Motivated by the problem of the coexistence on transmission links of
telecommunication networks of elastic and unresponsive traffic, we study in
this paper the impact on the busy period of an M/M/1 queue of a small
perturbation in the server rate. The perturbation depends upon an independent
stationary process (X(t)) and is quantified by means of a parameter \eps \ll 1.
We specifically compute the two first terms of the power series expansion in
\eps of the mean value of the busy period duration. This allows us to study the
validity of the Reduced Service Rate (RSR) approximation, which consists in
comparing the perturbed M/M/1 queue with the M/M/1 queue where the service rate
is constant and equal to the mean value of the perturbation. For the first term
of the expansion, the two systems are equivalent. For the second term, the
situation is more complex and it is shown that the correlations of the
environment process (X(t)) play a key role
