Recensione dell'articolo: (Atar, Rami; Shaikhet, Gennady - "Critically loaded queueing models that are throughput suboptimal" - Ann.Appl.Probab. 19 (2009), no.2, 521–555.)

In this paper the authors study a multiclass queuing network model. In this model customer classes are indexed by a finite set l and servers are heterogeneous with exponential service time distributions. The servers are grouped into pools, indexed by a finite set J, so that the rate µij at which a server from pool j ∈J serves customers from class i∈l depends on both i and j. The arrival of class-i customers is modelled as a renewal process with rate λi, i∈l. Servers are dynamically chosen to serve customers and global buffers are available to accommodate customers that wait to be served. The model is considered in a many-servers heavy traffic regime, in which the number of servers at each pool and the arrival rate are scaled up at a nearly fixed proportion and in such a way that the processes that represent the number of class-i customers in the system, i∈l, fluctuate about a certain static fluid model. This fluid model is assumed to be critically loaded in a standard sense. In particular, servers can be allocated in such a way that the total processing rate devoted to class-i customers is equal to arrival rate λi, for every i∈l, and this property does not hold if one of the arrival rates λi is replaced by some λ0 i strictly greater than λi. In this context the authors define the condition for the fluid model to be “throughput suboptimal” and prove their main result. They prove that when the fluid model is throughput suboptimal one can find a dynamic control policy for the queuing model that exhibits a strong form of efficiency. Under this policy, for every time T, the measure of the set of times prior to T at which at least one customer is in the buffer converges to zero in probability as the arrival rate and the number of servers go to infinity. Leonardo Pasini