Homework Answers
birth rates:
b_i = (i+1)*b
death rates:
d_i = i*d
net rate=dr/di=birth rate-death rate=(i+1)*b-i*d
Integrating
=> r(i)=Integral((i+1)*b-i*d)di)
=[(b-d)*i^2/2 +bi] +c
a)r(4)-r(0)= 8(b-d)+4b=12b-8d
b)r(5)-r(2)=12.5*(b-d)+5b-2(b-d)-2b=13.5b-10.5d
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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) Dete...
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...
4. Consider a Birth and Death process with birth rate λ + 1 and death rate...
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175-3. Consider the birth-and-death process with the following mean rates. The birth rates are Ao-2, A1...
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
Consider a birth/death process with . - 2 customers per time unit, n = 1, 2,...
Consider a birth/death process with . - 2 customers per time unit, n = 1, 2, ..., 10 = 3 arrivals per time unit, 24 = 2.12 =1, and 2n=0, n = 3, 4, ... (note that, when there are 3 customers in the systems, the arrival rate is zero). (a) (5 points) Display the transition rate diagram, including the transition rates. (b) ) Calculate the steady state probabilities. (©) Calculate L, LG, W, W, (d) (5 points) Use Kendall...
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...
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...
1. During stage 2 of the demographic transition model death rates and birth rates remain high...
1. During stage 2 of the demographic transition model death rates and birth rates remain high death rates decline, while birth rates remain high the population is typically decreasing in size death and birth rates are increasing birth rates decline rapidly 2. The population in a country with a total fertility rate of 4, a low death rate and a immigration rate that equals the emigration rate is unpredictable increasing stable decreasing
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Consider a birth-and-death process with infinitesimal parameters Ae-5 for k> 0 and μk-15 for k 1....
Consider a birth-and-death process with infinitesimal parameters Ae-5 for k> 0 and μk-15 for k 1. 1. Derive the limiting distribution of X(t), as t-oo 2. Find the limiting expectation of X(t), as t → oo. 3. Find the limiting variance of X (t)
Problem 10: 10 points Consider a birth-and-death process with infinitesimal parameters, λ,-5 for k20 and A4k-15...
Problem 10: 10 points Consider a birth-and-death process with infinitesimal parameters, λ,-5 for k20 and A4k-15 for k21. 1. Derive the limiting distribution of X(t), as t → oo. 2. Find the limiting expectation of X(t), as t → oo. 3. Find the limiting variance of X(t)
Consider a Markov chain with state space S = {0, 1, 2, 3} and transition probability...
Consider a Markov chain with state space S = {0, 1, 2, 3} and
transition probability matrix
P=
(a) Starting from state 1, determine the mean time that the
process spends in each transient state 1 and 2, separately, prior
to absorption.
(b) Determine the mean time to absorption starting from state
1.
(c) Starting from state 1, determine the probability for the
process to be absorbed in state 0. Which state is it then more
likely for the process...
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
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
Consider a birth/death process with . - 2 customers per time unit, n = 1, 2, ..., 10 = 3 arrivals per time unit, 24 = 2.12 =1, and 2n=0, n = 3, 4, ... (note that, when there are 3 customers in the systems, the arrival rate is zero). (a) (5 points) Display the transition rate diagram, including the transition rates. (b) ) Calculate the steady state probabilities. (©) Calculate L, LG, W, W, (d) (5 points) Use Kendall...
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...
Consider a birth-and-death process with infinitesimal parameters Ae-5 for k> 0 and μk-15 for k 1. 1. Derive the limiting distribution of X(t), as t-oo 2. Find the limiting expectation of X(t), as t → oo. 3. Find the limiting variance of X (t)
Problem 10: 10 points Consider a birth-and-death process with infinitesimal parameters, λ,-5 for k20 and A4k-15 for k21. 1. Derive the limiting distribution of X(t), as t → oo. 2. Find the limiting expectation of X(t), as t → oo. 3. Find the limiting variance of X(t)
Consider a Markov chain with state space S = {0, 1, 2, 3} and
transition probability matrix
P=
(a) Starting from state 1, determine the mean time that the
process spends in each transient state 1 and 2, separately, prior
to absorption.
(b) Determine the mean time to absorption starting from state
1.
(c) Starting from state 1, determine the probability for the
process to be absorbed in state 0. Which state is it then more
likely for the process...
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