Approximating the Randomized Hitting Time Distribution of a Non-stationary Gamma Process

The non-stationary gamma process is a non-decreasing stochastic process with independent increments. By this monotonic behavior this stochastic process serves as a natural candidate for modelling time-dependent phenomena such as degradation. In condition-based maintenance the first time such a process exceeds a random threshold is used as a model for the lifetime of a device or for the random time between two successive imperfect maintenance actions. Therefore there is a need to investigate in detail the cumulative distribution function (cdf) of this so-called randomized hitting time. We first relate the cdf of the (randomized) hitting time of a non-stationary gamma process to the cdf of a related hitting time of a stationary gamma process. Even for a stationary gamma process this cdf has in general no elementary formula and its evaluation is time-consuming. Hence two approximations are proposed in this paper and both have a clear probabilistic interpretation. Numerical experiments show that these approximations are easy to evaluate and their accuracy depends on the scale parameter of the non-stationary gamma process. Finally, we also consider some special cases of randomized hitting times for which it is possible to give an elementary formula for its cdf.Approximation;Condition based maintenance;First hitting time;Non-stationary gamma process;Random threshold

On Noncooperative Games, Minimax Theorems and Equilibrium Problems

In this chapter we give an overview on the theory of noncooperative games. In the first part we consider in detail for zero-sum (and constant-sum) noncooperative games under which necessary and sufficient conditions on the payoff function and different (extended) strategy sets for both players an equilibrium saddlepoint exists. This is done by using the most elementary proofs. One proof uses the separation result for disjoint convex sets, while the other proof uses linear programming duality and some elementary properties of compact sets. Also, for the most famous saddlepoint result given by Sion's minmax theorem an elementary proof using only the definition of connectedness is given. In the final part we consider n-person nonzero-sum noncooperative games and show by a simple application of the KKM lemma that a so-called Nash equilibrium point exists for compact strategy sets and concavity conditions on the payoff functions.KKM Lemma;Equilibrium Problems;Minimax Theorems;Nash Equilibrium Point;Non-Cooperative Game Theory

Renewal theory for random variables with a heavy tailed distribution and finite variance

Let X-1, X-2,... X-n be independent and identically distributed (i.i.d.) non-negative random variables with a common distribution function (d.f.) F with unbounded support and EX12 < infinity. We show that for a large class of heavy tailed random variables with a finite variance the renewal function U satisfies U(x) - x/mu - mu(2)/2 mu(2) similar to -1/mu x integral(infinity)(x) integral(infinity)(s) (1 - F(u))duds as x -> infinity

Approximating the randomized hitting time distribution of a non-stationary gamma process

The non-stationary gamma process is a non-decreasing stochasticprocess with independent increments. By this monotonic behavior thisstochastic process serves as a natural candidate for modellingtime-dependent phenomena such as degradation. In condition-basedmaintenance the first time such a process exceeds a random thresholdis used as a model for the lifetime of a device or for the randomtime between two successive imperfect maintenance actions. Thereforethere is a need to investigate in detail the cumulative distributionfunction (cdf) of this so-called randomized hitting time. We firstrelate the cdf of the (randomized) hitting time of a non-stationarygamma process to the cdf of a related hitting time of a stationarygamma process. Even for a stationary gamma process this cdf has ingeneral no elementary formula and its evaluation is time-consuming.Hence two approximations are proposed in this paper and both have aclear probabilistic interpretation. Numerical experiments show thatthese approximations are easy to evaluate and their accuracy dependson the scale parameter of the non-stationary gamma process. Finally,we also consider some special cases of randomized hitting times forwhich it is possible to give an elementary formula for its cdf.approximation;condition based maintencance;first hitting time;non-stationary gamma process;random threshold

On noncooperative games, minimax theorems and equilibrium problems

In this chapter we give an overview on the theory of noncooperative games. In the first part we consider in detail for zero-sum (and constant-sum) noncooperative games under which necessary and sufficient conditions on the payoff function and different (extended) strategy sets for both players an equilibrium saddlepoint exists. This is done by using the most elementary proofs. One proof uses the separation result for disjoint convex sets, while the other proof uses linear programming duality and some elementary properties of compact sets. Also, for the most famous saddlepoint result given by Sion's minmax theorem an elementary proof using only the definition of connectedness is given. In the final part we consider n-person nonzero-sum noncooperative games and show by a simple application of the KKM lemma that a so-called Nash equilibrium point exists for compact strategy sets and concavity conditions on the payoff functions.

The rate of convergence to optimality of the LPT rule

The LPT rule is a heuristic method to distribute jobs among identical machines so as to minimize the makespan of the resulting schedule. If the processing times of the jobs are assumed to be independent identically distributed random variables, then (under a mild condition on the distribution) the absolute error of this heuristic is known to converge to 0 almost surely. In this note we analyse the asymptotic behaviour of the absolute error and its first and higher moments to show that under quite general assumptions the speed of convergence is proportional to appropriate powers of (log log n)/n and 1/n. Thus, we simplify, strengthen and extend earlier results obtained for the uniform and exponential distribution.

Application of a General Risk Management Model to Portfolio Optimization Problems with Elliptical Distributed Returns for Risk Neutral and Risk Averse Decision Makers

In this paper portfolio problems with linear loss functions and multivariate elliptical distributed returns are studied. We consider two risk measures, Value-at-Risk and Conditional-Value-at-Risk, and two types of decision makers, risk neutral and risk averse. For Value-at-Risk, we show that the optimal solution does not change with the type of decision maker. However, this observation is not true for Conditional-Value-at-Risk. We then show for Conditional-Value-at-Risk that the objective function can be approximated by Monte Carlo simulation using only a univariate distribution. To solve the equivalent Markowitz model, we modify and implement a finite step algorithm. Finally, a numerical study is conducted.Conditional value-at-risk;Disutility;Elliptical distributions;Linear loss functions;Portfolio optimization;Value-at-risk

Fractional location problems

In this paper we analyze some variants of the classical uncapacitated facility location problem with a ratio as an objective function. Using basic concepts and results of fractional programming, we identify a class of one-level fractional location problems which can be solved in polynomial time in terms of the size of the problem. We also consider the fractional two-echelon location problem, which is a special case of the general two-level fractional location problem. For this two-level fractional location problem we identify cases for which its solution involves decomposing the problem into several one-level fractional location problems.discrete location;fractional program

Modelling and Optimizing Imperfect Maintenance of Coatings on Steel Structures

Steel structures such as bridges, tanks and pylons are exposed to outdoor weathering conditions. In order to prevent them from corrosion they are protected by an organic coating system. Unfortunately, the coating system itself is also subject to deterioration. Imperfect maintenance actions such as spot repair and repainting can be done to extend the lifetime of the coating. In this paper we consider the problem of finding the set of actions that minimizes the expected maintenance costs over a bounded horizon. To this end we model the size of the area affected by corrosion by a non-stationary gamma process. An imperfect maintenance action is to be done as soon as a fixed threshold is exceeded. The direct effect of such an action on the condition of the coating is assumed to be random. On the other hand, maintenance may also change the parameters of the gamma deterioration process. It is shown that the optimal maintenance decisions related to this problem are a solution of a continuous-time renewal-type dynamic programming equation. To solve this equation time is discretized and it is verified theoretically that this discretization induces only a small error. Finally, the model is illustrated with a numerical example.non-stationary gamma process;condition-based maintenance;degradation modelling;imperfect maintenance;life-cycle management;renewal-type dynamic programming equation