Semi-Markov trial

A Semi-Markov Process (SMP), also known as a Markov Renewal Process , is a generalization of a Markov Process . In contrast to a Markov process, whose state changes take place at the same time intervals, the length of stay in a state is given by a further stochastic process.

definition

In the theory of stochastic processes , a semi-Markov process is given by a pair of processes . There is a Markov chain with state space and transition matrix (so-called controlling chain ). is a process that only depends on and . The distribution function is given by . ${\displaystyle Z}$ ${\displaystyle Z=(X,Y)}$ ${\displaystyle X}$ ${\displaystyle S}$ ${\displaystyle P}$ ${\displaystyle Y}$ ${\displaystyle Y(n)}$ ${\displaystyle r=X(n-1)}$ ${\displaystyle s=X(n)}$ ${\displaystyle F_{rs}}$

The semi-Markov process is then the process whose state is determined accordingly at the point in time from . The dwell time from to is then given by . ${\displaystyle Z}$ ${\displaystyle n}$ ${\displaystyle S}$ ${\displaystyle X(n)}$ ${\displaystyle X(n-1)}$ ${\displaystyle X(n)}$ ${\displaystyle Y(n)}$

properties

Since the properties depend on both the current state and the subsequent state , the Markov property is generally not fulfilled. Still, the process is a Markov process. This also explains the name Semi-Markov Process . ${\displaystyle Y}$ ${\displaystyle X(n-1)}$ ${\displaystyle X(n)}$ ${\displaystyle W(n)=(X(n),Y(n))}$

Applications

Systems in queuing theory , for example, have properties that cannot always be mapped using simple Markov processes . An example is the autocorrelation . To achieve this, semi-Markov processes are often used to model the arrival rates.

Individual evidence

1. ↑ Kempken, Sebastian: Modeling and verified analysis of time-correlated data traffic in the Internet VDI Verlag, Düsseldorf 2009, ISBN 978-3-18-380410-8