The analytical solution of a queueing network is an appreciable first option in performance studies of service systems. Under the so-called product-form conditions, Mean Value Analysis (MVA) is the standard algorithm still adopted, but the user has to face the exponential computational complexity in the number of customer classes. In the last three decades, some (pseudo) polynomial approximated variants to MVA have been proposed in literature. These approximations are based on the transformation of the recursive MVA equations into a system of nonlinear equations to be solved iteratively. They are consolidated only with reference to (fixed-rate) single-server stations and are used in practice even though theoretical convergence remains an open problem. In this paper we exploit the possibility of aggregating customer classes in order to replace the exact multi-class MVA by new approximated procedures where MVA has to be run under at the most two customer classes. The resulting procedures are developed around a nested fixed-point iteration schema and are especially suitable for solving large size multi-class networks with multi-server stations under a first-come-first-served discipline. Convergence and accuracy of our procedures are numerically assessed through a very large set of experiments against the exact solution by the multi-class MVA.
This paper considers a cognitive radio network system which has two classes of users that we call Primary Users and Secondary Users. We model this system using a single server retrial queueing model. In the conventional retrial queue, if a customer cannot find an idle channel, he joins the orbit and retries to occupy a channel. The new feature of our model is that every arriving Secondary User first enters the orbit. He starts to sense the channels and tries to find an idle channel. For this model, we obtain explicit expressions for the joint generating functions of the state of the server and the number of Secondary Users in the orbit. In addition, we derive the necessary stability condition. We obtain the distribution of the number of retrials by a Secondary User using simulation. Beside, we consider the multiserver model for which we obtain the average of the number of Secondary Users in the orbit and the distribution of the number of retrials by simulation.
We consider a multiclass multiserver retrial queuing system with classical retrial discipline: the customers, meeting server busy, are blocked on the corresponding (virtual) orbit and then retry to occupy server independently. The retrial times have general class-dependent distributions. The input process is renewal and a new arrival is class-i customer with a given probability $$p_i$$. We exploit a regenerative structure of a basic process describing the dynamics of the system to establish stability conditions. More exactly, we show that, provided the retrial times belong to the New-Better-Than-Used class, the convenient requirement that the mean load (traffic intensity) is less than the number of servers, is the stability criterion of the model. A few numerical results are included which, in particular, show that this condition ensures stability of the system with the New-Worse-Than-Used Weibull retrial times as well.
We consider a fluid flow model with infinite buffer. We compute the Laplace-Stieltjes transform of the sojourn time proceeding in two steps. We first compute the stationary distribution of the buffer at arrival instants, using a change of clock. Secondly, we compute the transform of the time spent to empty the buffer. Numerical examples of sojourn time in a fluid flow are finally examined.
