Question: Consider a factory where an array of N = 10 machines work in on parallel some raw materials to obtain some product. The processing of a block of raw material by any machinery takes an independent random time with exponential distribution with parameter µ = 1. The raw materials arrive at the machines according to a Poisson process with parameter λ and are dispatched to the first available machinery. If all machines are busy, the arriving raw materials have to be dropped. We are interested in the asymptotic working regime of the system.
a) Suggest a valuable QS to model this scenario.
b) Determine the value of λ for which the mean number of servers in use is N/2.
c) Determine the value of λ for which the mean number of servers in use is N × 0.9.
d) Determine the percentage of raw material dropped in the two previous cases.
e) Assume that a buffer place is added to the system, capable of holding a single unit of raw material. What is now the arrival rate λ needed to have, on average, N × 0.9 servers in use? What is the loss probability of raw material in this case?