Coupled processors with regularly varying service times

Abstract

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The impact of the service discipline on delay asymptotics

“…For general service requirement distributions, the joint distribution of the amounts of work of both classes has been obtained in [27] by solving a Wiener-Hopf problem (see [32,40] for the case of exponential service requirement distributions). The results of [27] have been exploited in [13,17,18]. In those papers, service requirements at Q 1 are either exponential or regularly varying; at Q 2 they are regularly varying.…”

“…Whether the service requirement tail behavior at Q 2 affects the workload tail at Q 1 is shown to depend crucially on whether or not Q 1 is able to handle all its offered work by working at the low speed that occurs while Q 2 is non-empty (i.e., whether or not ρ 1 < φ 1 ). The method employed in [13,17,18] starts from a, complicated, expression for the workload LST. In some cases Lemma 3.1 is applicable, but in other cases an extension of this lemma must be used.…”

Heavy Tails: The Effect of the Service Discipline

“…A key property of regularly varying probability distributions (and of the larger class of subexponential distributions), specified to the sum of residual service requirements in (8), is that…”

“…For general service requirement distributions, the joint distribution of the amounts of work of both classes has been obtained in [19] by solving a Wiener-Hopf problem (see [22] and [28] for the case of exponential service requirement distributions). The results of [19] have been exploited in [8,10,11]. In those papers, service requirements at Q 1 are either exponential or regularly varying; at Q 2 they are regularly varying.…”

“…Whether the service requirement tail behavior at Q 2 affects the workload tail at Q 1 is shown to depend crucially on whether or not Q 1 is able to handle all its offered work by working at the low speed that occurs while Q 2 is non-empty (i.e., whether or not ρ 1 < φ 1 ). The method employed in [8,10,11] starts from acomplicated -expression for the workload LST. In some cases Lemma 1 is applicable, but in other cases an extension of this lemma must be used.…”