Question 4 (a) Assume that the probability of rain tomorrow is 0.5 if it is raining today, and assume that the probability of its being clear (no rain) tomorrow is 0.9 if it is clear today. Also assume that these probabilities do not change if information is also provided about the weather before today. (i) Explain why the stated assumptions imply that the Markovian property holds for the evolution of the weather. (ii) Formulate the evolution of the weather as a Markov chain by defining its states and giving its (one-step) transition matrix b) i. 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
(a) Assume that the
assume that the probability of its being clear (no rain) tomorrow is 0.9 if it is
clear today. Also assume that these probabilities do not change if information
is also provided about the weather before today.
(i) Explain why the stated assumptions imply that the Markovian property holds
for the evolution of the weather.
(ii) Formulate the evolution of the weather as a Markov chain by defining its
states and giving its (one-step) transition matrix
b) i. 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
d) If an M/D/1 queue has utilization of 80% do you expect its mean queue length
and response time to be less, same, or greater than that of an M/M/1?
Explain your answer
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