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 prespecified 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.

It has long been recognized that the bullwhip effect in real life depends on a behavioral component. However, non-experimental research typically considers only structural causes in its analysis. In this article, we study the impact of behavioral biases on the performance of inventory/production systems modeled through an APVIOBPCS (Automatic Pipeline, Variable Inventory, Order-Based Production Control System) design using linear control theory. To explicitly model managerial behavior, we allow independent adjustments to inventory and pipeline feedback loops. We consider the biases of smoothing/over-reaction to inventory and pipeline mismatches and the under-/over-estimation of the pipeline. To quantify the performance of the system, we first develop a new procedure to determine the exact stability region of the system and we derive an asymptotic stability region that is independent of the lead time. Afterwards, we analyze the effect of different demand signals on order and inventory variations. Our findings suggest that normative policy recommendations must take demand structure explicitly into account. Finally, through extensive numerical experiments, we find that the performance of the system depends on the combination of the behavioral biases and the structure of the demand stream.

Numerical evaluation of performance measures in heavy-tailed risk models is an important and challenging problem. In this paper, we construct very accurate approximations of such performance measures that provide small absolute and relative errors. Motivated by statistical analysis, we assume that the claim sizes are a mixture of a phase-type and a heavy-tailed distribution and with the aid of perturbation analysis we derive a series expansion for the performance measure under consideration. Our proposed approximations consist of the first two terms of this series expansion, where the first term is a phasetype approximation of our measure. We refer to our approximations collectively as corrected phase-type approximations. We show that the corrected phase-type approximations exhibit a nice behavior both in finite and infinite time horizon, and we check their accuracy through numerical experiments.

In this paper, we study the influence of seasonal demands and forecasts on 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 that uses a seasonal forecast based on a no-trend, additive-seasonality exponential-smoothing model, and compare its performance to an equivalent system using (non-seasonal) simple exponential smoothing. To quantify the performance of these systems, we first conduct extensive numerical experimentation to calculate the Bullwhip Effect for orders and inventory under a number of different behavioral parametrizations and seasonality conditions. Then, we run simulation experiments under different stochastic seasonal demands to characterize the robustness of each of the systems to behavioral biases and mispecification of parameters. Finally, we perform an analysis of the response of the system under real-life demand streams taken from the recent M5 forecasting competition dataset. We find that the system with seasonal forecasting significantly outperforms the system with simple exponential smoothing under certain demand assumptions. With optimal parameter settings, the forecast error of the seasonal model can be up to 40% lower. However, we also find that the forecast superiority does not necessarily translate to the performance of the system measured through the bullwhip metrics. In addition, the seasonal forecasting model is very sensitive to the demand frequency and smoothing parameters, while the simple exponential smoothing model is very robust. This implies that the real life benefits of implementing a seasonal forecasting model are not obvious and depend on the particular situation; under a large number of settings (e.g., low sea-

We consider in this paper a risk reserve process where the claims and gains arrive according to two independent Poisson processes. While the gain sizes are phase-type distributed, we assume instead that the claim sizes are phase-type perturbed by a heavy-tailed component; that is, the claim size distribution is formally chosen to be phase-type with large probability 1 − ǫ and heavy-tailed with small probability ǫ.We analyze the seminal Gerber-Shiu function coding the joint distribution of the time to ruin, the surplus immediately before ruin, and the deficit at ruin. We derive its value as an expansion with respect to powers of ǫ with known coefficients and we construct approximations from the first two terms of the aforementioned series. The main idea is based on the so-called fluid embedding that allows to put the considered risk process into the framework of spectrally negative Markov-additive processes and use its fluctuation theory developed in [18].