Mathematics Poisson processes and Markov Chain problems

Question #1 Consider a coffee vending machine that breaks down according to a Poisson process with an

average of once a year. When it fails, it must complete a repair procedure consisting of four

phases (Phases 1,2,3 and 4 in that order). We assume that the time required to complete each

phase follows an exponential distribution with rate:

µ1 = 365 for phase 1 (one day)

µ2 = 52 for phase 2 (one week)

µ3 = 730 for phase 3 (half a day)

µ4 = 26 for phase 4 (two weeks)

a) Draw the graph of the Continuous-time Markov chain {X(t) t≥0} where

X(t) = 0, if the machine is working

X(t) = i, if the machine is in the I phase of repair (i being 1,2,3 or 4)

b) Write and solve the equations of equilibrium of the Markov chain

c) In what proportion of time is the machine shut down?

d) In what proportion of time is the machine functional?

Question #2 Data packets arrive at the switch of a computer system according to a Poisson process with an

average rate of 10 packets per second.

a) What is the probability that no packet arrives in a period of one second?

b) What is the probability that 8 or less packets arrive within a period of two seconds?

c) Note T the random variable giving the time elapsed until the arrival of the hundredth

package after 8:00, knowing that there was 553 packets between 7:59 and 8:00.

Calculate the Expected value of T.

Question #3 Consider a bus passing randomly at a stop where passengers are waiting. We assume that the

passengers arrive at the stop according to a Poisson process of rate λ = 17 passengers per hour.

Furthermore, it is assumed that the bus pass at the bus stop according to a Poisson process of

rate µ = 2 times per hour. The bus always has enough room to take all the passengers waiting at

the stop. We consider the Continuous-time Markov chain {X(t) t≥0} where X(t) denotes the

number of passengers waiting at the bus stop at a time t, measured in hours. We talk about a

process of birth and death. The transition probabilities are given by:

a) Draw the graph associated with this process

b) Write the equilibrium equations associated with this process and determine the limit

probabilities Pn , for n ≥ 1.

c) Determine the average number of passengers waiting at the bus stop.