Some exact results for dams with Markovian inputs

1976 ◽  
Vol 13 (02) ◽  
pp. 329-337
Author(s):  
Pyke Tin ◽  
R. M. Phatarfod
Keyword(s):  
Probability Distribution ◽  
Stationary Distribution ◽  
Special Form ◽  
Geometric Distribution ◽  
Transition Probabilities ◽  
Exact Results ◽  
Input Process ◽  
Largest Eigenvalue ◽  
The Matrix ◽  
Finite Dam

In the theory of dams with Markovian inputs explicit results are not usually obtained, as the theory depends very heavily on the largest eigenvalue of the matrix (pijzj ) where p ij are the transition probabilities of the input process. In this paper we show that explicit results can be obtained if one considers an input process of a special form. The probability distribution of the time to first emptiness is obtained for both the finite and the infinite dam, as well as the stationary distribution of the dam content for the finite dam. Explicit results are given for the case where the stationary distribution of the input process has a geometric distribution.

Some exact results for dams with Markovian inputs

1976 ◽  
Vol 13 (2) ◽  
pp. 329-337 ◽  
Author(s):  
Pyke Tin ◽  
R. M. Phatarfod
Keyword(s):  
Probability Distribution ◽  
Stationary Distribution ◽  
Special Form ◽  
Geometric Distribution ◽  
Transition Probabilities ◽  
Exact Results ◽  
Input Process ◽  
Largest Eigenvalue ◽  
The Matrix ◽  
Finite Dam

In the theory of dams with Markovian inputs explicit results are not usually obtained, as the theory depends very heavily on the largest eigenvalue of the matrix (pijzj) where pij are the transition probabilities of the input process. In this paper we show that explicit results can be obtained if one considers an input process of a special form. The probability distribution of the time to first emptiness is obtained for both the finite and the infinite dam, as well as the stationary distribution of the dam content for the finite dam. Explicit results are given for the case where the stationary distribution of the input process has a geometric distribution.

A Markov chain associated with the minimal quasi-stationary distribution of birth–death chains

1995 ◽  
Vol 32 (01) ◽  
pp. 25-38
Author(s):  
Servet Martínez ◽  
Maria Eulália Vares
Keyword(s):  
Markov Chain ◽  
Stationary Distribution ◽  
Conditional Distribution ◽  
Transition Probabilities ◽  
The Matrix ◽  
Quasi Stationary Distribution ◽  
Birth Death

We show that if the limiting conditional distribution for an absorbed birth–death chain exists, then the chain conditioned to non-absorption converges to a Markov chain with transition probabilities given by the matrix associated with the minimal quasi-stationary distribution.

On dams with Markovian inputs

1973 ◽  
Vol 10 (02) ◽  
pp. 317-329 ◽  
Author(s):  
A. G. Pakes
Keyword(s):  
Markov Chain ◽  
Generating Functions ◽  
Transition Probabilities ◽  
Heavy Traffic ◽  
General Situation ◽  
Input Process ◽  
Heavy Traffic Limit ◽  
Special Cases ◽  
The Matrix ◽  
Special Case

Some recent work on discrete time dam models has been concerned with special cases in which the input process is a Markov chain whose transition probabilities, p ij , are given by where A(·) and B(·) are probability generating functions (p.g.f.'s). In this paper we obtain some results for the general situation. The convergence norm of the matrix [p ij xj] is found and the results are used to obtain the p.g.f. of the first emptiness time. Distributions of the dam content are obtained and conditions are found for the existence of their limits. The p.g.f. of this distribution is so complicated that its identification in any special case is extremely difficult, or even impossible. Thus useful approximations are needed; we obtain a ‘heavy traffic’ limit theorem which suggests that under certain circumstances the limiting distribution can be approximated by an exponential distribution.

On dams with Markovian inputs

1973 ◽  
Vol 10 (2) ◽  
pp. 317-329 ◽  
Author(s):  
A. G. Pakes
Keyword(s):  
Markov Chain ◽  
Generating Functions ◽  
Transition Probabilities ◽  
Heavy Traffic ◽  
General Situation ◽  
Input Process ◽  
Heavy Traffic Limit ◽  
Special Cases ◽  
The Matrix ◽  
Special Case

Some recent work on discrete time dam models has been concerned with special cases in which the input process is a Markov chain whose transition probabilities, pij, are given by where A(·) and B(·) are probability generating functions (p.g.f.'s). In this paper we obtain some results for the general situation. The convergence norm of the matrix [pijxj] is found and the results are used to obtain the p.g.f. of the first emptiness time. Distributions of the dam content are obtained and conditions are found for the existence of their limits. The p.g.f. of this distribution is so complicated that its identification in any special case is extremely difficult, or even impossible. Thus useful approximations are needed; we obtain a ‘heavy traffic’ limit theorem which suggests that under certain circumstances the limiting distribution can be approximated by an exponential distribution.

A Markov chain associated with the minimal quasi-stationary distribution of birth–death chains

1995 ◽  
Vol 32 (1) ◽  
pp. 25-38 ◽  
Author(s):  
Servet Martínez ◽  
Maria Eulália Vares
Keyword(s):  
Markov Chain ◽  
Stationary Distribution ◽  
Conditional Distribution ◽  
Transition Probabilities ◽  
The Matrix ◽  
Quasi Stationary Distribution ◽  
Birth Death

We show that if the limiting conditional distribution for an absorbed birth–death chain exists, then the chain conditioned to non-absorption converges to a Markov chain with transition probabilities given by the matrix associated with the minimal quasi-stationary distribution.

Perturbation theory and finite Markov chains

1968 ◽  
Vol 5 (2) ◽  
pp. 401-413 ◽  
Author(s):  
Paul J. Schweitzer
Keyword(s):  
Perturbation Theory ◽  
Markov Chain ◽  
Markov Chains ◽  
Stationary Distribution ◽  
Fundamental Matrix ◽  
Transition Probabilities ◽  
Semi Group ◽  
Partial Derivatives ◽  
Finite Markov Chains ◽  
Derivatives Of

A perturbation formalism is presented which shows how the stationary distribution and fundamental matrix of a Markov chain containing a single irreducible set of states change as the transition probabilities vary. Expressions are given for the partial derivatives of the stationary distribution and fundamental matrix with respect to the transition probabilities. Semi-group properties of the generators of transformations from one Markov chain to another are investigated. It is shown that a perturbation formalism exists in the multiple subchain case if and only if the change in the transition probabilities does not alter the number of, or intermix the various subchains. The formalism is presented when this condition is satisfied.

The reverse Galton-Watson process

1975 ◽  
Vol 12 (03) ◽  
pp. 574-580 ◽  
Author(s):  
Warren W. Esty
Keyword(s):  
Markov Process ◽  
Stationary Distribution ◽  
Probability Function ◽  
Transition Probabilities ◽  
Initial State ◽  
Reverse Process ◽  
Watson Process ◽  
Step Transition

Consider the following path, Zn (w), of a Galton-Watson process in reverse. The probabilities that ZN–n = j given ZN = i converge, as N → ∞ to a probability function of a Markov process, Xn , which I call the ‘reverse process’. If the initial state is 0, I require that the transition probabilities be the limits given not only ZN = 0 but also ZN –1 > 0. This corresponds to looking at a Galton-Watson process just prior to extinction. This paper gives the n-step transition probabilities for the reverse process, a stationary distribution if m ≠ 1, and a limit law for Xn/n if m = 1 and σ 2 < ∞. Two related results about Zcn, 0 < c < 1, for Galton-Watson processes conclude the paper.

A comparison of convergence rates for three models in the theory of dams

1997 ◽  
Vol 34 (01) ◽  
pp. 74-83
Author(s):  
Robert Lund ◽  
Walter Smith
Keyword(s):  
Convergence Rate ◽  
Convergence Rates ◽  
Sample Path ◽  
Jump Process ◽  
Limiting Distribution ◽  
Regenerative Processes ◽  
Input Process ◽  
The Moment ◽  
Finite Dam ◽  
General Conditions

This paper compares the convergence rate properties of three storage models (dams) driven by time-homogeneous jump process input: the infinitely high dam, the finite dam, and the infinitely deep dam. We show that the convergence rate of the infinitely high dam depends on the moment properties of the input process, the finite dam always approaches its limiting distribution exponentially fast, and the infinitely deep dam approaches its limiting distribution exponentially fast under very general conditions. Our methods make use of rate results for regenerative processes and several sample path orderings.

Stationary distributions for dams with additive input and content-dependent release rate

1977 ◽  
Vol 9 (03) ◽  
pp. 645-663 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Continuous Function ◽  
Probability Measure ◽  
Stationary Distribution ◽  
Release Rate ◽  
Zero Set ◽  
Stationary Distributions ◽  
Special Cases ◽  
Necessary And Sufficient ◽  
Limiting Case ◽  
Finite Dam

Conditions are derived under which a probability measure on the Borel subsets of [0, ∞) is a stationary distribution for the content {Xt } of an infinite dam whose cumulative input {At } is a pure-jump Lévy process and whose release rate is a non-decreasing continuous function r(·) of the content. The conditions are used to find stationary distributions in a number of special cases, in particular when and when r(x) = x α and {A t } is stable with index β ∊ (0, 1). In general if EAt , < ∞ and r(0 +) > 0 it is shown that the condition sup r(x)>EA 1 is necessary and sufficient for a stationary distribution to exist, a stationary distribution being found explicitly when the conditions are satisfied. If sup r(x)>EA 1 it is shown that there is at most one stationary distribution and that if there is one then it is the limiting distribution of {Xt } as t → ∞. For {At } stable with index β and r(x) = x α , α + β = 1, we show also that complementing results of Brockwell and Chung for the zero-set of {Xt } in the cases α + β < 1 and α + β > 1. We conclude with a brief treatment of the finite dam, regarded as a limiting case of infinite dams with suitably chosen release functions.

scholarly journals On Truncation of the Matrix-Geometric Stationary Distributions

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 798 ◽  
Author(s):  
Naumov ◽  
Gaidamaka ◽  
Samouylov
Keyword(s):  
Finite Number ◽  
Stationary Distribution ◽  
Queueing Systems ◽  
Death Process ◽  
Birth And Death Process ◽  
Loss System ◽  
Stationary Distributions ◽  
Open Network ◽  
The Matrix

In this paper, we study queueing systems with an infinite and finite number of waiting places that can be modeled by a Quasi-Birth-and-Death process. We derive the conditions under which the stationary distribution for a loss system is a truncation of the stationary distribution of the Quasi-Birth-and-Death process and obtain the stationary distributions of both processes. We apply the obtained results to the analysis of a semi-open network in which a customer from an external queue replaces a customer leaving the system at the node from which the latter departed.

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