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Person# Gianluca Rizzo

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Related research domains (10)

Computer network

A computer network is a set of computers sharing resources located on or provided by network nodes. Computers use common communication protocols over digital interconnections to communicate with eac

Leaky bucket

The leaky bucket is an algorithm based on an analogy of how a bucket with a constant leak will overflow if either the average rate at which water is poured in exceeds the rate at which the bucket le

Dynamical system

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical mode

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Jean-Yves Le Boudec, Gianluca Rizzo

Aggregate scheduling is one of the most promising solutions to the issue of scalability in networks, like DiffServ networks and high speed switches, where hard QoS guarantees are required. For networks of FIFO aggregate schedulers, the main existing sufficient conditions for stability (the possibility to derive bounds to delay and backlog at each node) are of little practical utility, as they are either relative to specific topologies, or based on strong ATM- like assumptions on the network (the so- called ”RIN” result), or they imply an extremely low node utilization. We use a deterministic approach to this problem. We identify a non linear operator on a vector space of finite (but large) dimension, and we derive a first sufficient condition for stability, based on the super-additive closure of this operator. Second, we use different upper bounds of this operator to obtain practical results. We find new sufficient conditions for stability, valid in an heterogeneous environment and without any of the restrictions of existing results. We present a polynomial time algorithm to test our sufficient conditions for stability. We show that with leaky-bucket constrained flows, the inner bound to the stability region derived with our algorithm is always larger than the one determined by all existing results. We prove that all the main existing results can be derived as special cases of our results. We also present a method to compute delay bounds in practical cases.

2008Jean-Yves Le Boudec, Gianluca Rizzo

We demonstrate that, contrary to what is generally believed, the existing end-to-end delay bounds apply only to GR nodes that are FIFO per flow. We show this by exhibiting a counter-example. Then we show that the proof of the existing bounds has a subtle, but important, dependency on the FIFO assumption, which was never noticed before. Finally, we give a tight delay bound that is valid in the non-FIFO case; it is noticeably higher that the existing one. In particular, the phenomenon known as ?pay bursts only once? does not apply to non-FIFO nodes. These findings are important in the context of differentiated services. Indeed the existing bounds have been applied to cases where a flow (in the sense of the GR definition) is an aggregate of end-user microflows, and it is not generally true that a router is FIFO per aggregate; thus the GR node model of a differentiated services router cannot always be assumed to be FIFO per flow.

We study networks of FIFO nodes, where flows are constrained by arrival curves. A crucial question with these networks is: Can we derive a bound to the maximum delay that a packet can experience when traversing the network, and to the maximum queue size at each node? For a generic FIFO network these are still open issues: Some examples show that, contrary to common sense, no matter how low the maximum node utilization is in the network, it is possible to build an example of an unstable FIFO network. The importance of this issue lies in the necessity of hard bounds on packet delay and queue size, in order to enable QoS guarantees in these networks. For this reason we choose to tackle this problem through a deterministic approach, based on worst-case behavior. Our first result is the determination of a general method to derive sufficient conditions for the stability of a network: We show how, with a proper choice of the observed variables in the network and with the use of network calculus results, it is possible to derive the expression of an operator whose properties are associated to the stability of the network. Exploiting this method on a simple example, we first derive a generalization of the RIN result to heterogeneous settings and to leaky bucket constrained flows. Through some realistic examples, we show how this method allows networks to achieve a level of utilization which is more than three times larger than the best existing result. By applying the general method to three different variable classes, we derive some new sufficient conditions for stability, that perform largely better than all the main existing results, and we show how they can all be derived from the new sufficient conditions. Finally, we present a new formula for the computation of end-to-end delay bounds in a network of GR nodes.