5 Questions on Stochastics and Mathematics

Q1) A pure birth process starting from X(0)=0 has birth parameters λ₀=1, λ₁=3, λ₂=2, λ₃=5. Determine P_n(t) for n=0,1,2,3.

Q2) A pure death process starting from X(0)=3 has death parameters µ₀=0, µ1=3, µ2=2, µ3=5.  Determine P_n(t) for n=0,1,2,3.

6.13) A small barbershop, operated by a single barber, has room for at most two customers. Potential customers arrive at a Poisson rate of three per hour, and the successive service times are independent exponential random variables with mean 4 hour. What is

a)      the average number of customers in the shop?

b)     the proportion of potential customers that enter the shop?

c)      If the barber could work twice as fast, how much more business would he do?

6.15) A service center consists of two servers, each working at an exponential rate of two services per hour. If customers arrive at a Poisson rate of three per hour, then, assuming a system capacity of at most three customers,

a)      what fraction of potential customers enter the system?

b)     what would the value of part (a) be if there was only a single server, and his rate was twice as fast (that is, µ = 4)?

Q5) A factory has five machines and a single repairman. The operating time until failure of a machine is exponentially distributed random variable with parameter (rate) 0.20 per hour. The repair time of a failed machine is exponentially distributed random variable with parameter (rate) 0.50 per hour. Up to five machines may be operating at any given time, their failures being independent of one another, but at most one machine may be in repair at any time. In the long run, what fraction of time is the repairman idle?