1. The weather in a certain locale consists of alternating wet and dry spells. Suppose that the number of days in each rainy spell is a Poisson distribution with mean 2, and that a dry spell follows a geometric distribution with mean 7. Assume that the successive durations of rainy and dry spells are independent. What is the long-run fraction of time that it rains?

2. Counter processes. Suppose that particles arrive at a counter according to a Poisson process with rate λ. An arriving particle that finds the counter free gets registered and then locks the counter for an amount of time τ . Particles that arrive while the counter is locked have no effect (i.e., are not counted).
(a) Find the limiting probability the counter is locked at time t.
(b) Compute the limiting fraction of particles that get registered

3. A young doctor is working at night in an emergency room. Emergencies come in at times of a Poisson process with rate 0.5 per hour. The doctor can only get to sleep when it has been 36 minutes (.6 hours) since the last emergency. For example, if there is an emergency at 1:00 and a second one at 1:17 then she will not be able to get to sleep until at least 1:53, and it will be even later if there is another emergency before that time.
(a) Compute the long-run fraction of time she spends sleeping, by formulating a renewal reward process in which the reward in the ith interval is the amount of time she gets to sleep in that interval.
(b) The doctor alternates between sleeping for an amount of time Si and being awake for an amount of time Ui . Use the result from (a) to compute E[Ui].

4. A scientist has a machine for measuring ozone in the atmosphere that is located in the mountains just north of Los Angeles. At times of a Poisson process with rate 1, storms or animals disturb the equipment so that it can no longer collect data. The scientist comes every L units of time to check the equipment. If the equipment has been disturbed then she can usually fix it quickly so we will assume the the repairs take 0 time.
(a) What is the limiting fraction of time the machine is working?
(b) Suppose that the data that is being collected is worth a dollars per unit time, while each inspection costs c < a. Find the best value of the inspection time L.

5. Each time the frozen yogurt machine at the mall breaks down, it is replaced by a new one of the same type.
(a)What is the limiting age distribution for the machine in use if the lifetime of a machine has a Erlang(2, λ) distribution, i.e., the sum of two exponentials with mean 1/λ.
(b)Find the answer to (a) by thinking about a rate one Poisson process in which arrivals are alternately colored red and blue.

6. A shuttle service at a certain beach resort transports passengers between two different locations, A and B. Shuttles are always available at location A, but to make it worth the trip a shuttle never leaves until it has five passengers. Passengers in need of transportation arrive according to a Poisson process with rate λ. Once a shuttle leaves, another one immediately starts accepting passengers. A hotel guest who does not need to take the shuttle approaches location A. What is the distribution of the amount of time this guest needs to wait until the next shuttle leaves?

1. The weather in a certain locale consists of alternating wet and...