On the accuracy of phase-type approximations of heavy-tailed risk models

Numerical evaluation of ruin probabilities in the classical risk model is an important problem. If claim sizes are heavy-tailed, then such evaluations are challenging. To overcome this, an attractive way is to approximate the claim sizes with a phase-type distribution. What is not clear though is how many phases are enough in order to achieve a specific accuracy in the approximation of the ruin probability. The goals of this paper are to investigate the number of phases required so that we can achieve a pre-specified accuracy for the ruin probability and to provide error bounds. Also, in the special case of a completely monotone claim size distribution we develop an algorithm to estimate the ruin probability by approximating the excess claim size distribution with a hyperexponential one. Finally, we compare our approximation with the heavy traffic and heavy tail approximations.Comment: 24 pages, 13 figures, 8 tables, 38 reference

Corrected phase-type approximations of heavy-tailed queueing models in a Markovian environment

Significant correlations between arrivals of load-generating events make the numerical evaluation of the workload of a system a challenging problem. In this paper, we construct highly accurate approximations of the workload distribution of the MAP/G/1 queue that capture the tail behavior of the exact workload distribution and provide a bounded relative error. Motivated by statistical analysis, we consider the service times as a mixture of a phase-type and a heavy-tailed distribution. With the aid of perturbation analysis, we derive our approximations as a sum of the workload distribution of the MAP/PH/1 queue and a heavy-tailed component that depends on the perturbation parameter. We refer to our approximations as corrected phase-type approximations, and we exhibit their performance with a numerical study.Comment: Received the Marcel Neuts Student Paper Award at the 8th International Conference on Matrix Analytic Methods in Stochastic Models 201

The Class of Semi-Markov Accumulation Processes

In conjunction with the 15th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM 2017)International audienceIn this paper, we introduce a new accumulation process, the Semi-Markov Accumulation Process (SMAP). This class of processes extends the framework of continuous-time Markov Additive Processes (MAPs) by allowing the underlying environmental component to be a semi-Markov process instead of a Markov process. Next, we follow an analytic approach to derive a Master Equation formula of the Renewal type that describes the evolution of SMAPs in time. We show that under exponential holding times, a matrix exponential form analogous to the matrix exponent of a MAP is attained. Finally, we consider an application of our results where closed-form solutions are rather easy to achieve

Finite-time ruin probabilities under large-claim reinsurance treaties for heavy-tailed claim sizes

We investigate the probability that an insurance portfolio gets ruined within a finite time period under the assumption that the r largest claims are (partly) reinsured. We show that for regularly varying claim sizes the probability of ruin after reinsurance is also regularly varying in terms of the initial capital, and derive an explicit asymptotic expression for the latter. We establish this result by leveraging recent developments on sample-path large deviations for heavy tails. Our results allow, on the asymptotic level, for an explicit comparison between two well-known large-claim reinsurance contracts, namely LCR and ECOMOR. We finally assess the accuracy of the resulting approximations using state-of-the-art rare event simulation techniqu

A control theoretic analysis of the Bullwhip effect under triple exponential smoothing forecasts

In this paper, we study the performance of an Automatic Pipeline, Variable Inventory, Order-Based Production Control System (APVIOBPCS) using linear control theory. In particular, we consider a system with independent adjustments for the inventory and pipeline feedback loops and the use of triple exponential smoothing (the Holt-Winters no-trend, additive seasonality model) as a forecasting strategy. To quantify the performance of the system, we derive the transfer functions of the system and plot the frequency response of the system under a number of different parametrizations. We find that the system using Holt-Winters forecasting (the HW-model) significantly outperforms the system using simple exponential smoothing (the SES model), commonly found in the literature, under certain demand assumptions. However, we find that the HW-model is very sensitive to the demand frequency, while the SES is very robust. Thus, the performance range is substantially narrower for the SES model. Finally, we show that previous insights related to behavioral biases are not affected by the choice of forecasting strategy

Finite-time ruin probabilities under large-claim reinsurance treaties for heavy-tailed claim sizes

We investigate the probability that an insurance portfolio gets ruined within a finite time period under the assumption that the r largest claims are (partly) reinsured. We show that for regularly varying claim sizes the probability of ruin after reinsurance is also regularly varying in terms of the initial capital, and derive an explicit asymptotic expression for the latter. We establish this result by leveraging recent developments on sample-path large deviations for heavy tails. Our results allow, on the asymptotic level, for an explicit comparison between two well-known large-claim reinsurance contracts, namely LCR and ECOMOR. We finally assess the accuracy of the resulting approximations using state-of-the-art rare event simulation techniqu

Asymptotic error bounds for truncated buffer approximations of a 2-node tandem queue.

We consider the queue lengths of a tandem queueing network. The number of customers in the system can be modelled as QBD with a doubly-innite state-space. Due to the innite phase-space, this system does not have a product-form solution. A natural approach to nd a numerical solution with the aid of matrix analytic methods is by truncating the phase-space; however, this approach imposes approximation errors. The goal of this paper is to study these approximation errors mathematically, using large deviations and extreme value theory. We obtain a simple asymptotic error bound for the approximations that depends on the truncation level. We test the accuracy of our bound numerically