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64
Stochastic Network Calculus
, 2008
"... A basic calculus is presented for stochastic service guarantee analysis in communication networks. Central to the calculus are two definitions, maximum(virtual)backlogcentric (m.b.c) stochastic arrival curve and stochastic service curve, which respectively generalize arrival curve and service c ..."
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Cited by 116 (23 self)
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A basic calculus is presented for stochastic service guarantee analysis in communication networks. Central to the calculus are two definitions, maximum(virtual)backlogcentric (m.b.c) stochastic arrival curve and stochastic service curve, which respectively generalize arrival curve and service curve in the deterministic network calculus framework. With m.b.c stochastic arrival curve and stochastic service curve, various basic results are derived under the (min, +) algebra for the general case analysis, which are crucial to the development of stochastic network calculus. These results include (i) superposition of flows, (ii) concatenation of servers, (iii) output characterization, (iv) perflow service under aggregation, and (v) stochastic backlog and delay guarantees. In addition, to perform independent case analysis, stochastic strict server is defined, which uses an ideal service process and an impairment process to characterize a server. The concept of stochastic strict server not only allows us to improve the basic results (i) – (v) under the independent case, but also provides a convenient way to find the stochastic service curve of a serve. Moreover, an approach is introduced to find the m.b.c stochastic arrival curve of a flow and the stochastic service curve of a server.
An EndtoEnd Probabilistic Network Calculus with Moment Generating Functions
, 2006
"... Network calculus is a minplus system theory for performance evaluation of queuing networks. Its elegance stems from intuitive convolution formulas for concatenation of deterministic servers. Recent research dispenses with the worstcase assumptions of network calculus to develop a probabilistic equi ..."
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Cited by 66 (5 self)
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Network calculus is a minplus system theory for performance evaluation of queuing networks. Its elegance stems from intuitive convolution formulas for concatenation of deterministic servers. Recent research dispenses with the worstcase assumptions of network calculus to develop a probabilistic equivalent that benefits from statistical multiplexing. Significant achievements have been made, owing for example to the theory of effective bandwidths, however, the outstanding scalability set up by concatenation of deterministic servers has not been shown. This paper establishes a concise, probabilistic network calculus with moment generating functions. The presented work features closedform, endtoend, probabilistic performance bounds that achieve the objective of scaling linearly in the number of servers in series. The consistent application of moment generating functions put forth in this paper utilizes independence beyond the scope of current statistical multiplexing of flows. A relevant additional gain is demonstrated for tandem servers with independent crosstraffic.
Theories and Models for Internet Quality of Service
, 2002
"... 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 serv ..."
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Cited by 64 (1 self)
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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.
A Network Calculus with Effective Bandwidth
, 2003
"... We present a statistical network calculus in a setting where both arrivals and service are specified interms of probabilistic bounds. We provide explicit bounds on delay, backlog, and output burstiness in a network. By formulating wellknown effective bandwidth expressions in terms of envelope func ..."
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Cited by 61 (10 self)
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We present a statistical network calculus in a setting where both arrivals and service are specified interms of probabilistic bounds. We provide explicit bounds on delay, backlog, and output burstiness in a network. By formulating wellknown effective bandwidth expressions in terms of envelope functions,we are able to apply our calculus to a wide range of traffic source models, including Fractional Brownian Motion. We present probabilistic lower bounds on the service for three scheduling algorithms: Static Priority (SP), Earliest Deadline First (EDF), and Generalized Processor Sharing (GPS).
Sequencing and routing in multiclass queueing networks part I: Feedback regulation
 SIAM J. Control Optim
"... Abstract. Part II continues the development of policy synthesis techniques for multiclass queueing networks based upon a linear fluid model. The following are shown: (i) A relaxation of the fluid model based on workload leads to an optimization problem of lower dimension. An analogous workloadrelax ..."
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Cited by 56 (12 self)
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Abstract. Part II continues the development of policy synthesis techniques for multiclass queueing networks based upon a linear fluid model. The following are shown: (i) A relaxation of the fluid model based on workload leads to an optimization problem of lower dimension. An analogous workloadrelaxation is introduced for the stochastic model. These relaxed control problems admit pointwise optimal solutions in many instances. (ii) A translation to the original fluid model is almost optimal, with vanishing relative error as the networkload ρ approaches one. It is pointwise optimal after a short transient period, provided a pointwise optimal solution exists for the relaxed control problem. (iii) A translation of the optimal policy for the fluid model provides a policy for the stochastic networkmodel that is almost optimal in heavy traffic, over all solutions to the relaxed stochastic model, again with vanishing relative error. The regret is of order  log(1 − ρ).
A network service curve approach for the stochastic analysis of networks
 IN PROCEEDINGS OF ACM SIGMETRICS
, 2005
"... The stochastic network calculus is an evolving new methodology for backlog and delay analysis of networks that can account for statistical multiplexing gain. This paper advances the stochastic network calculus by deriving a network service curve, which expresses the service given to a flow by the ne ..."
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Cited by 51 (3 self)
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The stochastic network calculus is an evolving new methodology for backlog and delay analysis of networks that can account for statistical multiplexing gain. This paper advances the stochastic network calculus by deriving a network service curve, which expresses the service given to a flow by the network as a whole in terms of a probabilistic bound. The presented network service curve permits the calculation of statistical endtoend delay and backlog bounds for broad classes of arrival and service distributions. The benefits of the derived service curve are illustrated for the exponentially bounded burstiness (EBB) traffic model. It is shown that endtoend performance measures computed with a network service curve are bounded by O (H log H), where H is the number of nodes traversed by a flow. Using currently available techniques that compute endtoend bounds by adding single node results, the corresponding performance measures are bounded by O(H³).
Scaling Properties of Statistical Endtoend Bounds in the Network Calculus
"... The stochastic network calculus is an evolving new methodology for backlog and delay analysis of networks that can account for statistical multiplexing gain. This paper advances the stochastic network calculus by deriving a network service curve, which expresses the service given to a flow by the n ..."
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Cited by 43 (21 self)
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The stochastic network calculus is an evolving new methodology for backlog and delay analysis of networks that can account for statistical multiplexing gain. This paper advances the stochastic network calculus by deriving a network service curve, which expresses the service given to a flow by the network as a whole in terms of a probabilistic bound. The presented network service curve permits the calculation of statistical endtoend delay and backlog bounds for broad classes of arrival and service distributions. The benefits of the derived service curve are illustrated for the exponentially bounded burstiness (EBB) traffic model. It is shown that endtoend performance measures computed with a network service curve are bounded by O (H log H), where H is the number of nodes traversed by a flow. Using currently available techniques, which compute endtoend bounds by adding single node results, the corresponding performance measures are bounded by O (H³).
A Calculus for Endtoend Statistical Service Guarantees
, 2001
"... The deterministic network calculus offers an elegant framework for determining delays and backlog in a network with deterministic service guarantees to individual traffic flows. A drawback of the deterministic network calculus is that it only provides worstcase bounds. Here we present a network cal ..."
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Cited by 38 (6 self)
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The deterministic network calculus offers an elegant framework for determining delays and backlog in a network with deterministic service guarantees to individual traffic flows. A drawback of the deterministic network calculus is that it only provides worstcase bounds. Here we present a network calculus for statistical service guarantees, which can exploit the statistical multiplexing gain of sources. We introduce the notion of an effective service curve as a probabilistic bound on the service received by an individual flw, and construct an effective service curve for a network where capacities are provisioned exclusively to aggregates of flows. Numerical examples demonstrate that the calculus is able to extract a significant amount of multiplexing gain in networks with a large number of flows.
A minplus calculus for endtoend statistical service guarantees
 IEEE TRANSACTION ON INFORMATION THEORY
, 2006
"... The network calculus offers an elegant framework for determining worstcase bounds on delay and backlog in a network. This paper extends the network calculus to a probabilistic framework with statistical service guarantees. The notion of a statistical service curve is presented as a probabilistic b ..."
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Cited by 35 (5 self)
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The network calculus offers an elegant framework for determining worstcase bounds on delay and backlog in a network. This paper extends the network calculus to a probabilistic framework with statistical service guarantees. The notion of a statistical service curve is presented as a probabilistic bound on the service received by an individual flow or an aggregate of flows. The problem of concatenating pernode statistical service curves to form an endtoend (network) statistical service curve is explored. Two solution approaches are presented that can each yield statistical network service curves. The first approach requires the availability of time scale bounds at which arrivals and departures at each node are correlated. The second approach considers a service curve that describes service over time intervals. Although the latter description of service is less general, it is argued that many practically relevant service curves may be compliant to this description.
Delay Bounds under Arbitrary Multiplexing
, 2007
"... Network calculus has proven as a valuable and versatile methodology for worstcase analysis of communication networks. One issue in which it is still lacking is the treatment of aggregate multiplexing, in particular if the FIFO property cannot be assumed when flows are merged. In this report, we add ..."
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Cited by 28 (8 self)
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Network calculus has proven as a valuable and versatile methodology for worstcase analysis of communication networks. One issue in which it is still lacking is the treatment of aggregate multiplexing, in particular if the FIFO property cannot be assumed when flows are merged. In this report, we address the problem of bounding the delay of individual traffic flows in feedforward networks under arbitrary multiplexing. Somewhat surprisingly, we find that direct application of network calculus results in loose bounds even in seemingly simple scenarios. The reasons for this failure of network calculus are discussed in detail and a method to arrive at tight delay bounds for arbitrary (aggregate) multiplexing is presented. This method is based on the solution of an optimization problem. For the special case of sinktree networks this optimization problem is solved explicitly, thus arriving at a closedform expression for the delay bound. Numerical experiments illustrate that in sinktree networks the improvement over bounds based on direct application of network calculus can be considerable.