please solve

4. (Submit, 10pts) Ratty is a farm dog. His master keeps him because he is very gd at catching rats if they enter the hen house, where the hens lay eggs. He prefers to chase a rat and play with it before killing it, so that when there is a single rat in the hen house it takes Ratty 5 minutes on average to (catch and) kill it. If there are two live rats in the hen house Ratty takes on average 3 minutes to kill the first of them, but if there are more than two rats running around in the hen house Ratty gets confused and the first kill takes him 10 minutes. Rats enter the hen house randomly and independently at an average rate of one every ten minutes, but some shared instinct means that there will never be more than four live rats in the hen house at any time. On the other hand there is no way out for a rat except as a dead rat! 2 (a) For what fraction of his time in the hen house is Ratty able to rest? (b) Suppose a rat has just entered the hen house and observes with horror that Ratty is in the hen house and is currently occupied with another rat. On average how long can this rat expect to have to wait before starting to receive Ratty's attention? (Note that if this rat was allowed to enter the hen house, there must have been 0,1,2,3 rats ahead of it but not 4; if there were 4 ahead of it then it would not have entered the hen house.)