This article aims to derive the Laplace-Stieltjes transform matrix for the total increment of a one-level process during the first passage of another level process to level zero in so-called the two-dimensional Markov modulated Brownian motion. The process comprises an irreducible continuous-time Markov process with a finite state space, alongside two level processes modulated by the Markov process. These paired level processes can be viewed as a two-dimensional Brownian motion, with Brownian parameters varying based on the Markov process. Due to the infeasibility of explicit computation, we formulate a nonsymmetric algebraic Riccati equation with a minimal nonnegative solution that represents the transform matrix through a matrix exponential function. To our knowledge, this achievement is innovative within the context of the two- dimensional Markov modulated Brownian motion.

The goal of this paper is to find a nonsymmetric algebraic Riccati equation(NARE) of which the minimal nonnegative solution can represent the Laplace transform of the total increment of one component during the first passage time of the other in the two-dimensional Brownian motion. For that purpose, we construct a sequence of two-dimensional Markov modulated fluid flow which converges to the two-dimensional Brownian motion and then derive various approximation results relevant to the NARE of our interest. This is the preliminary research for investigating first-passage-related quantities in the two-dimensional Markov modulated Brownian motion in which the parameters vary according to the states of an underlying Markov process.

The Markov modulated Brownian motion is a substantial generalization of the classical Brownian Motion.On the other hand, the Markovian arrival process (MAP) is a point process whose family is dense for any stochastic point process and is used to approximate complex stochastic counting processes.In this paper, we consider a superposition of the Markov modulated Brownian motion (MMBM) and the Markovian arrival process of jumps which are distributed as the bilateral ph-type distribution, the class of which is also dense in the space of distribution functions defined on the whole real line.In the model, we assume that the inter-arrival times of the MAP depend on the underlying Markov process of the MMBM.One of the subjects of this paper is introducing how to obtain the first passage probabilities of the superposed process using a stochastic doubling algorithm designed for getting the minimal solution of a nonsymmetric algebraic Riccatti equation.The other is to provide eigenvalue and eigenvector results on the superposed process to make it possible to apply the GTH-like algorithm, which improves the accuracy of the doubling algorithm.

We consider a Markov-modulated fluid flow model with server maintenance period. As soon as the fluid level reaches zero, the server begins a maintenance period of a random length. During the maintenance period, fluid arrives from outside depending on the state of the background Markov process and the level increases either vertically or linearly. This model can be applied to various real-world systems such as inventory systems and production systems. We first derive the distribution of the fluid level and the mean performance measures. Then, we present some numerical examples to show the effect of the maintenance time.

A Markov modulated Brownian motion (MMBM) is a substantial generalization of the classical Brownian motion and is obtained by allowing the Brownian parameters to be modulated by an underlying Markov chain of environments. As in Brownian motion, the stationary analysis of the MMBM becomes easy once the distributions of the first passage time between levels are determined. Asmussen (Stochastic Models, 1995) proved that such distributions can be obtained by solving a suitable quadratic matrix equation (QME), while, more recently, Ahn and Ramaswami (Stochastic Models, 2017) derived the distributions from the solution of a suitable algebraic Riccati equation (NARE). In this paper we provide an explicit algebraic relation between the QME and the NARE, based on a linearization of a matrix polynomial. Moreover, we discuss the doubling algorithms such as the structure-preserving doubling algorithm (SDA) and alternating-directional doubling algorithm (ADDA), with shifting technique, which are used for finding the sought of the NARE.

In this paper, we intend to generalize the well-known reflection principle, one of the most interesting properties of the Brownian motion. The essence of our generalization lies in its ability to stochastically eliminate arbitrary number of partial maximums (or minimums) in the joint events associated with the Brownian motion, thereby allowing us to express the joint probabilities in terms of the multivariate normal distribution functions. Due to the simplicity and versatility, our generalized reflection principle can be used to solve many probabilistic problems pertaining to the Brownian motion. To illustrate, we consider evaluating barrier options and autocallable structured product. Using the basic inclusion-exclusion principle, we obtain integrated pricing formulas for various barrier options under the Black-Scholes model, and derive an explicit pricing formula for the autocallable product, which is not known yet despite its popularity. These formulas are explored through numerical examples. The method of Esscher transform demonstrates its time-honored value during the derivation process.

In this study, we consider an M/M/1 queuing model with an attached continuous-type inventory. Customers arrive in the system according to a Poisson process and are served individually on a first-come, first-served basis. The service times of customers are assumed to be independent and identically distributed exponential random variables. Along with the queue, there is an internal finite storage for the inventory and each service requires an exponentially distributed random amount H of inventory from the storage. Therefore, a customer leaves the system with H amount of item at his/her service completion time. The inventory is replenished by an outside supplier with a random lead time under an (s,Q) inventory control policy. We assume that the customers who arrive during stock-out periods are lost (lost sales). For this queuing-inventory system, we derive the stationary joint probability distribution of queue length and inventory level in explicit product form. Numerical examples followed by a cost model are also presented.

A number of analytical models have been proposed to estimate the write amplification of the Flash storage to obtain the expected lifespan. This work is dedicated to examining the practical implication of the four existing analytical models for estimating the write amplification: Coupon Collector, Uniform Distribution, Expected Value and Markov model. Since the models assume uniform random workload in full utilization of an SSD to predict write amplification, they are not applicable in predicting write amplification in general workload. Moreover, the existing models have not been verified with the real SSD. In this work, we compare the write amplification of the models with that of a real SSD. When we use 0.147 as the overprovisioning factor of an SSD while running uniform random workload, the write amplification of Uniform Distribution, Expected Value, Markov model is 3.90, 4.08, and 4.08, respectively. However, write amplification of the real SSD shows 1.19, which is very different from that of the prediction models. Through experiment, we found that write amplification is closely related to the value of overprovisioning factor. To improve the accuracy of existing prediction models, we update the overprovisioning factor to take account of the ratio of a hot file and the utilization of the storage. We also find that by setting the overprovisioning factor to 1.15, we can obtain write amplification of 1.2 which is close to the write amplification of general workload in a real SSD.

A Markov-modulated Brownian motion (MMBM) is a substantial generalization of the classical BrownianMotion and is obtainedby allowing the Brownian parameters to be modulated by an underlying Markov chain of environments. As with Brownian Motion, the time-dependent analysis of the MMBM becomes easy once the first passage times between levels are determined. However, in the MMBM those distributions cannot be obtained explicitly, and we need efficient algorithms to compute them. In this article, we provide a powerful approach based on approximating the MMBM with a sequence of scaled Markov-modulated fluid flows without Brownian components that weakly converge to the MMBM. Our main result is a Riccati equation for an associated matrix of transforms that satisfies conditions for the Newton scheme to have quadratic convergence and thus yields a very practical tool. The solution of that Riccati equation determines needed first passage times in the MMBM without much additional work. The success of our approach, which is based essentially on first-order fluid flows and a stochastic limit process, is argued to be due to thewaywe have isolated certain terms involving the quadratic variation effects of the Brownian. As an illustration of our algorithm, we present a numerical example of timedependent results for aMMBMconsidered byAsmussen for which he determined (only) the eventual first return probabilities which we use here as an accuracy check.

Inspired by the claim reserving problem in non-life insurance, this paper proposes to study the insurer’s surplus process under a micro-level framework, with particular focus on modeling the Incurred But Not Reported (IBNR) and the Reported But Not Settled (RBNS) claims. It is assumed that accidents occur according to a Poisson point process, and each accident is accompanied by a claim developmental mark that contains the reporting time, the settlement time, and the size of (possibly multiple) payments between these two times. Under exponential reporting and settlement delays, we show that our model can be represented as a Markovian risk process with countably infinite number of states. This can in turn be transformed to an equivalent fluid flow model when the payments are phase-type distributed. As a result, classical measures such as ruin probability or more generally the Gerber–Shiu expected discounted penalty function follow directly. The joint Laplace transform and the pairwise joint moments involving the ruin time and the aggregate payments of different types (with and without claim settlement) are further derived. Numerical illustrations are given at the end, including the use of a real insurance dataset.

In estimating the parametersof the two-parameter Pareto distributionit is well known that the performance of the maximum likelihood estimator deteriorates when sample sizes are small or the underlying model is contaminated. In this paper we propose a new parameter estimator that utilizes a pivotal quantity based on the regression framework, allowing separate estimation of the two parameters in a straightforward manner. The consistency of the estimator is also established. Simulation studies show that the proposed estimator is a competitive, well-rounded robust estimator for both Pareto and contaminated Pareto datasets when the sample sizes are small.

This article describes our study of the total shift during the first passages (one-sided and two-sided exit times) of Markov-modulated Brownian motion with bilateral ph-type jumps, which is referred to as MMBM. The total shift is defined as the value of a so-called shift process at the first passage epochs of the MMBM. The shift process, introduced by Bean and O'Reilly, behaves like a continuous Markovian fluid process; that is, it increases or decreases linearly with slopes regulated by the underlying Markov process that determines the path of the MMBM. Hence, the notion of total shift, which includes the first passage times of the MMBM as special cases, is useful for describing various performance measures of systems modeled by the MMBM. In this article, we present formulas for the Laplace-Stieltjes transform matrices of the total shift during various first passages of the MMBM. In particular, a Riccati equation is derived so that a matrix associated with the Laplace-Stieltjes transform of the total shift during the first return time of the MMBM is its minimal non-negative solution matrix. With this solution matrix, the Laplace-Stieltjes transform matrices can be obtained without much additional work. Furthermore, it is shown that the Riccati equation satisfies the conditions for the Newton scheme to have quadratic convergence, which enables us to use algorithms with quadratic convergence, such as Newton's method and the Stochastic Doubling Algorithm, to compute the presented matrix-driven formulas. For the analyses, we take an approach based on approximating the MMBM with a sequence of scaled Markov-modulated fluid flows with bilateral ph-type jumps, referred to as MMFF, that weakly converge to the MMBM. Another contribution of this article is that duality results are derived in relation to the MMBM, which is an extension of the duality theorems developed by Ahn and Ramaswami for an MMFF without a jump.

A Markov-modulated Brownian motion with bilateral ph-type jumps, referred to as MMBM, is a generalization of the Lévy process. In this paper, we study the time-dependent behavior of the two-sided reflected MMBM (TR-MMBM) with boundaries 0 and β>0. In contrast to previous research on the subject, we propose a different approach based on the observation that the TR-MMBM can be realized as the limit of a sequence of two-sided reflected Markov-modulated fluid flows with bilateral ph-type jumps (TR-MMFF), which are MMBMs without a Brownian component. Therefore, the TR-MMBM can be analyzed via methods for the TR-MMFFs, through limiting arguments based on the weak convergence and continuous mapping theorems. Along these lines, we first analyze time-dependent behaviors of the sequence of TR-MMFFs using a new methodology that adopts the so-called completed graph and also using Markov renewal and skip-free level crossing arguments. Then, relying on the appropriate stochastic limit arguments, we finally present the Laplace transform of the time-dependent distribution of the TR-MMBM with respect to time. In addition, we show that the stationary distribution of the TR-MMBM can be obtained directly from the Laplace transform.

In this paper, we consider a diffusion risk process, in which, its surplus process behaves like a Brownian motion in-between adjacent epochs of claims. We assume that the claims occur following a Poisson process and their sizes are independent and exponentially distributed with the same intensity. Our main goal is to derive the exact formula of the joint moment generating function of the ruin time and the total amount of aggregated claim sizes until ruin in the diffusion risk process. We also provide a method for computing the related first and second moments using the joint moment generating function and the augmented matrix exponential function.

Time-dependent solutions to queuing models are very useful for evaluating the performance of real-world systems. However, because of their mathematical complexity, few available results exist. In this paper, we derive the time-dependent performance measures for an M/D/1 queue starting with a positive number of initial customers. Using the limiting property of an Erlang distribution, we obtain closed-form time-dependent formulas for the queue length and the waiting time. Furthermore, the time-dependent queue length probability in a busy period is derived.

The steady-state distribution of the two-sided reflected Markov Modulated Brownian motion is derived through an alternative fluid approximation approach. In this paper, we have shown that the distribution can be obtained by taking limit on the steady-state distribution of a weakly convergent Markov modulated fluid model. In addition, we present how the duality theorem developed by Ahn and Ramaswami is applied to get the limit result. For computation, we introduce a quadratically convergent algorithm and its related theories which rely on the fluid approximation approach. Through numerical examples, we finally illustrate behaviors of the steady-state distribution and also some performances of the algorithm.

In this paper, we analyze Markov modulated fluid flow processes with one-sided ph-type jumps using the completed graph and also through the limits of coupled queueing processes to be constructed. For the models, we derive various results on time-dependent distributions and distributions of first passage times, and present the Riccati equations that transform matrices of the first return times to 0 satisfy. The Riccati equations enable us to compute the transform matrices using Newton’s method which is known very fast and stable. Finally,wepresent some duality results between the model with ph-type downward jumps and the model with ph-type upward jumps. This paper contains extended results of Ahn (2009) and probabilistic interpretations given by the completed graphs.

We consider a MAP-modulated fluid flow queueing model with multiple vacations. As soon as the fluid level reaches zero, the server leaves for repeated vacations of random length V until the server finds any fluid in the system. During the vacation period, fluid arrives from outside according to the MAP (Markovian Arrival Process) and the fluid level increases vertically at the arrival instance. We first derive the vector Laplace–Stieltjes transform (LST) of the fluid level at an arbitrary point of time in steady-state and show that the vector LST is decomposed into two parts, one of which the vector LST of the fluid level at an arbitrary point of time during the idle period. Then we present a recursive moments formula and numerical examples.

Many insurance loss data are known to be heavy-tailed. In this article we study the class of Log phase-type (LogPH) distributions as a parametric alternative in fitting heavy tailed data. Transformed from the popular phase-type distribution class, the LogPH introduced by Ramaswami exhibits several advantages over other parametric alternatives. We analytically derive its tail related quantities including the conditional tail moments and the mean excess function, and also discuss its tail thickness in the context of extreme value theory. Because of its denseness proved herein, we argue that the LogPH can offer a rich class of heavy-tailed loss distributions without separate modeling for the tail side, which is the case for the generalized Pareto distribution (GPD). As a numerical example we use the well-known Danish fire data to calibrate the LogPH model and compare the result with that of the GPD. We also present fitting results for a set of insurance guarantee loss data.