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65
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 118 (22 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
 in Proc. IEEE 14th International Workshop on Quality of Servic (IWQoS
"... Abstract — 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 probabi ..."
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Cited by 70 (5 self)
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Abstract — 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. I.
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 54 (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 46 (23 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 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 38 (9 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.
Analysis of stochastic service guarantees in communication networks: A server model
 In Proc. of the International Workshop on Quality of Service (IWQoS 2005
, 2005
"... Abstract. The arrival curve has been used as a powerful concept for deterministic service guarantee analysis in communication networks. Since many applications and networks do not require or provide deterministic service guarantees, stochastic service guarantee analysis is becoming increasingly impo ..."
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Cited by 29 (10 self)
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Abstract. The arrival curve has been used as a powerful concept for deterministic service guarantee analysis in communication networks. Since many applications and networks do not require or provide deterministic service guarantees, stochastic service guarantee analysis is becoming increasingly important and has attracted a lot of research attention in recent years. For this, several probabilistic versions of the arrival curve have been proposed in the literature. They extend the concept of arrival curve to the stochastic case based on its traffic amount property. In this paper, we explore another property, called the virtual backlog property, of an arrival curve. Based on the virtual backlog property, we introduce the concept of virtualbacklogcentric (v.b.c) stochastic arrival curve to facilitate the analysis of stochastic service guarantees. We prove that a v.b.c stochastic arrival curve has a similar duality as a (deterministic) arrival curve. With the concept of v.b.c stochastic arrival curve, we derive results for stochastic service guarantee analysis of systems with the timevarying setting. In addition, we prove that many wellknown types of traffic can be readily represented using v.b.c stochastic arrival curves.
Network calculus delay bounds in queueing networks with exact solutions
 In 20th International Teletraffic Congress (ITC
, 2007
"... Abstract. The purpose of this paper is to shed light on the accuracy of probabilistic delay bounds obtained with network calculus. In particular, by comparing calculus bounds with exact results in a series of M/M/1 queues with cross traffic, we show that reasonably accurate bounds are achieved when ..."
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Cited by 23 (11 self)
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Abstract. The purpose of this paper is to shed light on the accuracy of probabilistic delay bounds obtained with network calculus. In particular, by comparing calculus bounds with exact results in a series of M/M/1 queues with cross traffic, we show that reasonably accurate bounds are achieved when the percentage of cross traffic is low. We use recent results in network calculus and, in addition, propose novel bounds based on Doob’s maximal inequality for supermartingales. In the case of single M/M/1 and �when M/D/1 queues, our results improve existing bounds by the utilization factor ρ converges to one. 1 Ω�log(1−ρ) −1
On Superlinear Scaling of Network Delays
"... We investigate scaling properties of endtoend delays in packet networks for a flow that traverses a sequence of H nodes and that experiences cross traffic at each node. When the traffic flow and the cross traffic do not satisfy independence assumptions, we find that delay bounds scale faster than ..."
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Cited by 14 (7 self)
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We investigate scaling properties of endtoend delays in packet networks for a flow that traverses a sequence of H nodes and that experiences cross traffic at each node. When the traffic flow and the cross traffic do not satisfy independence assumptions, we find that delay bounds scale faster than linearly. More precisely, for exponentially bounded packetized traffic we show that delays grow with Θ(H log H) in the number of nodes on the network path. This superlinear scaling of delays is qualitatively different from the scaling behavior predicted by a worstcase analysis or by a probabilistic analysis assuming independence of traffic arrivals at network nodes.
Delay bounds in communication networks with heavytailed and selfsimilar traffic
 IEEE Transactions on Information Theory
, 2012
"... Traffic with selfsimilar and heavytailed characteristics has been widely reported in communication networks, yet, the stateoftheart of analytically predicting the delay performance of such networks is lacking. We address a particularly difficult type of heavytailed traffic where only the first ..."
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Cited by 12 (3 self)
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Traffic with selfsimilar and heavytailed characteristics has been widely reported in communication networks, yet, the stateoftheart of analytically predicting the delay performance of such networks is lacking. We address a particularly difficult type of heavytailed traffic where only the first moment can be computed, and present nonasymptotic endtoend delay bounds for such traffic. The derived performance bounds are nonasymptotic in that they do not assume a steady state, large buffer, or many sources regime. The analysis follows a network calculus approach where traffic is characterized by envelope functions and service is described by service curves. Our analysis considers a multihop path of fixedcapacity links with heavytailed selfsimilar cross traffic at each node. A key contribution of the analysis is a novel probabilistic samplepath bound for heavytailed arrival and service processes, which is based on a scalefree sampling method. We explore how delays scale as a function of the length of the path, and compare them with lower bounds. A comparison with simulations illustrates pitfalls when simulating selfsimilar heavytailed traffic, providing further evidence for the need of analytical bounds. I.
Network calculus and queueing theory: two sides of one coin
 in Proc. 4th International Conference on Performance Evaluation Methodologies and Tools (VALUETOOLS
, 2009
"... Network calculus is a theory dealing with queueing type problems encountered in computer networks, with particular focus on quality of service guarantee analysis. Queueing theory is the mathematical study of queues, proven to be applicable to a wide area of problems, generally concerning about the ( ..."
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Cited by 11 (5 self)
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Network calculus is a theory dealing with queueing type problems encountered in computer networks, with particular focus on quality of service guarantee analysis. Queueing theory is the mathematical study of queues, proven to be applicable to a wide area of problems, generally concerning about the (average) quantities in an equilibrium state. Since both network calculus and queueing theory are analytical tools for studying queues, a question arises naturally as is if and where network calculus and queueing theory meet. In this paper, we explore queueing principles that underlie network calculus and exemplify their use. Particularly, based on the network calculus queueing principles, we show that for GI/GI/1, similar inequalities in the theory of queues can be derived. In addition, we prove that the endtoend performance of a tandem network is independent of the order of servers in the network even under some general settings. Through these, we present a network calculus perspective on queues and relate network calculus to queueing theory. 1.