The behavior of a connection transmitting packets into a network according to a general additive-increase multiplicative-decrease (AIMD) algorithm is investigated. It is assumed that loss of packets occurs in clumps. When a packet is lost, a certain number of subsequent packets are also lost (correlated losses). The stationary behavior of this algorithm is analyzed when the rate of occurrence of clumps becomes arbitrarily small. From a probabilistic point of view, it is shown that exponential functionals associated to compound Poisson processes play a key role. A formula for the fractional moments and some density functions are derived. Analytically, to get the explicit expression of the distributions involved, the natural framework of this study turns out to be the $q$-calculus. Different loss models are then compared using concave ordering. Quite surprisingly, it is shown that, for a fixed loss rate, the correlated loss model has a higher throughput than an uncorrelated loss model.

It is shown that peak cell rate policing can be combined with fast reservation protocols for achieving statistical multiplexing in asynchronous transfer mode (ATM) networks. A cell spacer-controller architecture which smooths out cell clusters according to the original period both at the user network interface (UNI) and at network node interface (NNI) is proposed. A performance evaluation of the spacer-controller is presented. The evaluation involves modeling the spacer-controller, dimensioning a virtual scheduling algorithm, and the impact on cell delay variation (CDV). It is shown that large CDVs may induce severe degradation in the accuracy of usage parameter control/network parameter control mechanisms.

The additive-increase multiplicative-decrease (AIMD) schemes designed to control congestion in communication networks are investigated from a probabilistic point of view. Functional limit theorems for a general class of Markov processes that describe these algorithms are obtained. The asymptotic behaviour of the corresponding invariant measures is described in terms of the limiting Markov processes. For some special important cases, including TCP congestion avoidance, an important autoregressive property is proved. As a consequence, the explicit expression of the related invariant probabilities is derived. The transient behaviour of these algorithms is also analysed.

We consider a system consisting of N parallel servers, where jobs with different resource requirements arrive and are assigned to the servers for processing. Each server has a finite resource capacity and therefore can serve only a finite number of jobs at a time. We assume that different servers have different resource capacities. A job is accepted for processing only if the resource requested by the job is available at the server to which it is assigned. Otherwise, the job is discarded or blocked. We consider randomized schemes to assign jobs to servers with the aim of reducing the average blocking probability of jobs in the system. In particular, we consider a scheme that assigns an incoming job to the server having maximum available vacancy or unused resource among d randomly sampled servers. We consider the system in the limit where both the number of servers and the arrival rates of jobs are scaled by a large factor. This gives rise to a mean field analysis. We show that in the limiting system the servers behave independentlya property termed as propagation of chaos. Stationary tail probabilities of server occupancies are obtained from the stationary solution of the mean field which is shown to be unique and globally attractive. We further characterize the rate of decay of the stationary tail probabilities. Numerical results suggest that the proposed scheme significantly reduces the average blocking probability of jobs as compared to static schemes that probabilistically route jobs to servers independently of their states.

The authors address peak rate enforcement in asynchronous transfer mode (ATM) networks by pick-up policing mechanisms. Since the network cannot rely on user's compliance when declaring his traffic parameters, the policing function has been introduced to monitor the traffic characteristics of any connection. Policing mechanisms proposed in the literature are actually pick-up mechanisms since cells can pass transparently through unless a policing action was performed according to some counting process. The authors analyze the impact of the cell multiplexing jitter on policing mechanism dimensioning. Recognizing that dimensioning is a critical point for such mechanisms. It is shown that network buffer overflow cannot be avoided in certain conditions. Therefore, a new policing scheme, called cell spacing, is introduced, and two spacing algorithms are presented.