Renewal reward processes with heavy-tailed inter-renewal times and heavy-tailed rewards. (English) Zbl 0954.60071

Levy, Joshua B.; Taqqu, Murad S.

Bernoulli 6, No. 1, 23-44 (2000).

This paper studies a superposition \(W^*\) of renewal-reward processes having inter-renewal time and reward distributions with heavy tails of exponents \(\alpha\) and \(\beta\), respectively, where \(1< \alpha< 2\), \(0<\beta<2\). In the case \(\beta\leq\alpha\), \(W^*\) suitably normalized converges to Lévy stable motion with index \(\beta\); see the authors [Ann. Sci. Math. Qué. 11, 95-110 (1987; Zbl 0646.60091)]. The present work re-iterates this result and establishes a limit for the case \(\beta>\alpha\), which is also a self-similar, symmetric \(\beta\)-stable process with stationary increments; here, however, the increments are dependent.

Reviewer: J.Preater (Keele)

MSC:

60K05	Renewal theory
60K30	Applications of queueing theory (congestion, allocation, storage, traffic, etc.)

Keywords:

computer networks; infinite variance; self-similar processes; stable processes; telecommunications