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Question 4 Consider a birth death model with birth rates 2, in states n = 0, 1, ... and death rates u in states n = 1, 2, .. (a) Give the detailed balance conditions for the equilibrium probability of being in state n, for n= 0, 1, ... and explain how they are derived. (b) Using the detailed balance conditions derive an expression p, and state any condition needed to ensure that the equilibrium distribution exists. (c) i Now consider the M/M/1 queue with arrival rate A and service rate u and explain how it can be modelled as a birth death model. ii. For the M/M/1 queue derive the form of p, , the equilibrium distribution of the number of customers present, and give any conditions needed to ensure the existence of the equilibrium distribution. Derive the mean value of the equilibrium distribution

Question 4 Consider a birth death model with birth rates 2, in states n = 0, 1, ... and death rates u in states n = 1, 2, .. (a) Give the detailed balance conditions for the equilibrium probability of being in state n, for n= 0, 1, ... and explain how they are derived. (b) Using the detailed balance conditions derive an expression p, and state any condition needed to ensure that the equilibrium distribution exists. (c) i Now consider the M/M/1 queue with arrival rate A and service rate u and explain how it can be modelled as a birth death model. ii. For the M/M/1 queue derive the form of p, , the equilibrium distribution of the number of customers present, and give any conditions needed to ensure the existence of the equilibrium distribution. Derive the mean value of the equilibrium distribution

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Question 4 Consider a birth death model with birth rates 2, in states n = 0, 1, ... and death rates u in states n = 1, 2, .. (a) Give the detailed balance conditions for the equilibrium probability of being in state n, for n= 0, 1, ... and explain how they are derived. (b) Using the detailed balance conditions derive an expression p, and state any condition needed to ensure that the equilibrium distribution exists. (c) i Now consider the M/M/1 queue with arrival rate l and service rate u and explain how it can be modelled as a birth death model. ii. For the M/M/1 queue derive the form of p,, the equilibrium distribution of the number of customers present, and give any conditions needed to ensure the existence of the equilibrium distribution. Derive the mean value of the equilibrium distribution
Transcribed Image Text:Question 4 Consider a birth death model with birth rates 2, in states n = 0, 1, ... and death rates u in states n = 1, 2, .. (a) Give the detailed balance conditions for the equilibrium probability of being in state n, for n= 0, 1, ... and explain how they are derived. (b) Using the detailed balance conditions derive an expression p, and state any condition needed to ensure that the equilibrium distribution exists. (c) i Now consider the M/M/1 queue with arrival rate l and service rate u and explain how it can be modelled as a birth death model. ii. For the M/M/1 queue derive the form of p,, the equilibrium distribution of the number of customers present, and give any conditions needed to ensure the existence of the equilibrium distribution. Derive the mean value of the equilibrium distribution
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