Joshua B. Levy

Renewal reward processes with heavy-tailed inter-renewal times and heavy-tailed rewards

It is well known that fractional Brownian motion can be obtained as the limit of a superposition of renewal reward processes with inter-renewal times that have infinite variance (heavy tails with exponent a) and with rewards that have finite variance. We show here that if the rewards also have infinite variance (heavy tails with exponent P) then the limit Z# is a #-stable self-similar process. If / - a, then Zp is a stable process with dependent increments and self-similarity parameter H = (f a + 1)/3.

A Characterization of the Asymptotic Behavior of Stationary Stable Processes

Let X(t) be a stationary stable process and let r(t) be the difference between the joint characteristic function of (X(t), X(0)) and the product of the characteristic functions of X(t) and X(0). The function r(t) as t → ∞ is a measure of asymptotic dependence. We analyze the behavior of r(t) as t → ∞ for various stable stochastic processes.

DEPENDENCE STRUCTURE OF A RENEWAL-REWARD PROCESS WITH INFINITE VARIANCE

The on-off renewal-reward process used to explain long-range dependence in Ethernet traffic can be extended to the case where, not only the inter-renewal times but also the rewards have infinite variance. The covariation and the codifference, which generalize the covariance to the infinite variance case, are computed for the limiting process. It is shown that they decay like a power function. The exponent of that power is the same as for fractional stable noise, even though the increments of the limiting process are different from fractional stable noise.

Analysis of project scheduling strategies in a client-contractor environment

In this article we consider a cost-minimization model to investigate scheduling strategies for multistaged projects in a client-contractor environment. This type of environment is symptomatic of temporal changes in project definition and scope. At prespecified epochs the client conducts an external evaluation of the project and either accepts or rejects the contractors current work. The resulting uncertainty from the clients review is modeled via monotonically varying acceptance probabilities. The model is designed primarily to address the interaction between earliest-, intermediate-, and latest-start options and project-crashing stragies for a broad range of penalty costs. Theoretical results are introduced, while numerical examples for both exponentially and polynomially based acceptance probabilities are discussed.