Download 1 Birth Death Process 2 Machine

1
Birth Death Process
Given is the following Birth Death process X(t) with 5 states {0,1,2,3,4}. Births occur according to the rates
λ0 = 1,λ1 = 0.75,λ2 = 0.5 and λ3 = 0.25, deaths occur with the same rate µi = µ = 0.5 for all i = 0, 1, 2, 3.
• Draw a state diagram of the process and mark the arcs with the appropriate transition rates.
• Find the steady state probability to be in each state, i.e. find p0 , p1 , p2 , p3 and p4 . (Hint: Use the
Flow-In Rate = Flow-Out Rate Principle)
• What is the expected value for X after reaching the steady state? (Hint: Based on the probabilities
above compute the expected value.)
• What real-life application fits a birth & death process with the above transition rates? Give an example.
2
Machine-Repairmen Problem
A jobshop consists of three machines and two repairmen. Each machine is in use whenever it is working.
The amount of time a machine works before breaking down is exponentially distributed with mean 12 days.
If the amount of time it takes a single repairman to fix a machine is exponentially distributed with mean 8
days, then
1. Draw a state diagram of the number of broken machines.
2. What is the probability that none of the three machines is broken?
3. What is the average number of machines not in use?
4. What proportion of time are both repairmen busy?