(N\right)}r{}_{k+1}^{\left(i-1\right)}}{b_k+{}_{\left(N\right)}r{}_{k+1}^{\left(n\right)}}\left(i=2,\cdots ,n;k=N,N-1,\cdots ,0\right)\end{array}$ (11)

with initial values ${}_{\left(N\right)}r{}_{N+1}^{\left(i\right)}=0$ for $i=1,\cdots ,n$ . Hence

$\frac{A_{N+n+1}^{\left(i\right)}}{B_{N+n+1}}={}_{\left(N\right)}r{}_0^{\left(i\right)}\left(i=1,\cdots ,n\right)$

and

${\xi }^{\left(i\right)}=\underset{N\to \infty }{\lim }{}_{\left(N\right)}r{}_0^{\left(i\right)}\left(i=1,\cdots ,n\right).$ (12)

Alternative procedures for computing GCFs are described in [13] .

The relations between GCFs and linear difference equations have been first recognized by Perron [14] . Perron’s results were generalized by Van der Cruyssen [13] , Hanschke [15] [16] and Levrie and Bultheel [17] . The following theorem is due to Van der Cruyssen [13] .

Theorem 1. A GCF $\left(a^{\left(1\right)}\mathrm{,}\cdots \mathrm{,}{a}^{\left(n\right)}\mathrm{,}b\right)$ converges iff there are $n+1$ linearly independent solutions $x^{\left(1\right)}\mathrm{,}\cdots \mathrm{,}{x}^{\left(n\right)}\mathrm{,}y$ of the recurrence relation (6) satisfying

$\underset{k\to \infty }{\lim }\frac{x_k^{\left(i\right)}}{y_k}=0\left(i=1,\cdots ,n\right)$ (13)

and

$\left|\begin{array}{ccc}{x}_0^{\left(1\right)}& \cdots & x_{n-1}^{\left(1\right)}\\ ⋮& & ⋮\\ x_0^{\left(n\right)}& \cdots & x_{n-1}^{\left(n\right)}\end{array}\right|\ne 0.$ (14)

As pointed out by Gautschi [4] [5] and Van der Cruyssen [13] , forward computation of a solution $x\in \text{span}\left(x^{\left(1\right)},\cdots ,x^{\left(n\right)}\right)$ is numerically unstable. Van der Cruyssen [13] establishes that if the limits

$\left[\begin{array}{c}{\xi }_l^{\left(1\right)}\\ ⋮\\ {\xi }_l^{\left(n\right)}\end{array}\right]=\left[\begin{array}{cccc}{a}_l^{\left(1\right)}& a_{l+1}^{\left(1\right)}& a_{l+2}^{\left(1\right)}& \cdots \\ ⋮& ⋮& ⋮& \\ a_l^{\left(n\right)}& a_{l+1}^{\left(n\right)}& a_{l+2}^{\left(n\right)}& \cdots \\ b_l& b_{l+1}& b_{l+2}& \cdots \end{array}\right]$ (15)

exist for all $l\ge 0$ , then $x\in \text{span}\left(x^{\left(1\right)},\cdots ,x^{\left(n\right)}\right)$ iff

$x_{l+n}=-{\xi }_l^{\left(n\right)}{x}_{l+n-1}-\cdots -{\xi }_l^{\left(1\right)}{x}_l\left(l=0,1,2,\cdots \right).$ (16)

Combining (11), (12) and (16), one obtains an efficient algorithm (which we will refer to as Van der Cruyssen’s algorithm) for approximating the first $L+1$ components $x_0\mathrm{,}\cdots \mathrm{,}{x}_L$ of an element $x\in \text{span}\left(x^{\left(1\right)},\cdots ,x^{\left(n\right)}\right)$ with prescribed values $x_0\mathrm{,}\cdots \mathrm{,}{x}_{n-1}$ :

Step 1: Select $N>L-n$ and define ${}_{\left(N\right)}r{}_l^{\left(i\right)}$ for $i=1,\cdots ,n;l=0,1,\cdots ,N+1$ by

${}_{\left(N\right)}r{}_{N+1}^{\left(1\right)}=\cdots ={}_{\left(N\right)}r{}_{N+1}^{\left(n\right)}=0,$

${}_{\left(N\right)}r{}_l^{\left(1\right)}=\frac{a_l^{\left(1\right)}}{b_l+{}_{\left(N\right)}r{}_{l+1}^{\left(n\right)}}\left(l=N,N-1,\cdots ,0\right),$

${}_{\left(N\right)}r{}_l^{\left(i\right)}=\frac{a_l^{\left(i\right)}+{}_{\left(N\right)}r{}_{l+1}^{\left(i-1\right)}}{b_l+{}_{\left(N\right)}r{}_{l+1}^{\left(n\right)}}\left(i=2,\cdots ,n;l=N,N-1,\cdots ,0\right)$ .

Step 2: Set ${}_{\left(N\right)}x{}_0=x_0,\cdots ,{}_{\left(N\right)}x{}_{n-1}=x_{n-1}$ and

${}_{\left(N\right)}x{}_{l+n}=-{}_{\left(N\right)}r{}_l^{\left(n\right)}{}_{\left(N\right)}x{}_{l+n-1}-\cdots -{}_{\left(N\right)}r{}_l^{\left(1\right)}{}_{\left(N\right)}x{}_l$

for $l=0,1,\cdots ,L-n$ .

Step 3: Increase N until the accuracy of the iterates is within prescribed limits.

For any $l$ , the vector ${\left({}_{\left(N\right)}r{}_l^{\left(1\right)}\mathrm{,}\cdots \mathrm{,}{}_{\left(N\right)}r{}_l^{\left(n\right)}\right)}^{\text{T}}$ is an approximant of a GCF. Hence, convergence of the algorithm is related to convergence of GCFs. For the latter, we cite two helpful results.

Theorem 2. (Levrie [18] , Perron [12] ). The GCF

$\left[\begin{array}{cccc}{a}_0^{\left(1\right)}& a_1^{\left(1\right)}& a_2^{\left(1\right)}& \cdots \\ ⋮& ⋮& ⋮& \\ a_0^{\left(n\right)}& a_1^{\left(n\right)}& a_2^{\left(n\right)}& \cdots \\ b_0& b_1& b_2& \cdots \end{array}\right]$

converges if it satisfies the so-called dominance condition, i.e.

$\left|a_k^{\left(1\right)}\right|+\left|a_k^{\left(2\right)}\right|+\cdots +\left|a_k^{\left(n\right)}\right|+1\le \left|b_k\right|\text{for}\text{all}k\ge 0.$

Theorem 3. (De Bruin [19] , Perron [12] ). Consider a GCF of the form (8) with nth approximant numerators $A_{\nu +n+1}^{\left(i\right)}\left(i=\mathrm{1,}\cdots \mathrm{,}n\right)$ and denominators $B_{\nu +n+1}$ and the GCF given by

$\left[\begin{array}{ccc}{a}_0^{\left(1\right)}{\rho }_0{\rho }_{-1}\cdots {\rho }_{-n}& a_1^{\left(1\right)}{\rho }_1{\rho }_0\cdots {\rho }_{1-n}\cdots & a_{\nu }^{\left(1\right)}{\rho }_{\nu }{\rho }_{\nu -1}\cdots {\rho }_{\nu -n}\cdots \\ a_0^{\left(2\right)}{\rho }_0{\rho }_{-1}\cdots {\rho }_{1-n}& a_1^{\left(2\right)}{\rho }_1{\rho }_0\cdots {\rho }_{2-n}\cdots & a_{\nu }^{\left(2\right)}{\rho }_{\nu }{\rho }_{\nu -1}\cdots {\rho }_{\nu -n+1}\\ ⋮& ⋮& ⋮\\ a_0^{\left(n\right)}{\rho }_0{\rho }_{-1}& a_1^{\left(n\right)}{\rho }_1{\rho }_0& a_{\nu }^{\left(n\right)}{\rho }_{\nu }{\rho }_{\nu -1}\\ b_0{\rho }_0& b_1{\rho }_1& b_{\nu }{\rho }_{\nu }\end{array}\right]$ (17)

with nth numerators ${\tilde{A}}_{\nu +n+1}^{\left(i\right)}\left(i=\mathrm{1,}\cdots \mathrm{,}n\right)$ and denominators ${\tilde{B}}_{\nu +n+1}$ where the real numbers ${\rho }_{-n}\mathrm{,}{\rho }_{-n+1}\mathrm{,}\cdots \mathrm{,}{\rho }_0\mathrm{,}{\rho }_1\mathrm{,}\cdots$ are all different from zero. Then

$\begin{array}{l}{\tilde{A}}_{\nu +n+1}^{\left(i\right)}={\rho }_{\nu }{\rho }_{\nu -1}\cdots {\rho }_1{\rho }_0\cdots {\rho }_{-n+i-1}{A}_{\nu +n+1}^{\left(i\right)}\left(i=1,\cdots ,n\right)\hfill \\ {\tilde{B}}_{\nu +n+1}={\rho }_{\nu }{\rho }_{\nu -1}\cdots {\rho }_0{B}_{\nu +n+1}\hfill \end{array}\}\nu \in {\mathbb{N}}_0$

In other words, if one of these two GCFs converges so does the other.

Notice that the GCF (17) may be interpreted as an equivalence transformation of the GCF (12).

3. Main Results

To make (4) congruent with (6) we put

$a_k^{\left(i\right)}=-\frac{P_{k+i-1,k+n}}{P_{k+n+1,k+n}}\left(i=1,\cdots ,n,k=0,1,2,\cdots \right),$ (18)

$b_k=-\frac{P_{k+n,k+n}-1}{P_{k+n+1,k+n}}\left(k=0,1,2,\cdots \right).$ (19)

Theorem 4. Let $a_k^{\left(i\right)}\mathrm{,}{b}_k$ be defined as in (18) and (19). The limits

$\left[\begin{array}{c}{\xi }_l^{\left(1\right)}\\ ⋮\\ {\xi }_l^{\left(n\right)}\end{array}\right]=\left[\begin{array}{cccc}{a}_l^{\left(1\right)}& a_{l+1}^{\left(1\right)}& a_{l+2}^{\left(1\right)}& \cdots \\ ⋮& ⋮& ⋮& \\ a_l^{\left(n\right)}& a_{l+1}^{\left(n\right)}& a_{l+2}^{\left(n\right)}& \cdots \\ b_l& b_{l+1}& b_{l+2}& \cdots \end{array}\right]$ (20)

exist for all $l\ge 0$ if the corresponding Markov chain is irreducible and recurrent. In case of convergence the invariant measure is a dominated solution of (6) and satisfies the decoupled recursion

$x_{l+n}=-{\xi }_l^{\left(n\right)}{x}_{l+n-1}-\cdots -{\xi }_l^{\left(1\right)}{x}_l\left(l=0,1,2,\cdots \right).$ (21)

Proof. Denote by ${}_{\left(N\right)}P={\left(P_{ik}\right)}_{0\le i\mathrm{,}k\le N}$ the $\left(N+1\right)\times \left(N+1\right)$ northwest corner truncation of $P$ . Since $P$ is assumed to be irreducible and recurrent we conclude from Seneta [20] [21] [22] that the finite system

${}_{\left(N\right)}h\left({}_{\left(N\right)}I-{}_{\left(N\right)}P\right)=e_0$ (22)

where ${}_{\left(N\right)}I$ is the unit matrix and ${}_{\left(N\right)}e{}_0$ is the vector with unity in the first position and zeros elsewhere has a unique solution ${}_{\left(N\right)}h=\left({}_{\left(N\right)}h{}_0\mathrm{,}{}_{\left(N\right)}h{}_1\mathrm{,}\cdots \mathrm{,}{}_{\left(N\right)}h{}_N\right)$ satisfying

$\underset{N\to \infty }{lim}\frac{{}_{\left(N\right)}h{}_k}{{}_{\left(N\right)}h{}_0}=\frac{x_k}{x_0}\left(k=\mathrm{0,1,2,}\cdots \right)\mathrm{.}$

The vector ${}_{\left(N\right)}h$ satisfies the homogeneous linear difference Equation (6) for $k=0,1,\cdots ,N-n$ and is subject to the boundary conditions

$\begin{array}{l}\left(1-P_{00}\right){}_{\left(N\right)}h{}_0-P_{10}{}_{\left(N\right)}h{}_1=1\hfill \\ -P_{01}{}_{\left(N\right)}h{}_0+\left(1-P_{11}\right){}_{\left(N\right)}h{}_1-P_{21}{}_{\left(N\right)}h{}_2=0\hfill \\ ⋮\hfill \\ -P_{0,n-1}{}_{\left(N\right)}h{}_0-P_{1,n-1}{}_{\left(N\right)}h{}_1-\cdots -P_{n,n-1}{}_{\left(N\right)}h{}_n=0\hfill \end{array}\}$ (23)

and

${}_{\left(N\right)}h{}_{N+1}=0.$ (24)

Notice that the numerators $A_l^{\left(1\right)}\mathrm{,}\cdots \mathrm{,}{A}_l^{\left(n\right)}$ and the denumerators $B_l$ of the GCF (20) satisfy

$x_{k+n+1}=b_{k+l}{x}_{k+n}+a_{k+l}^{\left(n\right)}{x}_{k+n-1}+\cdots +a_{k+l}^{\left(1\right)}{x}_k\left(k=0,1,2,\cdots \right)$ (25)

and build up a fundamental system of solutions to Equation (6) for $k\ge l$ . Hence any solution of the recurrence relation (6) can be expressed in terms of these functions. In view of their initial values

$\{\begin{array}{l}{A}_{l,j}^{\left(i\right)}={\delta }_{i,j+1}=\{\begin{array}{l}1\text{for}i=j+1\hfill \\ 0\text{for}i\ne j+1\hfill \end{array}\left(i=1,\cdots ,n;j=0,\cdots ,n\right)\\ B_{l,j}=0\text{for}j=0,\cdots ,n-1\text{and}{B}_{l,l+n}=1;\end{array}$ (26)

we get

${}_{\left(N\right)}h{}_k={}_{\left(N\right)}h{}_l{A}_{l,k-l}^{\left(1\right)}+\cdots +{}_{\left(N\right)}h{}_{l+n-1}{A}_{l,k-l}^{\left(n\right)}+{}_{\left(N\right)}h{}_{l+n}{B}_{l,k-l}\left(k=l,l+1,\cdots \right).$ (27)

Choosing $k=N+1$ and utilizing (24) equation (27) reduces to

${}_{\left(N\right)}h{}_{l+n}{B}_{l,N-l+1}=-{}_{\left(N\right)}h{}_l{A}_{l,N-l+1}^{\left(1\right)}-\cdots -{}_{\left(N\right)}h{}_{l+n-1}{A}_{l,N-l+1}^{\left(n\right)}.$ (28)

Dividing both sides of (28) by $B_{l\mathrm{,}N-l+1}$ and ${}_{\left(N\right)}h{}_0$ we formally arrive at

$\frac{{}_{\left(N\right)}h{}_{l+n}}{{}_{\left(N\right)}h{}_0}=-\frac{{}_{\left(N\right)}h{}_l}{{}_{\left(N\right)}h{}_0}\frac{A_{l,N-l+1}^{\left(1\right)}}{B_{l,N-l+1}}-\cdots -\frac{{}_{\left(N\right)}h{}_{l+n-1}}{{}_{\left(N\right)}h{}_0}\frac{A_{l,N-l+1}^{\left(n\right)}}{B_{l,N-l+1}}.$ (29)

Passing to the limit $N\to \infty$ we get the decoupled recursion (21) provided that the limits $li{m}_{N\to \infty }{A}_{l\mathrm{,}N-l+1}^{\left(i\right)}/B_{l\mathrm{,}N-l+1}$ do exist for $i=0,1,\cdots ,n$ .

To prove our assertion we utilize that the invariant measure ${\left(x_k\right)}_{k\ge 0}$ of the irreducible and recurrent Markov chain (1) is strictly positive and satisfies the recurrence relation (4) which can be rewritten as

$\begin{array}{l}\frac{P_{k,k+n}\cdot x_k}{P_{k+n+1,k+n}\cdot x_{k+n+1}}+\frac{P_{k+1,k+n}\cdot x_{k+1}}{P_{k+n+1,k+n}\cdot x_{k+n+1}}+\cdots +\frac{P_{k+n-1,k+n}\cdot x_{k+n-1}}{P_{k+n+1,k+n}\cdot x_{k+n+1}}+1\\ =\frac{\left(1-P_{k+n,k+n}\right)x_{k+n}}{P_{k+n+1,k+n}\cdot x_{k+n+1}}\left(k=0,1,2,\cdots \right).\end{array}$ (30)

Define ${\rho }_k=\frac{x_{k+n}}{x_{k+n+1}}$ for $k=0,1,2,\cdots$ Then (30) becomes equivalent to

$\begin{array}{l}\left|a_k^{\left(1\right)}{\rho }_k{\rho }_{k-1}\cdots {\rho }_{k-n}\right|+\left|a_k^{\left(2\right)}{\rho }_k{\rho }_{k-1}\cdots {\rho }_{k-n+1}\right|\\ +\cdots +\left|a_k^{\left(n\right)}{\rho }_k{\rho }_{k-1}\right|+1=\left|b_k{\rho }_k\right|\left(k=0,1,2,\cdots \right)\end{array}$

implying that the GCFs

$\left[\begin{array}{ccc}{a}_l^{\left(1\right)}{\rho }_l{\rho }_{l-1}\cdots {\rho }_{l-n}& a_{l+1}^{\left(1\right)}{\rho }_{l+1}{\rho }_l\cdots {\rho }_{l-n+1}& \cdots \\ a_l^{\left(2\right)}{\rho }_l{\rho }_{l-1}\cdots {\rho }_{l-n+1}& a_{l+1}^{\left(2\right)}{\rho }_{l+1}{\rho }_l\cdots {\rho }_{l-n+2}& \cdots \\ ⋮& ⋮& \\ a_l^{\left(n\right)}{\rho }_l{\rho }_{l-1}& a_{l+1}^{\left(n\right)}{\rho }_{l+1}{\rho }_l& \cdots \\ b_l{\rho }_l& b_{l+1}{\rho }_l& \cdots \end{array}\right]$

converge for all $l\ge 0$ by means of Theorem 2. From Theorem 3, we then conclude that the same is true for the GCFs (8). ,

Theorem 2 says that the numerical calculation of the invariant measures of X can be reduced to Van der Cruyssen’s algorithm with (23) as initial conditions.

Remark. It is important that the statement of Theorem 4 consists of two parts:

・ The invariant measure is dominated by other solutions of the difference Equation (4).

・ By means of GCFs, we are able to construct a decoupled recursion which is not affected to numerical instability.

Our main goal was the derivation of the first statement. Although the existence of dominating solutions has been pointed out in specific examples (e.g. ( [8] , p. 167) where the exact solutions of the difference equations are known), to our knowledge, no statement in this generality has been published. Since the desired solution being dominated directly implies numerical instability, the system $x=xP$ should not be solved by forward computation. Note, that up to some slight modifications concerning the boundary conditions, the truncated system (22) yields the same difference Equation (4) for $k=0,1,\cdots ,N-n-1$ . Thus, forward computation could be applied to (22), too, but at least for large N, the result of numerical instability and hence, the recommendation not to use forward computation, holds for (22) as well.

The observation of inherent numerical instability of (4) is essential since the transition structure under consideration arises in many practical applications, e.g. state-dependent queueing systems with bulk arrivals, and it is also valid in a more general context, e.g. as an approximation to state-dependent variants of the traditional G/M/1 queueing model.

Basically, we have exploited that the solutions of the truncated system (22) converge to invariant measures as $N\to \infty$ . For Markov chains with a general transition structure, there is no way of solving $x=xP$ directly (in particular, forward computation cannot be applied if $P_{kl}>0$ for some $k<l-1$ ). In this situation, Seneta [20] [21] [22] as well as Golub and Seneta [23] already recommended to use the convergence of the solutions of the truncated system (22) to the invariant measure for numerical issues. Furthermore, Seneta [22] discussed numerical aspects (in terms of the condition number) of the finite system (22) and stated numerical stability of Gaussian elimination or LU decomposition. The above comment concerning forward computation as a solution method for (22) emphasizes on the fact that it is important to solve this finite system in an appropriate way. The GCF-based algorithm can be interpreted as a combination of building up the truncated system (22), and then applying a modification of Gaussian elimination procedure. Therefore, it follows both steps recommended in the literature cited above and represents the complete procedure in a combined mathematical form, which is interesting from a structural point of view.

4. Continuous-Time Markov Chains

The results of the preceding chapter can easily be extended to continuous-time Markov chains $Y={\left(Y_t\right)}_{t\ge 0}$ generated by a conservative, irreducible and regular Q-matrix of the form

$Q=\left[\begin{array}{ccccccccc}{q}_{00}& q_{01}& q_{02}& \cdots & q_{0n}& O& O& \cdots & \cdots \\ q_{10}& q_{11}& q_{12}& \cdots & q_{1n}& q_{1,n+1}& O& \cdots & \cdots \\ O& q_{21}& q_{22}& \cdots & q_{2n}& q_{2,n+1}& q_{2,n+2}& O& \cdots \\ O& O& q_{32}& \cdots & q_{3n}& q_{3,n+1}& q_{3,n+2}& q_{3,n+3}& \\ \ddots & \ddots & \ddots & & \ddots & \ddots & \ddots & \ddots & \ddots \end{array}\right]$ (31)

where $n\ge 1$ and $q_{k\mathrm{,}k+n}\ne 0$ for at least one $k\in {\mathbb{N}}_0$ . Notice that the irreducibility of Y implies

$q_{k,k-1}\ne 0\text{for}k=1,2,\cdots$

If Y is positive recurrent, the limits

${\pi }_k=\underset{t\to \infty }{\lim }P\left(Y_t=k|Y_0=i\right)\left(k=0,1,2,\cdots \right)$

exist independently of the initial state and build up a probability distribution which is strictly positive. For the construction of a continuous-time Markov process from its infinitesimal properties the reader is referred to Reuter [24] and Kendall and Reuter [25] . Now let ${}_{\left(N\right)}Q$ be the $\left(N+1\right)\times \left(N+1\right)$ northwest corner truncation of Q. If Y is regular and positive recurrent, then the finite system

${}_{\left(N\right)}z{}_{\left(N\right)}Q={}_{\left(N\right)}e{}_0$ (32)

has a unique solution ${}_{\left(N\right)}z=\left({}_{\left(N\right)}z{}_0\mathrm{,}{}_{\left(N\right)}z{}_1\mathrm{,}\cdots \mathrm{,}{}_{\left(N\right)}z{}_N\right)$ satisfying

$\underset{N\to \infty }{\lim }\frac{{}_{\left(N\right)}z{}_k}{{}_{\left(N\right)}z{}_0}=\frac{{\pi }_k}{{\pi }_0}\left(k=0,1,2,\cdots \right),$

see Tweedie [26] . Now let

$c_k^{\left(i\right)}=-\frac{q_{k+i-1,k+n}}{q_{k+n+1,k+n}}\left(i=1,\cdots ,n,k=0,1,2,\cdots \right),$ (33)

$d_k=-\frac{q_{k+n,k+n}}{q_{k+n+1,k+n}}\left(k=0,1,2,\cdots \right).$ (34)

By substituting (31) into (32) it is seen that ${}_{\left(N\right)}z$ satisfies the $\left(n+1\right)\text{th}$ order homogeneous linear difference equation

${\pi }_{k+n+1}=d_k{\pi }_{k+n}+c_k^{\left(n\right)}{\pi }_{k+n-1}+\cdots +c_k^{\left(1\right)}{\pi }_k\left(k=0,1,2,\cdots \right)$ (35)

for $k=0,1,\cdots ,N-n$ and the boundary conditions

$\begin{array}{l}{q}_{00}{}_{\left(N\right)}z{}_0+q_{10}{}_{\left(N\right)}z{}_1=1\\ q_{01}{}_{\left(N\right)}z{}_0+q_{11}{}_{\left(N\right)}z{}_1+q_{21}{}_{\left(N\right)}z{}_2=0\\ ⋮\\ q_{0,n-1}{}_{\left(N\right)}z{}_0+q_{1,n-1}{}_{\left(N\right)}z{}_1+\cdots +q_{n,n-1}{}_{\left(N\right)}z{}_n=0\end{array}\}$ (36)

${}_{\left(N\right)}z{}_{N+1}=0.$ (37)

By replacing (33) with (18) and (34) with (19), it is readily seen that the scheme (35)-(37) formally coincides with that of the discrete time case. Hence (26)-(29) also hold for ${}_{\left(N\right)}z{}_l$ . To prove convergence of the associated GCFs

$\left[\begin{array}{c}{\Lambda }_l^{\left(1\right)}\\ ⋮\\ {\Lambda }_l^{\left(n\right)}\end{array}\right]=\left[\begin{array}{cccc}{c}_l^{\left(1\right)}& c_{l+1}^{\left(1\right)}& c_{l+2}^{\left(1\right)}& \cdots \\ ⋮& ⋮& ⋮& \\ c_l^{\left(n\right)}& c_{l+1}^{\left(n\right)}& c_{l+2}^{\left(n\right)}& \cdots \\ d_l& d_{l+1}& d_{l+2}& \cdots \end{array}\right]\left(l=0,1,2,\cdots \right)$ (38)

we use the same arguments as in the proof of Theorem 4. Notice that ${\left({\pi }_k\right)}_{k=0}^{\infty }$ satisfies (35). Rearranging yields

$-c_k^{\left(1\right)}\frac{{\pi }_k}{{\pi }_{k+n+1}}-\cdots -c_k^{\left(n\right)}\frac{{\pi }_{k+n-1}}{{\pi }_{k+n+1}}+1=d_k\frac{{\pi }_{k+n}}{{\pi }_{k+n+1}}\left(k=0,1,2,\cdots \right).$ (39)

Define ${\gamma }_k=\frac{{\pi }_{k+n}}{{\pi }_{k+n+1}}$ for $k=0,1,2,\cdots$ . Since $-c_k^{\left(i\right)}>0$ for $i=1,\cdots ,n,\mathrm{<mi>\; </mi>}k\ge 0$ and $d_k>0$ for $k\ge 0$ , (39) is equivalent to

$\begin{array}{l}\left|c_k^{\left(i\right)}{\gamma }_k{\gamma }_{k-1}\cdots {\gamma }_{k-n}\right|+\left|c_k^{\left(2\right)}{\gamma }_k{\gamma }_{k-1}\cdots {\gamma }_{k-n+1}\right|+\cdots +\left|c_k^{\left(n\right)}{\gamma }_k{\gamma }_{k-1}\right|+1\\ =\left|d_k{\gamma }_k\right|\left(k=\mathrm{0,1,2,}\cdots \right)\mathrm{.}\end{array}$

By Theorem 3, the GCF

$\left[\begin{array}{ccc}{c}_l^{\left(1\right)}{\gamma }_l{\gamma }_{l-1}\cdots {\gamma }_{l-n}& c_{l+1}^{\left(1\right)}{\gamma }_{l+1}{\gamma }_l\cdots {\gamma }_{l-n+1}& \dots \\ c_l^{\left(2\right)}{\gamma }_l{\gamma }_{l-1}\cdots {\gamma }_{l-n+1}& c_{l+1}^{\left(2\right)}{\gamma }_{l+1}{\gamma }_l\cdots {\gamma }_{l-n+2}& \dots \\ ⋮& ⋮& \\ c_l^{(n)}{\gamma }_l{\gamma }_{l-1}& c_{l+1}^{\left(n\right)}{\gamma }_{l+1}{\gamma }_l& \dots \\ d_l{\gamma }_l& d_{l+1}{\gamma }_l& \cdots \end{array}\right]$

then converges for all $l\ge 0$ . Hence the same is true for the GCFs (38). Summarizing we arrive at the following theorem.

Theorem 5. Let $c_k^{\left(i\right)}\mathrm{,}{d}_k$ be defined as in (33) and (34), respectively. The limits (38) exist for all $l\ge 0$ if the corresponding Markov process is irreducible, regular and positive recurrent. In case of convergence the limiting distribution ${\left({\pi }_k\right)}_{k=0}^{\infty }$ is a dominated solution of (35) and satisfies the decoupled recursion.

${\pi }_{l+n}=-{\Lambda }_l^{\left(n\right)}{\pi }_{l+n-1}-\cdots -{\Lambda }_l^{\left(1\right)}{\pi }_l\left(l=0,1,2,\cdots \right).$ (40)