An animal shelter has a small no-kill canine adoption facility. Dogs are brought to the facility the evening before and then it is opened to the public for adoptions in the morning. The facility has limited space for the animals and can only house 3 at a time. The following distribution represents the number of dogs that are turned into the shelter each day:

0.3   = 1
0.7   = 2
As the public comes in to adopt animals the following distribution represents the dogs that leave the shelter each day:

0.1   = 0
0.4   = 1
0.5   = 2

Let Xn be the number of dogs available for adoption at the beginning of day n.

a. prepare an expression for Xn+1.

b. Find the transition probability matrix for the shelter.

c. If there are currently 3 dogs in the shelter, what is the distribution of the number of dogs that will be in the shelter in two weeks?

d. What is the expected number of dogs in the shelter after 2 weeks given that there are currently no dogs in the shelter?

Assume now that one of the dogs needs to be kept in a holding center, making the capacity of the holding center 1 and the adoption facility 2. Dogs put in the holding center will not be moved into the adoption center until there is room. Meaning, if at the end of the day the adoption facility is full, and two more dogs arrive the next morning they are placed in the holding center. These dogs are not adoptable until they are moved to the adoption facility. If there is room in the adoption facility dogs are moved directly there. Let Yn be the number of dogs in the holding center at the beginning of day n.

e. prepare an expression for [Yn+1, Xn+1] in this case.
f. Find the transition probability matrix for this situation.