David A. Stanford

Risk processes analyzed as fluid queues

This paper presents the Laplace transform of the time until ruin for a fairly general risk model. The model includes both the classical and most Sparre-Andersen risk models with phase-distributed claim amounts as special cases. It also allows for correlated arrival processes, and claim sizes that depend upon environmental factors such as periods of contagion. The paper exploits the relationship between the surplus process and fluid queues, where a number of recent developments have provided the basis for our analysis.

Phase-type approximations to finite-time ruin probabilities in the Sparre-Andersen and stationary renewal risk models

The present paper extends the “Erlangization” idea introduced by Asmussen, Avram, and Usabel (2002) to the Sparre-Andersen and stationary renewal risk models. Erlangization yields an asymptotically-exact method for calculating finite time ruin probabilities with phase-type claim amounts. The method is based on finding the probability of ruin prior to a phase-type random horizon, independent of the risk process. When the horizon follows an Erlang-l distribution, the method provides a sequence of approximations that converges to the true finite-time ruin probability as l increases. Furthermore, the random horizon is easier to work with, so that very accurate probabilities of ruin are obtained with comparatively little computational effort. An additional section determines the phase-type form of the deficit at ruin in both models. Our work exploits the relationship to fluid queues to provide effective computational algorithms for the determination of these quantities, as demonstrated by the numerical examples.

On the Distribution of the Deficit at Ruin when Claims are Phase-type

We consider the distribution of the deficit at ruin in the Sparre Andersen renewal risk model given that ruin occurs. We show that if the individual claim amounts have a phase-type distribution, then there is a simple phase-type representation for the distribution of the deficit. We illustrate the application of this result with several examples.

Matrix Analytic Methods

This chapter shows how to find the equilibrium probabilities in processes of GI/M/1 type, and M/G/1 type, and GI/G/1 type by matrix analytic methods. GI/M/1-type processes are Markov chains with transition matrices having the same structure as the imbedded Markov chain of a GI/M/1 queue, except that the entries are matrices rather than scalars. Similarly, M/G/1 type processes have transition matrices of the same form as the imbedded Markov chain of the M/G/1 queue, except that the entries are matrices. In the imbedded Markov chain of the GI/M/1 queue, all columns but the first have the same entries, except that they are displaced so that the diagonal block entry is common to all. Similarly, in the M/G/1 queue, all rows except the first one are equal after proper centering.

A stochastic forest fire growth model

We consider a stochastic fire growth model, with the aim of predicting the behaviour of large forest fires. Such a model can describe not only average growth, but also the variability of the growth. Implementing such a model in a computing environment allows one to obtain probability contour plots, burn size distributions, and distributions of time to specified events. Such a model also allows the incorporation of a stochastic spotting mechanism.

The erlangization method for Markovian fluid flows

For applications of stochastic fluid models, such as those related to wildfire spread and containment, one wants a fast method to compute time dependent probabilities. Erlangization is an approximation method that replaces various distributions at a time t by the corresponding ones at a random time with Erlang distribution having mean t. Here, we develop an efficient version of that algorithm for various first passage time distributions of a fluid flow, exploiting recent results on fluid flows, probabilistic underpinnings, and some special structures. Some connections with a familiar Laplace transform inversion algorithm due to Jagerman are also noted up front.

The Bilingual Server System: A Queueing Model Featuring Fully And Partially Qualified Servers

AbstractNumerous organizations are required to provide service in two languages in the same local area. Within each such region, one language (the majority language) predominates, but adequate service in the second (the minority language) must be maintained. The servers hired by the organization either speak both languages (bilingual), or only speak the majority language (unilingual). This paper presents a server provisioning scenario for handling this mix of traffic. The queueing model for this problem is then presented. Performance measures that are derived include the average queueing delays, the queue length distribution, and the probability of a language mismatch for a minority language customer. From these, it is possible to find the minimal number of bilingual servers required to maintain “satisfactory” service for the minority-language customers. Numerical examples are given.

Erlangian approximation to finite time ruin probabilities in perturbed risk models

In this paper, we consider a class of perturbed risk processes that have an underlying Markov structure, including Markov-modulated risk processes, and Sparre–Andersen risk processes when both inter-claim times and claim sizes are phase-type. We apply the Erlangization method to the risk process in the class in order to obtain an accurate approximation of the finite time ruin probability. In addition, we develop an efficient recursive procedure by recognizing a repeating structure in the probability matrices we work with. We believe the present work is among the first to either compute or approximate finite time ruin probabilities in the perturbed risk model.

The interdeparture-time distribution for each class in the ∑ i M i /G i /1 queue

A stationary queueing system is described in which a single server handles several competing Poisson arrival streams on a first-come first-served basis. Each class has its own generally distributed service time characteristics. The principal result is the Laplace-Stieltjes transform, for each class, of the interdeparture time distribution function. Examples are given and applications are discussed.

The surplus prior to ruin and the deficit at ruin for a correlated risk process

This paper presents an explicit characterization for the joint probability density function of the surplus immediately prior to ruin and the deficit at ruin for a general risk process, which includes the Sparre-Andersen risk model with phase-type inter-claim times and claim sizes. The model can also accommodate a Markovian arrival process which enables claim sizes to be correlated with the inter-claim times. The marginal density function of the surplus immediately prior to ruin is specifically considered. Several numerical examples are presented to illustrate the application of this result.