Recensione dell'articolo: (Kitsio, V.; Yechiali, U. - " Multi-server queues with intermediate buffer and delayed information on service completions " - Stoch.Models 24 (2008), no.2, 212–245.)

In this paper the authors study a multi-server queue model with delayed information on service completion. The model studied is defined as a modification of the classical M/M/c model with Poisson arrival rate λ and exponential service time with parameter µ. The model is characterized by the fact that customers arriving in the system are queued into an unlimited buffer which operates as a controller for the downstream customer flow to the servers. Controller behaviour is based on a feed-back information flow from the servers. In fact, an information signal is generated from the servers at each service completion event. The generated information signal reaches the controller after some random duration exponentially distributed with parameter γ. The model architecture includes also an intermediate buffer of size c, for customers, which is placed in front of the c servers. When the controller gets the information that a service has been completed, and there are less than c waiting customers in the intermediate buffer, it dispatches a waiting customer, if any, from its buffer into the intermediate buffer. When a server completes its service of a customer and the intermediate buffer is non-empty, it starts serving one of the customers there with no further delay. The authors denote such a system by M(λ)/intermediate:M(µ)+M(γ)/c. They define the state of the system as a triplet (N,J,X) where N denotes the number of customers waiting in the controller’s overall buffer, X counts the combined number of customers in the intermediate buffer or in service, and J denotes the total number of customers whose service has already been completed, but the corresponding information signal has not yet reached the controller. In this framework the authors study the balance equations and calculate the stationary condition of the system for any number of servers c≥1. Leonardo Pasini