Heavy traffic scaling limits for shortest remaining processing time queues with heavy tailed processing time distributions

We study a single server queue operating under the shortest remaining processing time (SRPT) scheduling policy; that is, preemptively serves job with first. Since one needs to keep track of times all jobs in system order describe evolution, natural state descriptor for an SRPT is measure valued process which at given finite nonnegative Borel on real line puts unit atom each system. In this work we are interested studying asymptotic behavior suitably scaled descriptors sequence queuing systems. Gromoll, Kruk and Puha (Stoch. Syst. 1 (2011) 1–16) have studied problem diffusive scaling (time by r2 mass normalized r, where r parameter approaching infinity). setting distributions bounded support, suitable conditions, they show converge distribution any located right edge support size fluctuating randomly time. unbounded diffusion identically zero. (Ann. Appl. Probab. 25 (2015) 3381–3404) light tails, nonstandard length shown give rise form space collapse results nonzero limit. current consider case second moments regularly varying tails. Results suggest governed cr as certain inverse function related tails first moment distribution. Using novel factor r2, cr/r (representing times) 1/cr. converges (in paths measures). sharp contrast tailed service distributions, there no limiting measures not concentrated atom. Nevertheless, description limit simple explicitly terms R+ random field determined from Brownian motion. Along way establish convergence workload processes. also tail becomes lighter appropriate fashion, difference between zero, thereby collapse.