The probability density for value and distinct rates is a linear combination of exponentials for and zero for. Together, these parameters determine the overall shape of the probability density function PDF and, depending on their values, the PDF may be monotonic decreasing or unimodal. In addition, the tails of the PDF are "thin" in the sense that the PDF decreases exponentially rather than decreasing algebraically for large values of. This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution. While the foundations of Coxian distributions originate with the work of mathematician D. Cox in the s, much of the current corpus of knowledge was established through work on generalizations of hyperexponential distributions dating from the s.

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The probability density for value and distinct rates is a linear combination of exponentials for and zero for. Together, these parameters determine the overall shape of the probability density function PDF and, depending on their values, the PDF may be monotonic decreasing or unimodal.

In addition, the tails of the PDF are "thin" in the sense that the PDF decreases exponentially rather than decreasing algebraically for large values of. This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution. While the foundations of Coxian distributions originate with the work of mathematician D.

Cox in the s, much of the current corpus of knowledge was established through work on generalizations of hyperexponential distributions dating from the s. A number of real-world phenomena behave in a way naturally modeled by a Coxian distribution, including teletraffic in mobile cellular networks, durations of stay among patients in geriatric facilities, and queueing systems of various types.

RandomVariate can be used to give one or more machine- or arbitrary-precision the latter via the WorkingPrecision option pseudorandom variates from a Coxian distribution. The mean, median, variance, raw moments, and central moments may be computed using Mean , Median , Variance , Moment , and CentralMoment , respectively.

DistributionFitTest can be used to test if a given dataset is consistent with a Coxian distribution, EstimatedDistribution to estimate a Coxian parametric distribution from given data, and FindDistributionParameters to fit data to a Coxian distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Coxian distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Coxian distribution.

TransformedDistribution can be used to represent a transformed Coxian distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a Coxian distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Coxian distributions.

The Coxian distribution is related to a number of other distributions.

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## COXIAN DISTRIBUTION PDF

Details This is the distribution of the time to reach state 3 in a continuous-time Markov model with three states and transitions permitted from state 1 to state 2 with intensity lambda1 state 1 to state 3 intensity mu1 and state 2 to state 3 intensity mu2. States 1 and 2 are the two "phases" and state 3 is the "exit" state. Quantiles are calculated by numerically inverting the distribution function. Value d2phase gives the density, p2phase gives the distribution function, q2phase gives the quantile function, r2phase generates random deviates, and h2phase gives the hazard. This can be useful for choosing intuitively reasonable initial values for procedures to fit these models to data. This is the minimum hazard ratio for decreasing hazards.

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## Phase-type distribution

Zololl I have a suggestion. It is usually assumed the probability of process starting in the absorbing state is zero i. The actuar R package implements a general n-phase distribution defined by the time to absorption of a general continuous-time Markov chain with a single absorbing state, where the process starts in one of the transient states with a given probability. Modelling Techniques and Tools. Density, distribution, quantile functions and other utilities for the Coxian phase-type distribution with two phases.

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## CoxianDistribution

Tutilar Views Read Edit View history. The parameter of the phase-type distribution are: Performance Modeling and Design of Computer Systems. The Coxian distribution is a generalisation of the hypoexponential distribution. The actuar R package implements a general n-phase distribution defined by the time to absorption of a general continuous-time Markov chain with a single absorbing state, where the process starts in one of the transient states with a given probability.

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## COXIAN PHASE TYPE DISTRIBUTION PDF

Instead of only being able to enter the absorbing state from state k it can be reached from any phase. Sol Morales The hypoexponential distribution is a generalisation of the Erlang distribution by having different rates for each transition the non-homogeneous case. The Coxian distribution is extremely important as any acyclic phase-type distribution has an equivalent Coxian representation. Mathematics Stack Exchange works best with JavaScript enabled.