# Mix of Poisson processes

Suppose we had a machine where two types of jobs arrive. Jobs of type 1 arrive according to a Poisson process with a rate of $\lambda_1 = 45$ jobs per hour and need an exponential service time with a mean of $\frac{1}{2}$ minutes (so the service time of type 1 jobs is exponentially distributed with parameter $\mu_1 = 2$). Jobs of type two also arrive according to a Poisson process but with a rate of $\lambda_2 = 15$ jobs per hour and have service times $B_2$ which are distribted $\text{exp}(\mu_2)$ with $E(B_2) = 1 \Longrightarrow \mu_2 = 1$. The arrival- and service times are iid.

I'm trying to figure out how to model this process. Am I right in saying that with probability $\frac{3}{4}$ jobs of type 1 come arrive, since $\frac{\lambda_1}{\lambda_1 + \lambda_2} = \frac{45}{60} = \frac{3}{4}$? And does this imply that the arrival- and service times are both hyperexponentially distributed?

• Ok, so the arrival times are a Poisson process with rate $\lambda = 45 + 15 = 60$ jobs per hour and the service time has a probability density function $f_{B}(t) = \frac{3}{4}2e(-2t) + \frac{1}{4}e(-t)$, the mix of two exponential pdfs? May 12, 2011 at 16:04