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This paper is concerned with the asymptotic analysis of sojourn times of random fields with continuous sample paths. Under a very general framework we show that there is an interesting relationship between tail asymptotics of sojourn times and that of supremum. Moreover, we establish the uniform double-sum method to derive the tail asymptotics of sojourn times. In the literature, based on the pioneering research of S. Berman the sojourn times have been utilised to derive the tail asymptotics of supremum of Gaussian processes. In this paper we show that the opposite direction is even more fruitful, namely knowing the asymptotics of supremum of random processes and fields (in particular Gaussian) it is possible to establish the asymptotics of their sojourn times. We illustrate our findings considering i) two dimensional Gaussian random fields, ii) chi-process generated by stationary Gaussian processes and iii) stationary Gaussian queueing processes. [ABSTRACT FROM AUTHOR]

For a non-negative separable random field Z(t), $t\in \mathbb{R}^d$ , satisfying some mild assumptions, we show that $ H_Z^\delta =\lim_{{T} \to \infty} ({1}/{T^d}) \mathbb{E}\{{\sup_{ t\in [0,T]^d \cap \delta \mathbb{Z}^d } Z(t) }\} for $\delta \ge 0$ , where $0 \mathbb{Z}^d\,:\!=\,\mathbb{R}^d$ , and prove that $H_Z^0$ can be approximated by $H_Z^\delta$ if $\delta$ tends to 0. These results extend the classical findings for Pickands constants $H_{Z}^\delta$ , defined for $Z(t)= \exp(\sqrt{ 2} B_\alpha (t)- \lvert {t} \rvert^{2\alpha })$ , $t\in \mathbb{R}$ , with $B_\alpha$ a standard fractional Brownian motion with Hurst parameter $\alpha \in (0,1]$. The continuity of $H_{Z}^\delta$ at $\delta=0$ is additionally shown for two particular extensions of Pickands constants. [ABSTRACT FROM AUTHOR]

The ruin probability in the classical Brownian risk model can be explicitly calculated for both finite and infinite time horizon. This is not the case for the simultaneous ruin probability in the two-dimensional Brownian risk model. Relying on asymptotic theory, we derive in this contribution approximations for both simultaneous ruin probability and simultaneous ruin time for the two-dimensional Brownian risk model when the initial capital increases to infinity. [ABSTRACT FROM AUTHOR]

In this article, we analyze the models quantifying the bullwhip effect in supply chains with stochastic lead times and find advantages and disadvantages of their approaches to the bullwhip problem. Moreover, using computer simulation, we find interesting insights into the bullwhip behavior for a particular instance of a multi-echelon supply chain with constant customer demands and random lead times. We confirm the recent finding of Michna and Nielsen that under certain circumstances lead time signal processing is by itself a fundamental cause of bullwhip effect just like demand-signal processing is. The simulation also shows that in this supply chain the delay parameter of demand forecasting smooths the bullwhip effect at the manufacturer level much faster than the delay parameter of lead time forecasting. Additionally, in the supply chain with random demands, the reverse behavior is observed, that is, the delay parameter of lead time forecasting smooths bullwhip effect at the retailer stage much faster than the delay parameter of demand forecasting. At the manufacturer level, the delay parameter of demand forecasting and the delay parameter of lead time forecasting dampen the effect with a similar strength. [ABSTRACT FROM AUTHOR]

In this article, we review the research state of the bullwhip effect in supply chains with stochastic lead times. We analyze problems arising in a supply chain when lead times are not deterministic. Using real data from a supply chain, we confirm that lead times are stochastic and can be modeled by a sequence of independent identically distributed random variables. This underlines the need to further study supply chains with stochastic lead times and model the behavior of such chains. [ABSTRACT FROM AUTHOR]

Lead times and their nature have received limited interest in literature despite their large impact on the performance and the management of supply chains. This paper presents a method and a case implementation of the same, to establish the behavior of real lead times in supply chains. The paper explores the behavior of lead times and illustrates how in one particular case they can and should be considered to be independent and identically distributed (i.i.d.). The conclusion is also that the stochastic nature of the lead times contributes more to lead time demand variance than demand variance. [ABSTRACT FROM AUTHOR]

In this note we derive the famous formula of F. Chen, Z. Drezner, J.K. Ryan and D. Simchi-Levi (2000a) for the bullwhip effect measure in a simple two-stage supply chain under assumption that demands constitute autoregressive structure of order 1. Our approach is a little different than in Chen et. al (2000a) and therefore we obtain the formula as an equality unlike Chen et. al (2000a) where they have it as a lower bound. Moreover we analyze the bullwhip effect measure formula and in some cases we have different conclusions than in Chen et. al (2000a). [ABSTRACT FROM AUTHOR]

This article aims to present the applications of Lévy processes for the stochastic modeling of storage resources. Two cases were considered. In the first one, the volume of supplies to the storehouse is described by a random process (Lévy process), while issuing the products is described by a deterministic and linear function. The second case is reversed: the delivery to the storehouse is described by a linear function (variable: time), while issuing the goods is described by a Lévy process. For both cases the form of the stock level process and examples of its trajectories, when the net supply is a Lévy process, are given. We investigated the following net supply processes: gamma process, α-stable Lévy process with α = 0.5, Cauchy process, Wiener process. [ABSTRACT FROM AUTHOR]

In this article, we consider a Levy process under condition Γ1 = x where {Γk} is a sequence of arrivals of a Poisson process with unit arrival rate. We show that under condition Γ1 = x Levy process can be decomposed into a simple process and a Levy process. These two processes are independent. As an application of this decomposition we consider α-stable Levy processes. We give a closed form of the process [image omitted] and evaluate its expected value where Z is an α-stable Levy process with 0 < α < 2. We show that X(t) = 0 for 1 < α < 2 and this expectation is equal infinity for α ≤ 1. [ABSTRACT FROM AUTHOR]

In this paper we discuss decomposition principle for a-stable Lévy pro- cesses. We investigate asymptotic properties of components and stochastic integrals driven by such processes providing an important class of anoma- lous diffusions. We consider two case studies with integrands being fractional Brownian motion and gamma process. [ABSTRACT FROM AUTHOR]

In this paper we consider a jump-diffusion type approximation of the classical risk process by a gamma Lévy process. We derive here the asymptotic behavior (lower and upper bounds) of the finite time ruin probability for any gamma Lévy process. [ABSTRACT FROM AUTHOR]

Let Z(t) be a stationary centered Gaussian process with a Markovian structure. In some fluid models, the stationary buffer content V can be expressed as supt≥0( ∫0tZ(s) ds−ct) and P(V>u)=Ce−γu(1+o(1)). The asymptotic constant C can be expressed by the so called generalized Pickands constants H. In most cases no formula or approximation for C are known. In this paper we show a method of simulation of C by the use of change of measure technique. The method is applicable when Z(t) is a stationary Ornstein-Uhlenbeck process or Z(t)= ∑j=1nXj(t), where (X1(t),…,Xn(t)) is a Gauss-Markov process. Two examples of simulations are included. Moreover we give a formula for a lower bound for generalized Pickands constants. [ABSTRACT FROM AUTHOR]