We consider a fluid queue fed by sessions, arriving according to a Poisson process; a session has a heavy-tailed duration, during which traffic is sent at a constant rate. We scale Poisson input rate Λ, buffer space B, and link rate C by n, such that we get nλ,nb , and nc, respectively. Then we let n grow large. In this regime, the overflow probability decays exponentially in the number of sources n; we examine the specific situation in which b is also large.
In Duffield (Queueing Syst. 28 (1998) 245–266) this setting is considered. A crucial role was played by the function ≔v(t)≔−logP(D★>t) for large t,D★ being the residual session duration. Duffield covers the case that v(·) is regularly varying of index strictly between 0 and 1 (e.g., Weibull); this note treats slowly varying v(·) (e.g., Pareto, Lognormal).
The proof adds insight into the way overflow occurs. If v(·) is slowly varying then, during the trajectory to overflow, the input rate will exceed the link rate only slightly. Consequently, the buffer will fill ‘slowly’, and the typical time to overflow will grow ‘faster than linearly’ in the buffer size. This is essentially different from the ‘Weibull case’, where the input rate will significantly exceed the link rate, and the time to overflow is essentially proportional to the buffer size. In both cases there is a substantial number of sessions that remain in the system during the entire path to overflow.