Activity rates with very heavy tails

Consider a data network model in which sources begin to transmit at renewal time points {Sn}{Sn}. Transmissions proceed for random durations of time {Tn}{Tn} and transmissions are assumed to proceed at fixed rate unity. We study M(t)M(t), the number of active sources at time tt, a process we term the activity rate process  , since M(t)M(t) gives the overall input rate into the network at time tt. Under a variety of heavy-tailed assumptions on the inter-renewal times and the duration times, we can give results on asymptotic behavior of M(t)M(t) and the cumulative input process A(t)=∫0tM(s)ds.