Discrete phase-type distribution

<h2 id="definition">Definition</h2>
<p>A terminating Markov chain is a <a href="/facts/Markov_chain/5oP5hntk">Markov chain</a> where all states are transient, except one which is absorbing.
Reordering the states, the <a href="/facts/Transition_probability_matrix/2cRWAqC3">transition probability matrix</a> of a terminating Markov chain with 
  
    
      
        m
      
    
    {\displaystyle m}
  
 transient states is
</p>

P
        
        =
        
          [
          
            
              
                
                  
                    T
                  
                
                
                  
                    
                      T
                    
                    
                      0
                    
                  
                
              
              
                
                  
                    
                      0
                    
                    
                      
                        T
                      
                    
                  
                
                
                  1
                
              
            
          
          ]
        
        ,
      
    
    {\displaystyle {P}=\left[{\begin{matrix}{T}&\mathbf {T} ^{0}\\\mathbf {0} ^{\mathsf {T}}&1\end{matrix}}\right],}

<p>where 
  
    
      
        
          T
        
      
    
    {\displaystyle {T}}
  
 is a 
  
    
      
        m
        ×
        m
      
    
    {\displaystyle m\times m}
  
 matrix, 
  
    
      
        
          
            T
          
          
            0
          
        
      
    
    {\displaystyle \mathbf {T} ^{0}}
  
 and 
  
    
      
        
          0
        
      
    
    {\displaystyle \mathbf {0} }
  
 are column vectors with 
  
    
      
        m
      
    
    {\displaystyle m}
  
 entries, and 
  
    
      
        
          
            T
          
          
            0
          
        
        +
        
          T
        
        
          1
        
        =
        
          1
        
      
    
    {\displaystyle \mathbf {T} ^{0}+{T}\mathbf {1} =\mathbf {1} }
  
. The transition matrix is characterized entirely by its upper-left block 
  
    
      
        
          T
        
      
    
    {\displaystyle {T}}
  
.
</p><p>Definition. A distribution on 
  
    
      
        {
        0
        ,
        1
        ,
        2
        ,
        .
        .
        .
        }
      
    
    {\displaystyle \{0,1,2,...\}}
  
 is a discrete phase-type distribution if it is the distribution of the <a href="/facts/First_passage_time/smY5bheV">first passage time</a> to the absorbing state of a terminating Markov chain with finitely many states.
</p>
<h2 id="characterization">Characterization</h2>
<p>Fix a terminating Markov chain. Denote 
  
    
      
        
          T
        
      
    
    {\displaystyle {T}}
  
 the upper-left block of its transition matrix and 
  
    
      
        τ
      
    
    {\displaystyle \tau }
  
 the initial distribution.
The distribution of the first time to the absorbing state is denoted 
  
    
      
        
          
            P
            H
          
          
            d
          
        
        (
        
          τ
        
        ,
        
          T
        
        )
      
    
    {\displaystyle \mathrm {PH} _{d}({\boldsymbol {\tau }},{T})}
  
 or 
  
    
      
        
          D
          P
          H
        
        (
        
          τ
        
        ,
        
          T
        
        )
      
    
    {\displaystyle \mathrm {DPH} ({\boldsymbol {\tau }},{T})}
  
.
</p><p>Its cumulative distribution function is
</p>

F
        (
        k
        )
        =
        1
        −
        
          τ
        
        
          
            T
          
          
            k
          
        
        
          1
        
        ,
      
    
    {\displaystyle F(k)=1-{\boldsymbol {\tau }}{T}^{k}\mathbf {1} ,}

<p>for 
  
    
      
        k
        =
        1
        ,
        2
        ,
        .
        .
        .
      
    
    {\displaystyle k=1,2,...}
  
, and its density function is
</p>

f
        (
        k
        )
        =
        
          τ
        
        
          
            T
          
          
            k
            −
            1
          
        
        
          
            T
            
              0
            
          
        
        ,
      
    
    {\displaystyle f(k)={\boldsymbol {\tau }}{T}^{k-1}\mathbf {T^{0}} ,}

<p>for 
  
    
      
        k
        =
        1
        ,
        2
        ,
        .
        .
        .
      
    
    {\displaystyle k=1,2,...}
  
. It is assumed the probability of process starting in the absorbing state is zero. The <a href="/facts/Factorial_moment/RDIIdnd8">factorial moments</a> of the distribution function are given by,
</p>

E
        [
        K
        (
        K
        −
        1
        )
        .
        .
        .
        (
        K
        −
        n
        +
        1
        )
        ]
        =
        n
        !
        
          τ
        
        (
        I
        −
        
          T
        
        
          )
          
            −
            n
          
        
        
          
            T
          
          
            n
            −
            1
          
        
        
          1
        
        ,
      
    
    {\displaystyle E[K(K-1)...(K-n+1)]=n!{\boldsymbol {\tau }}(I-{T})^{-n}{T}^{n-1}\mathbf {1} ,}

<p>where 
  
    
      
        I
      
    
    {\displaystyle I}
  
 is the appropriate dimension <a href="/facts/Identity_matrix/Glbcf6ol">identity matrix</a>.
</p>
<h2 id="special-cases">Special cases</h2>
<p>Just as the continuous time distribution is a generalisation of the exponential distribution, the discrete time distribution is a generalisation of the geometric distribution, for example:
</p>
<ul><li><a href="/facts/Degenerate_distribution/Tj5b5h2z">Degenerate distribution</a>, point mass at zero or the empty phase-type distribution – 0 phases.</li>
<li><a href="/facts/Geometric_distribution/c8wjfaoV">Geometric distribution</a> – 1 phase.</li>
<li><a href="/facts/Negative_binomial_distribution/znzj365Y">Negative binomial distribution</a> – 2 or more identical phases in sequence.</li>
<li>Mixed Geometric distribution – 2 or more non-identical phases, that each have a probability of occurring in a mutually exclusive, or parallel, manner. This is the discrete analogue of the <a href="/facts/Hyperexponential_distribution/vxa9JHSY">Hyperexponential distribution</a>, but it is not called the <a href="/facts/Hypergeometric_distribution/g3mQdYwy">Hypergeometric distribution</a>, since that name is in use for an entirely different type of discrete distribution.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Phase-type_distribution/LGDbCMKh">Phase-type distribution</a></li>
<li><a href="/facts/Queueing_model/8NDl56EM">Queueing model</a></li>
<li><a href="/facts/Queueing_theory/8NDl56EM">Queueing theory</a></li></ul>

<ul><li>M. F. Neuts. Matrix-Geometric Solutions in Stochastic Models: an Algorithmic Approach, Chapter 2: Probability Distributions of Phase Type; Dover Publications Inc., 1981.</li>
<li>G. Latouche, V. Ramaswami. <i>Introduction to Matrix Analytic Methods in Stochastic Modelling</i>, 1st edition. Chapter 2: PH Distributions; ASA SIAM, 1999.</li></ul>

Discrete phase-type distribution open-in-new

Discrete phase-type distribution