Distribution phase-type

The fully developed fire is affected by (a) the size and shape of the enclosure, (b) the amount, distribution and type of fuel in the enclosure, (c) the amount, distribution and form of ventilation of the enclosure and (d) the form and type of construction materials comprising the roof (or ceiling), walls and floor of the enclosure. The significance of each phase of an enclosure fire depends on the fire safety system component under consideration. For components such as detectors or sprinklers, the fire development phase will have a great influence on the time at which they activate. The fully developed fire and its decay phase are significant for the integrity of the structural elements. 

The amide local anesthetics are widely distributed after intravenous bolus administration. There is also evidence that sequestration can occur in lipophilic storage sites (eg, fat). After an initial rapid distribution phase, which consists of uptake into highly perfused organs such as the brain, liver, kidney, and heart, a slower distribution phase occurs with uptake into moderately well-perfused tissues, such as muscle and the gastrointestinal tract. As a result of the extremely short plasma half-lives of the ester type agents, their tissue distribution has not been extensively studied. 

In the above configurations for interlayering and interlayer ion distribution, only type (c) represents an assemblage of two phases. The other forms are solid solutions. 

This section proposes the use of a semi-Markov model with Erlang- and phase-type retention-time distributions as a generic model for the kinetics of systems with inhomogeneous, poorly stirred compartments. These distributions are justified heuristically on the basis of their shape characteristics. The overall objective is to find nonexponential retention-time distributions that adequately describe the flow within a compartment (or pool). These distributions are then combined into a more mechanistic (or physiologically based) model that describes the pattern of drug distribution between compartments. The new semi-Markov model provides a generalized compartmental analysis that can be applied to compartments that are not well stirred. 

A more general yet tractable approach to semi-Markov models is the phase-type distribution developed by Neuts [363], who showed that any nondegenerate distribution / (a) of a retention time A with nonnegative support can be approximated, arbitrarily closely, by a distribution of phase type. Consequently, all semi-Markov models in the recent literature are special phase-type distribution models. However, the phase-type representation is not unique, and in any case it will be convenient to consider some restricted class of phase-type distributions. 

The phase-type distribution has an interpretation in terms of the compartmental model. Indeed, if the phenomenological compartment in the model, which is associated with a nonexponential retention-time distribution, is considered as consisting of a number of pseudocompartments (phases) with movement... 

Figure 9.6 Pseudocompartment configurations generating Erlang (A), generalized Erlang (B), and phase-type (C) distributions for retention times in phenomenological compartments. Retention times are distributed according to A Exp(Ai) and Ai Exp(A2).

The phase-type distributions are designed to serve as retention-time distributions in semi-Markov models. To obtain the equations of the model for a phenomenological compartmental configuration, one has to follow the following procedure ... 

Express the retention-time distribution for each phenomenological compartment by using phase-type distributions. However, the phase-type distributions for these sites are determined empirically. There is no assurance of finding the best phase-type distribution. This step leads to the expanded model involving pseudocompartments generating the desired phase-type distribution. 

For the resulting model with phase-type distributions, find the expanded... 

Erlang- and phase-type distributions provide a versatile class of distributions, and are shown to fit naturally into a Markovian compartmental system, where particles move between a series of compartments, so that phase-type compartmental retention-time distributions can be incorporated simply by increasing the size of the system. This class of distributions is sufficiently rich to allow for a wide range of behaviors, and at the same time offers computational convenience for data analysis. Such distributions have been used extensively in theoretical studies (e.g., [366]), because of their range of behavior, as well as in experimental work (e.g., [367]). Especially for compartmental models, the phase-type distributions were used by Faddy [364] and Matis [301,306] as examples of long-tailed distributions with high coefficients of variation. 

We propose the use of the phase-type distributions previously developed as retention-time distributions associated with the peripheral compartment. The... 

Equation (9.32) is a linear Fredholm integral equation of the first kind. It is also known as an unfolding or deconvolution equation. One can preanalyze the data and try to solve this first-kind integral equation. Besides the complexity of this equation, there is a paucity of numerical methods for determining the unknown function / (h) [208,379] with special emphasis on methods based on the principle of maximum entropy [207,380]. The so-obtained density function may be approximated by several models, gamma, Weibull, Erlang, etc., or by phase-type distributions. 

Faddy, M., Compartmental models with phase-type residence-time distributions, Applied Stochastic Models and Data Analysis, Vol. 6, 1990, pp. 121-127. 

In addition to the ion-clustered gel morphology and microcrystallinity, other structural features includes pore-size distribution, void type, compaction and hydrolysis resistance, capacity and charge density. The functional parameters of interest in this instance include permeability, diffusion coefficients, temperature-time, pressure, phase boundary solute concentrations, cell resistance, ionic fluxes, concentration profiles, membrane potentials, transference numbers, electroosmotic volume transfer and finally current efficiency. 

Detailed Analysis of Kinetic Energy Release Distributions for Type I Surfaces using Phase Space Theory. The model for the statistical phase space theory calculations(S) begins with Equation 1, where is the flux through the... 

Initial reaction rate, Vo 10, S- Volume fraction of dispersed phase, % Type of particle size distribution Average size of particles, D, pm... 

In the laboratory, very reproducible W/0 emulsions of monodispersed size distributions can be prepared when all the variables for emulsification are controlled. The variables for a beneh laboratory study are emul-sifier type and concentration, energy of mixing, time of mixing, method of mixing, volume fraetions of oil and water phases, type and viseosity of oil, quality of water, and temperature. The mixtiue is blended in specific vessels, usually with rest intervals to eontrol the rigidity of the film. The conditions are reprodueed from batch to batch. In real production this is not often the case. The immiseible phases are subject to variable high shear for 2-8 min in offshore production and 40-50 min in the oil sands extraetion process. Emulsion size distributions therefore vary with different systems. 

In summary, when both the liquid- and vapor-equilibrated transport modes occur in the membrane they are assumed to occur in parallel. In other words, there are two separate contiguous pathways through the membrane, one with liquid-filled channels and another that is a one-phase-type region with collapsed channels. To determine how much of the overall water flux is distributed between the two transport modes, the fraction of expanded channels is used. As a final note, at the limits of S = 1 and S = 0, Eqs. (5.17) and (5.18) or their effective property analogs collapse to the respective equations for the single transport mode, as expected. 

The RAM tool comprises several approximation and simplifications. An important issue to treat is the procedure to merge states with equal capacity into one state where the Markov property does not hold. The use of phase-type distributions is considered to improve the accuracy of the current modeling, see e.g. Neuts (1981). 

Here, i corresponds to the number of failed cells. The time required to reach k failed cells is now given by the time to reach state k when starting in state 0. This is known as a phase-type distribution (Neuts, 1981), but the inter-repair time distribution in this particular case is also known as a hypoexponential distribution ... 

Dohnal G., Meca M. Computational notes on Phase-type Distribution. In Proceedings of conference Risk, Quality and Reliability, CQR (2007), 33-38. 

ABSTRACT A k-out-of-n system G system whose components follow Weibull distributions is studied by using the approximation of these distributions for phase-type distributions. The model imder phase-type distributions is constructed. The survival function and the rate of occurrence of failures for the components and system are calculated. The case in which the system is replaced by a new and identical one when the system fails is considered by calculating the probabilities associated to the coimting process governing the replacements. The results are applied to a system under Weibull distributions, and we approximate these distributions for phase-type... 

The calculation of the rehabUity function for these systems when the components do not foUow exponential distributions has not been solved in a suitable way. The operations involving the rehabUity function in terms of the hfetime of the components make that the final expression of the reliabihty is in general unmanageable. When the components foUow phase-type distributions it has been solved in Perez-Ocon et al. (2006). 

The paper is organized as follows. In Section 2 we introduce the k-out-of-n G systems under phase-type distributions and calculate the Markov process governing the system. The lifetime and the rate of occurrence of the system are calculated. In Section 3 we consider a k-out-of-n G system governed by WeibuU distributions, approach these by phase-type distributions, and obtain numerical results. 

---