Table of Contents Author Guidelines Submit a Manuscript
Journal of Probability and Statistics
Volume 2010 (2010), Article ID 730543, 23 pages
http://dx.doi.org/10.1155/2010/730543
Research Article

Some Comments on Quasi-Birth-and-Death Processes and Matrix Measures

Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germany

Received 12 August 2009; Revised 15 April 2010; Accepted 25 May 2010

Academic Editor: Nikolaos E. Limnios

Copyright © 2010 Holger Dette and Bettina Reuther. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We explore the relation between matrix measures and quasi-birth-and-death processes. We derive an integral representation of the transition function in terms of a matrix-valued spectral measure and corresponding orthogonal matrix polynomials. We characterize several stochastic properties of quasi-birth-and-death processes by means of this matrixmeasure and illustrate the theoretical results by several examples.

1. Introduction

Let be a continuous-time two-dimensional homogeneous Markov process with state space and infinitesimal generator where The transition rate from state to state is given by the element in the position of the matrix Markov processes with an infinitesimal generator matrix of the form (1.2) are known as continuous-time quasi-birth-and-death processes. These models have many applications in the evaluation of communicating systems and queueing systems (see, e.g., [13]) and have been analyzed by many authors (see, e.g., [46]). The case corresponds to a “classical” birth-and-death process with a tridiagonal infinitesimal generator which has been investigated in great detail using the theory of orthogonal polynomials by Karlin and McGregor [7, 8]. Since this pioneering work several authors have used these techniques to derive interesting properties of birth-and-death processes in terms of orthogonal polynomials and the corresponding measure of orthogonality (see, e.g., [9, 10]).

It is the purpose of the present paper to extend some of these results to quasi-birth-and-death processes with a generator of the form (1.2) using the theory of matrix measures and corresponding orthogonal matrix polynomials.

We associate to a matrix of the form of (1.2) a sequence of matrix polynomials, recursively defined by with initial conditions and A matrix measure on the real line is a function for which is a symmetric and nonnegative definite matrix in for each Borel set where the entries are finite signed measures. In Section 2 we formulate sufficient conditions on the infinitesimal generator (1.2) such that there exists a matrix measure on the real line with that is, the matrix polynomials are orthonormal with respect to the matrix measure (see [11]). In this case we derive an integral representation for the blocks of the transition function in terms of the orthogonal matrix polynomials and the matrix measure , which generalize the representation of Karlin and McGregor [7] to the case . We also investigate relations between the Stieltjes transforms of random walk measures corresponding to two quasi-birth-and-death processes, where only a few blocks differ. In Section 3 we discuss several examples to illustrate the theory. Finally, in Section 4 the theoretical results are used to characterize -recurrence of quasi-birth-and-death processes.

2. Quasi-Birth-and-Death Processes and Matrix Polynomials

The moments of the matrix measure are defined by the matrices and throughout this paper we will only consider matrix measures with existing moments of all order. The “left’’ inner product with respect to of two matrix polynomials and is defined by If is a sequence of matrices such that the block Hankel matrices, are positive definite, then there exists a matrix measure with moments , and a sequence of matrix polynomials which is orthogonal with respect to (see [12]). The following theorem characterizes the existence of a matrix measure such that there is a sequence of matrix polynomials which is orthogonal with respect to . The proof follows by similar arguments as presented in Theorem of [13] and is therefore omitted.

Theorem 2.1. Let the matrices , and in (1.2) be nonsingular and , and assume that is a sequence of matrix polynomials defined by recursion (1.3).
There exists a matrix measure with positive definite block Hankel matrices , such that the sequence of matrix polynomials is orthogonal with respect to if and only if there is a sequence of nonsingular matrices with Moreover, and the matrices where , are orthogonal matrices and also satisfy condition (2.4).

Note that condition (2.4) is crucial for our approach and is always satisfied in the case . If it has to be checked in concrete examples, but—to our best knowledge—there do not exist any general conditions which imply (2.4). Some examples where (2.4) is satisfied are presented in Section 3. Several other examples can be found in the papers of Grünbaum [14, 15], Grünbaum et al. [16], and Cantero et al. [17]. If condition (2.4) is satisfied, the corresponding measure is called a spectral measure corresponding to and the matrix in (2.4), respectively. The infinitesimal generator matrix (1.2) is called conservative if where and (see [18]). In this case there exists a transition function with block matrices , which satisfies the Kolmogorov forward differential equation and the Kolmogorov backward differential equation The probability of going from state to in time is given by the element in the position of the matrix

Note that there always exists a transition function such that the Kolmogorov forward differential equation (2.9) is satisfied. The infinitesimal generator is called regular if there exists only one such transition function (see [18]). If additionally a spectral measure corresponding to the generator matrix (1.2) exists, we can derive an integral representation for the block of the transition function in the position in terms of the spectral measure and the corresponding matrix orthogonal polynomials, which generalizes the famous Karlin and McGregor representation.

Theorem 2.2. Assume that the conditions for the existence of the measure in Theorem 2.1 are satisfied and that there exists a transition function which satisfies the Kolmogorov forward equation (2.9) for all Then the following representation holds for the block in the position of the transition function :

Proof. Let denote the vector of orthogonal matrix polynomials with respect to the spectral measure . Then the recursive relation (1.3) is equivalent to the matrix equation Defining we obtain the differential equation and the condition yields Hence, it follows that which implies (integrating with respect to ) that Because of the orthogonality of the matrix polynomials , we obtain for the blocks of the transition function the representation which completes the proof of Theorem 2.2.

In what follows we present two results, which relate the Stieltjes transforms of the spectral measures of two quasi-birth-and-death processes, which have an infinitesimal generator of similar structure. The first result refers to the case where the entry has been replaced by the matrix . The proof is similar to a corresponding result in [13] and is therefore omitted.

Theorem 2.3. Consider the infinitesimal generator defined by (1.2) and the matrix Let be a spectral measure corresponding to the infinitesimal generator with positive definite block Hankel matrices such that the matrix is symmetric and such that is a sequence of matrix polynomials which satisfies condition (2.4). Then there exists a spectral measure corresponding to If the spectral measures and are determined by their moments, then the Stieltjes transforms of the measures satisfy

Given a sequence of matrix polynomials defined by recursion (1.3), the corresponding associated sequence of matrix polynomials of order is defined by a recursion of the form of (1.3), in which the matrices , and have been replaced by the matrices , and respectively (see [19]). The following result gives a relation between the Stieltjes transform of the spectral measure corresponding to the sequence of matrix polynomials and the Stieltjes transform of the spectral measure corresponding to The associated quasi-birth-and-death process will be denoted by with state space defined by (1.1) (throughout this paper we use the notation ).

Theorem 2.4. Consider the infinitesimal generator defined by (1.2) and the matrix The matrix is called the associated matrix of order corresponding to Assume that is a spectral measure corresponding to with positive definite block Hankel matrices, that is, there exists a sequence of nonsingular matrices, which satisfies condition (2.4) of Theorem 2.1. Then there exists a spectral measure corresponding to with positive definite block Hankel matrices. If the measures are determined by their moments, then the Stieltjes transforms of the measures are related by where and the Stieltjes transforms of the matrix measures and are related by

Proof. Let the sequence of polynomials be defined by recursion (1.3) with corresponding spectral measure Then the polynomials are orthonormal with respect to the matrix measure and satisfy the three-term recurrence relation with initial conditions and . From Theorem and Lemma in [20] it follows that Assume that the sequence of polynomials is defined by recursion (1.3), where the matrices , and have been replaced by the matrices , and respectively, that is with and . Define , , and , From Theorem 2.1 we obtain the symmetry of the matrices and the equation Therefore, from Theorem 2.1 it follows that there exists a spectral measure with positive definite block Hankel matrices corresponding to the sequence of polynomials
The polynomials are orthonormal with respect to the measure and satisfy the recursion where Therefore, it follows from Theorem and Lemma in [20] that A combination of (2.26) and (2.32) yields and from (2.32) and (2.23) we obtain which completes the proof of the theorem.

Remark 2.5. Note that in the literature, many queueing models are considered, where the matrices do not have full rank (see [21]). Following the arguments used in Remark in [13] the conditions are sufficient for the existence of a spectral measure corresponding to where is a sequence of symmetric matrices and In other words, the assumption of nonsingularity of the matrices can be relaxed. The same arguments as those used in Theorem 2.2 then imply that

3. Examples

Example 3.1. Dayar and Quessette [3] considered a queuing system consisting of an -system and an -system. Both queues have Poisson arrival processes with rate , and exponential service distributions with rate , and it was assumed that The level represents the length of queue 1, which is unbounded, and the phase represents the length of queue 1, which can range from 0 to . The process is of interest because of its level geometric stationary distribution. This system can be described by a homogeneous Markov process with state space where and denote the length of the first queue at time and the length of the second queue at time , respectively. The entries of the corresponding infinitesimal generator (1.2) have the form , and , It is easy to see that is conservative. A straightforward calculation shows that the conditions of Theorem 2.1 are satisfied with the matrices This implies the existence of a spectral measure.

Example 3.2. In general, the spectral distribution can only be identified in special cases. Even if the Stieltjes transform can be determined, its inversion is usually difficult (see, e.g., [22, Chapter 3]). We now present an example where the spectral measure can be found explicitly. To be precise consider a homogeneous Markov process with infinitesimal generator (1.2), where , and , A generator matrix of this form can be associated to a queueing model which consists of different -systems. Each -system has a Poisson arrival process with rate and an exponential service time distribution with rate If the customer is situated in system , then it changes to the system and with the rate and respectively. This model can be described by the two-dimensional homogeneous Markov process with state space where , denotes the number of customers in the whole model at time , and denotes the number of the system at time

If and the conditions of Theorem 2.1 are satisfied with This implies the existence of a spectral measure corresponding to In order to determine the measure explicitly, note that the matrices in (2.23) have the form The eigenvalues of the matrix are given by with corresponding eigenvectors given by where With the notations and , it follows that Let be the infinitesimal generator obtained from by replacing the first diagonal block by block (which coincides with all other blocks , ), and denote by the spectral measure corresponding to From [23] we obtain for the Stieltjes transform of the matrix measure and Theorem 2.3 gives the Stieltjes transform of the measure . Moreover, the results in [23, page 318] also show that the support of the spectral measure is given by Note that if

4. -Recurrence

The decay parameter of continuous-time quasi-birth-and-death processes was introduced by van Doorn [19]. To be precise assume that is an irreducible quasi-birth-and-death process with state space (1.1) and infinitesimal generator defined by (1.2), where Then the decay parameter of the process is defined by A state is called -recurrent where denotes the th unit vector. The process is called -recurrent if and only if some state (and then all states in ) is -recurrent. The process is called -positive if and only if for some state (and then for all states in ) The following results characterize -recurrence of the process in terms of the spectral measure , the corresponding orthogonal polynomials , and the blocks of the infinitesimal generator. Throughout this section it will be assumed that condition (2.4) of Theorem 2.1 is satisfied.

Theorem 4.1. Assume that the conditions of Theorem 2.1 are satisfied with a spectral measure supported in the interval and that there exists a transition function, which satisfies the Kolmogorov forward differential equation (2.9). The process is -recurrent if and only if for some state (and then for all states in )

Proof. With representation (2.11) and Fubini's Theorem, condition (4.3) is equivalent to which implies (4.5).

In the following we define for a matrix measure with existing moments the matrices and , where and denote the Schur complement of and in the matrix and respectively (see [24]). The next result gives a representation of the Stieltjes transform of the spectral measure in terms of the quantities and the blocks of the generator matrix (1.2).

Note that is crucial for our approach and in general difficult to check. Consider for example the case of recurrence (i.e., ), then it follows from the results of Duran and Lopez-Rodriguez [25] that the spectral measure can be found as weak accumulation points of a sequence of discrete measures with support precisely on A straightforward calculation shows that the set coincides with the eigenvalues of the matrix and consequently all bounds on eigenvalues of these matrices will yield bounds on the support of spectral measure.

Corollary 4.2. Assume that conditions (2.4) of Theorem 2.1 are satisfied. Let denote the corresponding orthogonal matrix polynomials defined by recursion (1.3). Assume that the corresponding spectral measure is supported in the interval and that it is determined by its moments. Then the Stieltjes transform of the measure can be represented as In particular, the following representations hold: where ,

Proof. From Lemma in [24] it follows that the monic orthogonal matrix polynomials with respect to a matrix measure supported in satisfy the recursive relation with , and where the matrices are positive definite. Then the polynomials are orthonormal with respect to the matrix measure and satisfy the recursion with , and From Theorem in [20] it follows that where denote the first associated polynomials for defined by recursion (4.13) with initial conditions , . An application of Markov's Theorem (see [26]), (4.17), and (4.18) now yields If then we obtain from (4.19) and (1.3) in [27] where , , and An induction argument yields and the first representation in (4.11) follows. For the second part we note that the polynomials have leading coefficient and because of (1.3) they satisfy the recursion A comparison with the polynomials in (4.13) now yields Define Then (4.23) imply Therefore, we can define the polynomials From (1.3) it follows that these polynomials satisfy the recurrence relation with and Consequently we obtain from (4.23) that and hence Equation (4.11) finally yields which completes the proof of the theorem.

In the following, the -recurrence condition will be represented in terms of properties of the spectral measure, the corresponding orthogonal matrix polynomials, and the blocks of the infinitesimal generator (1.2). For this purpose, consider the process with state space defined in (1.1) and infinitesimal generator matrix where The corresponding sequence of matrix polynomials satisfies the recurrence relation with initial conditions , If conditions (2.4) of Theorem 2.1 are satisfied, then the matrix can be symmetrized with the matrices An induction argument shows the representation and therefore where the matrix measure is defined by If representation (2.11) holds, it is easy to see that and the following remark is a consequence of Theorem 4.1.

Remark 4.3. Assume that the conditions of Theorem 4.1 are satisfied and that is a corresponding spectral measure supported in the interval The process is -recurrent if and only if for some . The process is -positive if for some . This is the case if and only if the measure has a jump in the point

Theorem 4.4. Assume that the conditions of Theorem 2.1 are satisfied and that the corresponding matrix measure is supported in the interval and determined by its moments. The process is -recurrent if and only if for some state (and then for all states in ) where ,

Proof. Because condition (2.4) holds for the polynomials , this condition is also fulfilled for the polynomials with . From (4.34) it follows that for all Therefore we obtain with (4.12) From the representation , , it follows from Remark 4.3 that the state is -recurrent if and only if where

Remark 4.5. In the case the results of Theorems 4.1 and 4.4 reduce to known results in the scalar case (see Theorem (ii), (iii), (vii) in [10]).

Remark 4.6. Assume that the conditions of Theorem 4.4 are satisfied, and let be a spectral measure corresponding to the sequence of associated matrix polynomials (1)The state is -recurrent if and only if (2) The state is -positive if and only if Note that conditions (4.3) and (4.4) reduce to recurrence and positive recurrence if Therefore, with Theorem we obtain the following conditions for recurrence and positive recurrence of a quasi-birth-and-death process.

Corollary 4.7. Assume that the conditions of Theorem 2.1 are satisfied and that the corresponding matrix measure is supported in the interval and determined by its moments. The following statements hold. (1)The state is recurrent if and only if where In particular, the state is recurrent if and only if (2)The state is recurrent if and only if with (3)The state is positive recurrent if and only if the matrix measure has a jump in the point

Remark 4.8. (1) Let be a spectral measure supported in corresponding to the associated polynomials introduced in Theorem 2.4. Then, a combination of Theorem 2.4 and Corollary 4.7 shows that the state is recurrent if and only if An induction argument shows that where are the first associated polynomials corresponding to and are the associated polynomials of order corresponding to Therefore it follows for the Stieltjes transform of the spectral measure corresponding to the associated orthogonal polynomials that where
(2) A straightforward calculation yields From Theorem 2.4 it follows that the state is positive recurrent if the condition holds.

Acknowledgments

The work of the authors was supported by the Deutsche Forschungsgemeinschaft (De 502/22-1/2). The authors would like to thank Martina Stein, who typed parts of this paper with considerable technical expertise. They are also grateful to two anonymous referees for their constructive comments on an earlier version of this paper.

References

  1. M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models. An Algorithmic Approach, vol. 2 of Johns Hopkins Series in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, Md, USA, 1981. View at MathSciNet
  2. A. Ost, Performance of Communication Systems: A Model-Based Approach with Matrix-Geometric Methods, Springer, Berlin, Germany, 2001.
  3. T. Dayar and F. Quessette, “Quasi-birth-and-death processes with level-geometric distribution,” SIAM Journal on Matrix Analysis and Applications, vol. 24, no. 1, pp. 281–291, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. L. Bright and P. G. Taylor, “Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes,” Communications in Statistics, vol. 11, no. 3, pp. 497–525, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. V. Ramaswami and P. G. Taylor, “Some properties of the rate operators in level dependent quasi-birth-and-death processes with a countable number of phases,” Communications in Statistics. Stochastic Models, vol. 12, no. 1, pp. 143–164, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. G. Latouche, C. E. M. Pearce, and P. G. Taylor, “Invariant measures for quasi-birth-and-death processes,” Communications in Statistics. Stochastic Models, vol. 14, no. 1-2, pp. 443–460, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. S. Karlin and J. L. McGregor, “The differential equations of birth-and-death processes, and the Stieltjes moment problem,” Transactions of the American Mathematical Society, vol. 85, pp. 489–546, 1957. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. S. Karlin and J. L. McGregor, “The classification of birth and death processes,” Transactions of the American Mathematical Society, vol. 86, pp. 366–400, 1957. View at Google Scholar
  9. E. A. van Doorn, “Representations for the rate of convergence of birth-death processes,” Theory of Probability and Mathematical Statistics, vol. 65, pp. 37–43, 2002. View at Google Scholar
  10. E. A. van Doorn, “On associated polynomials and decay rates for birth-death processes,” Journal of Mathematical Analysis and Applications, vol. 278, no. 2, pp. 500–511, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. A. Sinap and W. Van Assche, “Orthogonal matrix polynomials and applications,” Journal of Computational and Applied Mathematics, vol. 66, no. 1-2, pp. 27–52, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. F. Marcellán and G. Sansigre, “On a class of matrix orthogonal polynomials on the real line,” Linear Algebra and Its Applications, vol. 181, pp. 97–109, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. H. Dette, B. Reuther, W. J. Studden, and M. Zygmunt, “Matrix measures and random walks with a block tridiagonal transition matrix,” SIAM Journal on Matrix Analysis and Applications, vol. 29, no. 1, pp. 117–142, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  14. F. A. Grünbaum, “Random walks and orthogonal polynomials: some challenges,” in Probability, Geometry and Integrable Systems, vol. 55 of Publications of the Research Institute for Mathematical Science, pp. 241–260, Cambridge University Press, Cambridge, UK, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. F. A. Grünbaum, “The Karlin-McGregor formula for a variant of a discrete version of Walsh's spider,” Journal of Physics. A, vol. 42, no. 45, Article ID 454010, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. F. A. Grünbaum and Manuel D. de la Iglesia, “Matrix valued orthogonal polynomials arising from group representation theory and a family of quasi-birth-and-death processes,” SIAM Journal on Matrix Analysis and Applications, vol. 30, no. 2, pp. 741–761, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. M. J. Cantero, F. A. Grünbaum, L. Moral, and L. Velázquez, “Matrix-valued Szegő polynomials and quantum random walks,” Communications on Pure and Applied Mathematics, vol. 63, no. 4, pp. 464–507, 2010. View at Google Scholar · View at MathSciNet
  18. W. J. Anderson, Continuous-Time Markov Chains, Springer Series in Statistics: Probability and Its Applications, Springer, New York, NY, USA, 1991. View at Zentralblatt MATH · View at MathSciNet
  19. E. A. van Doorn, “On the α-classification of birth-death and quasi-birth-death processes,” Stochastic Models, vol. 22, no. 3, pp. 411–421, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  20. M. J. Zygmunt, “Matrix Chebyshev polynomials and continued fractions,” Linear Algebra and Its Applications, vol. 340, pp. 155–168, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, ASA-SIAM Series on Statistics and Applied Probability, SIAM, Philadelphia, Pa, USA, 1999. View at MathSciNet
  22. H. Dette and W. J. Studden, The Theory of Canonical Moments with Applications in Statistics, Probability, and Analysis, Wiley Series in Probability and Statistics: Applied Probability and Statistics, John Wiley & Sons, New York, NY, USA, 1997. View at MathSciNet
  23. A. J. Duran, “Ratio asymptotics for orthogonal matrix polynomials,” Journal of Approximation Theory, vol. 100, no. 2, pp. 304–344, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. H. Dette and W. J. Studden, “Matrix measures, moment spaces and Favard's theorem for the interval [0,1] and [0,),” Linear Algebra and its Applications, vol. 345, pp. 163–193, 2002. View at Google Scholar
  25. A. J. Duran and P. Lopez-Rodriguez, “Orthogonal matrix polynomials: zeros and Blumenthal's theorem,” Journal of Approximation Theory, vol. 84, no. 1, pp. 96–118, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. A. J. Duran, “Markov's theorem for orthogonal matrix polynomials,” Canadian Journal of Mathematics, vol. 48, no. 6, pp. 1180–1195, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. W. Fair, “Noncommutative continued fractions,” SIAM Journal on Mathematical Analysis, vol. 2, pp. 226–232, 1971. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet