A Note on the First Emptiness Problem of a Finite Dam with Poisson Type Inputs

1969 ◽  
Vol 6 (1) ◽  
pp. 227-230 ◽  
Author(s):  
R. M. Phatarfod
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Finite Capacity ◽  
Small Interval ◽  
Continuous Release ◽  
No Release ◽  
Emptiness Problem ◽  
Image Position ◽  
Finite Dam ◽  
Poisson Type

This paper is concerned with the problem of first emptiness in a continuous time dam model formulated by Gani and Prabhu (1959) based on Moran's (1954) discrete time dam model. Briefly the dam model is as follows: The dam is of finite capacity K, whose content 0 ≦ Z(t) ≦ K is defined in continuous time t (0 ≦ t < ∞) by the equation where ηδt is the time the dam is empty in (t, t + δt). X(t) represents the input into the dam during time t, a Poisson process with parameter λ, such that in a small interval of time δt, the quantity δX(t) = 0 or h (< K) may be added to the dam content; min{Z(t) + δX(t),K} indicates that there will be an overflow whenever Z(t) + δX(t) > K, leaving only the amount K in the dam, and (1-η)δt represents a continuous release occurring at a steady unit rate except when z(t) = 0, when there is no release.

A Note on the First Emptiness Problem of a Finite Dam with Poisson Type Inputs

1969 ◽  
Vol 6 (01) ◽  
pp. 227-230 ◽  
Author(s):  
R. M. Phatarfod
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Finite Capacity ◽  
Small Interval ◽  
Continuous Release ◽  
No Release ◽  
Emptiness Problem ◽  
Image Position ◽  
Finite Dam ◽  
Poisson Type

This paper is concerned with the problem of first emptiness in a continuous time dam model formulated by Gani and Prabhu (1959) based on Moran's (1954) discrete time dam model. Briefly the dam model is as follows: The dam is of finite capacity K, whose content 0 ≦ Z(t) ≦ K is defined in continuous time t (0 ≦ t &lt; ∞) by the equation where ηδt is the time the dam is empty in (t, t + δt). X(t) represents the input into the dam during time t, a Poisson process with parameter λ, such that in a small interval of time δt, the quantity δX(t) = 0 or h (&lt; K) may be added to the dam content; min{Z(t) + δX(t),K} indicates that there will be an overflow whenever Z(t) + δX(t) &gt; K, leaving only the amount K in the dam, and (1-η)δt represents a continuous release occurring at a steady unit rate except when z(t) = 0, when there is no release.

Equivalence of two absorption problems with Markovian transitions and continuous or discrete time parameters

1959 ◽  
Vol 55 (2) ◽  
pp. 177-180 ◽  
Author(s):  
R. A. Sack
Keyword(s):  
Finite Number ◽  
Markov Processes ◽  
Discrete Time ◽  
Continuous Time ◽  
Time Parameter ◽  
Rate Of Change ◽  
Matrix Form ◽  
Constant Matrix ◽  
Time Parameters ◽  
Image Position

1. Introduction. Ledermann(1) has treated the problem of calculating the asymptotic probabilities that a system will be found in any one of a finite number N of possible states if transitions between these states occur as Markov processes with a continuous time parameter t. If we denote by pi(t) the probability that at time t the system is in the ith state and by aij ( ≥ 0) the constant probability per unit time for transitions from the jth to the ith state, the rate of change of pi is given bywhere the sum is to be taken over all j ≠ i. This set of equations can be written in matrix form aswhere P(t) is the vector with components pi(t) and the constant matrix A has elements

A central limit theorem for processes defined on a finite Markov chain

1964 ◽  
Vol 60 (3) ◽  
pp. 547-567 ◽  
Author(s):  
J. Keilson ◽  
D. M. G. Wishart
Keyword(s):  
Markov Chain ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Discrete Time ◽  
Continuous Time ◽  
Probability Measures ◽  
Finite Markov Chain ◽  
Marginal Process ◽  
Vector Distribution ◽  
Image Position

We shall be concerned in this paper with a class of temporally homogeneous Markov processes, {R(t), X(t)}, in discrete or continuous time taking values in the spaceThe marginal process {X(t)} in discrete time is, in the terminology of Miller (10), a sequence of random variables defined on a finite Markov chain. Probability measures associated with these processes are vectors of the formwhereWe shall call a vector of the form of (0·2) a vector distribution.

scholarly journals On continuous time models in the theory of dams

1963 ◽  
Vol 3 (4) ◽  
pp. 480-487 ◽  
Author(s):  
J. F. C. Kingman
Keyword(s):  
Stochastic Process ◽  
Discrete Time ◽  
Continuous Time ◽  
Continuous Release ◽  
Time Problem ◽  
Continuous Time Stochastic Process ◽  
Realistic Formulation ◽  
Number Of Authors ◽  
Continuous Time Models

The problem of storage in an infinite dam with a continuous release has been studied by a number of authors ([5], [3], [2]), who have formulated it in probabilistic terms by supposing the input to be a continuous time stochastic process. These authors have encountered difficulties which they have overcome by regarding the continuous time problem as a limit of discrete time analogues. analogues. The purpose of this paper is to suggest that these difficulties are the result of an unfortunate specification of the problem, and to show that the adoption of a slightly different (and more realistic) formulation avoids the difficulties and allows a treatment which does not have recourse to discrete time analogues.

Design of Programmable Wideband Low Pass Filter with Continuous-Time/Discrete-Time Hybrid Architecture

2017 ◽  
Vol E100.C (10) ◽  
pp. 858-865 ◽  
Author(s):  
Yohei MORISHITA ◽  
Koichi MIZUNO ◽  
Junji SATO ◽  
Koji TAKINAMI ◽  
Kazuaki TAKAHASHI
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Hybrid Architecture ◽  
Pass Filter ◽  
Low Pass Filter ◽  
Low Pass

scholarly journals Time to Intervene: A Continuous-Time Approach to Network Analysis and Centrality

Psychometrika ◽  
2021 ◽  
Author(s):  
Oisín Ryan ◽  
Ellen L. Hamaker
Keyword(s):  
Network Analysis ◽  
Discrete Time ◽  
Continuous Time ◽  
Centrality Measures ◽  
Time Interval ◽  
Direct Effects ◽  
Network Approach ◽  
Var Models ◽  
Auto Regressive ◽  
Conceptual Shift

AbstractNetwork analysis of ESM data has become popular in clinical psychology. In this approach, discrete-time (DT) vector auto-regressive (VAR) models define the network structure with centrality measures used to identify intervention targets. However, VAR models suffer from time-interval dependency. Continuous-time (CT) models have been suggested as an alternative but require a conceptual shift, implying that DT-VAR parameters reflect total rather than direct effects. In this paper, we propose and illustrate a CT network approach using CT-VAR models. We define a new network representation and develop centrality measures which inform intervention targeting. This methodology is illustrated with an ESM dataset.

On quasi-stationary distributions in absorbing continuous-time finite Markov chains

1967 ◽  
Vol 4 (1) ◽  
pp. 192-196 ◽  
Author(s):  
J. N. Darroch ◽  
E. Seneta
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Continuous Time ◽  
Present Note ◽  
Time Parameter ◽  
Stationary Distributions ◽  
Finite State ◽  
Absorbing Markov Chains ◽  
Finite Markov Chains ◽  
Finite State Space

In a recent paper, the authors have discussed the concept of quasi-stationary distributions for absorbing Markov chains having a finite state space, with the further restriction of discrete time. The purpose of the present note is to summarize the analogous results when the time parameter is continuous.

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