Question 4 Consider a birth death model with birth rates in states n = 0, 1, . . . and death rates 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 and state any condition needed to ensure that the equilibrium distribution exists. (c) i Now consider the M/M/1 queue with arrival rate λ and service rate µ and explain how it can be modelled as a birth death model. ii. For the M/M/1 queue derive the form of , 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 in states n = 0, 1, . . .
and death rates 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 and state any
condition needed to ensure that the equilibrium distribution exists.
(c) i Now consider the M/M/1 queue with arrival rate λ and service rate µ and
explain how it can
be modelled as a birth death model.
ii. For the M/M/1 queue derive the form of , 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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