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25
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.
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.
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.
A Calculus for Stochastic QoS Analysis
, 2006
"... The issue of Quality of Service (QoS) performance analysis in packetswitched networks has drawn a lot of attention in the networking community. There is a lot of work including an elegant theory under the name of network calculus, which focuses on analysis of deterministic worst case QoS performanc ..."
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Cited by 9 (7 self)
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The issue of Quality of Service (QoS) performance analysis in packetswitched networks has drawn a lot of attention in the networking community. There is a lot of work including an elegant theory under the name of network calculus, which focuses on analysis of deterministic worst case QoS performance bounds. In the meantime, researchers have studied stochastic QoS performance for specific schedulers. However, most previous works on deterministic QoS analysis or stochastic QoS analysis have only considered a server that provides deterministic service, i.e. deterministically bounded rate service. Few have considered the behavior of a stochastic server that provides input flows with variable rate service, for example wireless links. In this paper, we propose a stochastic network calculus to analyze the endtoend stochastic QoS performance of a system with stochastically bounded input traffic over a series of deterministic and stochastic servers. We also prove that a server serving an aggregate of flows can be regarded as a stochastic server for individual flows within the aggregate. Based on this, the proposed framework is further applied to analyze perflow stochastic QoS performance under aggregate scheduling.
A (min, ×) Network Calculus for MultiHop Fading Channels
, 2013
"... A fundamental problem for the delay and backlog analysis across multihop paths in wireless networks is how to account for the random properties of the wireless channel. Since the usual statistical models for radio signals in a propagation environment do not lend themselves easily to a description ..."
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Cited by 7 (1 self)
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A fundamental problem for the delay and backlog analysis across multihop paths in wireless networks is how to account for the random properties of the wireless channel. Since the usual statistical models for radio signals in a propagation environment do not lend themselves easily to a description of the available service rate, the performance analysis of wireless networks has resorted to higherlayer abstractions, e.g., using Markov chain models. In this work, we propose a network calculus that can incorporate common statistical models of fading channels and obtain statistical bounds on delay and backlog across multiple nodes. We conduct the analysis in a transfer domain, which we refer to as the SNR domain, where the service process at a link is characterized by the instantaneous signaltonoise ratio at the receiver. We discover that, in the transfer domain, the network model is governed by a dioid algebra, which we refer to as (min, ×) algebra. Using this algebra we derive the desired delay and backlog bounds. An application of the analysis is demonstrated for a simple multihop network with Rayleigh fading channels.
Nonasymptotic Delay Bounds for Networks with HeavyTailed Traffic
, 2010
"... Traffic with selfsimilar and heavytailed characteristics has been widely reported in networks, yet, only few analytical results are available for predicting the delay performance of such networks. We address a particularly difficult type of heavytailed traffic where only the first moment can be ..."
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Cited by 6 (5 self)
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Traffic with selfsimilar and heavytailed characteristics has been widely reported in networks, yet, only few analytical results are available for predicting the delay performance of such networks. We address a particularly difficult type of heavytailed traffic where only the first moment can be computed, and present the first 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. 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 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.
Scaling Properties in the Stochastic Network Calculus
, 2007
"... Modern networks have become increasingly complex over the past years in terms of control algorithms, applications and service expectations. Since classical theories for the analysis of telephone networks were found inadequate to cope with these complexities, new analytical tools have been conceived ..."
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Cited by 6 (2 self)
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Modern networks have become increasingly complex over the past years in terms of control algorithms, applications and service expectations. Since classical theories for the analysis of telephone networks were found inadequate to cope with these complexities, new analytical tools have been conceived as of late. Among these, the stochastic network calculus has given rise to the optimism that it can emerge as an elegant mathematical tool for assessing network performance. This thesis argues that the stochastic network calculus can provide new analytical insight into the scaling properties of network performance metrics. In this sense it is shown that endtoend delays grow as Θ(H log H) in the number of network nodes H, as opposed to the Θ(H) order of growth predicted by other theories under simplifying assumptions. It is also shown a comparison between delay bounds obtained with the stochastic network calculus and exact results available in some productform queueing networks. The main technical contribution of this thesis is a construction of a statistical network service curve that expresses the service given to a flow by a network as if the flow traversed a single node only. This network service curve enables the proof of the O(H log H) scaling
On the flowlevel delay of a spatial multiplexing MIMO wireless channel
 In Proc. IEEE ICC
, 2011
"... Abstract—The MIMO wireless channel offers a rich ground for quality of service analysis. In this work, we present a stochastic network calculus analysis of a MIMO system, operating in spatial multiplexing mode, using moment generating functions (MGF). We quantify the spatial multiplexing gain, achie ..."
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Cited by 6 (2 self)
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Abstract—The MIMO wireless channel offers a rich ground for quality of service analysis. In this work, we present a stochastic network calculus analysis of a MIMO system, operating in spatial multiplexing mode, using moment generating functions (MGF). We quantify the spatial multiplexing gain, achieved through multiple antennas, for flow level quality of service (QoS) performance. Specifically we use GilbertElliot model to describe individual spatial paths between the antenna pairs and model the whole channel by an NState Markov Chain, where N depends upon the degrees of freedom available in the MIMO system. We derive probabilistic delay bounds for the system and show the impact of increasing the number of antennas on the delay bounds under various conditions, such as channel burstiness, signal strength and fading speed. Further we present results for multihop scenarios under statistical independence. I.
Stochastic service guarantee analysis based on timedomain models
 In Proc. MASCOTS
, 2009
"... Abstract—Stochastic network calculus is a theory for stochastic service guarantee analysis of computer communication networks. In the current stochastic network calculus literature, its traffic and server models are typically based on the cumulative amount of traffic and cumulative amount of service ..."
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Cited by 6 (4 self)
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Abstract—Stochastic network calculus is a theory for stochastic service guarantee analysis of computer communication networks. In the current stochastic network calculus literature, its traffic and server models are typically based on the cumulative amount of traffic and cumulative amount of service respectively. However, there are network scenarios where the applicability of such models is limited, and hence new ways of modeling traffic and service are needed to address this limitation. This paper presents timedomain models and results for stochastic network calculus. Particularly, we define traffic models, which are based on probabilistic lowerbounds on cumulative packet interarrival time, and server models, which are based on probabilistic upperbounds on cumulative packet service time. In addition, examples demonstrating the use of the proposed timedomain models are provided. On the basis of the proposed models, the five basic properties of stochastic network calculus are also proved, which implies broad applicability of the proposed timedomain approach. I.