  The performance analysis of peer-to-peer (P2P) networks calls for a new kind
of queueing model, in which jobs and service stations arrive randomly. Except
in some simple special cases, in general, the queueing model with varying
service rate is mathematically intractable. Motivated by the P-K formula for
M/G/1 queue, we developed a limiting analysis approach based on the connection
between the fluctuation of service rate and the mean queue length. Considering
the two extreme service rates, we proved the conjecture on the lower bound and
upper bound of mean queue length previously postulated. Furthermore, an
approximate P-K formula to estimate the mean queue length is derived from the
convex combination of these two bounds and the conditional mean queue length
under the overload condition. We confirmed the accuracy of our approximation by
extensive simulation studies with different system parameters. We also verified
that all limiting cases of the system behavior are consistent with the
predictions of our formula.
