Birth - death processes part 1 Flashcards
(14 cards)
What is a birth-death process?
A continuous-time Markov process where transitions occur one step up (birth) or one step down (death) in the population size. Births and deaths happen independently and at different rates.
What is the state space of a birth-death process?
The state space is S = {0, 1, 2, …}, representing the size of the population at any time.
What is the rate of birth (λn) in a birth-death process?
λn is the birth rate at population size n, and it can depend on n.
What is the rate of death (μn) in a birth-death process?
μn is the death rate at population size n, and it can also depend on n. For n=0, μ0 = 0.
What is the rate at which the process leaves state n?
The rate at which the process leaves state n is the sum of the birth and death rates: qn = λn + μn.
What is the distribution of the time until the next event in a birth-death process?
The time until the next event (birth or death) is exponentially distributed with rate (λn + μn).
What is the probability of a birth when the process is in state n?
The probability of a birth when the process is in state n is λn / (λn + μn).
What is the probability of a death when the process is in state n?
The probability of a death when the process is in state n is μn / (λn + μn).
What is the transition matrix of the embedded jump chain in a birth-death process?
The transition matrix of the embedded jump chain has non-zero entries for moving from state n to n+1 (birth) with probability λn / (λn + μn), and from state n to n-1 (death) with probability μn / (λn + μn).
What is the generator matrix (Q) of a birth-death process?
The generator matrix Q for a birth-death process has diagonal entries of - (λn + μn) and off-diagonal entries of λn for a birth and μn for a death.
What is the transition probability for a birth in a small time interval h?
The transition probability for a birth from state n to n+1 in a small time interval h is p(n,n+1)(h) = λn h + o(h).
What is the transition probability for a death in a small time interval h?
The transition probability for a death from state n to n-1 in a small time interval h is p(n,n-1)(h) = μn h + o(h).
What does the generator matrix for a birth-death process look like?
The generator matrix Q for a birth-death process has entries: diagonal entries are - (λn + μn), off-diagonal entries are λn for births and μn for deaths.
What is the transition probability for no event occurring in a small time interval h?
The probability for no event in a small time interval h is p(n,n)(h) = 1 - (λn + μn)h + o(h).
