scholarly journals Minimal waiting times in static traffic control

2003 ◽  
Vol 7 (1) ◽  
pp. 11-28
Author(s):  
O. Moeschlin ◽  
C. Poppinga
Keyword(s):  
Waiting Time ◽  
Traffic Control ◽  
Law Of Large Numbers ◽  
Waiting Times ◽  
Strong Law ◽  
Poisson Arrival ◽  
Large Numbers ◽  
Arrival Processes ◽  
The Mean ◽  
Traffic Organization

The paper discusses the question of the optimal control of an unsymmetric bottleneck system with Poisson arrival processes having the minimization of the mean individual waiting time as objective. The setup allows the straightforward generalization to more complicated forms of traffic organization. The notion of the mean individual waiting time is based on a theorem of the Little type, which is derived by a strong law of large numbers. The proof makes use of McNeil's formula, which connects the expected total waiting time with the expected queue length.

scholarly journals Means as Improper Integrals

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 284
Author(s):  
John Gray ◽  
Andrew Vogt
Keyword(s):  
Law Of Large Numbers ◽  
Cauchy Distribution ◽  
Cumulative Distribution ◽  
Tail Probabilities ◽  
Strong Law ◽  
Improper Integral ◽  
Heavy Tailed Distributions ◽  
Large Numbers ◽  
The Mean ◽  
Heavy Tailed

The aim of this work is to study generalizations of the notion of the mean. Kolmogorov proposed a generalization based on an improper integral with a decay rate for the tail probabilities. This weak or Kolmogorov mean relates to the weak law of large numbers in the same way that the ordinary mean relates to the strong law. We propose a further generalization, also based on an improper integral, called the doubly-weak mean, applicable to heavy-tailed distributions such as the Cauchy distribution and the other symmetric stable distributions. We also consider generalizations arising from Abel–Feynman-type mollifiers that damp the behavior at infinity and alternative formulations of the mean in terms of the cumulative distribution and the characteristic function.

Asymptotics of First-Passage Percolation on One-Dimensional Graphs

2015 ◽  
Vol 47 (01) ◽  
pp. 182-209
Author(s):  
Daniel Ahlberg
Keyword(s):  
Law Of Large Numbers ◽  
Nearest Neighbour ◽  
First Passage Percolation ◽  
First Passage ◽  
Strong Law ◽  
One Dimensional ◽  
Minimal Weight ◽  
Large Numbers ◽  
Mean And Variance ◽  
The Mean

In this paper we consider first-passage percolation on certain one-dimensional periodic graphs, such as thenearest neighbour graph ford,K≥ 1. We expose a regenerative structure within the first-passage process, and use this structure to show that both length and weight of minimal-weight paths present a typical one-dimensional asymptotic behaviour. Apart from a strong law of large numbers, we derive a central limit theorem, a law of the iterated logarithm, and a Donsker theorem for these quantities. In addition, we prove that the mean and variance of the length and weight of minimizing paths are monotone in the distance between their end-points, and further show how the regenerative idea can be used to couple two first-passage processes to eventually coincide. Using this coupling we derive a 0–1 law.

Asymptotics of First-Passage Percolation on One-Dimensional Graphs

2015 ◽  
Vol 47 (1) ◽  
pp. 182-209 ◽  
Author(s):  
Daniel Ahlberg
Keyword(s):  
Law Of Large Numbers ◽  
Nearest Neighbour ◽  
First Passage Percolation ◽  
First Passage ◽  
Strong Law ◽  
One Dimensional ◽  
Minimal Weight ◽  
Large Numbers ◽  
Mean And Variance ◽  
The Mean

In this paper we consider first-passage percolation on certain one-dimensional periodic graphs, such as the nearest neighbour graph for d, K ≥ 1. We expose a regenerative structure within the first-passage process, and use this structure to show that both length and weight of minimal-weight paths present a typical one-dimensional asymptotic behaviour. Apart from a strong law of large numbers, we derive a central limit theorem, a law of the iterated logarithm, and a Donsker theorem for these quantities. In addition, we prove that the mean and variance of the length and weight of minimizing paths are monotone in the distance between their end-points, and further show how the regenerative idea can be used to couple two first-passage processes to eventually coincide. Using this coupling we derive a 0–1 law.

scholarly journals On Little’s Formula in Multiphase Queues

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2282
Author(s):  
Saulius Minkevičius ◽  
Igor Katin ◽  
Joana Katina ◽  
Irina Vinogradova-Zinkevič
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Queuing Theory ◽  
Law Of Large Numbers ◽  
Strong Law ◽  
Large Numbers ◽  
Virtual Waiting Time ◽  
Multiphase Systems ◽  
Two Stages ◽  
Little's Formula

The structure of this work in the field of queuing theory consists of two stages. The first stage presents Little’s Law in Multiphase Systems (MSs). To obtain this result, the Strong Law of Large Numbers (SLLN)-type theorems for the most important MS probability characteristics (i.e., queue length of jobs and virtual waiting time of a job) are proven. The next stage of the work is to verify the result obtained in the first stage.

scholarly journals Means as Improper Integrals

2019 ◽  
Author(s):  
John E. Gray ◽  
Andrew Vogt
Keyword(s):  
Law Of Large Numbers ◽  
Cauchy Distribution ◽  
Cumulative Distribution ◽  
Tail Probabilities ◽  
Strong Law ◽  
Improper Integral ◽  
Heavy Tailed Distributions ◽  
Large Numbers ◽  
The Mean ◽  
Heavy Tailed

The aim of this work is to study generalizations of the notion of mean. Kolmogorov proposed a generalization based on an improper integral with a decay rate for the tail probabilities. This weak or Kolmogorov mean relates to the Weak Law of Large Numbers in the same way that the ordinary mean relates to the Strong Law. We propose a further generalization, also based on an improper integral, called the doubly weak mean, applicable to heavy-tailed distributions such as the Cauchy distribution and the other symmetric stable distributions, We also consider generalizations arising from Abel-Feynman type mollifiers that damp the behavior at infinity and alternative formulations of the mean in terms of the cumulative distribution and the characteristic function.

scholarly journals On Some Conditions for Strong Law of Large Numbers for Weighted Sums of END Random Variables under Sublinear Expectations

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Xiaochen Ma ◽  
Qunying Wu
Keyword(s):  
Probability Space ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Weighted Sums ◽  
Dependent Random Variables ◽  
Strong Law ◽  
Sublinear Expectation ◽  
Negatively Dependent Random Variables ◽  
Large Numbers ◽  
End Random Variables

In this article, we research some conditions for strong law of large numbers (SLLNs) for weighted sums of extended negatively dependent (END) random variables under sublinear expectation space. Our consequences contain the Kolmogorov strong law of large numbers and the Marcinkiewicz strong law of large numbers for weighted sums of extended negatively dependent random variables. Furthermore, our results extend strong law of large numbers for some sequences of random variables from the traditional probability space to the sublinear expectation space context.

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