Markov renewal processes are a class of random processes in probability and statistics that generalize the class of Markov jump processes. Other classes of random processes, such as Markov chains and Poisson processes, can be derived as special cases among the class of Markov renewal processes, while Markov renewal processes are special cases among the more general class of renewal processes.
 In the context of a jump process that takes states in a state space 
  
    
      
        
          S
        
      
    
    {\displaystyle \mathrm {S} }
  
, consider the set of random variables 
  
    
      
        (
        
          X
          
            n
          
        
        ,
        
          T
          
            n
          
        
        )
      
    
    {\displaystyle (X_{n},T_{n})}
  
, where 
  
    
      
        
          T
          
            n
          
        
      
    
    {\displaystyle T_{n}}
  
 represents the jump times and 
  
    
      
        
          X
          
            n
          
        
      
    
    {\displaystyle X_{n}}
  
 represents the associated states in the sequence of states (see Figure). Let the sequence of inter-arrival times 
  
    
      
        
          τ
          
            n
          
        
        =
        
          T
          
            n
          
        
        −
        
          T
          
            n
            −
            1
          
        
      
    
    {\displaystyle \tau _{n}=T_{n}-T_{n-1}}
  
. In order for the sequence 
  
    
      
        (
        
          X
          
            n
          
        
        ,
        
          T
          
            n
          
        
        )
      
    
    {\displaystyle (X_{n},T_{n})}
  
 to be considered a Markov renewal process the following condition should hold:
 
  
    
      
        
          
            
              
              
                
                Pr
                (
                
                  τ
                  
                    n
                    +
                    1
                  
                
                ≤
                t
                ,
                
                  X
                  
                    n
                    +
                    1
                  
                
                =
                j
                ∣
                (
                
                  X
                  
                    0
                  
                
                ,
                
                  T
                  
                    0
                  
                
                )
                ,
                (
                
                  X
                  
                    1
                  
                
                ,
                
                  T
                  
                    1
                  
                
                )
                ,
                …
                ,
                (
                
                  X
                  
                    n
                  
                
                =
                i
                ,
                
                  T
                  
                    n
                  
                
                )
                )
              
            
            
              
                =
                

                
              
              
                
                Pr
                (
                
                  τ
                  
                    n
                    +
                    1
                  
                
                ≤
                t
                ,
                
                  X
                  
                    n
                    +
                    1
                  
                
                =
                j
                ∣
                
                  X
                  
                    n
                  
                
                =
                i
                )
                
                ∀
                n
                ≥
                1
                ,
                t
                ≥
                0
                ,
                i
                ,
                j
                ∈
                
                  S
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}&\Pr(\tau _{n+1}\leq t,X_{n+1}=j\mid (X_{0},T_{0}),(X_{1},T_{1}),\ldots ,(X_{n}=i,T_{n}))\\[5pt]={}&\Pr(\tau _{n+1}\leq t,X_{n+1}=j\mid X_{n}=i)\,\forall n\geq 1,t\geq 0,i,j\in \mathrm {S} \end{aligned}}}