Matrix Analytic Methods Research Articles

The matrices R and G of matrix analytic methods and the time-inhomogeneous periodic Quasi-Birth-and-Death process

We solve for the asymptotic periodic distribution of the continuous time quasi-birth-and-death process with time-varying periodic rates in terms of $\hat{\mathbf{R}}$ and $\hat{\mathbf{G}}$ matrix functions which are analogues of the R and G matrices of matrix analytic methods. We evaluate these QBDs numerically by solving for $\hat{\mathbf{R}}$ numerically.

Pricing kth-to-default swaps under default contagion: the matrix analytic approach

We study a model for default contagion in intensity-based credit risk and its consequences for pricing portfolio credit derivatives. The model is specified through default intensities that are assumed to be constant between defaults, but that can jump at the times of defaults. The model is translated into a Markov jump process that represents the default status in the credit portfolio. This makes it possible to use matrix-analytic methods to derive computationally tractable closed-form expressions for single-name credit default swap (CDS) spreads and kth-to-default swap spreads. We “semicalibrate” the model for portfolios (of up to 15 obligors) against market CDS spreads and compute the corresponding kth-to-default spreads. In a numerical study based on a synthetic portfolio of 15 telecom bonds we study a number of questions: How do spreads depend on the amount of default interaction?; How do the values of the underlying market CDS prices used for calibration influence kth-th-to default spreads?; How does a portfolio with inhomogeneous recovery rates compare with a portfolio that satisfies the standard assumption of identical recovery rates?; How well can kth-thto default spreads in a non-symmetric portfolio be approximated by spreads in a symmetric portfolio?

Multi-server queueing systems with cooperation of the servers

In this paper, we consider an N-server queueing model with homogeneous servers in which customers arrive according to a stationary Poisson arrival process. The service times are exponentially distributed. Two new customer’s service disciplines assuming simultaneous service of arriving customer by all currently idle servers are discussed. The steady state analysis of the queue length and sojourn time distribution is performed by means of the matrix analytic methods. Numerical examples, which illustrate advantage of introduced disciplines comparing to the classical one, are presented.

Two classes of time-inhomogeneous Markov chains: Analysis of the periodic case

We consider the M/G/1 and GI/M/1 types of Markov chains for which their one step transitions depend on the times of the transitions. These types of Markov chains are encountered in several stochastic models, including queueing systems, dams, inventory systems, insurance risk models, etc. We show that for the cases when the time parameters are periodic the systems can be analyzed using some extensions of known results in the matrix-analytic methods literature. We have limited our examples to those relating to queueing systems to allow us a focus. An example application of the model to a real life problem is presented.

Markovian trees: properties and algorithms

In this paper we introduce a structure called the Markovian tree (MT). We define the MT and explore its alternative representation as a continuous-time Markovian Multitype Branching Process. We then develop two algorithms, the Depth and Order algorithms to determine the probability of eventual extinction of the MT process. We show that both of these algorithms have very natural physically intuitive interpretations and are analogues of the Neuts and U algorithms in Matrix-analytic Methods. Furthermore, we show that a special case of the Depth algorithm sheds new light on the interpretation of the sample paths of the Neuts algorithm.

Coxian approximations of matrix-exponential distributions

In this paper, we study the approximation of matrix-exponential distributions by Coxian distributions. Based on the spectral polynomial algorithm, we develop an algorithm for computing Coxian representations of Coxian distributions that are approximations of matrix-exponential distributions. As a specialization, we show that phase-type (PH) distributions can be approximated by Coxian distributions. We also show that any phase-type generator with only real eigenvalues is PH-majorized by ordered Coxian generators. Consequently, the algorithm is modified for computing ordered Coxian representations of any phase-type distribution whose Laplace-Stieltjes transform has only real poles. Numerical examples are presented to show the efficiency of the algorithm and the accuracy of the Coxian approximations. Keywords: Matrix-exponential distribution, Coxian distribution, phase-type distribution, matrix analytic methods, Perron-Frobenius theory Mathematics subject classification (2000): Primary 60A99, Secondary 15A18

Algorithmic Analysis of the Sparre Andersen Model in Discrete Time

In this paper, we show that the delayed Sparre Andersen insurance risk model in discrete time can be analyzed as a doubly infinite Markov chain. We then describe how matrix analytic methods can be used to establish a computational procedure for calculating the probability distributions associated with fundamental ruin-related quantities of interest, such as the time of ruin, the surplus immediately prior to ruin, and the deficit at ruin. Special cases of the model, namely the ordinary and stationary Sparre Andersen models, are considered in several numerical examples.

Algorithmic Analysis of the Sparre Andersen Model in Discrete Time

In this paper, we show that the delayed Sparre Andersen insurance risk model in discrete time can be analyzed as a doubly infinite Markov chain. We then describe how matrix analytic methods can be used to establish a computational procedure for calculating the probability distributions associated with fundamental ruin-related quantities of interest, such as the time of ruin, the surplus immediately prior to ruin, and the deficit at ruin. Special cases of the model, namely the ordinary and stationary Sparre Andersen models, are considered in several numerical examples.

An Application of the Pump-to-Fill Policy for Management of Urban Stormwater

We consider the management of urban stormwater in two connected dams. Stormwater generated by local rainfall flows into a capture dam and is subsequently pumped into a similar sized holding dam. We assume random gross inflow and constant demand. If we wish to minimise overflow from the system then the optimal management policy is to pump as much water as possible each day from the capture dam to the holding dam without allowing the holding dam to overflow. We shall refer to this policy as the pump-to-fill policy. The model is based on the Parafield stormwater management system in the City of Salisbury (CoS) but assumes constant demand instead of level dependent outflow. If there is insufficient water in the holding dam to meet the desired daily demand then all water in the holding dam is used and the shortfall is obtained from other sources. CoS, in suburban Adelaide in South Australia, is recognised in local government circles as a world leader in urban stormwater management. The water is supplied to local industry to replace regular mains water and is also used to restore and maintain urban wetlands. In mathematical terms the pump-to-fill policy defines a Markov chain with a large transition matrix and a characteristic regular block structure. We use specialised Matrix Analytic Methods to decompose the event space and find simplified equations for the steady state probability vector. In this way we enable an elementary solution procedure which we illustrate by solving the modified Parafield problem. The optimal nature of the pump-to-fill policy is established in a recent paper by Pearce et al. (JIMO 3(2):313–320, 2007). The purpose of the current study is to find optimal management policies for urban stormwater systems.

Modelling Default Contagion Using Multivariate Phase-Type Distributions

We model dynamic credit portfolio dependence by using default contagion in an intensity-based framework. Two different portfolios (with 10 obligors), one in the European auto sector, the other in the European financial sector, are calibrated against their market CDS spreads and the corresponding CDS-correlations. After the calibration, which are perfect for the banking portfolio, and good for the auto case, we study several quantities of importance in active credit portfolio management. For example, implied multivariate default and survival distributions, multivariate conditional survival distributions, implied default correlations, expected default times and expected ordered defaults times. The default contagion is modelled by letting individual intensities jump when other defaults occur, but be constant between defaults. This model is translated into a Markov jump process, a so called multivariate phase type distribution, which represents the default status in the credit portfolio. Matrix-analytic methods are then used to derive expressions for the quantities studied in the calibrated portfolios.

Pricing kth-to-Default Swaps Under Default Contagion: The Matrix-Analytic Approach

We study a model for default contagion in intensity-based credit risk and its consequences for pricing portfolio credit derivatives. The model is specified through default intensities which are assumed to be constant between defaults, but which can jump at the times of defaults. The model is translated into a Markov jump process which represents the default status in the credit portfolio. This makes it possible to use matrix-analytic methods to derive computationally tractable closed-form expressions for single-name credit default swap spreads and kth-to-default swap spreads. We semi-calibrate the model for portfolios (of up to 15 obligors) against market CDS spreads and compute the corresponding kth-to-default spreads. In a numerical study based on a synthetic portfolio of 15 telecom bonds we study a number of questions: how spreads depend on the amount of default interaction; how the values of the underlying market CDS-prices used for calibration influence kth-to-default spreads; how a portfolio with inhomogeneous recovery rates compares with a portfolio which satisfies the standard assumption of identical recovery rates; and, finally, how well kth-to-default spreads in a nonsymmetric portfolio can be approximated by spreads in a symmetric portfolio.

Management of water storage in two connected dams

We consider the management of urban stormwater in two connected dams. Stormwater generated by local rainfall flows into a large capture dam and is subsequently pumped to a smaller supply dam. We assume random supply and constant demand. A simple management policy is to pump as much water as possible each day from the capture dam to the supply dam without allowing the supply dam to overflow. If there is insufficient water in the supply dam to meet the desired daily demand then all water in the supply dam is used. The policy defines a large block transition matrix with repeating entries. We use Matrix Analytic Methods to decompose the event space for the various state transitions and subsequently construct simplified equations for the invariant state probability vector. In this way we enable an elementary solution procedure to find the invariant probability. This paper extends previous work by Piantadosi [12] by allowing higher levels of demand and prefaces more recent work by Piantadosi et al [13].

On the analysis of the Gerber–Shiu discounted penalty function for risk processes with Markovian arrivals

In this paper, we consider an insurance risk model governed by a Markovian arrival claim process and by phase-type distributed claim amounts, which also allows for claim sizes to be correlated with the inter-claim times. A defective renewal equation of matrix form is derived for the Gerber–Shiu discounted penalty function and solved using matrix analytic methods. The use of the busy period distribution for the canonical fluid flow model is a key factor in our analysis, allowing us to obtain an explicit form of the Gerber–Shiu discounted penalty function avoiding thus the use of Lundberg’s fundamental equation roots. As a special case, we derive the triple Laplace transform of the time to ruin, surplus prior to ruin, and deficit at ruin in explicit form, further obtaining the discounted joint and marginal moments of the surplus prior to ruin and the deficit at ruin.

Analyzing priority queues with 3 classes using tree-like processes

In this paper we demonstrate how tree-like processes can be used to analyze a general class of priority queues with three service classes, creating a new methodology to study priority queues. The key result is that the operation of a 3-class priority queue can be mimicked by means of an alternate system that is composed of a single stack and queue. The evolution of this alternate system is reduced to a tree-like Markov process, the solution of which is realized through matrix analytic methods. The main performance measures, i.e., the queue length distributions and loss rates, are obtained from the steady state of the tree-like process through a censoring argument. The strength of our approach is demonstrated via a series of numerical examples.

A matrix-analytic solution for the DBMAP/PH/1 priority queue

Priority queueing models have been commonly used in telecommunication systems. The development of analytically tractable models to determine their performance is vitally important. The discrete time batch Markovian arrival process (DBMAP) has been widely used to model the source behavior of data traffic, while phase-type (PH) distribution has been extensively applied to model the service time. This paper focuses on the computation of the DBMAP/PH/1 queueing system with priorities, in which the arrival process is considered to be a DBMAP with two priority levels and the service time obeys a discrete PH distribution. Such a queueing model has potential in performance evaluation of computer networks such as video transmission over wireless networks and priority scheduling in ATM or TDMA networks. Based on matrix-analytic methods, we develop computation algorithms for obtaining the stationary distribution of the system numbers and further deriving the key performance indices of the DBMAP/PH/1 priority queue.

Queues with Correlated Service and Inter-Arrival Times and Their Application to Optical Buffers

This paper presents an algorithmic procedure to calculate the queue length and delay distribution of customers in a discrete time D-MAP/PH/1 queue, where the service time distribution of a customer depends on the inter-arrival time between himself and his predecessor. Setting up a Markov chain that keeps track of the contents of such a queue will result in a state space explosion as the inter-arrival times of all customers present in the system must be remembered. We avoid these difficulties by making use of the age process, a process that keeps track of the “age” of the customer in the service facility. From this process, which we solve by means of matrix analytic methods, we compute the queue length and sojourn time distribution by means of a simple formula and obtain an expression for the stability of the system. We also demonstrate that the D-MAP arrival process can be easily replaced by the more general semi-Markovian arrival process, without any additional computational costs. Queueing systems of this type arise in the domain of synchronous optical buffers. Based on the numerical analysis of such a queueing system, some guidelines for the design of optical buffers are presented. We also show the impact on the numerical results when the cross-correlation that exists between the service and inter-arrival times is neglected.

Solving Ck/Cm/1/N queues by using characteristic roots in matrix analytic methods

In this paper, we study a C k / C m /1/ N open queueing system with finite capacity. We investigate the property which shows that a product of the Laplace Stieltjes Transforms of interarrival and service times distributions satisfies an equation of a simple form. According to this equation, we present that the stationary probabilities on the unboundary states can be written as a linear combination of vector product-forms. Each component of these products is expressed in terms of roots of an associated characteristic polynomial. As a result, we carry out an algorithm for solving stationary probabilities in C k / C m /1/ N systems, which is independent of N , hence greatly reducing the computational complexity.

Linking priority queues and tree-like processes

This paper links the analysis of a general class of priority queues with three service classes with tree-like processes. This is realized by demonstrating that the operation of a 3-class priority queue can be mimicked by means of an alternate system that is composed of a single stack and queue. Some insights on how to reduce the evolution of this alternate system to a tree-like process are provided, allowing an efficient solution of the priority system through matrix analytic methods.

A bibliographical guide to the analysis of retrial queues through matrix analytic techniques

This paper provides a bibliographical guide to researchers who are interested in the analysis of retrial queues through matrix analytic methods. It includes an author index and a subject index of research papers written in English and published in journals or collective publications, as well as some papers accepted for a forthcoming publication.

Bridging ETAQA and Ramaswami’s formula for the solution of M/G/1-type processes

M/G/1-type processes are often evaluated using matrix analytic methods. Specifically, Ramaswami’s recursive formula has been established as the numerically stable solution tool. Additionally, the ETAQA method, proposed previously, offers a more efficient alternative for the exact computation of a general class of metrics for M/G/1-type processes. However, the theoretical stability of ETAQA and its relation to Ramaswami’s method were not well understood. In this paper, we derive a new formulation that improves the numerical stability and computational performance of ETAQA. The resulting new method solves a homogeneous system of equations to obtain the aggregate probability of a finite set of classes of states from the state space of the underlying Markov chain. The new method, constructs this system of linear equations in a way similar to the method of Ramaswami, decoupling the computation of the probabilities of the first two initial classes of states from the computation of the aggregate probability. Because direct methods are used to solve this system, the decoupling implies an often significant speed-up over ETAQA. In addition, we show that the coefficients of the homogeneous system of equations define an M-matrix, which under certain conditions, is also diagonally dominant and thus can be factored stably. More importantly, we show that the new method is just an efficient way to implement Ramaswami’s method. We also discuss alternative normalization conditions for Ramaswami’s method. Our numerical experiments demonstrate the stability of our method for both stiff and well-behaved processes, and for both low and high system utilizations.