## Product-form queueing networks with negative and positive customers.(English)Zbl 0741.60091

The model considered is a generalization of standard queueing network models widely used in computer and communication system performance modelling which have only “positive” customers. The paper considers an open network with $$n$$ servers and i.i.d. exponential service times and given arrival rates. Two types of customers circulate in the network: “positive” and “negative”. The external arrivals can either be positive customers which arrive to the $$i$$-th queue according to a Poisson process of rate $$\Lambda(i)$$, or negative customers constituting a Poisson process of rate $$\lambda(i)$$ to the $$i$$-th queue. A customer leaving the $$i$$-th queue after having completed service with given probabilities heads for queue $$j$$ as a positive or a negative customer, or it leaves the system. A negative customer reduces the queue length by 1 and does not require service, or has no effect if the corresponding queue is empty. A positive customer adds 1 to the queue length. The queue length is constituted only by the positive customers and their service is carried out in the usual manner. The positive customers can be considered to be resource requests, while negative customers can correspond to decisions to cancel requests for resources. It is shown that this system has a specific kind of product form: for the open network the stationary distribution of its state can be written as the product of marginal probabilities of the state of each queue.

### MSC:

 60K20 Applications of Markov renewal processes (reliability, queueing networks, etc.) 60K25 Queueing theory (aspects of probability theory)
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