Math and Medicine: Homework Assignment Week 10
Due on March 26
1. Without deaths, we found that the probability pi of there being exactly i people on a waiting
list where people arrive at rate λ and receive transplants at rate σ is
pi =
λ
σ
!i−1
(1 −
λ
)
σ
Suppose the list runs on a “first come, first serve” basis.
a. Suppose the list has length 1. How long will that person wait on average before getting
a transplant?
b. What if the list has length 2? How long will the last person to enter the list wait on
average?
c. Use this to find the overall average waiting time (these times weighted by the pi ). You
should get something a lot like
average waiting time =
mean list length
σ
d. Suppose there is a small rate µ of death. What is the average probability of death?
e. If the region gets m times larger, λ and σ will increase by a factor of m, while µ remains
the same. How should the probability of death depend on m?
2. Get all files from the website in the “queueing” folder and read the README file.
a. Start with the program “Qdriver.R”. Set µ = 0 and see what happens to the list when
λ is just a tiny bit smaller than σ. What about if λ is a bit bigger than σ? What might
happen in real life in this case?
b. Now pick µ > 0 (but not too big). What happens when λ is just a bit smaller, or just
a bit larger, than σ? Does the number of deaths jump up when λ passes σ?
c. Use “Qloop.R” to compare different values of the size parameter sz in some case with
µ > 0. Does the probability of death decrease proportional to sz as predicted in
problem 1?