Estimating the input of a Lévy-driven queue by Poisson sampling of the workload process
Estimating the input of a Lévy-driven queue by Poisson sampling of the workload process
复制标题
通过工作负载过程的泊松采样来估计 Lévy 驱动队列的输入
作者：
L. Ravner;O. Boxma;M. Mandjes
本文旨在通过在泊松时刻对工作量过程进行采样，半参数地估计一个由列维（Lévy）过程驱动的队列的输入过程。我们为列维过程的特征指数构建了一个基于矩方法的估计量。该方法利用了在指数时间采样的工作量的已知分布，从而考虑了后续样本之间的相关性。给出了一致性和渐近正态性的可验证条件，以及渐近方差的明确表达式。该方法需要一个中间估计步骤，即估计一个常数（与输入分布和采样率都有关）；这个常数在渐近分析中也起作用。对于从属过程列维输入，为中间步骤构建了一个部分极大似然估计量（partial MLE），并且我们表明它满足一致性和渐近正态性条件。对于一般的谱正列维输入，提出了一个有偏估计量，它仅使用高于某个阈值的工作量观测值；通过适当选择阈值，可以使偏差任意小。
This paper aims at semi-parametrically estimating the input process to a L\'evy-driven queue by sampling the workload process at Poisson times. We construct a method-of-moments based estimator for the L\'evy process' characteristic exponent. This method exploits the known distribution of the workload sampled at an exponential time, thus taking into account the dependence between subsequent samples. Verifiable conditions for consistency and asymptotic normality are provided, along with explicit expressions for the asymptotic variance. The method requires an intermediate estimation step of estimating a constant (related to both the input distribution and the sampling rate); this constant also features in the asymptotic analysis. For subordinator L\'evy input, a partial MLE is constructed for the intermediate step and we show that it satisfies the consistency and asymptotic normality conditions. For general spectrally-positive L\'evy input a biased estimator is proposed that only uses workload observations above some threshold; the bias can be made arbitrarily small by appropriately choosing the threshold.