A Functional Central Limit Theorem for The
نویسنده
چکیده
In this paper, we present a functional fluid limit theorem and a functional central limit theorem for a queue with an infinity of servers M/GI/∞. The system is represented by a point-measure valued process keeping track of the remaining processing times of the customers in service. The convergence in law of a sequence of such processes after rescaling is proved by compactness-uniqueness methods , and the deterministic fluid limit is the solution of an integrated equation in the space S ′ of tempered distributions. We then establish the corresponding central limit theorem, that is, the approximation of the normalized error process by a S ′-valued diffusion. We apply these results to provide fluid limits and diffusion approximations for some performance processes. 1. Introduction. The queues with an infinite reservoir of servers are classical models in queueing theory. In such cases, all the customers are immediately taken care of upon arrival, and spend in the system a sojourn time equal to their service time. Beyond its interest to represent telecommunica-tion networks or computers architectures in which the number of resources is extremely large, this model (commonly referred to as pure delay queue) has been often used for comparison to other ones whose dynamics are formally much more complicated, but close in some sense. Then the performances of the pure delay system may give good estimators, or bounds, of that of the other system. The studies proposed in the literature mainly focused on classical descrip-tors, such as the length of the queue: among others, its stationary regime under Markovian assumptions ([11]), the transient behavior and law of hitting times of given levels ([14]), the fluid limit and diffusion approximations
منابع مشابه
The Local Limit Theorem: A Historical Perspective
The local limit theorem describes how the density of a sum of random variables follows the normal curve. However the local limit theorem is often seen as a curiosity of no particular importance when compared with the central limit theorem. Nevertheless the local limit theorem came first and is in fact associated with the foundation of probability theory by Blaise Pascal and Pierre de Fer...
متن کاملCentral Limit Theorem in Multitype Branching Random Walk
A discrete time multitype (p-type) branching random walk on the real line R is considered. The positions of the j-type individuals in the n-th generation form a point process. The asymptotic behavior of these point processes, when the generation size tends to infinity, is studied. The central limit theorem is proved.
متن کاملDensity Estimators for Truncated Dependent Data
In some long term studies, a series of dependent and possibly truncated lifetime data may be observed. Suppose that the lifetimes have a common continuous distribution function F. A popular stochastic measure of the distance between the density function f of the lifetimes and its kernel estimate fn is the integrated square error (ISE). In this paper, we derive a central limit theorem for t...
متن کاملThe Central Limit Theorem for Random Perturbations of Rotations
We prove a functional central limit theorem for stationary random sequences given by the transformations T ;! (x; y) = (2x; y + ! + x) mod 1 on the two-dimensional torus. This result is based on a functional central limit theorem for ergodic stationary martingale diierences with values in a separable Hilbert space of square integrable functions.
متن کاملConvergence of averages of scaled functions of I(1) linear processes
Econometricians typically make use of functional central limit theorems to prove results for I(1) processes. For example, to establish the limit distributions of unit root tests such as the Phillips–Perron and Dickey–Fuller tests, the functional central limit theorem plays a crucial role. In this paper, it is pointed out that for linear processes, minimal conditions that ensure that only a cent...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2008