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.

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 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.

scholarly journals Phase Enlargement of Semi-Markov Systems without Determining Stationary Distribution of Embedded Markov Chain

2020 ◽  
Vol 19 (3) ◽  
pp. 539-563
Author(s):  
Vadim Kopp ◽  
Mikhail  Zamoryonov ◽  
Nikita Chalenkov ◽  
Ivan Skatkov
Keyword(s):  
Distribution Function ◽  
Markov Chain ◽  
State Space ◽  
Stationary Distribution ◽  
Phase State ◽  
Random Variables ◽  
Embedded Markov Chain ◽  
Discrete State ◽  
Markov Systems ◽  
The Difference

A phase enlargement of semi-Markov systems that does not require determining stationary distribution of the embedded Markov chain is considered. Phase enlargement is an equivalent replacement of a semi-Markov system with a common phase state space by a system with a discrete state space.  Finding the stationary distribution of an embedded Markov chain for a system with a continuous phase state space is one of the most time-consuming and not always solvable stage, since in some cases it leads to a solution of integral equations with kernels containing sum and difference of variables. For such equations there is only a particular solution and there are no general solutions to date. For this purpose a lemma on a type of a distribution function of the difference of two random variables, provided that the first variable is greater than the subtracted variable, is used. It is shown that the type of the distribution function of difference of two random variables under the indicated condition depends on one constant, which is determined by a numerical method of solving the equation presented in the lemma. Based on the lemma, a theorem on the difference of a random variable and a complicated recovery flow is built up. The use of this method is demonstrated by the example of modeling a technical system consisting of two series-connected process cells, provided that both cells cannot fail simultaneously. The distribution functions of the system residence times in enlarged states, as well as in a subset of working and non-working states, are determined. The simulation results are compared by the considered and classical method proposed by V. Korolyuk, showed the complete coincidence of the sought quantities.

Criteria for the non-ergodicity of stochastic processes: application to the exponential back-off protocol

1987 ◽  
Vol 24 (02) ◽  
pp. 347-354 ◽  
Author(s):  
Guy Fayolle ◽  
Rudolph Iasnogorodski
Keyword(s):  
Stochastic Process ◽  
Markov Chain ◽  
Stochastic Processes ◽  
State Space ◽  
Discrete Time ◽  
Random Variables ◽  
Countable State Space ◽  
Countable State ◽  
New Criteria

In this paper, we present some simple new criteria for the non-ergodicity of a stochastic process (Yn ), n ≧ 0 in discrete time, when either the upward or downward jumps are majorized by i.i.d. random variables. This situation is encountered in many practical situations, where the (Yn ) are functionals of some Markov chain with countable state space. An application to the exponential back-off protocol is described.

A storage model in which the net growth-rate is a Markov chain

1972 ◽  
Vol 9 (01) ◽  
pp. 129-139 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Markov Chain ◽  
Continuous Time ◽  
Rate Of Change ◽  
Continuous Time Markov Chain ◽  
Finite Capacity ◽  
Storage Model ◽  
Finite State ◽  
Net Growth Rate ◽  
The Times ◽  
Finite State Space

The distribution of the times to first emptiness and first overflow, together with the limiting distribution of content are determined for a dam of finite capacity. It is assumed that the rate of change of the level of the dam is a continuous-time Markov chain with finite state-space (suitably modified when the dam is full or empty).

A storage model in which the net growth-rate is a Markov chain

1972 ◽  
Vol 9 (1) ◽  
pp. 129-139 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Markov Chain ◽  
Continuous Time ◽  
Rate Of Change ◽  
Continuous Time Markov Chain ◽  
Finite Capacity ◽  
Storage Model ◽  
Finite State ◽  
Net Growth Rate ◽  
The Times ◽  
Finite State Space

The distribution of the times to first emptiness and first overflow, together with the limiting distribution of content are determined for a dam of finite capacity. It is assumed that the rate of change of the level of the dam is a continuous-time Markov chain with finite state-space (suitably modified when the dam is full or empty).

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