Wolfgang Stadje

Optimal Maintenance Strategies for Repairable Systems with General Degree of Repair

Abstract : Our project focuses on a single unit system which is subject to random failure. The age of the system in service is maintained as a control variable. Upon failure, an age-dependent maintenance action specifying the degree of repair is taken by a controller. By employing analytical tools and numerical procedures, we investigate and characterize the structures of the optimal repair policy under a discounted cost optimality criterion. Special analytical and numerical effort is directed throughout the study for the development of efficient computational procedures for the optimal strategy.

The exact probability distribution of a two-dimensional random walk

A calculation is made of the exact probability distribution of the two-dimensional displacement of a particle at timet that starts at the origin, moves in straight-line paths at constant speed, and changes its direction after exponentially distributed time intervals, where the lengths of the straight-line paths and the turn angles are independent, the angles being uniformly distributed. This random walk is the simplest model for the locomotion of microorganisms on surfaces. Its weak convergence to a Wiener process is also shown.

Leveraging the Global Information Revolution for Economic Development: Singapore's Evolving Information Industry Strategy

Despite her small size and late industrialization, Singapore has managed to capture a significant share of the global information industry over the last three decades. This paper analyzed the structure and growth dynamics of Singapores information industries using an analytical framework that integrates the four key components of an information economy: ICT goods industry, content industry, network infrastructure, and informatization. The paper identifies four generic stages in the development path of Singapores information economy and highlights policy implications for other small, open economies.

A generalized impulse control model of cash management

Abstract This paper presents a general model of cash management, viewed as an impulse control problem for a stochastic money flow process. Generalizing classical approaches, we represent this process by a superposition of a Brownian motion and a compound Poisson process, controlled by two-sided target-trigger policies. For phase-type distributions for the upward and downward jumps we determine all pertinent cost functionals explicitly. Moreover, the controlled process is studied in steady state. The closed-form results can be used to determine optimal values for the target and trigger values numerically.

First-exit times for compound poisson processes for some types of positive and negative jumps

We consider the one-sided and the two-sided first-exit problem for a compound Poisson process with linear deterministic decrease between positive and negative jumps. This process (X(t)) t≥0 occurs as the workload process of a single-server queueing system with random workload removal, which we denote by M/G u /G d /1, where G u (G d ) stands for the distribution of the upward (downward) jumps; other applications are to cash management, dams, and several related fields. Under various conditions on G u and G d (assuming e.g. that one of them is hyperexponential, Erlang or Coxian), we derive the joint distribution of τ y =inf{t≥0|X(t)∉(0,y)}, y>0, and X(τ y ) as well as that of T=inf{t≥0|X(t)≤0} and X(T). We also determine the distribution of sup{X(t)|0≤t≤T}.

The busy period of the queueing system M / G /∞

For the queueing system M/G/oo some distributions connected with the associated busy periods are derived. QUEUES WITH UNLIMITED SERVICE

Clearing Models for M/G/1 Queues

We consider M/G/1-type queueing systems with ‘disasters’, occurring at certain random times and causing an instantaneous removal of the entire residual workload from the system. After such a clearing, the system is assumed to be ready to start working again immediately. We consider clearings at deterministic equidistant times, at random times and at crossings of some prespecified level, and derive the stationary distribution of the workload process for these clearing times and some of their combinations.

Optimal repair policies with general degree of repair in two maintenance models

Two maintenance models for repairable systems with exponential lifetimes are studied. A maintenance strategy in the proposed framework specifies the degree of repair to be taken as a function of the virtual age of the system upon failure. In Model I we restrict attention to maintenance strategies in which only minimal or perfect repair can be employed upon failure. In Model II the degree of repair may vary. For some age-dependent cost functions, optimal maintenance strategies (minimizing the expected discounted costs over an infinite horizon) are determined analytically.

Exact probability distributions for noncorrelated random walk models

A stochastic model for the idealized locomotion of cells is studied. The cell is assumed to cover a polygonal line in ℝn, the times between turns are exponentially distributed and independent of the directions, and the density of thenth directione does not depend on the (n−1)th directione′. The resulting Markov process (X(t), D(t)) for position and direction of the motion at timet is studied by using the integrodifferential equation for the transition function. For example, the joint distribution of (X(t), D(t)) is derived in closed form ifn=2 orn=3 and all chosen directions (including the initial one) are uniformly distributed. For higher dimensions the combined Fourier-Laplace transform ofX(t) is given. The case of a fixed initial direction is also considered.

UPPER FIRST-EXIT TIMES OF COMPOUND POISSON PROCESSES REVISITED

For a compound Poisson process (CPP) with only positive jumps, an elegant formula connects the density of the hitting time for a lower straight line with that of the process itself at time t, h(x; t), considered as a function of time and position jointly. We prove an analogous (albeit more complicated) result for the first time the CPP crosses an upper straight line. We also consider the conditional density of the CPP at time t, given that the upper line has not been reached before t. Finally, it is shown how to compute certain moment integrals of h.