Birth - death processes part 2 Flashcards
(19 cards)
Do all birth-death processes have an equilibrium distribution?
No, not all birth-death processes have an equilibrium distribution.
What does Theorem 11.1 state about equilibrium distributions?
Theorem 11.1 states that if an irreducible continuous-time Markov chain has an invariant distribution, it is unique and the distribution converges to the equilibrium distribution as time approaches infinity.
Are all birth-death processes irreducible?
No, not all birth-death processes are irreducible (e.g., Poisson process).
What does irreducibility imply for equilibrium distributions?
If the process is irreducible, the equilibrium and invariant distributions coincide.
What is the equation to find the invariant distribution for a birth-death process?
Solving πQ = 0 gives π1 = λ0/μ1 * π0; πj = (λj-1…λ0)/(μj…μ1) * π0.
How is the invariant distribution related to the total probability?
For the invariant distribution to be a probability distribution, the sum of all probabilities must equal 1, i.e., Σπj = 1.
When does an invariant distribution exist?
An invariant distribution exists if the sum Σ (λj-1…λ0) / (μj…μ1) converges to a finite value.
What does it mean if the sum in the invariant distribution is divergent?
If the sum diverges, the invariant distribution and equilibrium distribution do not exist.
How does the probability of extinction relate to the equilibrium distribution?
For linear birth-death processes, if extinction happens with probability 1, then π = (1, 0, 0, …) is an equilibrium distribution.
What is the generating function G(s,t) for a linear birth-death process?
G(s,t) = E[s^N(t)] = Σs^n P(N(t)=n). For λ≠μ, G(s,t) = (μ(1-s) - (μ-λ)s * e^(-(λ-μ)t)) / (λ(1-s) - (μ-λ)s * e^(-(λ-μ)t)).
What is the probability of extinction for a linear birth-death process?
P(N(t)=0) gives the extinction probability, and as t → ∞, P(eventual extinction) = 1 if λ ≤ μ, or (μ/λ)^N(0) if λ > μ.
Under what condition does a linear birth-death process have an equilibrium distribution?
A linear birth-death process has an equilibrium distribution if λ ≤ μ. In this case, the equilibrium distribution is (1, 0, 0, …).
What happens if λ > μ in a linear birth-death process?
If λ > μ, the population size increases to infinity as t → ∞, and there is no equilibrium distribution.
What happens when a birth-death process has an equilibrium distribution?
When a birth-death process has an equilibrium distribution, it either becomes extinct or grows indefinitely, depending on the rates of birth and death.
When does a birth-death process with immigration have an equilibrium distribution?
A birth-death process with immigration has an equilibrium distribution if the invariant distribution exists.
Does a linear death process have an equilibrium distribution?
A linear death process has an equilibrium distribution if λ ≤ μ. The equilibrium distribution is (1, 0, 0, …).
When does a linear birth with immigration process have an equilibrium distribution?
A linear birth with immigration process has an equilibrium distribution if the invariant distribution exists.
Does an emigration process have an equilibrium distribution?
An emigration process has an equilibrium distribution if the invariant distribution exists.
Does a linear death with emigration process have an equilibrium distribution?
A linear death with emigration process has an equilibrium distribution if the invariant distribution exists.
