MATH 5040/6810: Homework 5
Due on Thursday, Dec. 1st, by the end of lecture.
Problem 1 Consider a birth and death chain (continuous time) with immigration, i.e. the
birth rate λn = nλ + ν and death rate µ. The rate ν is the rate that outsiders come in.
(a) Find conditions on λ, ν, µ that guarantee that the population will never go extinct.
(b) Find conditions on λ, ν, µ that guarantee that the population will eventually go
extinct (and when new immigrants come in they find a ghost town!)
(c) Find conditions on λ, ν, µ so that the chain is null recurrent.
Problem 2 Let Xt be a Poisson processes with arrival rate µ, that model the arrivals of
customers at a certain mall that has only two stores Xanadu and Yliaster. After a customer
arrives at the mall, with probability p they go to Xanadu and with probability 1 − p, they
go to Yliaster.
(1) What is the probability that a customer visits Xanadu before a customer visits
Yliaster?
(2) What is the probability that in the first two hours a total of 7 customers arrived at
the two stores. For this part, assume that time is measured in quarters of an hour
(so we expect µ customers per 15 minutes.)
(3) Given that exactly 5 customers arrived at the two stores, what is the probability
that they all went to Yliaster?
(4) Assume T is the time when the first customer arrived in Xanadu. Find P{YT = k},
i.e. the probability that k customers arrived in Yliaster by the time one customer
arrived in Xanadu.
(5) What is the probability that k customers arrived at the mall, if j of them went to
Xanadu?
Problem 3 Let Xt a continuous time birth and death process with death rate µ = 1.
Decide whether the chain is transient, positive recurrent or null recurrent when:
(1)
(2)
(3)
(4)
The
The
The
The
birth
birth
birth
birth
rate
rate
rate
rate
λn
λn
λn
λn
1
.
= 1 + n+1
1
= 1 − n+2 .
= (n + 1) log(n + 1).
= (n + 1)(log(n + 1))2 .
Now decide whether explosion occurs in each of the above cases.
1
2
Problem 4
(1) Problem 6.1, p. 151 from your book.
(2) Problem 6.3, p. 152 from your book.
Problem 5 Find the distribution of the queue length in equilibrium of the M/M/1 and
M/M/∞ models with service rates µ and customer arrival rates λ. Make sure to state the
relationship between the parameters that guarantee that an equilibrium distribution exists.
Problem 6 Suppose you have a renewal process Nt with renewal times Ti that have density
1 µ−1 −x
fT (x) =
x
e , x ≥ 0.
Γ(µ)
Nt
• Find lim
.
t→∞ t
√
• If µ = 2, give an approximate value for the probability that Nt > t/2 + t when t
is large.