On a generalized storage model with moment assumptions

1981 ◽  
Vol 18 (02) ◽  
pp. 473-481 ◽  
Author(s):  
Prem S. Puri ◽  
Samuel W. Woolford
Keyword(s):  
Distribution Function ◽  
Renewal Process ◽  
Random Variables ◽  
Positive Zero ◽  
Markov Renewal Process ◽  
Random Input ◽  
Storage Model ◽  
Limit Distributions ◽  
Second Moments

This paper considers a semi-infinite storage model, of the type studied by Senturia and Puri [13] and Balagopal [2], defined on a Markov renewal process, {(Xn, Tn ), n = 0, 1, ·· ·}, with 0 = T 0 < T 1 < · ··, almost surely, where Xn takes values in the set {1, 2, ·· ·}. If at Tn, Xn = j, then there is a random ‘input' Vn (j) (a negative input implying a demand) of ‘type' j, having distribution function Fj (·). We assume that {Vn (j)} is an i.i.d. sequence of random variables, taken to be independent of {(Xn, Tn )} and of {Vn (k)}, for k ≠ j, and that Vn (j) has first and second moments. Here the random variables Vn (j) represent instantaneous ‘inputs' (a negative value implying a demand) of type j for our storage model. Under these assumptions, we establish certain limit distributions for the joint process (Z(t), L(t)), where Z(t) (defined in (2)) is the level of storage at time t and L(t) (defined in (3)) is the demand lost due to shortage of supply during [0, t]. Different limit distributions are obtained for the cases when the ‘average stationary input' ρ, as defined in (5), is positive, zero or negative.

On a generalized storage model with moment assumptions

1981 ◽  
Vol 18 (2) ◽  
pp. 473-481 ◽  
Author(s):  
Prem S. Puri ◽  
Samuel W. Woolford
Keyword(s):  
Distribution Function ◽  
Renewal Process ◽  
Random Variables ◽  
Positive Zero ◽  
Markov Renewal Process ◽  
Random Input ◽  
Storage Model ◽  
Limit Distributions ◽  
Second Moments

This paper considers a semi-infinite storage model, of the type studied by Senturia and Puri [13] and Balagopal [2], defined on a Markov renewal process, {(Xn, Tn), n = 0, 1, ·· ·}, with 0 = T0 < T1 < · ··, almost surely, where Xn takes values in the set {1, 2, ·· ·}. If at Tn, Xn = j, then there is a random ‘input' Vn (j) (a negative input implying a demand) of ‘type' j, having distribution function Fj(·). We assume that {Vn (j)} is an i.i.d. sequence of random variables, taken to be independent of {(Xn, Tn)} and of {Vn (k)}, for k ≠ j, and that Vn (j) has first and second moments. Here the random variables Vn (j) represent instantaneous ‘inputs' (a negative value implying a demand) of type j for our storage model. Under these assumptions, we establish certain limit distributions for the joint process (Z(t), L(t)), where Z(t) (defined in (2)) is the level of storage at time t and L(t) (defined in (3)) is the demand lost due to shortage of supply during [0, t]. Different limit distributions are obtained for the cases when the ‘average stationary input' ρ, as defined in (5), is positive, zero or negative.

On a generalized finite-capacity storage model

1983 ◽  
Vol 20 (3) ◽  
pp. 663-674 ◽  
Author(s):  
Samuel W. Woolford
Keyword(s):  
Distribution Function ◽  
Markov Chain ◽  
State Space ◽  
Random Variables ◽  
Time Dependent ◽  
Finite Capacity ◽  
Random Input ◽  
Storage Model ◽  
Limit Distributions

This paper considers a finite-capacity storage model defined on a Markov chain {Xn; n = 0, 1, ·· ·}, having state space J ⊆ {1, 2, ·· ·}. If Xn = j, then there is a random ‘input' Vn(j) (a negative input implying a demand) of ‘type' j, having a distribution function Fj(·). We assume that {Vn(j)} is an i.i.d. sequence of random variables, taken to be independent of {Xn} and of {Vn (k)}, for k ≠ j. Here, the random variables Vn(j) represent instantaneous ‘inputs' of type j for our storage model. Within this framework, we establish certain limit distributions for the joint processes (Zn, Xn) and (Zn, Qn, Ln), where Zn (defined in (1.2)) is the level of storage at time n, Qn (defined in (1.3)) is the cumulative overflow at time n, and Ln (defined in (1.4)) is the cumulative demand lost due to shortage of supply up to time n. In addition, an expression for the time-dependent distribution of (Zn, Xn) is obtained.

On a generalized finite-capacity storage model

1984 ◽  
Vol 16 (1) ◽  
pp. 23-23
Author(s):  
Samuel W. Woolford
Keyword(s):  
Distribution Function ◽  
Markov Chain ◽  
State Space ◽  
Random Variables ◽  
Time Dependent ◽  
Finite Capacity ◽  
Random Input ◽  
Storage Model ◽  
Limit Distributions

This paper considers a finite-capacity storage model defined on a Markov chain {Xn; n = 0, 1, …}, having state space J ⊆ {1, 2, …}. If Xn, = j, then there is a random. ‘input’ Vn(j) (a negative input implying a demand) of ‘type’ j, having a distribution function Fj (·). We assume that {Vn (j)} is an i.i.d. sequence of random variables, taken to be independent of {Xn} and of {Vn(k)}, for k ≠ = j. Here, the random variables Vn(j) represent instantaneous ‘inputs’ of type j for our storage model. Within this framework, we establish certain limit distributions for the joint processes (zn, Xn) and (Zn, OnLn), where Zn, is the level of storage at time n, Qn is the cumulative overflow at time n, and Ln is the cumulative demand lost due to shortage of supply up to time n. In addition, an expression for the time-dependent distribution of (Zn, Xn) is obtained.

On a generalized finite-capacity storage model

1983 ◽  
Vol 20 (03) ◽  
pp. 663-674
Author(s):  
Samuel W. Woolford
Keyword(s):  
Distribution Function ◽  
Markov Chain ◽  
State Space ◽  
Random Variables ◽  
Time Dependent ◽  
Finite Capacity ◽  
Random Input ◽  
Storage Model ◽  
Limit Distributions

This paper considers a finite-capacity storage model defined on a Markov chain {Xn ; n = 0, 1, ·· ·}, having state space J ⊆ {1, 2, ·· ·}. If Xn = j, then there is a random ‘input' Vn (j) (a negative input implying a demand) of ‘type' j, having a distribution function Fj (·). We assume that {Vn (j)} is an i.i.d. sequence of random variables, taken to be independent of {Xn } and of {Vn (k)}, for k ≠ j. Here, the random variables Vn (j) represent instantaneous ‘inputs' of type j for our storage model. Within this framework, we establish certain limit distributions for the joint processes (Zn, Xn ) and (Zn, Qn, Ln ), where Zn (defined in (1.2)) is the level of storage at time n, Qn (defined in (1.3)) is the cumulative overflow at time n, and Ln (defined in (1.4)) is the cumulative demand lost due to shortage of supply up to time n. In addition, an expression for the time-dependent distribution of (Zn, Xn ) is obtained.

On a generalized finite-capacity storage model

1984 ◽  
Vol 16 (01) ◽  
pp. 23
Author(s):  
Samuel W. Woolford
Keyword(s):  
Distribution Function ◽  
Markov Chain ◽  
State Space ◽  
Random Variables ◽  
Time Dependent ◽  
Finite Capacity ◽  
Random Input ◽  
Storage Model ◽  
Limit Distributions

This paper considers a finite-capacity storage model defined on a Markov chain {Xn ; n = 0, 1, …}, having state space J ⊆ {1, 2, …}. If Xn , = j, then there is a random. ‘input’ Vn (j) (a negative input implying a demand) of ‘type’ j, having a distribution function Fj (·). We assume that {Vn (j)} is an i.i.d. sequence of random variables, taken to be independent of {Xn } and of {Vn(k)}, for k ≠ = j. Here, the random variables Vn (j) represent instantaneous ‘inputs’ of type j for our storage model. Within this framework, we establish certain limit distributions for the joint processes (zn , Xn ) and (Zn , On Ln ), where Zn , is the level of storage at time n, Qn is the cumulative overflow at time n, and Ln is the cumulative demand lost due to shortage of supply up to time n. In addition, an expression for the time-dependent distribution of (Zn , Xn ) is obtained.

The convergence of a sum of independent random variables

1963 ◽  
Vol 59 (2) ◽  
pp. 411-416
Author(s):  
G. De Barra ◽  
N. B. Slater
Keyword(s):  
Distribution Function ◽  
Euclidean Space ◽  
Rate Of Convergence ◽  
Radial Distribution Function ◽  
Random Variables ◽  
Covariance Matrices ◽  
Independent Random Variables ◽  
Second Moments ◽  
Image Position ◽  
Real Euclidean Space

Let Xν, ν= l, 2, …, n be n independent random variables in k-dimensional (real) Euclidean space Rk, which have, for each ν, finite fourth moments β4ii = l,…, k. In the case when the Xν are identically distributed, have zero means, and unit covariance matrices, Esseen(1) has discussed the rate of convergence of the distribution of the sumsIf denotes the projection of on the ith coordinate axis, Esseen proves that ifand ψ(a) denotes the corresponding normal (radial) distribution function of the same first and second moments as μn(a), thenwhere and C is a constant depending only on k. (C, without a subscript, will denote everywhere a constant depending only on k.)

Limit distributions of the number of renewals and waiting times

1976 ◽  
Vol 13 (2) ◽  
pp. 301-312
Author(s):  
N. R. Mohan
Keyword(s):  
Distribution Function ◽  
Limit Distribution ◽  
Infinite Sequence ◽  
Waiting Times ◽  
Random Variables ◽  
Stable Laws ◽  
Domains Of Attraction ◽  
Limit Distributions

Let {Xn} be an infinite sequence of independent non-negative random variables. Let the distribution function of Xi, i = 1, 2, …, be either F1 or F2 where F1 and F2 are distinct. Set Sn = X1 + X2 + … + Xn and for t > 0 define and Zt = SN(t)+1 – t. The limit distributions of N(t), Yt and Zt as t → ∞ are obtained when F1 and F2 are in the domains of attraction of stable laws with exponents α1 and α2, respectively and Sn properly normalised has the composition of these two stable laws as its limit distribution.

Limit distributions of the number of renewals and waiting times

1976 ◽  
Vol 13 (02) ◽  
pp. 301-312
Author(s):  
N. R. Mohan
Keyword(s):  
Distribution Function ◽  
Limit Distribution ◽  
Infinite Sequence ◽  
Waiting Times ◽  
Random Variables ◽  
Stable Laws ◽  
Domains Of Attraction ◽  
Limit Distributions

Let {X n} be an infinite sequence of independent non-negative random variables. Let the distribution function of Xi , i = 1, 2, …, be either F 1 or F 2 where F 1 and F 2 are distinct. Set Sn = X 1 + X 2 + … + Xn and for t &gt; 0 define and Zt = SN (t)+1 – t. The limit distributions of N(t), Yt and Zt as t → ∞ are obtained when F 1 and F 2 are in the domains of attraction of stable laws with exponents α 1 and α 2 , respectively and Sn properly normalised has the composition of these two stable laws as its limit distribution.

scholarly journals Limit Distributions for the Extreme Order Statistics

1978 ◽  
Vol 21 (4) ◽  
pp. 447-459 ◽  
Author(s):  
D. Mejzler
Keyword(s):  
Distribution Function ◽  
Asymptotic Behaviour ◽  
Order Statistics ◽  
Random Variables ◽  
Independent Random Variables ◽  
Maximal Term ◽  
Limit Distributions ◽  
Extreme Order Statistics

Let Xl, …, Xn be independent random variables with the same distribution function (df) F(x) and let Xln≤X2n≤…≤nr be the corresponding order statistics. The (df) of Xkn will be denoted always by Fkn(x). Many authors have investigated the asymptotic behaviour of the maximal term Xnn as n → ∞. Gnedenko [3] proved the following

II.—Asymptotic Renewal Theorems

1953 ◽  
Vol 64 (1) ◽  
pp. 9-48 ◽  
Author(s):  
Walter L. Smith
Keyword(s):  
Distribution Function ◽  
Renewal Process ◽  
Age Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Renewal Theorem ◽  
Discrete Process ◽  
Main Emphasis ◽  
Image Position ◽  
Independent And Identically Distributed

SynopsisA sequence of non-negative random variables {Xi} is called a renewal process, and if the Xi may only take values on some sequence it is termed a discrete renewal process. The greatest k such that X1 + X2 + … + Xk ≤ x(> o) is a random variable N(x) and theorems concerning N(x) are renewal theorems. This paper is concerned with the proofs of a number of renewal theorems, the main emphasis being on processes which are not discrete. It is assumed throughout that the {Xi} are independent and identically distributed.If H(x) = Ɛ{N(x)} and K(x) is the distribution function of any non-negative random variable with mean K > o, then it is shown that for the non-discrete processwhere Ɛ{Xi} need not be finite; a similar result is proved for the discrete process. This general renewal theorem leads to a number of new results concerning the non-discrete process, including a discussion of the stationary “age-distribution” of “renewals” and a discussion of the variance of N(x). Lastly, conditions are established under whichThese new conditions are much weaker than those of previous theorems by Feller, Täcklind, and Cox and Smith.

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