Heavy-tailed limits for medium size jobs and comparison scheduling

We study the conditional sojourn time distributions of processor sharing (PS), foreground background processor sharing (FBPS) and shortest remaining processing time first (SRPT) scheduling disciplines on an event where the job size of a customer arriving in stationarity is smaller than exactly k≥0 out of the preceding m≥k arrivals. Then, conditioning on the preceding event, the sojourn time distribution of this newly arriving customer behaves asymptotically the same as if the customer were served in isolation with a server of rate (1−ρ)/(k+1) for PS/FBPS, and (1−ρ) for SRPT, respectively, where ρ is the traffic intensity. Hence, the introduced notion of conditional limits allows us to distinguish the asymptotic performance of the studied schedulers by showing that SRPT exhibits considerably better asymptotic behavior for relatively smaller jobs than PS/FBPS.