Ehrenfest urn models

1965 ◽  
Vol 2 (02) ◽  
pp. 352-376 ◽  
Author(s):  
Samuel Karlin ◽  
James McGregor
Keyword(s):  
Exponential Distribution ◽  
Continuous Time ◽  
Random Variables ◽  
Independent Random Variables ◽  
Urn Models ◽  
Time Intervals ◽  
Negative Exponential Distribution ◽  
Negative Exponential ◽  
Random Times

In the Ehrenfest model with continuous time one considers two urns and N balls distributed in the urns. The system is said to be in stateiif there areiballs in urn I, N −iballs in urn II. Events occur at random times and the time intervals T between successive events are independent random variables all with the same negative exponential distributionWhen an event occurs a ball is chosen at random (each of theNballs has probability 1/Nto be chosen), removed from its urn, and then placed in urn I with probabilityp, in urn II with probabilityq= 1 −p, (0 <p< 1).

Ehrenfest urn models

1965 ◽  
Vol 2 (2) ◽  
pp. 352-376 ◽  
Author(s):  
Samuel Karlin ◽  
James McGregor
Keyword(s):  
Exponential Distribution ◽  
Continuous Time ◽  
Random Variables ◽  
Independent Random Variables ◽  
Urn Models ◽  
Time Intervals ◽  
Negative Exponential Distribution ◽  
Negative Exponential ◽  
Random Times

In the Ehrenfest model with continuous time one considers two urns and N balls distributed in the urns. The system is said to be in state i if there are i balls in urn I, N − i balls in urn II. Events occur at random times and the time intervals T between successive events are independent random variables all with the same negative exponential distributionWhen an event occurs a ball is chosen at random (each of the N balls has probability 1/N to be chosen), removed from its urn, and then placed in urn I with probability p, in urn II with probability q = 1 − p, (0 < p < 1).

A characterization of the negative exponential distribution with application to reliability theory

1981 ◽  
Vol 18 (3) ◽  
pp. 652-659 ◽  
Author(s):  
M. J. Phillips
Keyword(s):  
Exponential Distribution ◽  
Random Variables ◽  
Reliability Theory ◽  
The Other ◽  
Independent Random Variables ◽  
System Availability ◽  
Failure Times ◽  
Negative Exponential Distribution ◽  
Negative Exponential

The negative exponential distribution is characterized in terms of two independent random variables. Only one of the random variables has a negative exponential distribution whilst the other can belong to a wide class of distributions. This result is then applied to two models for the reliability of a system of two modules subject to revealed and unrevealed faults to show when the models are equivalent. It is also shown, under certain conditions, that the system availability is only independent of the distribution of revealed failure times in one module when unrevealed failure times in the other module have a negative exponential distribution.

A characterization of the negative exponential distribution with application to reliability theory

1981 ◽  
Vol 18 (03) ◽  
pp. 652-659 ◽  
Author(s):  
M. J. Phillips
Keyword(s):  
Exponential Distribution ◽  
Random Variables ◽  
Reliability Theory ◽  
The Other ◽  
Independent Random Variables ◽  
System Availability ◽  
Failure Times ◽  
Negative Exponential Distribution ◽  
Negative Exponential

The negative exponential distribution is characterized in terms of two independent random variables. Only one of the random variables has a negative exponential distribution whilst the other can belong to a wide class of distributions. This result is then applied to two models for the reliability of a system of two modules subject to revealed and unrevealed faults to show when the models are equivalent. It is also shown, under certain conditions, that the system availability is only independent of the distribution of revealed failure times in one module when unrevealed failure times in the other module have a negative exponential distribution.

A note on passage times and infinitely divisible distributions

1967 ◽  
Vol 4 (2) ◽  
pp. 402-405 ◽  
Author(s):  
H. D. Miller
Keyword(s):  
Random Walk ◽  
Markov Process ◽  
Continuous Time ◽  
Random Variables ◽  
Independent Random Variables ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible ◽  
Time Intervals ◽  
Mutually Independent ◽  
Passage Times

Let X(t) be the position at time t of a particle undergoing a simple symmetrical random walk in continuous time, i.e. the particle starts at the origin at time t = 0 and at times T1, T1 + T2, … it undergoes jumps ξ1, ξ2, …, where the time intervals T1, T2, … between successive jumps are mutually independent random variables each following the exponential density e–t while the jumps, which are independent of the τi, are mutually independent random variables with the distribution . The process X(t) is clearly a Markov process whose state space is the set of all integers.

Arithmetic Research of Computer Simulation of the Bus Junction Stations

2010 ◽  
Vol 108-111 ◽  
pp. 441-445
Author(s):  
Yong Luo ◽  
Xiu Chun Guo
Keyword(s):  
Computer Simulation ◽  
Normal Distribution ◽  
Exponential Distribution ◽  
Poisson Distribution ◽  
Computer Model ◽  
Time Distribution ◽  
Random Variables ◽  
Step Length ◽  
Negative Exponential Distribution ◽  
Negative Exponential

This paper will construct a discrete system's computer model, include the time distribution of buses arrive, stop and passengers’ get down and get off. The emulation clock advanced with the method of incident step length. Through the inversion produce the random variables of Poisson distribution, negative exponential distribution and normal distribution, simulate the conditions of bus arrive, leave and pick up passengers. Finally calculate with the relative data and then get a series of indicators to evaluate the crowing degree of the bus station.

A note on passage times and infinitely divisible distributions

1967 ◽  
Vol 4 (02) ◽  
pp. 402-405 ◽  
Author(s):  
H. D. Miller
Keyword(s):  
Random Walk ◽  
Markov Process ◽  
Continuous Time ◽  
Random Variables ◽  
Independent Random Variables ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible ◽  
Time Intervals ◽  
Mutually Independent ◽  
Passage Times

Let X(t) be the position at time t of a particle undergoing a simple symmetrical random walk in continuous time, i.e. the particle starts at the origin at time t = 0 and at times T 1, T 1 + T 2, … it undergoes jumps ξ 1, ξ 2, …, where the time intervals T 1, T 2, … between successive jumps are mutually independent random variables each following the exponential density e–t while the jumps, which are independent of the τi , are mutually independent random variables with the distribution . The process X(t) is clearly a Markov process whose state space is the set of all integers.

scholarly journals ON USE OF GAMMA DISTRIBUTION FOR EVALUATION OF RELIABILITY AND AVAILABILITY OF A SINGLE UNIT SYSTEM SUBJECT TO ARRIVAL TIME OF THE SERVER

2019 ◽  
pp. 93-102
Author(s):  
N. Nandal ◽  
S.C. Malik
Keyword(s):  
Exponential Distribution ◽  
Gamma Distribution ◽  
Single Unit ◽  
Arrival Time ◽  
Random Variables ◽  
Repairable Systems ◽  
Unit System ◽  
Negative Exponential Distribution ◽  
Negative Exponential ◽  
Reliability And Availability

The preference to the use of single unit systems over the redundant systems has been given due to their intrinsic reliability and affordability. And, stochastic modeling of repairable systems of one or more unit has been done by assuming negative exponential distribution for failure and repair times. In fact, the repairable systems may or may not have constant failure and repair rates. In such situations some other distributions possessing monotonic nature of the random variables associated with different time points may be considered. Gamma distribution is one of the distributions that may offer a good fit to some set of failure data. Also, negative exponential distribution is a special case of this distribution. Hence, in this paper reliability and availability of a single unit system by considering Gamma distribution for the random variables associated with failure and repair times of the system have been evaluated. A single server is employed to carry out the repair activities. The server is allowed to take some time to arrive at the system (called arrival time). The system has all the transit points as regenerative and so regenerative point has been used to derive the expressions for reliability measures. The values of reliability and availability are obtained for particular situations of the parameters. The behavior of these measures has been observed for the arbitrary values of the parameters.  

The Comparison Between the Bayes Estimator and the Maximum Likelihood Estimator of the Reliability Function for Negative Exponential Distribution

2018 ◽  
pp. 439
Author(s):  
Hazim Mansour Gorgees ◽  
Bushra Abdualrasool Ali ◽  
Raghad Ibrahim Kathum
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimator ◽  
Exponential Distribution ◽  
Reliability Function ◽  
Bayes Estimator ◽  
Likelihood Estimator ◽  
Monte Carlo Simulation Technique ◽  
Negative Exponential Distribution ◽  
Negative Exponential ◽  
Better Than

     In this paper, the maximum likelihood estimator and the Bayes estimator of the reliability function for negative exponential distribution has been derived, then a Monte –Carlo simulation technique was employed to compare the performance of such estimators. The integral mean square error (IMSE) was used as a criterion for this comparison. The simulation results displayed that the Bayes estimator performed better than the maximum likelihood estimator for different samples sizes.

On the distribution of the number of cyclically occurring random events

1972 ◽  
Vol 9 (3) ◽  
pp. 681-683
Author(s):  
Leon Podkaminer
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Time Intervals ◽  
Time Period ◽  
Random Events ◽  
Mutually Independent

The probabilities of the occurrence of n events in a certain time period are calculated under the assumptions that the time intervals between the neighbouring events are mutually independent random variables, satisfying some analytic conditions.

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