Activity rates with very heavy tails

Consider a data network model in which sources begin to transmit at renewal time points { S n } . Transmissions proceed for random durations of time { T n } and transmissions are assumed to proceed at fixed rate unity. We study M ( t ) , the number of active sources at time t , a process we term the activity rate process, since M ( t ) gives the overall input rate into the network at time t . 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 ) and the cumulative input process A ( t ) = ∫ 0 t M ( s ) d s .