Question

6. You have 3 independent companies (A, B, and C ) that are concurrently (i.e. at the same time) sending deliveries to you. Suppose that the Times between (consecutive) A’s deliveries to you each have independent Exp(λ=3.5 per hour) distributions. The Times between B’s deliveries to you each have independent Exp(λ=1.8) per hour) distributions. While, The Times between C’s deliveries to you each have independent Exp(λ=1.8) per hour) distributions.

What is the distribution for the time between any two (consecutive) deliveries, regardless of which company or companies they came from?

Answer 1

For eah company, the time between deliveries are exponentially distributed, so the number of arrivals are Poisson distributed. We know for a distribution which is sum of 2 Poisson distributions, the expected value (or the parameter) is the sum of the parameters of the two individual distributions.

We have 3 Poisson processes, with parameters 60/3.5, 60/1.8 and 60/1.8 arrivals per hour. So, in effect, we have

60/3.5 + 60/1.8 + 60/1.8 = 83.80 arrivals per hour.

Time between consecutive deliveries = 60/83.80 = 0.72 hour, eqv to 42.96 seconds.