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Proportional fairness and its relationship with multi-class queueing networks. (English) Zbl 1198.60039
The connections between three different models of multi-class single-server queueing network are determined. The first model concerns a single-server multiclass queueing network and is treated as a microscopic model of a macroscopic model. The second, i.e., the macroscopic model, is a specific stochastic flow level model and the models document transfer across a packet switched network. The third, known as the teleological model, refers to the descriptions of the network.
The first result of this paper shows that a sequence of multi-class queueing network modeling of document transfer across a packet switching network weakly converges in the Skorokhod topology of these networks with a specific stochastic flow level model. The author referred to the resulting stochastic flow level model as the spinning network. The second result of this paper is concerned with connecting both microscopic and macroscopic models with the teleological model. This paper provides proofs of the mathematical relationship between multi-class networks of single-server queues and proportional fairness. It was observed that the constraints of a network may include transfer rates. By applying the Contraction Principle a new rate function $$\alpha(.)$$, expressed as a convex optimization problem, is gained. In its primal form $$\alpha(.)$$ is interpreted as maximizing entropy subject to a constraints. The determined dual form of $$\alpha(.)$$ is up to constant, proportional to the observed throughput of these networks. The final observation is that the multi-class queueing networks considered here have no prescribed optimization structure. It is surprising to see that asymptotically these networks implicitly solve a utility optimization problem.

##### MSC:
 60K25 Queueing theory (aspects of probability theory) 60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.) 90B22 Queues and service in operations research
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