Rate modulation in dams and ruin problems

We consider a dam in which the release rate depends both on the state and some modulating process. Conditions for the existence of a limiting distribution are established in terms of an associated risk process. The case where the release rate is a product of the state and the modulating process is given special attention, and in particular explicit formulas are obtained for a finite state space Markov modulation.

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Rate modulation in dams and ruin problems

Journal of Applied Probability ◽

10.1017/s0021900200099940 ◽

1996 ◽

Vol 33 (02) ◽

pp. 523-535 ◽

Cited By ~ 5

Author(s):

Søren Asmussen ◽

Offer Kella

Keyword(s):

State Space ◽

Release Rate ◽

Risk Process ◽

The State ◽

Limiting Distribution ◽

Process Conditions ◽

Explicit Formulas ◽

Rate Modulation ◽

Finite State ◽

Finite State Space

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Aggregated Markov processes with negative exponential time interval omission

Advances in Applied Probability ◽

10.1017/s0001867800023144 ◽

1990 ◽

Vol 22 (04) ◽

pp. 802-830 ◽

Cited By ~ 3

Author(s):

Frank Ball

Keyword(s):

State Space ◽

Continuous Time ◽

Single Channel ◽

The State ◽

Time Interval ◽

Continuous Time Markov Chain ◽

Negative Exponential Distribution ◽

Finite State ◽

Negative Exponential ◽

Finite State Space

We consider a time reversible, continuous time Markov chain on a finite state space. The state space is partitioned into two sets, termed open and closed, and it is only possible to observe whether the process is in an open or a closed state. Further, short sojourns in either the open or closed states fail to be detected. We consider the situation when the length of minimal detectable sojourns follows a negative exponential distribution with mean μ–1. We show that the probability density function of observed open sojourns takes the form , where n is the size of the state space. We present a thorough asymptotic analysis of f O(t) as μ tends to infinity. We discuss the relevance of our results to the modelling of single channel records. We illustrate the theory with a numerical example.

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A Time-Inhomogeneous Prendiville Model with Failures and Repairs

Mathematics ◽

10.3390/math10020251 ◽

2022 ◽

Vol 10 (2) ◽

pp. 251

Author(s):

Virginia Giorno ◽

Amelia G. Nobile

Keyword(s):

Markov Chain ◽

State Space ◽

The State ◽

Random Time ◽

Death Process ◽

Inhomogeneous Markov Chain ◽

Finite State ◽

Intensity Functions ◽

Finite State Space ◽

Birth Death

We consider a time-inhomogeneous Markov chain with a finite state-space which models a system in which failures and repairs can occur at random time instants. The system starts from any state j (operating, F, R). Due to a failure, a transition from an operating state to F occurs after which a repair is required, so that a transition leads to the state R. Subsequently, there is a restore phase, after which the system restarts from one of the operating states. In particular, we assume that the intensity functions of failures, repairs and restores are proportional and that the birth-death process that models the system is a time-inhomogeneous Prendiville process.

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Aggregated Markov processes with negative exponential time interval omission

Advances in Applied Probability ◽

10.2307/1427563 ◽

1990 ◽

Vol 22 (4) ◽

pp. 802-830 ◽

Cited By ~ 6

Author(s):

Frank Ball

Keyword(s):

State Space ◽

Continuous Time ◽

Single Channel ◽

The State ◽

Time Interval ◽

Continuous Time Markov Chain ◽

Negative Exponential Distribution ◽

Finite State ◽

Negative Exponential ◽

Finite State Space

We consider a time reversible, continuous time Markov chain on a finite state space. The state space is partitioned into two sets, termed open and closed, and it is only possible to observe whether the process is in an open or a closed state. Further, short sojourns in either the open or closed states fail to be detected. We consider the situation when the length of minimal detectable sojourns follows a negative exponential distribution with mean μ–1. We show that the probability density function of observed open sojourns takes the form , where n is the size of the state space. We present a thorough asymptotic analysis of fO(t) as μ tends to infinity. We discuss the relevance of our results to the modelling of single channel records. We illustrate the theory with a numerical example.

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