Exact Methods for the Transient Analysis of Nonhomogeneous Continuous Time Markov Chains

Rindos, Andy; Woolet, S.; Viniotis, I.; Trivedi, Kishor S.

doi:10.1007/978-1-4615-2241-6_8

Cited by 30 publications

(23 citation statements)

References 8 publications

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“…In a nonhomogeneous continuous time Poisson process with intensity function λ ( t ), the interarrival times of the events are functions of m ( t ) where

m ((), t) = \int_{0}^{t} λ ((), τ) italicdτ

. For example, T 1 and T 2 are the first and the second interarrival times with probability density functions:

f_{T_{1}} ((), t) = λ ((), t) e^{- m ((), t)},

f_{T_{2}} ((), t) = \int_{0}^{\infty} λ ((), τ) λ ((), t + τ) e^{- m ((), t + τ)} italicdτ .

Evidently, if a perturbation occurs in the different time instances in [0, t ], the change in the values of

f_{T_{1}} ((), t)

and

f_{T_{2}} ((), t)

will be different. Considering what happens in a nonhomogeneous continuous time Poisson process, it can be deduced that the system unavailability change at time t , using the nonhomogeneous continuous time MC, is a function of the perturbation occurrence time; this assertion is proved as follows.…”

Section: Considering Perturbation Time In Importance Analysismentioning

confidence: 99%

Importance analysis considering time‐varying parameters and different perturbation occurrence times

Noroozian

Kazemzadeh

Zio

et al. 2019

Quality & Reliability Eng

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Importance measures are integral parts of risk assessment for risk‐informed decision making. Because the parameters of a risk model, such as the component failure rates, are functions of time and a perturbation (change) in their values can occur during the mission time, time dependence must be considered in the evaluation of the importance measures. In this paper, it is shown that the change in system performance at time t, and consequently the importance of the parameters at time t, depends on the parameters perturbation time and their value functions during the system mission time. We consider a nonhomogeneous continuous time Markov model of a series‐parallel system to propose the mathematical proofs and simulations, while the ideas are also shown to be consistent with general models having nonexponential failure rates. Two new measures of importance and a simulation scheme for their computation are introduced to account for the effect of perturbation time and time‐varying parameters.

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“…In a nonhomogeneous continuous time Poisson process with intensity function λ ( t ), the interarrival times of the events are functions of m ( t ) where

m ((), t) = \int_{0}^{t} λ ((), τ) italicdτ

. For example, T 1 and T 2 are the first and the second interarrival times with probability density functions:

f_{T_{1}} ((), t) = λ ((), t) e^{- m ((), t)},

f_{T_{2}} ((), t) = \int_{0}^{\infty} λ ((), τ) λ ((), t + τ) e^{- m ((), t + τ)} italicdτ .

Evidently, if a perturbation occurs in the different time instances in [0, t ], the change in the values of

f_{T_{1}} ((), t)

and

f_{T_{2}} ((), t)

Section: Considering Perturbation Time In Importance Analysismentioning

confidence: 99%

Importance analysis considering time‐varying parameters and different perturbation occurrence times

Noroozian

Kazemzadeh

Zio

et al. 2019

Quality & Reliability Eng

View full text Add to dashboard Cite

show abstract

“…The uniqueness of Y (t) is a technical matter which we do not consider, because in the phylogenetic case there are extra restrictions which led to the unique solution (17). Considering again the general case, we write…”

Section: Generalized Pulley Principlementioning

confidence: 99%

Using the tangle: A consistent construction of phylogenetic distance matrices for quartets

Sumner

Jarvis

2006

Mathematical Biosciences

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Distance based algorithms are a common technique in the construction of phylogenetic trees from taxonomic sequence data. The first step in the implementation of these algorithms is the calculation of a pairwise distance matrix to give a measure of the evolutionary change between any pair of the extant taxa. A standard technique is to use the log det formula to construct pairwise distances from aligned sequence data. We review a distance measure valid for the most general models, and show how the log det formula can be used as an estimator thereof. We then show that the foundation upon which the log det formula is constructed can be generalized to produce a previously unknown estimator which improves the consistency of the distance matrices constructed from the log det formula. This distance estimator provides a consistent technique for constructing quartets from phylogenetic sequence data under the assumption of the most general Markov model of sequence evolution. * Australian Postgraduate Award † Alexander von Humboldt Fellow

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“…In the 1990's some work has been published on the computation of transient state probabilities for inhomogeneous Markovian models without rewards [67,58,68]. A more recent paper [62] characterizes the performability distribution in inhomogeneous MRMs through a coupled system of partial differential equations that is solved through discretization, and used to derive systems of ordinary differential equations to determine moments of accumulated reward.…”

Section: Computing Distributions and Expected Valuesmentioning

confidence: 99%

Model-based energy analysis of battery powered systems

Jongerden¹

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The use of mobile devices is often limited by the lifetime of the included batteries. This lifetime naturally depends on the battery's capacity and on the rate at which the battery is discharged. However, it also depends on the usage pattern, i.e., the workload, of the battery. When a battery is continuously discharged, a high current will cause it to provide less energy until the end of its lifetime than a lower current. This effect is termed the rate-capacity effect. On the other hand, during periods of low or no discharge current, the battery can recover to a certain extent. This effect is termed the recovery effect. In order to investigate the influence of the device workload on the battery lifetime a battery model is needed that includes the above described effects.Many different battery models have been developed for different application areas. We make a comparison of the main approaches that have been taken. Analytical models appear to be the best suited to use in combination with a device workload model, in particular, the so-called kinetic battery model. This model is combined with a continuous-time Markov chain, which models the workload of a battery powered device in a stochastic manner. For this model, we have developed algorithms to compute both the distribution and expected value of the battery lifetime and the charge delivered by the battery. These algorithms are used to make comparisons between different workloads, and can be used to analyse their impact on the system lifetime.In a system where multiple batteries can be connected, battery scheduling can be used to "spread" the workload over the individual batteries. Two approaches have been taken to find the optimal schedule for a given load. In the first approach scheduling decisions are only taken when a change in the workload occurs. The kinetic battery model is incorporated into a priced-timed automata model, and we use the model checking tool Uppaal Cora to find schedules that lead to the longest system lifetime.The second approach is an analytical one, in which scheduling decisions can be made at any point in time, that is, independently of workload changes. The analysis of the equations of the kinetic battery model provides an upper bound for the battery lifetime. This upper bound can be approached with any type v Abstract of scheduler, as long as one can switch fast enough. Both the approaches show that battery scheduling can potentially provide a considerable improvement of the system lifetime. The actual improvement mainly depends on the ratio between the battery capacity and the average discharge current.vi

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Exact Methods for the Transient Analysis of Nonhomogeneous Continuous Time Markov Chains

Cited by 30 publications

References 8 publications

Importance analysis considering time‐varying parameters and different perturbation occurrence times

Importance analysis considering time‐varying parameters and different perturbation occurrence times

Using the tangle: A consistent construction of phylogenetic distance matrices for quartets

Model-based energy analysis of battery powered systems

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