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G-networks with multiple classes of negative and positive customers. (English) Zbl 0873.68010

Summary: In recent years a new class of queueing networks with “negative and positive” customers was introduced by one of the authors E. Gelenbe [J. Appl. Probab. 28, No. 3, 656-663 (1991; Zbl 0741.60091)], and shown to have a nonstandard product form. This model has undergone several generalizations to include triggers or signals which are special forms of customers whose role is to move other customers from some queue to another queue [E. Gelenbe, P. Glynn and K. Sigmann, ibid. 28, No. 1, 245-250 (1991; Zbl 0744.60110); E. Gelenbe and R. Schassberger, Probab. Eng. Inf. Sci. 6, 271-276 (1992); E. Gelenbe, J. Appl. Probab. 30, No. 3, 742-748 (1993; Zbl 0781.60088); E. Gelenbe, Probab. Eng. Inf. Sci. 7, 335-342 (1993)]. Positive customers are identical to the usual customers of a queueing network, while a negative customer which arrives to a queue simply destroys a positive customer. We call these generalized queueing networks G-networks. We extend the basic model of [Gelenbe (loc. cit.) to the case of multiple classes of positive customers, and multiple classes of negative customers. As in other multiple class queueing networks, a positive customer class is characterized by the routing probabilities and the service rate parameter at each service center while negative customers of different classes may have different “customer destruction” capabilities. In the present paper all service time distributions are exponential and the service centers can be of the following types: FIFO (first-in-first-out), LIFO/PR (last-in-first-out with preemption), PS (processor sharing), with class-dependent service rates.

MSC:

68M20 Performance evaluation, queueing, and scheduling in the context of computer systems
60K25 Queueing theory (aspects of probability theory)
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