By Ward Whitt
ISBN-10: 0387217487
ISBN-13: 9780387217482
ISBN-10: 0387953582
ISBN-13: 9780387953588
This publication is ready stochastic-process limits - limits during which a chain of stochastic methods converges to a different stochastic technique. those are necessary and engaging simply because they generate easy approximations for sophisticated stochastic techniques and in addition aid clarify the statistical regularity linked to a macroscopic view of uncertainty. This e-book emphasizes the continuous-mapping method of receive new stochastic-process limits from formerly proven stochastic-process limits. The continuous-mapping technique is utilized to acquire heavy-traffic-stochastic-process limits for queueing versions, together with the case during which there are unrivaled jumps within the restrict approach. those heavy-traffic limits generate uncomplicated approximations for sophisticated queueing approaches they usually exhibit the effect of variability upon queueing functionality.
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Additional info for Stochastic-process limits: an introduction to stochastic-process limits and their application to queues
Sample text
Recall that the cumulative distribution function (cdf) F of a random variable X is the function F (t) ≡ P (X ≤ t) for t ∈ R . 1. 7. Estimates of the probability density of the final position of the random walk, obtained from 10j independent samples of the centered partial sum S1000 − 500 for j = 2, . . , 5, for the case in which the steps Uk are uniformly distributed in the interval [0, 1], based on the nonparametric density estimator density from S. Similarly, the empirical cdf of a data set of size n is the proportion Fn (t) of the n data points that are less than or equal to t, as a function of t.
The limits in the SLLN and the FSLLN only involve the single parameter m, which is the mean step size in the IID case. Similarly, after centering is done, the limits in the CLT and FCLT only involve the single parameter σ 2 , which is the variance of the step size in the IID case. Thus these limit theorems are invariance principles. Moreover, the plots have an even stronger invariance property, because the limiting plots have no parameters at all! ) Assuming only that the mean is positive, the plots of the uncentered random walk (with arbitrary step-size distribution) approach the identity function e ≡ e(t) ≡ t, 0 ≤ t ≤ 1.
Thus it is natural to seek a continuous-time representation of the original discretetime random walk. , t denotes the greatest integer less than or equal to t. If we also want to introduce centering, then we do the centering first, and instead consider the centered process {S t − m t : t ≥ 0} for appropriate centering constant m, which here is 1/2. Thus the continuous-time representation of the random walk is a step function, which coincides with the random walk at integer arguments. However, the step function is not the only possible continuous-time representation of the random walk.
Stochastic-process limits: an introduction to stochastic-process limits and their application to queues by Ward Whitt
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