13 research outputs found

    Generalization of the RIN Result to Heterogeneous Networks of Aggregate Schedulers and Leaky Bucket Constrained Flows

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    We consider networks of FIFO aggregate schedulers. Quite surprisingly, the natural condition (node utilization inferior to one) in general is not sufficient in these networks to ensure stability (boundedness of delay and backlog at each node). Deriving good sufficient conditions for stability and delay bounds for these networks is of fundamental importance if we want to offer quality of service guarantees in such networks as DiffServ networks, high speed switches and network-on-chips. The main existing sufficient conditions for stability in these networks are the “DiffServ bound” [1] and the Route Interference Number (RIN) result [2]. We use an algebraic approach. First, we develop a model of the network as a dynamical system, and we show how the problem can be reduced to properties of the state transition function. Second, we obtain new sufficient conditions for stability valid without any of the restrictions of the “RIN result”. We show that in practical cases, when flows are leaky bucket constrained, the new sufficient conditions perform better than existing results. We also prove that the “RIN result” can be derived as a special case from our approach. We finally derive an expression for a bound to delay at all nodes

    Generalization of the RIN result to heterogeneous networks of aggregate schedulers and leaky bucket constrained flows

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    We consider networks of FIFO aggregate schedulers. Quite surprisingly, the natural condition (node utilization inferior to one) in general is not sufficient in these networks to ensure stability (boundedness of delay and backlog at each node). Deriving good sufficient conditions for stability and delay bounds for these networks is of fundamental importance if we want to offer quality of service guarantees in such networks as Diffserv networks, high speed switches and network-on-chips. The main existing sufficient conditions for stability in these networks are the "DiffServ bound" [1] and the Route Interference Number (RIN) result [2]. We use an algebraic approach. First, we develop a model of the network as a dynamical system, and we show how the problem can be reduced to properties of the state transition function. Second, we obtain new sufficient conditions for stability valid without any of the restrictions of the "RIN result". We show that in practical cases, when flows are leaky bucket constrained, the new sufficient conditions perform better than existing results. We also prove that the "RIN result" can be derived as a special case from our approach. We finally derive an expression for a bound to delay at all nodes

    Stability and Delay Bounds in Heterogeneous Networks of Aggregate Schedulers

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    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

    Application of Network Calculus To Guaranteed Service Networks

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    We use recent network calculus results to study some properties of lossless multiplexing as it may be used in guaranteed service networks. We call network calculus a set of results that apply min-plus algebra to packet networks. We provide a simple proof that shaping a traffic stream to conform with a burstiness constraint preserves the original constraints satisfied by the traffic stream We show how all rate based packet schedulers can be modeled with a simple rate latency service curve. Then we define a general form of deterministic effective bandwidth and equivalent capacity. We find that call acceptance regions based on deterministic criteria (loss or delay) are convex, in contrast to statistical cases where it the complement of the region which is convex. We thus find that, in general, the limit of the call acceptance region based on statistical multiplexing when the loss probability target tends to 0 may be strictly larger than the call acceptance region based on lossless multiplexing. Lastly, we consider the problem of determining the optimal parameters of a variable bit rate (VBR) connection when it is used as a trunk, or tunnel, given that the input traffic is known. We find that there is an optimal peak rate for the VBR trunk, essentially insensitive to the optimization criteria. For a linear cost function, we find an explicit algorithm for the optimal remaining parameters of the VBR trunk

    Some Properties Of Variable Length Packet Shapers

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    The min-plus theory of greedy shapers has been developed after Cruz`s results on the calculus of network delays. An example of greedy shaper is the buffered leaky bucket controller. The theory of greedy shapers establishes a number of properties such as the series decomposition of shapers or the conservation of arrival constraints by re-shaping. It applies in all rigor either to fluid systems, or to packets of constant size such as ATM. For variable length packets, the distortion introduced by packetization affects the theory, which is no longer valid. In this paper, we elucidate the relationship between shaping and packetization effects. We show a central result, namely, the min-plus representation of a packetized greedy shaper. We find a sufficient condition under which series decomposition of shapers and conservation of arrival constraints still hold in presence of packetization effects. This allows us to demonstrate the equivalence of implementing a buffered leaky bucket controller based on either virtual finish times or on bucket replenishment. However, we show on some examples that if the condition is not satisfied, then the property may not hold any more. This indicates that, for variable size packets, unlike for fluid systems, there is a fundamental difference between constraints based on leaky buckets, and constraints based on general arrival curves, such as spacing constraints. The latter are used in the context of ATM to obtain tight end-to-end delay bounds. In this paper, we use a min-plus theory, and obtain results on greedy shapers for variable length packets which are not readily explained with the max-plus theory of Chang

    Some properties of variable length packet shapers

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    Network Calculus

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    Network Calculus is a collection of results based on Min-Plus algebra, which applies to deterministic queuing systems found in communication networks. It can be used for example to understand - the computations for delays used in the IETF guaranteed service - why re-shaping delays can be ignored in shapers or spacer-controllers - a common model for schedulers - deterministic effective bandwidth and much more

    Theories and Models for Internet Quality of Service

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    We survey recent advances in theories and models for Internet Quality of Service (QoS). We start with the theory of network calculus, which lays the foundation for support of deterministic performance guarantees in networks, and illustrate its applications to integrated services, differentiated services, and streaming media playback delays. We also present mechanisms and architecture for scalable support of guaranteed services in the Internet, based on the concept of a stateless core. Methods for scalable control operations are also briefly discussed. We then turn our attention to statistical performance guarantees, and describe several new probabilistic results that can be used for a statistical dimensioning of differentiated services. Lastly, we review recent proposals and results in supporting performance guarantees in a best effort context. These include models for elastic throughput guarantees based on TCP performance modeling, techniques for some quality of service differentiation without access control, and methods that allow an application to control the performance it receives, in the absence of network support

    Advances in Internet Quality of Service

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    We describe recent advances in theories and architecture that support performance guarantees needed for quality of service networks. We start with deterministic computations and give applications to integrated services, differentiated services, and playback delays. We review the methods used for obtaining a scalable integrated services support, based on the concept of a stateless core. New probabilistic results that can be used for a statistical dimensioning of differentiated services are explained; some are based on classical queuing theory, while others capitalize on the deterministic results. Then we discuss performance guarantees in a best effort context; we review: methods to provide some quality of service in a pure best effort environment; methods to provide some quality of service differentiation without access control, and methods that allow an application to control the performance it receives, in the absence of network support

    Bits of Internet traffic control

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    In this work, we consider four problems in the context of Internet traffic control. The first problem is to understand when and why a sender that implements an equation-based rate control would be TCP-friendly, or not—a sender is said to be TCP-friendly if, under the same operating conditions, its long-term average send rate does not exceed that of a TCP sender. It is an established axiom that some senders in the Internet would need to be TCP-friendly. An equation-based rate control sender plugs-in some on-line estimates of the loss-event rate and an expected round-trip time in a TCP throughput formula, and then at some points in time sets its send rate to such computed values. Conventional wisdom held that if a sender adjusts its send rate as just described, then it would be TCP-friendly. We show exact analysis that tells us when we should expect an equation-based rate control to be TCP-friendly, and in some cases excessively so. We show experimental evidence and identify the causes that, in a realistic scenario, make an equation-based rate control grossly non-TCP-friendly. Our second problem is to understand the throughput achieved by another family of send rate controls—we termed these "increase-decrease controls," with additive-increase/multiplicative-decrease as a special case. One issue that we consider is the allocation of long-term average send rates among senders that adjust their send rates by an additive-increase/multiplicative-decrease control, in a network of links with arbitrary fixed routes, and arbitrary round-trip times. We show what the resulting send rate allocation is. This result advances the state-of-the-art in understanding the fairness of the rate allocation in presence of arbitrary round-trip times. We also consider the design of an increase-decrease control to achieve a given target loss-throughput function. We show that if we design some increase-decrease controls under a commonly used reference loss process—a sequence of constant inter-loss event times—then we know that these controls would overshoot their target loss-throughput function, for some more general loss processes. A reason to study the design problem is to construct an increase-decrease control that would be friendly to some other control, TCP, for instance. The third problem that we consider is how to obtain probabilistic bounds on performance for nodes that conform to the per-hop-behavior of Expedited Forwarding, a service of differentiated services Internet. Under the assumption that the arrival process to a node consists of flows that are individually regulated (as it is commonplace with Expedited Forwarding) and the flows are stochastically independent, we obtained probabilistic bounds on backlog, delay, and loss. We apply our single-node performance bounds to a network of nodes. Having good probabilistic bounds on the performance of nodes that conform to the per-hop-behavior of Expedited Forwarding, would enable a dimensioning of those networks more effectively, than by using some deterministic worst-case performance bounds. Our last problem is on the latency of an input-queued switch that implements a decomposition-based scheduler. With decomposition-based schedulers, we are given a rate demand matrix to be offered by a switch in the long-term between the switch input/output port pairs. A given rate demand matrix is, by some standard techniques, decomposed into a set of permutation matrices that define the connectivity of the input/output port pairs. The problem is how to construct a schedule of the permutation matrices such that the schedule offers a small latency for each input/output port pair of the switch. We obtain bounds on the latency for some schedulers that are in many situations smaller than a best-known bound. It is important to be able to design switches with bounds on their latencies in order to provide guarantees on delay-jitter
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