Birds arrive at four bird feeders according to a Poisson process with rate one b
ID: 3200034 • Letter: B
Question
Birds arrive at four bird feeders according to a Poisson process with rate one bird per minute. If all feeders are occupied, an arriving bird leaves, but otherwise it occupies a feeder and eats for a time that has an exponential distribution with mean 1 min. Consider this as a Markov chain where a “state” is the number of occupied feeders. The rate diagram is given below. (Please show work)
(a) Explain the rates in the diagram.
(b) Find the generator G.
(c) If three feeders are occupied, what is the expected time until this changes?
(d) If all feeders are occupied, what is the probability that a bird arrives before a feeder becomes free?
three feeders are ocaupied, what is the expected time until this changes? if all feeders are ocoupied, what is the probability that a bird arrives before a feedExplanation / Answer
Nature of Birds Arriving : Poisson Process where P() is the Probability of expecting x rate (expected rate ) and u is the average rate, e = 2.71.
P(x; ) = (e-) (x) / x!
Nature of Birds eating : Exponential Process i.e. all birds eat for average time 1 minute and then leave.
Since both of the above process of arrving and eating are random, total occupied feeders are also random.
question part c :
Since the arrival average time = 1 min and average time spent = 1 min, we can say there sould be only one bird at a time on average. But since everthing is random, some of the time we have more birds and some of the time we have birds spending more than 1 min eating, therefore lets find the probability of finding 3 birds at a time.
Probability calculation : P(x; ) = (e-) (x) / x!
Average rate u = 1, expected rate x = 3;
Calculating P from above 6.15 %, that is we have this probabity of finding 3 birds togather.
Therefore only 6.15 % chances that the state would be stay as it is, rest it would change.
Time taken to switch the state = 1 min * 0.0615 = 0.0615 min that is nearly 3.6 seconds.
question part d:
Probability of arriving more than 4 birds.
using the same formula on commutative basis that is more than 4 birds.
Plugging u=1, x= 4, P = 0.37 %
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