Steve Drekic

An analytical solution for a tandem queue with blocking

The model considered in this paper involves a tandem queue with two waiting lines, and as soon as the second waiting line reaches a certain upper limit, the first line is blocked. Both lines have exponential servers, and arrivals are Poisson. The objective is to determine the joint distribution of both lines in equilibrium. This joint distribution is found by using generalized eigenvalues. Specifically, a simple formula involving the cotangent is derived. The periodicity of the cotangent is then used to determine the location of the majority of the eigenvalues. Once all eigenvalues are found, the eigenvectors can be obtained recursively. The method proposed has a lower computational complexity than all other known methods.

ON THE DENSITY AND MOMENTS OF THE TIME OF RUIN WITH EXPONENTIAL CLAIMS

The probability density function of the time of ruin in the classical model with exponential claim sizes is obtained directly by inversion of the associated Laplace transform. This result is then used to obtain explicit closed-form expressions for the moments. The form of the density is examined for various parameter choices.

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.

On the analysis of a multi-threshold Markovian risk model

We consider a class of Markovian risk models perturbed by a multiple threshold dividend strategy in which the insurer collects premiums at rate c i whenever the surplus level resides in the i-th surplus layer, i=1, 2, …,n+1 where n<∞. We derive the Laplace-Stieltjes transform (LST) of the distribution of the time to ruin as well as the discounted joint density of the surplus prior to ruin and the deficit at ruin. By interpreting that the insurer, whose gross premium rate is c, pays dividends continuously at rate d i =c−c i whenever the surplus level resides in the i-th surplus layer, we also derive the expected discounted value of total dividend payments made prior to ruin. Our results are obtained via a recursive approach which makes use of an existing connection, linking an insurers surplus process to an embedded fluid flow process.

Analysis of a threshold dividend strategy for a MAP risk model

We consider a class of Markovian risk models in which the insurer collects premiums at rate c1(c2) whenever the surplus level is below (above) a constant threshold level b. We derive the Laplace-Stieltjes transform (LST) of the distribution of the time to ruin as well as the LST (with respect to time) of the joint distribution of the time to ruin, the surplus prior to ruin, and the deficit at ruin. By interpreting that the insurer pays dividends continuously at rate c1−c2 whenever the surplus level is above b, we also derive the expected discounted value of total dividend payments made prior to ruin. Our results are obtained by making use of an existing connection which links an insurers surplus process to an embedded fluid flow process.

Optimal dividends under a ruin probability constraint

ABSTRACT We consider a classical surplus process modified by the payment of dividends when the insurers surplus exceeds a threshold. We use a probabilistic argument to obtain general expressions for the expected present value of dividend payments, and show how these expressions can be applied for certain individual claim amount distributions. We then consider the question of maximising the expected present value of dividend payments subject to a constraint on the insurers ruin probability.

Dividend Moments in the Dual Risk Model: Exact and Approximate Approaches

In the classical compound Poisson risk model, it is assumed that a company (typically an insurance company) receives premium at a constant rate and pays incurred claims until ruin occurs. In contrast, for certain companies (typically those focusing on invention), it might be more appropriate to assume expenses are paid at a fixed rate and occasional random income is earned. In such cases, the surplus process of the company can be modelled as a dual of the classical compound Poisson model, as described in Avanzi et al. (2007). Assuming further that a barrier strategy is applied to such a model (i.e., any overshoot beyond a fixed level caused by an upward jump is paid out as a dividend until ruin occurs), we are able to derive integro-differential equations for the moments of the total discounted dividends as well as the Laplace transform of the time of ruin. These integro-differential equations can be solved explicitly assuming the jump size distribution has a rational Laplace transform. We also propose a discrete-time analogue of the continuous-time dual model and show that the corresponding quantities can be solved for explicitly leaving the discrete jump size distribution arbitrary. While the discrete-time model can be considered as a stand-alone model, it can also serve as an approximation to the continuous-time model. Finally, we consider a generalization of the so-called Dickson-Waters modification in optimal dividends problems by maximizing the difference between the expected value of discounted dividends and the present value of a fixed penalty applied at the time of ruin.

A preemptive priority queue with balking

Abstract This paper analyzes a 2-class, single-server preemptive priority queueing model with low priority balking customers. Arrivals to each class are assumed to follow a Poisson process with exponentially distributed service times. The decision to balk or not is made on the basis of queue length, and two specific forms of balking behaviour are considered. The system under consideration is a quasi-birth and death process, and the steady-state joint distribution of the number of high and low priority customers in the system is derived explicitly via the method of generalized eigenvalues.

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.

Threshold-based interventions to optimize performance in preemptive priority queues

This paper studies a single-server priority queueing model in which preemptions are allowed during the early stages of service. Once enough service effort has been rendered, however, further preemptions are blocked. The threshold where the change occurs is either a proportion of the service requirement, or time-based. The Laplace–Stieltjes transform and mean of each class sojourn time are derived for a model which employs this hybrid preemption policy. Both preemptive resume and preemptive repeat service disciplines are considered. Numerical examples show that it is frequently the case that a good combination of preemptible and nonpreemptible service performs better than both the standard preemptive and nonpreemptive queues. In a number of these cases, the thresholds that optimize performance measures such as overall average sojourn time are determined.