Question: Consider a Birth-Death process with birth rates {λn} and death rates {µn}, where µ0= 0....
Question:
Consider a Birth-Death process with birth rates {λn} and death rates {µn}, where µ0= 0.
Let Ti: the time, starting from state i, it takes for the process to enter i+1 for the first time, i ≥ 0.
Assume that it is allowed to go below i before reaching i+1. For i > 0, we have two scenarios:
• i -→ i +1,
• i -→ i -1
Denote the random variable 1i, to be the Indicator
Function, such that 1i=1 if i -→ i+1; and
1i=0 if i -→ i-1.
Compute E[Ti1(1i=1)] (notice
Ti multiplied by the indicator function) by giving
details. Hint: relate Ti to Tb;Td, where b and d stand for birth
and death.
Homework Answers
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Consider a Birth-Death process with birth rates {λn}
and death rates {µn}, where µ0= 0....
A birth and death process with parameters λn = 0 for all n ≥ 0 and...
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3. Consider a birth and death process with birth rates Ai-(i 1)A, i 2 0, and death rates (a) Determine the expected time to go from state 0 to state 4. (b) Determine the expected time...
175-3. Consider the birth-and-death process with the following mean rates. The birth rates are Ao-2, A1 3,A.: 2. A 3 1, and A,s() for " > 3. The death rates are μ.-3,Pc-4. μ.-1,and = 2 for n > 4. (a) Construct the rate diagram for this birth-and-death process. (h) Develop the balance equations. (c) Solve these equations to find the steady-state probability dis- (di Use the general formulas for the birth-and-death process to cal- Also calculate L. L W.and
4. Consider a Birth and Death process with birth rate λ + 1 and death rate μί i2 0. Show that the expected times to go from state i to i+1 satisfy ET = 1,Vi 0. for any
This homework deals with a birth and death process with a birth rate λ.-lit 1 and m = a 1. A plot of λί and μί are shown below. Give a one-sentence description of how the birth rate is a function of the state of the process, i. Do another one sentence description of the death rate. Then, explain how the birth and death rates will interact. That is, for what states should the process tend to grow, and grow...
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