On two integral equations of queueing theory

1967 ◽  
Vol 4 (02) ◽  
pp. 343-355 ◽  
Author(s):  
J. W. Cohen
Keyword(s):  
Stochastic Process ◽  
Integral Equations ◽  
Positive Constant ◽  
Mathematical Description ◽  
Queueing Theory ◽  
Image Position

In the present paper the solutions of two integral equations are derived. One of the integral equations dominates the mathematical description of the stochastic process { v n , n = 1,2, …}, recursively defined by K is a positive constant, τ 1, τ 2, …; Σ 1, Σ 2, …; are independent, non-negative variables, with τ 1, τ 2,…, identically distributed, similarly, the variables Σ 1, Σ 2, …, are identically distributed.

On two integral equations of queueing theory

1967 ◽  
Vol 4 (2) ◽  
pp. 343-355 ◽  
Author(s):  
J. W. Cohen
Keyword(s):  
Stochastic Process ◽  
Integral Equations ◽  
Positive Constant ◽  
Mathematical Description ◽  
Queueing Theory ◽  
Image Position

In the present paper the solutions of two integral equations are derived. One of the integral equations dominates the mathematical description of the stochastic process {vn, n = 1,2, …}, recursively defined by K is a positive constant, τ1, τ2, …; Σ1, Σ2, …; are independent, non-negative variables, with τ1, τ2,…, identically distributed, similarly, the variables Σ1, Σ2, …, are identically distributed.

scholarly journals Mean-square stability of a class of stochastic integral equations

1972 ◽  
Vol 7 (3) ◽  
pp. 337-352
Author(s):  
W.J. Padgett
Keyword(s):  
Stochastic Process ◽  
Integral Equations ◽  
Stochastic Integral ◽  
Second Order ◽  
Mean Square ◽  
Random Solution ◽  
Mean Square Stability ◽  
Stochastic Integral Equations ◽  
Exponentially Stable ◽  
Image Position

The object of this paper is to investigate under very general conditions the existence and mean-square stability of a random solution of a class of stochastic integral equations in the formfor t ≥ 0, where a random solution is a second order stochastic process {x(t; w) t ≥ 0} which satisfies the equation almost certainly. A random solution x(t; w) is defined to be stable in mean-square if E[|x(t; w)|2] ≤ p for all t ≥ 0 and some p > 0 or exponentially stable in mean-square if E[|x(t; w)|2] ≤ pe-at, t ≥ 0, for some constants ρ > 0 and α > 0.

scholarly journals Dual integral equations with a trigonometric Kernel

1979 ◽  
Vol 22 (3) ◽  
pp. 213-215 ◽  
Author(s):  
Brij M. Singh ◽  
Ranjit S. Dhaliwal
Keyword(s):  
Integral Equations ◽  
Positive Constant ◽  
Integrable Function ◽  
Crack Problem ◽  
Dual Integral Equations ◽  
Image Position

In this paper, we solve the following dual integral equationswhere δ is a real positive constant and f(x) is a continuous and integrable function of x in [0, a]. The dual integral equations (1) and (2) arise in a crack problem of elasticity.

scholarly journals Some dual series and triple integral equations

1969 ◽  
Vol 16 (4) ◽  
pp. 273-280 ◽  
Author(s):  
J. S. Lowndes
Keyword(s):  
Integral Equations ◽  
Jacobi Polynomial ◽  
Triple Integral Equations ◽  
Dual Series Equations ◽  
Image Position

In this paper we first of all solve the dual series equationswhere ƒ(ρ) and g(ρ) are prescribed functions,is the Jacobi polynomial (2).

scholarly journals The Distribution of Heights of Binary Trees and Other Simple Trees

1993 ◽  
Vol 2 (2) ◽  
pp. 145-156 ◽  
Author(s):  
Philippe Flajolet ◽  
Zhicheng Gao ◽  
Andrew Odlyzko ◽  
Bruce Richmond
Keyword(s):  
Asymptotic Formula ◽  
Positive Constant ◽  
Binary Trees ◽  
Asymptotic Results ◽  
Image Position

The number, , of rooted plane binary trees of height ≤ h with n internal nodes is shown to satisfyuniformly for δ−1(log n)−1/2 ≤ β ≤ δ(log n)1/2, where and δ is a positive constant. An asymptotic formula for is derived for h = cn, where 0 < c < 1. Bounds for are also derived for large and small heights. The methods apply to any simple family of trees, and the general asymptotic results are stated.

Functions regular in the unit circle

1956 ◽  
Vol 52 (1) ◽  
pp. 49-60 ◽  
Author(s):  
C. N. Linden ◽  
M. L. Cartwright
Keyword(s):  
Unit Circle ◽  
Positive Constant ◽  
Upper Bounds ◽  
Image Position

Letbe a function regular for | z | < 1. With the hypotheses f(0) = 0 andfor some positive constant α, Cartwright(1) has deduced upper bounds for |f(z) | in the unit circle. Three cases have arisen and according as (1) holds with α < 1, α = 1 or α > 1, the bounds on each circle | z | = r are given respectively byK(α) being a constant which depends only on the corresponding value of α which occurs in (1). We shall always use the symbols K and A to represent constants dependent on certain parameters such as α, not necessarily having the same value at each occurrence.

Analogue of a theorem of Khintchine in fields of formal power series

1966 ◽  
Vol 62 (4) ◽  
pp. 637-642 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Power Series ◽  
Positive Constant ◽  
Classical Result ◽  
Real Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
The Difference ◽  
Image Position ◽  
Formal Power ◽  
Positive Integers

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and letbe any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such thatfor all integers x1, x2, …, xm not all zero.

Central Limit Theorems for Interchangeable Processes

1958 ◽  
Vol 10 ◽  
pp. 222-229 ◽  
Author(s):  
J. R. Blum ◽  
H. Chernoff ◽  
M. Rosenblatt ◽  
H. Teicher
Keyword(s):  
Stochastic Process ◽  
Probability Measure ◽  
Central Limit ◽  
Joint Distribution ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Probability Measures ◽  
Image Position ◽  
Positive Integers

Let {Xn} (n = 1, 2 , …) be a stochastic process. The random variables comprising it or the process itself will be said to be interchangeable if, for any choice of distinct positive integers i 1, i 2, H 3 … , ik, the joint distribution of depends merely on k and is independent of the integers i 1, i 2, … , i k. It was shown by De Finetti (3) that the probability measure for any interchangeable process is a mixture of probability measures of processes each consisting of independent and identically distributed random variables.

scholarly journals The Approximate Solution of Dual Integral Equations by Variational Methods

1958 ◽  
Vol 11 (2) ◽  
pp. 115-126 ◽  
Author(s):  
B. Noble
Keyword(s):  
Approximate Solution ◽  
Variational Methods ◽  
Integral Equations ◽  
Separation Of Variables ◽  
Circular Disc ◽  
Cylindrical Coordinates ◽  
Dual Integral Equations ◽  
Particular Solution ◽  
Image Position

The classic application of dual integral equations occurs in connexion with the potential of a circular disc (e.g. Titchmarsh (9), p. 334). Suppose that the disc lies in z = 0, 0≤ρ≤1, where we use cylindrical coordinates (p, z). Then it is required to find a solution ofsuch that on z = 0Separation of variables in conjunction with the conditions that ø is finite on the axis and ø tends to zero as z tends to plus infinity yields the particular solution.

On Functional Properties of Incomplete Gaussian Sums

1991 ◽  
Vol 43 (1) ◽  
pp. 182-212 ◽  
Author(s):  
K. I. Oskolkov
Keyword(s):  
Functional Properties ◽  
Positive Constant ◽  
Absolute Constant ◽  
Special Function ◽  
Absolute Positive Constant ◽  
Image Position ◽  
Gaussian Sums

AbstractThe following special function of two real variables x2 and x1 is considered: and its connections with the incomplete Gaussian sums where ω are intervals of length |ω| ≤1. In particular, it is proved that for each fixed x2 and uniformly in X2 the function H(x2, x1) is of weakly bounded 2-variation in the variable x1 over the period [0, 1]. In terms of the sums W this means that for collections Ω = {ωk}, consisting of nonoverlapping intervals ωk ∪ [0,1) the following estimate is valid: where card denotes the number of elements, and c is an absolute positive constant. The exact value of the best absolute constant к in the estimate (which is due to G. H. Hardy and J. E. Littlewood) is discussed.

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