scholarly journals Asymptotic results for multiplexing subexponential on-off processes

1999 ◽  
Vol 31 (2) ◽  
pp. 394-421 ◽  
Author(s):  
Predrag R. Jelenković ◽  
Aurel A. Lazar
Keyword(s):  
Queueing System ◽  
Random Variable ◽  
Mean Value ◽  
Activity Period ◽  
Arrival Process ◽  
Asymptotic Results ◽  
Fluid Queue ◽  
Arrival Processes ◽  
Image Position

Consider an aggregate arrival process AN obtained by multiplexing N on-off processes with exponential off periods of rate λ and subexponential on periods τon. As N goes to infinity, with λN → Λ, AN approaches an M/G/∞ type process. Both for finite and infinite N, we obtain the asymptotic characterization of the arrival process activity period.Using these results we investigate a fluid queue with the limiting M/G/∞ arrival process At∞ and capacity c. When on periods are regularly varying (with non-integer exponent), we derive a precise asymptotic behavior of the queue length random variable QtP observed at the beginning of the arrival process activity periods where ρ = 𝔼At∞ < c; r (c ≤ r) is the rate at which the fluid is arriving during an on period. The asymptotic (time average) queue distribution lower bound is obtained under more general assumptions regarding on periods than regular variation.In addition, we analyse a queueing system in which one on-off process, whose on period belongs to a subclass of subexponential distributions, is multiplexed with independent exponential processes with aggregate expected rate 𝔼et. This system is shown to be asymptotically equivalent to the same queueing system with the exponential arrival processes being replaced by their total mean value 𝔼et.

scholarly journals Asymptotic results for multiplexing subexponential on-off processes

1999 ◽  
Vol 31 (02) ◽  
pp. 394-421 ◽  
Author(s):  
Predrag R. Jelenković ◽  
Aurel A. Lazar
Keyword(s):  
Queueing System ◽  
Random Variable ◽  
Mean Value ◽  
Activity Period ◽  
Arrival Process ◽  
Asymptotic Results ◽  
Fluid Queue ◽  
Arrival Processes ◽  
Image Position

Consider an aggregate arrival process A N obtained by multiplexing N on-off processes with exponential off periods of rate λ and subexponential on periods τon. As N goes to infinity, with λN → Λ, A N approaches an M/G/∞ type process. Both for finite and infinite N, we obtain the asymptotic characterization of the arrival process activity period. Using these results we investigate a fluid queue with the limiting M/G/∞ arrival process A t ∞ and capacity c. When on periods are regularly varying (with non-integer exponent), we derive a precise asymptotic behavior of the queue length random variable Q t P observed at the beginning of the arrival process activity periods where ρ = 𝔼A t ∞ &lt; c; r (c ≤ r) is the rate at which the fluid is arriving during an on period. The asymptotic (time average) queue distribution lower bound is obtained under more general assumptions regarding on periods than regular variation. In addition, we analyse a queueing system in which one on-off process, whose on period belongs to a subclass of subexponential distributions, is multiplexed with independent exponential processes with aggregate expected rate 𝔼e t . This system is shown to be asymptotically equivalent to the same queueing system with the exponential arrival processes being replaced by their total mean value 𝔼e t .

Numerical integration by systematic sampling

1950 ◽  
Vol 46 (1) ◽  
pp. 111-115 ◽  
Author(s):  
P. A. P. Moran
Keyword(s):  
Numerical Integration ◽  
Bounded Variation ◽  
Mathematical Problem ◽  
Random Variable ◽  
Mean Value ◽  
Lattice Points ◽  
Systematic Sampling ◽  
Image Position ◽  
Infinite Sum

Recent investigations by F. Yates (1) in agricultural statistics suggest a mathematical problem which may be formulated as follows. A function f(x) is known to be of bounded variation and Lebesgue integrable on the range −∞ < x < ∞, and its integral over this range is to be determined. In default of any knowledge of the position of the non-negligible values of the function the best that can be done is to calculate the infinite sumfor some suitable δ and an arbitrary origin t, where s ranges over all possible positive and negative integers including zero. S is evidently of period δ in t and ranges over all its values as t varies from 0 to δ. Previous writers (Aitken (2), p. 45, and Kendall (3)) have examined the resulting errors for fixed t. (They considered only symmetrical functions, and supposed one of the lattice points to be located at the centre.) Here we do not restrict ourselves to symmetrical functions and consider the likely departure of S(t) from J (the required integral) when t is a random variable uniformly distributed in (0, δ). It will be shown that S(t) is distributed about J as mean value, with a variance which will be evaluated as a function of δ, the scale of subdivision.

Characterization of matrix probability distributions by mean residual lifetime

1986 ◽  
Vol 100 (3) ◽  
pp. 583-589
Author(s):  
P. E. Jupp
Keyword(s):  
Symmetric Matrix ◽  
Probability Distributions ◽  
Random Variable ◽  
Residual Lifetime ◽  
Regularity Conditions ◽  
Mean Residual Lifetime ◽  
The Mean ◽  
Image Position ◽  
Vector Valued

The mean residual lifetime of a real-valued random variable X is the function e defined byOne of the more important properties of the mean residual lifetime function is that it determines the distribution of X. See, for example, Swartz [10]. References to related characterizations are given by Galambos and Kotz [3], pages 30–35. It was established by Jupp and Mardia[6] that this property holds also for vector-valued X. As (1·1) makes sense if X is a random symmetric matrix, it is natural to ask whether the property holds in this case also. The purpose of this note is to show that, under certain regularity conditions, the distributions of such matrices are indeed determined by their mean residual lifetimes.

The departure process from a queueing system

1976 ◽  
Vol 80 (2) ◽  
pp. 283-285 ◽  
Author(s):  
F. P. Kelly
Keyword(s):  
Queueing System ◽  
Arrival Process ◽  
Single Server ◽  
Departure Process ◽  
Common Distribution ◽  
Poisson Arrival ◽  
Queue Discipline ◽  
Common Distribution Function ◽  
Image Position ◽  
Poisson Arrival Process

Consider a single-server queueing system with a Poisson arrival process at rate λ and positive service requirements independently distributed with common distribution function B(z) and finite expectationwhere βλ < 1, i.e. an M/G/1 system. When the queue discipline is first come first served, or last come first served without pre-emption, the stationary departure process is Poisson if and only if G = M (i.e. B(z) = 1 − exp (−z/β)); see (8), (4) and (2). In this paper it is shown that when the queue discipline is last come first served with pre-emption the stationary departure process is Poisson whatever the form of B(z). The method used is adapted from the approach of Takács (10) and Shanbhag and Tambouratzis (9).

Analytic characterization of the optimal control of a queueing system

1970 ◽  
Vol 7 (03) ◽  
pp. 617-633 ◽  
Author(s):  
S. Zacks ◽  
M. Yadin
Keyword(s):  
Optimal Control ◽  
Queueing System ◽  
Control Policy ◽  
Random Variable ◽  
Management Policy ◽  
Homogeneous Poisson Process ◽  
Negative Exponential Distribution ◽  
Negative Exponential ◽  
Arrival Intensity

Summary In a recent paper [7] the authors studied the optimal control policy of the following queueing system. Customers arrive at a service station according to a time homogeneous Poisson process with a known arrival intensity, λ. The service time at the station is a random variable having a negative exponential distribution with intensity μ, which is under control and can be varied over a certain range, according to the management policy.

Mean values and classes of harmonic functions

1984 ◽  
Vol 96 (3) ◽  
pp. 501-505 ◽  
Author(s):  
Thomas Ramsey ◽  
Yitzhak Weit
Keyword(s):  
Unit Circle ◽  
Harmonic Functions ◽  
Borel Measure ◽  
Mean Value ◽  
Finite Complex ◽  
Mean Values ◽  
Complex Borel Measure ◽  
Generalized Mean ◽  
Image Position

Let μ be a finite complex Borel measure supported on the unit circle.In this paper, we are concerned with the characterization of the sets of functions satisfying the generalized mean value equation of the form.and for all ξ ∈ , | ξ | = R for some fixed R > 0.

scholarly journals Characterization of Model Uncertainty for the Vertical Pullout Capacity of Helical Anchors in Cohesive Soils

2020 ◽  
Vol 8 (10) ◽  
pp. 738
Author(s):  
Po Cheng ◽  
Jiang Tao Yi ◽  
Fei Liu ◽  
Jun Jie Dong
Keyword(s):  
Model Uncertainty ◽  
Lognormal Distribution ◽  
Prediction Error ◽  
Random Variable ◽  
Mean Value ◽  
Regression Equation ◽  
Cohesive Soils ◽  
Pullout Capacity ◽  
Input Parameters

This paper conducts coupled Eulerian–Lagrangian (CEL) analysis to characterize the model uncertainty of using the cylindrical shear method (CSM) to predict the pullout capacity of helical anchors in cohesive soils. The model factor M is adopted to represent the model uncertainty, which is equal to the value of measured capacity divided by estimated solution. The model factor Mcel can be considered to be a random variable with a lognormal distribution, and its mean value and coefficient of variation (COV) are 1.02 and 0.1, respectively. Correction factor η is introduced when comparing CSM and CEL, which is found to be influenced by input parameters. The dependence on input parameters is removed by performing regression analysis and the regression equation f is obtained. Substituting the regression equation f into the original CSM constitutes the modified CSM (MCSM), and the model factor of MCSM can be modeled as a random variable with a lognormal distribution, and its mean value and COV are 1.02 and 0.13, respectively. Finally, 13 filed tests are collected to compare the prediction accuracy, the results show that the prediction error range of MCSM is mostly within 15%. The present findings might be helpful for engineers and designers to estimate the pullout capacity of helical anchors in cohesive soils more confidently.

A note on the generalized Rao–Rubin condition and characterization of certain discrete distributions

1980 ◽  
Vol 17 (2) ◽  
pp. 563-569 ◽  
Author(s):  
Sheela Talwalker
Keyword(s):  
Discrete Distribution ◽  
Random Variable ◽  
Discrete Distributions ◽  
Binomial Probability ◽  
Poisson Distributions ◽  
Image Position ◽  
Original Observation

Assuming that the original observation from a discrete distribution is subject to damage according to a binomial probability law, it is shown here that a class of discrete distributions consisting of binomial, Poisson and mixed Poisson distributions, is characterized by the generalized Rao–Rubin condition 0 <π, π′ < 1 are constants. Y denotes the resulting or observed random variable, taking the values r = 0, 1, 2, ….

On the quasi-stationary distribution of the virtual waiting time in queues with Poisson arrivals

1971 ◽  
Vol 8 (3) ◽  
pp. 494-507 ◽  
Author(s):  
E. K. Kyprianou
Keyword(s):  
Waiting Time ◽  
Busy Period ◽  
Queueing System ◽  
Random Variable ◽  
Single Server ◽  
Virtual Waiting Time ◽  
Waiting Time Process ◽  
Poisson Arrivals ◽  
Image Position ◽  
Quasi Stationary Distribution

We consider a single server queueing system M/G/1 in which customers arrive in a Poisson process with mean λt, and the service time has distribution dB(t), 0 < t < ∞. Let W(t) be the virtual waiting time process, i.e., the time that a potential customer arriving at the queueing system at time t would have to wait before beginning his service. We also let the random variable denote the first busy period initiated by a waiting time u at time t = 0.

A note on the generalized Rao–Rubin condition and characterization of certain discrete distributions

1980 ◽  
Vol 17 (02) ◽  
pp. 563-569
Author(s):  
Sheela Talwalker
Keyword(s):  
Discrete Distribution ◽  
Random Variable ◽  
Discrete Distributions ◽  
Binomial Probability ◽  
Poisson Distributions ◽  
Image Position ◽  
Original Observation

Assuming that the original observation from a discrete distribution is subject to damage according to a binomial probability law, it is shown here that a class of discrete distributions consisting of binomial, Poisson and mixed Poisson distributions, is characterized by the generalized Rao–Rubin condition 0 &lt;π, π′ &lt; 1 are constants. Y denotes the resulting or observed random variable, taking the values r = 0, 1, 2, ….

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