Approximation of the exit probability of a stable Markov modulated constrained random walk

Abstract

Let X be the constrained random walk on \({\mathbb {Z}}_+^2\) having increments (1, 0), \((-\,1,1)\), \((0,-\,1)\) with jump probabilities \(\lambda (M_k)\), \(\mu _1(M_k)\), and \(\mu _2(M_k)\) where M is an irreducible aperiodic finite state Markov chain. The process X represents the lengths of two tandem queues with arrival rate \(\lambda (M_k)\), and service rates \(\mu _1(M_k)\), and \(\mu _2(M_k)\); the process M represents the random environment within which the system operates. We assume that the average arrival rate with respect to the stationary measure of M is less than the average service rates, i.e., X is assumed stable. Let \(\tau _n\) be the first time when the sum of the components of X equals n for the first time. Let Y be the random walk on \({{\mathbb {Z}}} \times {{\mathbb {Z}}}_+\) having increments \((-\,1,0)\), (1, 1), \((0,-\,1)\) with probabilities \(\lambda (M_k)\), \(\mu _1(M_k)\), and \(\mu _2(M_k)\). Supposing that the queues share a joint buffer of size n, \(p_n =P_{(x_n,m)}(\tau _n < \tau _0)\) is the probability that this buffer overflows during a busy cycle of the system. To the best of our knowledge, the only methods currently available for the approximation of \(p_n\) are classical large deviations analysis giving the exponential decay rate of \(p_n\) and rare event simulation. Let \(\tau \) be the first time the components of Y are equal. For \(x \in {{\mathbb {R}}}_+^2\), \(x(1) + x(2) < 1\), \(x(1) > 0\), and \(x_n = \lfloor nx \rfloor \), we show that \(P_{(n-x_n(1),x_n(2),m)}( \tau < \infty )\) approximates \(P_{(x_n,m)}(\tau _n < \tau _0)\) with exponentially vanishing relative error as \(n\rightarrow \infty \). For the analysis we define a characteristic matrix in terms of the jump probabilities of (XM). The 0-level set of the characteristic polynomial of this matrix defines the characteristic surface; conjugate points on this surface and the associated eigenvectors of the characteristic matrix are used to define (sub/super) harmonic functions which play a fundamental role both in our analysis and the computation/approximation of \(P_{(y,m)}(\tau < \infty )\).

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Notes

  1. 1.

    Note that \(\log ([(\beta ,\alpha ), y ]) = \langle (\log (\beta ),\log (\alpha )), ((y(1)-y(2)),y(2)) \rangle \) where \(\langle \cdot ,\cdot \rangle \) denotes the standard inner product in \({{\mathbb {R}}}^2\); the notation \([(\beta ,\alpha ), y ]\) was chosen with this connection to the inner product in mind. It also leads to much easier to read formulas because multiindex variables will be substituted for \(\beta \) and \(\alpha \), see, e.g., display (92).

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Two lemmas

Two lemmas

For a square matrix \({\varvec{G}}\), let \({\varvec{G}}^{i,j}\) denote the matrix obtained by removing the ith row and jth column of \({\varvec{G}}\).

Lemma A.1

For \(n_0 \in \{2,3,\ldots \}\), suppose \({\varvec{G}}\) is an \(n_0 \times n_0\) irreducible and aperiodic matrix with nonnegative entries. Then \(\det \left( (\Lambda _1({\varvec{G}}) {\varvec{I}} - {\varvec{G}})^{i,i}\right) > 0\) for all \(i \in \{1,2,\ldots ,n_0\}\), where \({\varvec{I}}\) is the \(n_0 \times n_0\) identity matrix.

Proof

The argument is the same for all \( i \in \{1,2,\ldots ,n_0\}\); so it suffices to argue for \(i=1\). Suppose the claim is not true and

$$\begin{aligned} \det \left( (\Lambda _1({\varvec{G}}){\varvec{I}} - {\varvec{G}})^{1,1}\right) \le 0. \end{aligned}$$
(101)

Consider the function \(u \mapsto g(u) = \det \left( ( u{\varvec{I}} -{\varvec{G}})^{1,1} \right) \), \(u \ge 0\). The multilinearity and continuity of \(\det \) implies \(\lim _{u\nearrow \infty } g(u) = \infty \). This implies that if (101) is true there must be \(u_0 \ge \Lambda _1({\varvec{G}})\) such that

$$\begin{aligned} \det \left( (u_0 {\varvec{I}} - {\varvec{G}})^{1,1} \right) = 0. \end{aligned}$$
(102)

The matrix \({\varvec{G}}^{1,1}\) is nonnegative, therefore, it has a largest eigenvalue \(\Lambda _1({\varvec{G}}^{1,1})\) with an eigenvector \({\varvec{v}}_1 \ge 0\). The equality (102) implies

$$\begin{aligned} \Lambda _1({\varvec{G}}^{1,1}) \ge u_0 \ge \Lambda _1({\varvec{G}}). \end{aligned}$$
(103)

That \({\varvec{G}}\) is irreducible and aperiodic implies that \({\varvec{G}}^{n_0}\) is strictly positive; its largest eigenvalue is

$$\begin{aligned} \Lambda _1({\varvec{G}}^{n_0}) = \Lambda _1({\varvec{G}})^{n_0}. \end{aligned}$$

The matrix \(({\varvec{G}}^{n_0})^{1,1}\) has strictly positive entries and therefore its largest eigenvalue \(\Lambda _1( ({\varvec{G}}^{n_0})^{1,1})\) has an eigenvalue \({\varvec{v}}_2\) with strictly positive entries. For two vectors \(x,y \in {{\mathbb {R}}}^d\), let \(x \ge y\) and \(x > y\) denote componentwise comparison. The inequality

$$\begin{aligned} ({\varvec{G}}^{n_0})^{1,1} \ge ({\varvec{G}}^{1,1})^{n_0} \end{aligned}$$

implies

$$\begin{aligned} ({\varvec{G}}^{n_0})^{1,1} {\varvec{v}}_1 \ge \Lambda _1({\varvec{G}}^{1,1})^{n_0} {\varvec{v}}_1. \end{aligned}$$
(104)

On the other hand

$$\begin{aligned} \Lambda _1(({\varvec{G}}^{n_0})^{1,1}) = \sup \{ c: \exists x \in {\mathbb R}^{n_0-1}_+, ({\varvec{G}}^{n_0})^{1,1}x \ge cx \}, \end{aligned}$$
(105)

(see Lax 1996, Proof of Theorem 1, Chapter 16). This and (104) imply

$$\begin{aligned} \Lambda _1(({\varvec{G}}^{n_0})^{1,1}) \ge \Lambda _1({\varvec{G}}^{1,1})^{n_0}. \end{aligned}$$
(106)

Define \({\varvec{v}}_3 = [ 1; {\varvec{v}}_2] \in {{\mathbb {R}}}^{n_0}\); it follows from \(({\varvec{G}}^{n_0})^{1,1} {\varvec{v}}_2 = \Lambda _1(({\varvec{G}}^{n_0}){1,1}) {\varvec{v}}_2\), the strict positivity of the components of \({\varvec{G}}^{n_0}\) and \({\varvec{v}}_2\) that one can choose \(\delta > 0\) small enough so that

$$\begin{aligned} {\varvec{G}}^{n_0} {\varvec{v}}_3 > \left( \Lambda _1(({\varvec{G}}^{n_0})^{1,1}+ \delta \right) {\varvec{v}}_3; \end{aligned}$$

This and

$$\begin{aligned} \Lambda _1({\varvec{G}}^{n_0})= \sup \{ c: \exists x \in {\mathbb R}^{n_0}_+, {\varvec{G}}^{n_0}x \ge cx \} \end{aligned}$$

imply

$$\begin{aligned} \Lambda _1({\varvec{G}}^{n_0}) > \Lambda _1\left( ({\varvec{G}}^{n_0})^{1,1}\right) . \end{aligned}$$

The last inequality, (106) and (103) imply

$$\begin{aligned} \Lambda _1({\varvec{G}})^{n_0}= \Lambda _1({\varvec{G}}^{n_0}) > \Lambda _1(({\varvec{G}}^{n_0})^{1,1}) \ge \Lambda _1({\varvec{G}}^{1,1})^{n_0} \ge \Lambda _1({\varvec{G}})^{n_0}, \end{aligned}$$

which is a contradiction. \(\square \)

In our analysis we need the following fact from Sezer (2009); its proof is elementary and follows from the multilinearity of the determinant function and the previous lemma.

Lemma A.2

Let \({\varvec{G}}\) be an aperiodic and irreducible transition matrix. Then the row vector whose ith component equals \(\det \left( ({\varvec{I}} - {\varvec{G}})^{i,i}\right) \) is the unique (upto scaling by a positive number) left eigenvector associated with the eigenvalue 1 of \({\varvec{G}}\).

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Başoğlu Kabran, F., Sezer, A.D. Approximation of the exit probability of a stable Markov modulated constrained random walk. Ann Oper Res (2020). https://doi.org/10.1007/s10479-020-03693-7

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Keywords

  • Markov modulation
  • Regime switch
  • Multidimensional constrained random walks
  • Exit probabilities
  • Rare events
  • Queueing systems
  • Characteristic surface
  • Superharmonic functions
  • Affine transformation

Mathematics Subject Classification

  • Primary 60G50
  • Secondary 60G40
  • 60F10
  • 60J45