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Queues and risk processes with dependencies. (English) Zbl 1306.60132

Summary: We study the generalization of the \(\mathrm{G}/\mathrm{G}/1\) queue obtained by relaxing the assumption of independence between inter-arrival times and service requirements. The analysis is carried out for the class of multivariate matrix exponential distributions introduced in [M. Bladt and B. F. Nielsen, Stoch. Models 26, No. 1, 1–26 (2010; Zbl 1196.60021)]. In this setting, we obtain the steady-state waiting time distribution, and we show that the classical relation between the steady-state waiting time and workload distributions remains valid when the independence assumption is relaxed. We also prove duality results with the ruin functions in an ordinary and a delayed ruin process. These extend several known dualities between queueing and risk models in the independent case. Finally, we show that there exist stochastic order relations between the waiting times under various instances of correlation.

MSC:

60K25 Queueing theory (aspects of probability theory)
60E15 Inequalities; stochastic orderings
91B30 Risk theory, insurance (MSC2010)

Citations:

Zbl 1196.60021
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