# 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?

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## 1 Answer

The superposition of independent Poisson processes is equivalent to a single Poisson process whose rate is the sum of the rates of the independent Poisson processes and where one affects independently each point to one of the Poisson processes with a probability proportional to its rate. Thus the global interarrival time is exponentially distributed with rate the sum of the rates and the distribution of the global service time is the barycenter of the distributions of the individual service times.

See here (pages 7-10) or Kingman's classical small book Poisson Processes.

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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? – Stijn May 12 '11 at 16:04
Indeed. And now you can add the tag (homework) to your question. – Did May 12 '11 at 16:05
Why would I do that if this isn't homework? Anyways thanks for the answer. – Stijn May 12 '11 at 16:08
Is it not?   – Did May 12 '11 at 16:12
Nope. It is an exercise from a reader I'm using but it is not part of any homework series. I'm just trying to do some exercises to get some, well... Exercise. – Stijn May 12 '11 at 16:44