A birth and death process with parameters λn = 0 for all n ≥ 0 and...
A birth and death process with parameters λn = 0 for
all n ≥ 0 and µn = µ
for all n ≥ 1 is called a pure death process. Find Pij (t) for this
process.
Homework Answers
The birth-death process is a special case of continuous time Markov process.
This is a birth-death process with λn = 0 and µn = µ. Thus, the Kolmogorov backward equations for Pij (t) is,
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A birth and death process with parameters λn = 0 for
all n ≥ 0 and...
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...
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)
3. Consider a birth and death process with birth rates Ai-(i 1)A, i 2 0, and death rates (a) Dete...
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 to go from state 2 to state 5.
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...
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...
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
Let A be an n × n real symmetric matrix with its row and column sums both equal to 0. Let λ1, . . . , λn be the eigenvalues of A, with λn = 0, and with corresponding eigenvectors v1,...,vn (these exis...
Let A be an n × n real symmetric matrix with its row and column sums both equal to 0. Let λ1, . . . , λn be the eigenvalues of A, with λn = 0, and with corresponding eigenvectors v1,...,vn (these exist because A is real symmetric). Note that vn = (1, . . . , 1). Let A[i] be the result of deleting the ith row and column. Prove that detA[i] = (λ1···λn-1)/n. Thus, the number of spanning...
1. Let {Xt;t >0} be a pure birth process with rate 1x > 0, for x...
1. Let {Xt;t >0} be a pure birth process with rate 1x > 0, for x € S = {0,1,2,...}. (a) Write the backward equations (KBE) and use it to solve for Prz(t). (b) Use the result to part (a) to show that the waiting time in state x, say Wx, is exponentially distributed (c) Suppose 1x = 1 is constant for all x E S. Prove by induction that Px-kx(t) = (at) ke Af/k! for k = 0,..., and...
Stochastic Processes 1. Let {Xt;t > 0} be a pure birth process with rate 1x >...
Stochastic Processes
1. Let {Xt;t > 0} be a pure birth process with rate 1x > 0, for x ES = {0,1,2,...}. (a) Write the backward equations (KBE) and use it to solve for Pxx(1).
Explain the following concepts: a) Explain the relation between a birth death process and a queui...
Explain the following concepts: a) Explain the relation between a birth death process and a queuing system b) Explain the notation of a queuing system c) Definc L, Ly,Lg and W,w,w d) Explain a birth dead process 1.
Explain the following concepts: a) Explain the relation between a birth death process and a queuing system b) Explain the notation of a queuing system c) Definc L, Ly,Lg and W,w,w d) Explain a birth dead process 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 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)
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 to go from state 2 to state 5.
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
1. Let {Xt;t >0} be a pure birth process with rate 1x > 0, for x € S = {0,1,2,...}. (a) Write the backward equations (KBE) and use it to solve for Prz(t). (b) Use the result to part (a) to show that the waiting time in state x, say Wx, is exponentially distributed (c) Suppose 1x = 1 is constant for all x E S. Prove by induction that Px-kx(t) = (at) ke Af/k! for k = 0,..., and...
Stochastic Processes
1. Let {Xt;t > 0} be a pure birth process with rate 1x > 0, for x ES = {0,1,2,...}. (a) Write the backward equations (KBE) and use it to solve for Pxx(1).
Explain the following concepts: a) Explain the relation between a birth death process and a queuing system b) Explain the notation of a queuing system c) Definc L, Ly,Lg and W,w,w d) Explain a birth dead process 1.
Explain the following concepts: a) Explain the relation between a birth death process and a queuing system b) Explain the notation of a queuing system c) Definc L, Ly,Lg and W,w,w d) Explain a birth dead process 1.
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