Limiting probabilities

Template:Welcome and expand The probability that a continuous-time Markov chain will be in state j at time t often converges to a limiting value which is independent of the intial state. We call this value Pj where Pj is equal to:

$\frac{({\lambda }_0)({\lambda }_1)({\lambda }_2)\cdot \cdot \cdot ({\lambda }_{n-1})}{({\mu }_1)({\mu }_2)\cdot \cdot \cdot ({\mu }_n)(1+\sum _{n=1}^{\mathrm{\infty }}\frac{({\lambda }_0)({\lambda }_1)({\lambda }_2)\cdot \cdot \cdot ({\lambda }_{n-1})}{({\mu }_1)({\mu }_2)\cdot \cdot \cdot ({\mu }_n)})},$

For a limiting probability to exist, it is necessary that

$\sum _{k=1}^{\mathrm{\infty }}\frac{({\lambda }_0)({\lambda }_1)({\lambda }_2)\cdot \cdot \cdot ({\lambda }_{n-1})}{({\mu }_1)({\mu }_2)\cdot \cdot \cdot ({\mu }_n)}<\mathrm{\infty },$

This condition may be shown to be sufficient.

We can determine the limiting probabilities for a birth and death process using these equations and equating the rate at which the process leaves a state with the rate at which it enters the state.