Download Homework 6 - Math 468/568, Spring 15 1. (from Lawler) The number

Homework 6 - Math 468/568, Spring 15
1. (from Lawler) The number of calls that come into an answering service
follows a Poisson process with a rate of 4 calls per hour.
(a) What is the probability that less than two calls come in during the first
hour?
(b) Suppose six calls arrive in the first hour. What is the probability that at
least two calls will arrive in the second hour?
(c) The person answering the phones waits until 15 calls have arrived to take
her lunch break. What is the mean time until the 15th call arrives?
(d) Suppose it is known that exactly eight calls arrived in the first two hours.
What is the probability that exactly five of them arrived in the first hour?
2.
(a) Let Xt and Yt be two independent Poisson processes with rates λ and µ.
Let Zt = Xt + Yt . Explain why Zt is also a Poisson process. You need not
prove your answer, just give a reasonable argument. What in the rate for
Zt ?
(b) Now let Zt be a Poisson process with rate λ. We have a coin with
probability p of heads. At each time that Zt increases, we flip the coing and
label the jump as heads or tails according to the outcome of the flip. Let Xt
be the number of heads jumps by time t, and Yt the number of tails jumps
by time t. So Zt = Xt + Y + t. Explain why Xt and Yt are Poisson processes
and give their rates.
3. Consider the continuous time
and infinitesimal generator

−3
 0
A=
 1
0
Markov chain with state space {1, 2, 3, 4}

1
1
1
−3 2
1 

2 −4 1 
0
1 −1
(a) Find the stationary or equilibrium distribution π.
(b) If we start in state 1, what is the expected value of the time when the
chain first changes state.
(c) If we start in state 1, what is the expected value of the time it takes to
reach state 4 for the first time?
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4. An irreducible continuous-time Markov P
chain on a finite state space has
rates α(x, y) for x 6= y. As always, α(x) = y6=x α(x, y). Let
p(x, y) =
α(x, y)
,
α(x)
x 6= y
and p(x, x) = 0 for all x. This is the transition matrix for a discrete time
Markov chain that corresponds to the continuous time chain “when it jumps”.
Let π be the stationary distribution for the continuous time chain. Find the
stationary distribution for the discrete time chain in terms of π and α.
5. (from Lawler) Consider the population model (example 3 in section 3.3
of Lawler). So it is a birth and death chain with λn = nλ and µn = nµ.
For which values of µ, λ is extinction certain, i.e. , when is the probability
of eventually reaching state 0 equal to 1?
6. (from Lawler) Consider the population model with immigration (example
4 in section 3.3 of Lawler). So it is a birth and death chain with λn = nλ + ν
and µn = nµ. For which values of µ, λ is the chain positive recurrent, null
recurrent, transient?
7. (from Lawler) Consider the birth and death process with λn = 1/(n + 1)
and µn = 1. Show the process is positive recurrent and find the stationary
distribution.
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