Markov renewal process

In probability and statistics a Markov renewal process is a random process that generalizes the notion of Markov jump processes. Other random processes like Markov chain, Poisson process, and renewal process can be derived as a special case of an MRP (Markov renewal process).

Definition

Consider a state space $\mathrm {S} .$ Consider a set of random variables $(X_{n},T_{n})$ , where $T_{n}$ are the jump times and $X_{n}$ are the associated states in the Markov chain (see Figure). Let the inter-arrival time, $\tau _{n}=T_{n}-T_{n-1}$ . Then the sequence $(X_{n},T_{n})$ is called a Markov renewal process if

{\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}}

Relation to other stochastic processes

If we define a new stochastic process $Y_{t}:=X_{n}$ for $t\in [T_{n},T_{n+1})$ , then the process $Y_{t}$ is called a semi-Markov process. Note the main difference between an MRP and a semi-Markov process is that the former is defined as a two-tuple of states and times, whereas the latter is the actual random process that evolves over time and any realisation of the process has a defined state for any given time. The entire process is not Markovian, i.e., memoryless, as happens in a continuous time Markov chain/process (CTMC). Instead the process is Markovian only at the specified jump instants. This is the rationale behind the name, Semi-Markov.^[1]^[2]^[3] (See also: hidden semi-Markov model.)
A semi-Markov process (defined in the above bullet point) where all the holding times are exponentially distributed is called a CTMC. In other words, if the inter-arrival times are exponentially distributed and if the waiting time in a state and the next state reached are independent, we have a CTMC.
${\begin{aligned}&\Pr(\tau _{n+1}\leq t,X_{n+1}=j\mid (X_{0},T_{0}),(X_{1},T_{1}),\ldots ,(X_{n}=i,T_{n}))\\[3pt]={}&\Pr(\tau _{n+1}\leq t,X_{n+1}=j\mid X_{n}=i)\\[3pt]={}&\Pr(X_{n+1}=j\mid X_{n}=i)(1-e^{-\lambda _{i}t}),{\text{ for all }}n\geq 1,t\geq 0,i,j\in \mathrm {S} ,i\neq j\end{aligned}}$
The sequence $X_{n}$ in the MRP is a discrete-time Markov chain. In other words, if the time variables are ignored in the MRP equation, we end up with a DTMC.
$\Pr(X_{n+1}=j\mid X_{0},X_{1},\ldots ,X_{n}=i)=\Pr(X_{n+1}=j\mid X_{n}=i)\,\forall n\geq 1,i,j\in \mathrm {S}$
If the sequence of $\tau$ s are independent and identically distributed, and if their distribution does not depend on the state $X_{n}$ , then the process is a renewal process. So, if the states are ignored and we have a chain of iid times, then we have a renewal process.
$\Pr(\tau _{n+1}\leq t\mid T_{0},T_{1},\ldots ,T_{n})=\Pr(\tau _{n+1}\leq t)\,\forall n\geq 1,\forall t\geq 0$

References and further reading

^ Medhi, J. (1982). Stochastic processes. New York: Wiley & Sons. ISBN 978-0-470-27000-4.
^ Ross, Sheldon M. (1999). Stochastic processes (2nd ed.). New York [u.a.]: Routledge. ISBN 978-0-471-12062-9.
^ Barbu, Vlad Stefan; Limnios, Nikolaos (2008). Semi-Markov chains and hidden semi-Markov models toward applications : their use in reliability and DNA analysis. New York: Springer. ISBN 978-0-387-73171-1.

[1] Medhi, J. (1982). Stochastic processes. New York: Wiley & Sons. ISBN 978-0-470-27000-4.

[2] Ross, Sheldon M. (1999). Stochastic processes (2nd ed.). New York [u.a.]: Routledge. ISBN 978-0-471-12062-9.

[3] Barbu, Vlad Stefan; Limnios, Nikolaos (2008). Semi-Markov chains and hidden semi-Markov models toward applications : their use in reliability and DNA analysis. New York: Springer. ISBN 978-0-387-73171-1.

[1]

[2]

[3]

Definition

Relation to other stochastic processes

See also

References and further reading