According to the Wikipedia article about M/M/1 queues, the rate at which new jobs arrive is a Poisson process with parameter $\lambda$, and the rate at which the jobs are finished is an exponential distribution process with mean service time $\mu$.
To quote directly:
Arrivals occur at rate λ according to a Poisson process and move the process from state i to i + 1. Service times have an exponential distribution with parameter 1/μ in the M/M/1 queue, where μ is the mean service rate.
This seems strange to me. Why not just be consistent and say they are both Poisson processes? After all, exponential distribution is essentially the same thing as a Poisson distribution. Why confuse people and call them by different variations of the same thing?