Limiting probabilities

Please help develop and classify this resource

Learn how you can develop this resource to teach participants about Limiting probabilities. Choose a subject, education level, and resource type to classify the resource by. You may learn more about the topic to help you develop and classify the resource from:

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:

( λ 0 ) ( λ 1 ) ( λ 2 ) ⋅ ⋅ ⋅ ( λ n − 1 ) ( μ 1 ) ( μ 2 ) ⋅ ⋅ ⋅ ( μ n ) ( 1 + ∑ n = 1 ∞ ( λ 0 ) ( λ 1 ) ( λ 2 ) ⋅ ⋅ ⋅ ( λ n − 1 ) ( μ 1 ) ( μ 2 ) ⋅ ⋅ ⋅ ( μ n ) ) , {\displaystyle {\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}^{\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

∑ k = 1 ∞ ( λ 0 ) ( λ 1 ) ( λ 2 ) ⋅ ⋅ ⋅ ( λ n − 1 ) ( μ 1 ) ( μ 2 ) ⋅ ⋅ ⋅ ( μ n ) < ∞ , {\displaystyle \sum _{k=1}^{\infty }{\frac {(\lambda _{0})(\lambda _{1})(\lambda _{2})\cdot \cdot \cdot (\lambda _{n-1})}{(\mu _{1})(\mu _{2})\cdot \cdot \cdot (\mu _{n})}}<\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.