Wiener–Hopf factorization, Erlangization and transient solutions in some Markov-modulated queueing models

Transient solutions for the buffer behavior in statistical multiplexing

In this paper, an analytical method for the transient solution of the semi-Markov model for availability analysis of repairable mechanical systems is explored. Currently, simulation software, such as Blocksim and RAPTOR, is being used mostly to solve such models. It is well known that for mechanical systems and their components, degradation rate increases with ageing process and the Weibull distribution for the time to failure is the most appropriate for such systems. Therefore, the Semi-Markov Process (SMP) model is developed for repairable mechanical systems. Since the involved integrals in the analytical solution of the SMP model with typical Weibull parameters could not be solved, the transient probability solution of states is obtained for a special case. These states, probability values lead to evaluation of the system availability. The transient solution is also obtained by the Monte Carlo simulation that matches with the analytical solution obtained from the SMP model.

In this paper we introduce a ten-parameter family of L\\'{e}vy processes for\nwhich we obtain Wiener-Hopf factors and distribution of the supremum process in\nsemi-explicit form. This family allows an arbitrary behavior of small jumps and\nincludes processes similar to the generalized tempered stable, KoBoL and CGMY\nprocesses. Analytically it is characterized by the property that the\ncharacteristic exponent is a meromorphic function, expressed in terms of beta\nand digamma functions. We prove that the Wiener-Hopf factors can be expressed\nas infinite products over roots of a certain transcendental equation, and the\ndensity of the supremum process can be computed as an exponentially converging\ninfinite series. In several special cases when the roots can be found\nanalytically, we are able to identify the Wiener-Hopf factors and distribution\nof the supremum in closed form. In the general case we prove that all the roots\nare real and simple, and we provide localization results and asymptotic\nformulas which allow an efficient numerical evaluation. We also derive a\nconvergence acceleration algorithm for infinite products and a simple and\nefficient procedure to compute the Wiener-Hopf factors for complex values of\nparameters. As a numerical example we discuss computation of the density of the\nsupremum process.\n

The Wiener–Hopf factorization plays a crucial role in studying various mathematical problems. Unfortunately, in many situations, the Wiener–Hopf factorization cannot provide closed form solutions and one has to employ some approximation techniques to find its solutions. This article provides several weak, approximation for a given Wiener–Hopf factorization problem. Application of our finding in spectral factorization and Levy processes have been given.

Both the physics and applications of fermionic symmetry-protected topological phases rely heavily on a principle known as bulk-boundary correspondence, which predicts the emergence of protected boundary-localized energy excitations (boundary states) if the bulk is topologically non-trivial. Current theoretical approaches formulate a bulk-boundary correspondence as an equality between a bulk and a boundary topological invariant, where the latter is a property of boundary states. However, such an equality does not offer insight into the stability or the sensitivity of the boundary states to external perturbations. To solve this problem, we adopt a technique known as the Wiener–Hopf factorization of matrix functions. Using this technique, we first provide an elementary proof of the equality of the bulk and the boundary invariants for one-dimensional systems with arbitrary boundary conditions in all Altland–Zirnbauer symmetry classes. This equality also applies to quasi-one-dimensional systems (e.g., junctions) formed by bulks belonging to the same symmetry class. We then show that only topologically non-trivial Hamiltonians can host stable zero-energy edge modes, where stability refers to continuous deformation of zero-energy excitations with external perturbations that preserve the symmetries of the class. By leveraging the Wiener–Hopf factorization, we establish bounds on the sensitivity of such stable zero-energy modes to external perturbations. Our results show that the Wiener–Hopf factorization is a natural tool to investigate bulk-boundary correspondence in quasi-one-dimensional fermionic symmetry-protected topological phases. Our results on the stability and sensitivity of zero-energy modes are especially valuable for applications, including Majorana-based topological quantum computing.

This paper establishes a link between the Wiener–Hopf factorization and the phase-type method for studying theGI/G/1 queue. Using the Wiener–Hopf factorization, infinite-matrix type results are established for theGI/G/1 queue. An iterative numerical procedure (‘Levinson&s method’) based on these results is described. This method does not always converge. For the situation where either the interarrival times or the service times are of the so-called almost phase type (APH) an alternative, probabilistic derivation of the same results is given. This alternative derivation shows that in the APH situation Levinson&s method converges, converges essentially monotonically, and converges to the correct values.The algorithm has been coded and examples of numerical results are included.

This paper establishes a link between the Wiener–Hopf factorization and the phase-type method for studying theGI/G/1 queue. Using the Wiener–Hopf factorization, infinite-matrix type results are established for theGI/G/1 queue. An iterative numerical procedure (‘Levinson&s method’) based on these results is described. This method does not always converge. For the situation where either the interarrival times or the service times are of the so-called almost phase type (APH) an alternative, probabilistic derivation of the same results is given. This alternative derivation shows that in the APH situation Levinson&s method converges, converges essentially monotonically, and converges to the correct values.The algorithm has been coded and examples of numerical results are included.

We present an analytic proof of the Pecherskii–Rogozin identity and the Wiener–Hopf factorization. The proof is rather general and requires only one mild restriction on the tail of the Lévy measure. The starting point of the proof of the Pecherskii–Rogozin identity is a two-dimensional integral equation satisfied by the joint distribution of the first passage time and the overshoot. This equation is reduced to a one-dimensional Wiener–Hopf integral equation, which is then solved using classical techniques from the theory of the Riemann boundary value problems. The Wiener–Hopf factorization is then derived as a corollary of the Pecherskii–Rogozin identity.

An equation with a Hardy space Toeplitz operator can be solved by Wiener–Hopf factorization. However, Wiener–Hopf factorization does not work for Bergman space Toeplitz operators. The only way we see to tackle equations with a Toeplitz operator on the Bergman space is to have recourse to approximation methods. The paper is intended as a review of and an illustrated tour through several such methods and thus through some beautiful topics in the intersection of operator theory, complex analysis, differential geometry, and numerical mathematics.

For a general Lévy process, it is possible to decompose its path into “excursions from the running maximum”. Conceptually, this decomposition is a priori somewhat tricky as, in principle, a general Lévy process may exhibit an infinite number of excursions from its maximum over any finite period of time. Nonetheless, when considered in the right mathematical framework, excursions from the maximum can be given a sensible definition in terms of a Poisson random measure. The theory of excursions presents one of the more mathematically challenging aspects of the theory of Lévy processes. This means that in order to keep to the level outlined in the preface of this text, there will be a number of proofs in the forthcoming sections which are excluded or discussed only at an intuitive level.Within a very broad spectrum of probabilistic literature, the Wiener–Hopf factorisation may be found as a common reference to a multitude of statements concerning the distributional decomposition of the path of any Lévy process, when sampled at an independent and exponentially distributed time, in terms of its excursions from the maximum. The collection of conclusions which fall under the umbrella of the Wiener–Hopf factorisation turns out to provide a robust tool with which one may analyse a number of problems concerning the fluctuations of Lévy processes, in particular, problems which have relevance to the applications we shall consider in later chapters. This chapter concludes with some special classes of Lévy processes for which the Wiener–Hopf factorisation may be exemplified in more detail.

In this paper, we study the Novikov–Shiryaev optimal stopping problem for Lévy processes. In particular, we are interested in finding the representing measure of the value function. It is seen that this can be expressed in terms of the Appell polynomials. For spectrally one-sided Lévy processes, the results are appealing and explicit. An important tool in our approach and computations is the Wiener–Hopf factorization.

In the paper, we develop a very fast and accurate method for pricing double barrier options with continuous monitoring in wide classes of Lévy models; the calculations are in the dual space, and the Wiener–Hopf factorization is used. For wide regions in the parameter space, the precision of the order of [Formula: see text] is achievable in seconds, and of the order of [Formula: see text]–[Formula: see text] — in fractions of a second. The Wiener–Hopf factors and repeated integrals in the pricing formulas are calculated using sinh-deformations of the lines of integration, the corresponding changes of variables and the simplified trapezoid rule. If the Bromwich integral is calculated using the Gaver–Wynn Rho acceleration instead of the sinh-acceleration, the CPU time is typically smaller but the precision is of the order of [Formula: see text]–[Formula: see text], at best. Explicit pricing algorithms and numerical examples are for no-touch options, digitals (equivalently, for the joint distribution function of a Lévy process and its supremum and infimum processes), and call options. Several graphs are produced to explain fundamental difficulties for accurate pricing of barrier options using time discretization and interpolation-based calculations in the state space.

We discuss an explicit algorithm for solving the Wiener–Hopf factorization problem for matrix polynomials. By an exact solution of the problem, we understand the one constructed by a symbolic computation. Since the problem is, generally speaking, unstable, this requirement is crucial to guarantee that the result following from the explicit algorithm is indeed a solution of the original factorization problem. We prove that a matrix polynomial over the field of Gaussian rational numbers admits the exact Wiener–Hopf factorization if and only if its determinant is exactly factorable. Under such a condition, we adapt the explicit algorithm to the exact calculations and develop the ExactMPF package realized within the Maple Software. The package has been extensively tested. Some examples are presented in the paper, while the listing is provided in the electronic supplementary material. If, however, a matrix polynomial does not admit the exact factorization, we clarify a notion of the numerical (or approximate) factorization that can be constructed by following the explicit factorization algorithm. We highlight possible obstacles on the way and discuss a level of confidence in the final result in the case of an unstable set of partial indices. The full listing of the package ExactMPF is given in the electronic supplementary material.

The last couple of years has seen a remarkable number of new, explicit examples of the Wiener–Hopf factorization for Lévy processes where previously there had been very few. We mention, in particular, the many cases of spectrally negative Lévy processes in [Sixth Seminar on Stochastic Analysis, Random Fields and Applications (2011) 119–146, Electron. J. Probab. 13 (2008) 1672–1701], hyper-exponential and generalized hyper-exponential Lévy processes [Quant. Finance 10 (2010) 629–644], Lamperti-stable processes in [J. Appl. Probab. 43 (2006) 967–983, Probab. Math. Statist. 30 (2010) 1–28, Stochastic Process. Appl. 119 (2009) 980–1000, Bull. Sci. Math. 133 (2009) 355–382], Hypergeometric processes in [Ann. Appl. Probab. 20 (2010) 522–564, Ann. Appl. Probab. 21 (2011) 2171–2190, Bernoulli 17 (2011) 34–59], β-processes in [Ann. Appl. Probab. 20 (2010) 1801–1830] and θ-processes in [J. Appl. Probab. 47 (2010) 1023–1033]. In this paper we introduce a new family of Lévy processes, which we call Meromorphic Lévy processes, or just M-processes for short, which overlaps with many of the aforementioned classes. A key feature of the M-class is the identification of their Wiener–Hopf factors as rational functions of infinite degree written in terms of poles and roots of the Laplace exponent, all of which are real numbers. The specific structure of the M-class Wiener–Hopf factorization enables us to explicitly handle a comprehensive suite of fluctuation identities that concern first passage problems for finite and infinite intervals for both the process itself as well as the resulting process when it is reflected in its infimum. Such identities are of fundamental interest given their repeated occurrence in various fields of applied probability such as mathematical finance, insurance risk theory and queuing theory.

Combinatorial approach to Markovian queueing models