Sojourn time tails in the single server queue with heavy-tailed service times

We consider the GI/GI/1 queue with regularly varying service requirement distribution of index −α. It is well known that, in the M/G/1 FCFS queue, the sojourn time distribution is also regularly varying, of index 1 − α, whereas in the case of LCFS or Processor Sharing, the sojourn time distribution is regularly varying of index −α. That raises the question whether there exist service disciplines that give rise to a regularly varying sojourn time distribution with any index −γ ∈ [−α, 1 − α]. In this paper that question is answered affirmatively.