MAKING SIMULATIONS MORE EFFICIENT WHEN ANALYZING POISSON ARRIVAL SYSTEMS AND MEANS OF MONOTONE FUNCTIONS

2006 ◽  
Vol 20 (2) ◽  
pp. 251-256
Author(s):  
Sheldon M. Ross
Keyword(s):  
Poisson Process ◽  
Random Variables ◽  
Random Variable ◽  
Expected Value ◽  
Stratified Sampling ◽  
Arrival Process ◽  
Monotone Functions ◽  
New Approach ◽  
Poisson Arrival ◽  
Increasing Function

For a system in which arrivals occur according to a Poisson process, we give a new approach for using simulation to estimate the expected value of a random variable that is independent of the arrival process after some specified time t. We also give a new approach for using simulation to estimate the expected value of an increasing function of independent uniform random variables. Stratified sampling is a key technique in both cases.

Average delay in queues with non-stationary Poisson arrivals

1978 ◽  
Vol 15 (03) ◽  
pp. 602-609 ◽  
Author(s):  
Sheldon M. Ross
Keyword(s):  
Poisson Process ◽  
Intensity Function ◽  
Arrival Process ◽  
Single Server ◽  
Actual Process ◽  
Average Delay ◽  
Average Value ◽  
Poisson Arrival ◽  
Poisson Arrivals ◽  
Customer Delay

One of the major difficulties in attempting to apply known queueing theory results to real problems is that almost always these results assume a time-stationary Poisson arrival process, whereas in practice the actual process is almost invariably non-stationary. In this paper we consider single-server infinite-capacity queueing models in which the arrival process is a non-stationary process with an intensity function ∧(t), t ≧ 0, which is itself a random process. We suppose that the average value of the intensity function exists and is equal to some constant, call it λ, with probability 1. We make a conjecture to the effect that ‘the closer {∧(t), t ≧ 0} is to the stationary Poisson process with rate λ ' then the smaller is the average customer delay, and then we verify the conjecture in the special case where the arrival process is an interrupted Poisson process.

PERFORMANCE MODELLING OF PIPELINED CIRCUIT SWITCHING UNDER MMPP TRAFFIC

2001 ◽  
Vol 02 (04) ◽  
pp. 471-483 ◽  
Author(s):  
GEYONG MIN ◽  
MOHAMED OULD-KHAOUA
Keyword(s):  
Poisson Process ◽  
Interconnection Networks ◽  
Arrival Process ◽  
Circuit Switching ◽  
Performance Modelling ◽  
Poisson Arrival ◽  
Proposed Model ◽  
Markov Modulated Poisson Process ◽  
Markov Modulated ◽  
Pipelined Circuit Switching

Performance properties of switching methods used in existing interconnection networks have been mainly examined under the assumption that network traffic follows the Poisson arrival process. However, the Poisson process is unable to capture the bursty nature of traffic. In this paper we propose the first analytical model for pipelined circuit switching under bursty traffic using the Markov-Modulated Poisson Process (MMPP). Simulation experiments reveal that the proposed model exhibits a good degree of accuracy for various networks sizes and under different traffic conditions.

An Application of Mean Square Calculus to Sliding Wear

2009 ◽  
Vol 77 (2) ◽  
Author(s):  
Cláudio R. Ávila da Silva ◽  
Giuseppe Pintaude ◽  
Hazim Ali Al-Qureshi ◽  
Marcelo Alves Krajnc
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Sliding Wear ◽  
Random Variables ◽  
Random Variable ◽  
Expected Value ◽  
Mean Square ◽  
Cauchy Problems ◽  
Numerical Examples ◽  
Correlation Models

In this paper the Archard model and classical results of mean square calculus are used to derive two Cauchy problems in terms of the expected value and covariance of the worn height stochastic process. The uncertainty is present in the wear and roughness coefficients. In order to model the uncertainty, random variables or stochastic processes are used. In the latter case, the expected value and covariance of the worn height stochastic process are obtained for three combinations of correlation models for the wear and roughness coefficients. Numerical examples for both models are solved. For the model based on a random variable, a larger dispersion in terms of worn height stochastic process was observed.

On a stochastic difference equation and a representation of non–negative infinitely divisible random variables

1979 ◽  
Vol 11 (04) ◽  
pp. 750-783 ◽  
Author(s):  
Wim Vervaat
Keyword(s):  
Difference Equation ◽  
Poisson Process ◽  
Existence Of Solutions ◽  
Borel Function ◽  
Random Variables ◽  
Random Variable ◽  
Stochastic Difference Equation ◽  
Infinitely Divisible ◽  
Uniform Random Variable

The present paper considers the stochastic difference equation Y n = A n Y n-1 + B n with i.i.d. random pairs (A n , B n ) and obtains conditions under which Y n converges in distribution. This convergence is related to the existence of solutions of and (A, B) independent, and the convergence w.p. 1 of ∑ A 1 A 2 ··· A n-1 B n . A second subject is the series ∑ C n f(T n ) with (C n ) a sequence of i.i.d. random variables, (T n ) the sequence of points of a Poisson process and f a Borel function on (0, ∞). The resulting random variable turns out to be infinitely divisible, and its Lévy–Hinčin representation is obtained. The two subjects coincide in case A n and C n are independent, B n = A n C n , A n = U 1/α n with U n a uniform random variable, f(x) = e −x/α.

On a stochastic difference equation and a representation of non–negative infinitely divisible random variables

1979 ◽  
Vol 11 (4) ◽  
pp. 750-783 ◽  
Author(s):  
Wim Vervaat
Keyword(s):  
Difference Equation ◽  
Poisson Process ◽  
Existence Of Solutions ◽  
Borel Function ◽  
Random Variables ◽  
Random Variable ◽  
Stochastic Difference Equation ◽  
Infinitely Divisible ◽  
Uniform Random Variable

The present paper considers the stochastic difference equation Yn = AnYn-1 + Bn with i.i.d. random pairs (An, Bn) and obtains conditions under which Yn converges in distribution. This convergence is related to the existence of solutions of and (A, B) independent, and the convergence w.p. 1 of ∑ A1A2 ··· An-1Bn. A second subject is the series ∑ Cnf(Tn) with (Cn) a sequence of i.i.d. random variables, (Tn) the sequence of points of a Poisson process and f a Borel function on (0, ∞). The resulting random variable turns out to be infinitely divisible, and its Lévy–Hinčin representation is obtained. The two subjects coincide in case An and Cn are independent, Bn = AnCn, An = U1/αn with Un a uniform random variable, f(x) = e−x/α.

Average delay in queues with non-stationary Poisson arrivals

1978 ◽  
Vol 15 (3) ◽  
pp. 602-609 ◽  
Author(s):  
Sheldon M. Ross
Keyword(s):  
Poisson Process ◽  
Intensity Function ◽  
Arrival Process ◽  
Single Server ◽  
Actual Process ◽  
Average Delay ◽  
Average Value ◽  
Poisson Arrival ◽  
Poisson Arrivals ◽  
Customer Delay

One of the major difficulties in attempting to apply known queueing theory results to real problems is that almost always these results assume a time-stationary Poisson arrival process, whereas in practice the actual process is almost invariably non-stationary. In this paper we consider single-server infinite-capacity queueing models in which the arrival process is a non-stationary process with an intensity function ∧(t), t ≧ 0, which is itself a random process. We suppose that the average value of the intensity function exists and is equal to some constant, call it λ, with probability 1.We make a conjecture to the effect that ‘the closer {∧(t), t ≧ 0} is to the stationary Poisson process with rate λ ' then the smaller is the average customer delay, and then we verify the conjecture in the special case where the arrival process is an interrupted Poisson process.

Notes on the variance of a widow's pension

1974 ◽  
Vol 101 (1) ◽  
pp. 103-107 ◽  
Author(s):  
P. P. Boyle
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Stochastic Approach ◽  
Higher Order ◽  
Expected Value ◽  
Confidence Limits ◽  
Higher Order Moments ◽  
Variable Time ◽  
Second Moments ◽  
Constant Rate

Pollard and Pollard have considered a stochastic approach to certain actuarial problems. In their notation ãx denotes the random variable whose expected value is ax. The random variable ãx depends on the random variable, time until death and the (assumed constant) rate of interest. The authors show how to calculate the second and higher-order moments of some common actuarial random variables such as ãx and Ãx. The second moments, in particular, are useful in estimating confidence limits for the total liability in respect of a group of contracts.

On some characterizations of the poisson process

1989 ◽  
Vol 26 (01) ◽  
pp. 176-181
Author(s):  
Wen-Jang Huang
Keyword(s):  
Poisson Process ◽  
Renewal Process ◽  
Arrival Time ◽  
Random Variables ◽  
Poisson Processes

In this article we give some characterizations of Poisson processes, the model which we consider is inspired by Kimeldorf and Thall (1983) and we generalize the results of Chandramohan and Liang (1985). More precisely, we consider an arbitrarily delayed renewal process, at each arrival time we allow the number of arrivals to be i.i.d. random variables, also the mass of each unit atom can be split into k new atoms with the ith new atom assigned to the process Di, i = 1, ···, k. We shall show that the existence of a pair of uncorrelated processes Di, Dj, i ≠ j, implies the renewal process is Poisson. Some other related characterization results are also obtained.

scholarly journals Stochastic Comparisons of Some Distances between Random Variables

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 981
Author(s):  
Patricia Ortega-Jiménez ◽  
Miguel A. Sordo ◽  
Alfonso Suárez-Llorens
Keyword(s):  
Financial Risk ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Stochastic Ordering ◽  
Random Variable ◽  
Distribution Functions ◽  
Stochastic Comparisons ◽  
Stochastic Orderings ◽  
Absolute Value ◽  
The Difference

The aim of this paper is twofold. First, we show that the expectation of the absolute value of the difference between two copies, not necessarily independent, of a random variable is a measure of its variability in the sense of Bickel and Lehmann (1979). Moreover, if the two copies are negatively dependent through stochastic ordering, this measure is subadditive. The second purpose of this paper is to provide sufficient conditions for comparing several distances between pairs of random variables (with possibly different distribution functions) in terms of various stochastic orderings. Applications in actuarial and financial risk management are given.

scholarly journals Fully degenerate Bell polynomials associated with degenerate Poisson random variables

2021 ◽  
Vol 19 (1) ◽  
pp. 284-296
Author(s):  
Hye Kyung Kim
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Combinatorial Identities ◽  
Bell Polynomials ◽  
Poisson Random Variable ◽  
Central Moments ◽  
Special Polynomials ◽  
Special Numbers And Polynomials ◽  
New Type ◽  
Two Parameters

Abstract Many mathematicians have studied degenerate versions of quite a few special polynomials and numbers since Carlitz’s work (Utilitas Math. 15 (1979), 51–88). Recently, Kim et al. studied the degenerate gamma random variables, discrete degenerate random variables and two-variable degenerate Bell polynomials associated with Poisson degenerate central moments, etc. This paper is divided into two parts. In the first part, we introduce a new type of degenerate Bell polynomials associated with degenerate Poisson random variables with parameter α > 0 \alpha \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the fully degenerate Bell polynomials. We derive some combinatorial identities for the fully degenerate Bell polynomials related to the n n th moment of the degenerate Poisson random variable, special numbers and polynomials. In the second part, we consider the fully degenerate Bell polynomials associated with degenerate Poisson random variables with two parameters α > 0 \alpha \gt 0 and β > 0 \beta \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the two-variable fully degenerate Bell polynomials. We show their connection with the degenerate Poisson central moments, special numbers and polynomials.

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