In an M/M/1 Queue, what does exponential distribution of service time mean?
In an exponential distribution, the probability is DECREASING exponentially -- hence little risk that service time will grow. On the contrary, the mean service time is fixed, at the inverse of the rate.
This distribution is actually very rare in practice, but is chosen because it is Markovian:
the probability to terminate servicing a client does not depend on the past, and in particular on how much time you already spent servicing that client. That yields very nice mathematical properties.