Nudge: Stochastically Improving upon FCFS

The First-Come First-Served (FCFS) scheduling policy is the most popular scheduling algorithm used in practice. Furthermore, its usage is theoretically validated: for light-tailed job size distributions, FCFS has weakly optimal asymptotic tail of response time. But what if we don't just care about the asymptotic tail? What if we also care about the 99th percentile of response time, or the fraction of jobs that complete in under one second? Is FCFS still best? Outside of the asymptotic regime, only loose bounds on the tail of FCFS are known, and optimality is completely open. In this paper, we introduce a new policy, Nudge, which is the first policy to provably stochastically improve upon FCFS. We prove that Nudge simultaneously improves upon FCFS at every point along the tail, for light-tailed job size distributions. As a result, Nudge outperforms FCFS for every moment and every percentile of response time. Moreover, Nudge provides a multiplicative improvement over FCFS in the asymptotic tail. This resolves a long-standing open problem by showing that, counter to previous conjecture, FCFS is not strongly asymptotically optimal.

First-passage time asymptotics over moving boundaries for random walk bridges

We study the asymptotic tail behavior of the first-passage time over a moving boundary for a random walk conditioned to return to zero, where the increments of the random walk have finite variance. Typically, the asymptotic tail behavior may be described through a regularly varying function with exponent -½, where the impact of the boundary is captured by the slowly varying function. Yet, the moving boundary may have a stronger effect when the tail is considered at a time close to the return point of the random walk bridge, leading to a possible phase transition depending on the order of the distance between zero and the moving boundary.

Estimating the coefficient of asymptotic tail independence

Many multivariate analyses require the account of extreme events. Correlation is an insufficient measure to quantify tail dependence. The most common tail dependence coefficients are based on the probability of simultaneous exceedances. The coefficient of asymptotic tail independence introduced in Ledford and Tawn (1996) is a bivariate measure often used in the tail modeling of data in finance, environment, insurance, among other fields of applications. It can be estimated as the tail index of the minimum component of a random pair with transformed unit Pareto marginals. The literature regarding the estimation of the tail index is extensive. Semi-parametric inference requires the choice of the number k of the largest order statistics that lead to the best estimate, where there is a tricky trade-off between variance and bias. Many methodologies have been developed to undertake this choice, most of them applied to the Hill estimator (Hill, 1975). We are going to analyze, through simulation, some of these methods within the estimation of the coefficient of asymptotic tail independence. We also compare with a minimum-variance reduced-bias Hill estimator presented in Caeiro et al. (2005). A pure heuristic procedure adapted from Frahm et al. (2005), used in a different context but with a resembling framework, will also be implemented. We will see that some of these simple tools should not be discarded in this context. Our study will be complemented by applications to real datasets.