Hypoexponential distribution

In probability theory the hypoexponential distribution or the generalized Erlang distribution is a continuous distribution, that has found use in the same fields as the Erlang distribution, such as queueing theory, teletraffic engineering and more generally in stochastic processes. It is called the hypoexponetial distribution as it has a coefficient of variation less than one, compared to the hyper-exponential distribution which has coefficient of variation greater than one and the exponential distribution which has coefficient of variation of one. In probability theory the hypoexponential distribution or the generalized Erlang distribution is a continuous distribution, that has found use in the same fields as the Erlang distribution, such as queueing theory, teletraffic engineering and more generally in stochastic processes. It is called the hypoexponetial distribution as it has a coefficient of variation less than one, compared to the hyper-exponential distribution which has coefficient of variation greater than one and the exponential distribution which has coefficient of variation of one. The Erlang distribution is a series of k exponential distributions all with rate λ {displaystyle lambda } . The hypoexponential is a series of k exponential distributions each with their own rate λ i {displaystyle lambda _{i}} , the rate of the i t h {displaystyle i^{th}} exponential distribution. If we have k independently distributed exponential random variables X i {displaystyle {oldsymbol {X}}_{i}} , then the random variable, is hypoexponentially distributed. The hypoexponential has a minimum coefficient of variation of 1 / k {displaystyle 1/k} . As a result of the definition it is easier to consider this distribution as a special case of the phase-type distribution. The phase-type distribution is the time to absorption of a finite state Markov process. If we have a k+1 state process, where the first k states are transient and the state k+1 is an absorbing state, then the distribution of time from the start of the process until the absorbing state is reached is phase-type distributed. This becomes the hypoexponential if we start in the first 1 and move skip-free from state i to i+1 with rate λ i {displaystyle lambda _{i}} until state k transitions with rate λ k {displaystyle lambda _{k}} to the absorbing state k+1. This can be written in the form of a subgenerator matrix, For simplicity denote the above matrix Θ ≡ Θ ( λ 1 , … , λ k ) {displaystyle Theta equiv Theta (lambda _{1},dots ,lambda _{k})} . If the probability of starting in each of the k states is then H y p o ( λ 1 , … , λ k ) = P H ( α , Θ ) . {displaystyle Hypo(lambda _{1},dots ,lambda _{k})=PH({oldsymbol {alpha }},Theta ).} Where the distribution has two parameters ( λ 1 ≠ λ 2 {displaystyle lambda _{1} eq lambda _{2}} ) the explicit forms of the probability functions and the associated statistics are CDF: F ( x ) = 1 − λ 2 λ 2 − λ 1 e − λ 1 x + λ 1 λ 2 − λ 1 e − λ 2 x {displaystyle F(x)=1-{frac {lambda _{2}}{lambda _{2}-lambda _{1}}}e^{-lambda _{1}x}+{frac {lambda _{1}}{lambda _{2}-lambda _{1}}}e^{-lambda _{2}x}} PDF: f ( x ) = λ 1 λ 2 λ 1 − λ 2 ( e − x λ 2 − e − x λ 1 ) {displaystyle f(x)={frac {lambda _{1}lambda _{2}}{lambda _{1}-lambda _{2}}}(e^{-xlambda _{2}}-e^{-xlambda _{1}})}