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Network calculus delay bounds in queueing networks with exact solutions (2007)

by F Ciucu
Venue:In LNCS 4516, ITC 2007
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Perspectives on Network Calculus -- No Free Lunch, but Still Good Value

by Florin Ciucu, Jens Schmitt , 2012
"... ACM Sigcomm 2006 published a paper [26] which was perceived to unify the deterministic and stochastic branches of the network calculus (abbreviated throughout as DNC and SNC) [39]. Unfortunately, this seemingly fundamental unification—which has raised the hope of a straightforward transfer of all re ..."
Abstract - Cited by 20 (11 self) - Add to MetaCart
ACM Sigcomm 2006 published a paper [26] which was perceived to unify the deterministic and stochastic branches of the network calculus (abbreviated throughout as DNC and SNC) [39]. Unfortunately, this seemingly fundamental unification—which has raised the hope of a straightforward transfer of all results from DNC to SNC—is invalid. To substantiate this claim, we demonstrate that for the class of stationary andergodic processes, whichis prevalentin traffic modelling, the probabilistic arrival model from [26] is quasideterministic, i.e., the underlying probabilities are either zero or one. Thus, the probabilistic framework from [26] is unable to account for statistical multiplexing gain, which is in fact the raison d’être of packet-switched networks. Other previous formulations of SNC can capture statistical multiplexing

On expressing networks with flow transformation in convolution-form

by Florin Ciucu, T-labs Tu Berlin, Jens Schmitt, Hao Wang, Tu Kaiserslautern - In Proceedings of IEEE INFOCOM , 1979
"... Abstract—Convolution-form networks have the property that the end-to-end service of network flows can be expressed in terms of a (min;+)-convolution of the per-node services. This property is instrumental for deriving end-to-end queueing results which fundamentally improve upon alternative results d ..."
Abstract - Cited by 11 (6 self) - Add to MetaCart
Abstract—Convolution-form networks have the property that the end-to-end service of network flows can be expressed in terms of a (min;+)-convolution of the per-node services. This property is instrumental for deriving end-to-end queueing results which fundamentally improve upon alternative results derived by a node-by-node analysis. This paper extends the class of convolution-form networks with stochastic settings to scenarios with flow transformations, e.g., by loss, dynamic routing or retransmissions. In these networks, it is shown that by using the tools developed in this paper end-to-end delays grow as O(n) in the number of nodes n; in contrast, by using the alternative node-by-node analysis, end-to-end delays grow as O (n2). I.

1 Characterizing the Impact of the Workload on the Value of Dynamic Resizing in Data Centers

by Kai Wang, Student Member, Minghong Lin, Student Member, Florin Ciucu, Adam Wierman, Chuang Lin, Senior Member
"... Abstract—Energy consumption imposes a significant cost for data centers; yet much of that energy is used to maintain excess service capacity during periods of predictably low load. Resultantly, there has recently been interest in developing designs that allow the service capacity to be dynamically r ..."
Abstract - Cited by 10 (2 self) - Add to MetaCart
Abstract—Energy consumption imposes a significant cost for data centers; yet much of that energy is used to maintain excess service capacity during periods of predictably low load. Resultantly, there has recently been interest in developing designs that allow the service capacity to be dynamically resized to match the current workload. However, there is still much debate about the value of such approaches in real settings. In this paper, we show that the value of dynamic resizing is highly dependent on statistics of the workload process. In particular, both slow time-scale non-stationarities of the workload (e.g., the peak-to-mean ratio) and the fast time-scale stochasticity (e.g., the burstiness of arrivals) play key roles. To illustrate the impact of these factors, we combine optimization-based modeling of the slow time-scale with stochastic modeling of the fast time scale. Within this framework, we provide both analytic and numerical results characterizing when dynamic resizing does (and does not) provide benefits.
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...hat the factor jρ − λ ¯ D is negative). The difference between the exact result (evaluated using a sophisticated numerical algorithm [20]) and the bound from Eq. (12) is almost indistinguishable. See =-=[12]-=- for a detailed evaluation of the tightness. Moreover, unlike the exact result, the closedform and explicit formula from Eq. (13) readily leads to qualitative properties on the scaling of the capacity...

Network calculus and queueing theory: two sides of one coin

by Yuming Jiang - 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 ( ..."
Abstract - Cited by 8 (3 self) - Add to MetaCart
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 end-to-end 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.
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...nerating function was initially made in [11] where a Chernoff delay bound for the single node case is implied. In addition, the analysis in Section 5.1 resembles much the single node case analysis in =-=[8]-=-, even though the final expression of the bounds and the condition for them have small difference in [11]. Nevertheless, there is a conceptual difference in deriving these bounds. In [8], the approach...

Non-asymptotic throughput and delay distributions in multi-hop wireless networks

by Florin Ciucu, Oliver Hohlfeld, Pan Hui - In Allerton Conference on Communications, Control and Computing , 2010
"... Abstract—The class of Gupta-Kumar results give the asymp-totic throughput in multi-hop wireless networks but cannot predict the throughput behavior in networks of typical size. This paper addresses the non-asymptotic analysis of the multi-hop wireless communication problem and provides, for the firs ..."
Abstract - Cited by 8 (4 self) - Add to MetaCart
Abstract—The class of Gupta-Kumar results give the asymp-totic throughput in multi-hop wireless networks but cannot predict the throughput behavior in networks of typical size. This paper addresses the non-asymptotic analysis of the multi-hop wireless communication problem and provides, for the first time, closed-form results on multi-hop throughput and delay distributions. The results are non-asymptotic in that they hold for any number of nodes and also fully account for transient regimes, i.e., finite time scales, delays, as well as bursty arrivals. Their accuracy is supported by the recovery of classical single-hop results, and also by simulations from empirical data sets with realistic mobility settings. Moreover, for a specific network scenario and a fixed pair of nodes, the results confirm Gupta-Kumar’s Ω
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...t can be computed as in Theorem 1. PROOF. Using Eq. (5) we have for t, σ 0 Pr (D(t) A(t) σ) Pr ( sup 0≤s≤t fA(s, t) t+ s+ V (s, t)g σ ) . (15) To bound the sample-path probability we follow =-=[10]-=-. First note that the processes A(s, t) and V (s, t) are statistically independent, since the definition of V (t) is independent of the actual arrival process A(t). Let r, θ > 0 such that E [ eθ(A(1)+...

On the Model Transform in Stochastic Network Calculus

by Kui Wu, Yuming Jiang, Jie Li
"... Stochastic network calculus requires special care in the search of proper stochastic traffic arrival models and stochastic service models. Tradeoff must be considered between the feasibility for the analysis of performance bounds, the usefulness of performance bounds, and the ease of their numerical ..."
Abstract - Cited by 5 (0 self) - Add to MetaCart
Stochastic network calculus requires special care in the search of proper stochastic traffic arrival models and stochastic service models. Tradeoff must be considered between the feasibility for the analysis of performance bounds, the usefulness of performance bounds, and the ease of their numerical calculation. In theory, transform between different traffic arrival models and transform between different service models are possible. Nevertheless, the impact of the model transform on performance bounds has not been thoroughly investigated. This paper is to investigate the effect of the model transform and to provide practical guidance in the model selection in stochastic network calculus.
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...trange that most papers on stochastic network calculus mainly focus on the theoretical development. Various types of calculus are proposed to analyze the performance bounds in a stochastic sense [4], =-=[3]-=-, [6], [12], [11], [17]. Different approaches have been used, for example, the effective bandwidth [4], [17], moment generating functions [6], Martingale inequality [11]. In [10], a stochastic network...

Stochastic Network Calculus for Performance Analysis of Internet Networks – An Overview and Outlook

by Yuming Jiang
"... Abstract—Stochastic network calculus is a theory for performance guarantee analysis of Internet networks. Originated in early 1990s, stochastic network calculus has its foundation on the min-plus convolution and max-plus convolution queueing principles. Although challenging, it has shown tremendous ..."
Abstract - Cited by 4 (1 self) - Add to MetaCart
Abstract—Stochastic network calculus is a theory for performance guarantee analysis of Internet networks. Originated in early 1990s, stochastic network calculus has its foundation on the min-plus convolution and max-plus convolution queueing principles. Although challenging, it has shown tremendous potential in dealing with queueing type problems encountered in Internet networks. By focusing on bounds, stochastic network calculus compliments the classical queueing theory. This paper provides an overview of stochastic network calculus from the queueing principle perspective and presents an outlook by discussing crucial yet still open challenges in the area. I.
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...nes. 1) GI/GI/c Analysis: Recall that snetcal only provides bounds for the results. In the classical queueing theory, GI/GI/c has been thoroughly studied. For GI/GI/1, it has been shown (e.g. in [23] =-=[34]-=-) that snetcal analysis can lead to bounds that are similar to the Kingman’s inequality [35] in queueing theory, and that the snetcal delay bound for M/M/1 is tight in the sense that it matches with t...

A Case for Decomposition of FIFO Networks

by Florin Ciucu, Jörg Liebeherr
"... Recent studies showing that the output of traffic flows at packet switches has similar characteristics as the corresponding input enables a decomposition analysis of a network where nodes can be studied in isolation, thus simplifying an endto-end analysis of networks. However, network decomposition ..."
Abstract - Cited by 3 (2 self) - Add to MetaCart
Recent studies showing that the output of traffic flows at packet switches has similar characteristics as the corresponding input enables a decomposition analysis of a network where nodes can be studied in isolation, thus simplifying an endto-end analysis of networks. However, network decomposition results available today are mostly many-sources asymptotics. In this paper we explore the viability of network decomposition in a non-asymptotic regime with a finite number of flows. For traffic with Exponentially Bounded Burstiness (EBB) we derive statistical bounds for the output traffic at a FIFO buffer and compare them with bounds on the input. Evaluating the accuracy of the output bounds with exact results available for special cases and by numerical illustrations we find that conditions for network decomposition appear favorable even if the number of flows is relatively small.
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...y to a suitable constructed supermartingale. More exactly, we adapt the proof for the derivation of backlog bounds in GI/GI/1 queues from [19] to a network calculus setting, in a similar manner as in =-=[8]-=-. PROOF. Let J be the set of cross flows, and let the entire arrival process at the node AN (t) = A(t) + ∑ j∈J Aj(t). Following [13], the process S(s, t) = Nc(t − s) − ∑ Aj(s, t) j∈J is a statistical ...

Sharp Per-Flow Delay Bounds for Bursty Arrivals: The Case of FIFO, SP, and EDF Scheduling

by Florin Ciucu, Felix Poloczek, Jens Schmitt
"... The practicality of the stochastic network calculus (SNC) is often questioned on grounds of potential looseness of its performance bounds. In this paper, it is uncovered that for bursty arrival processes (specifically Markov-Modulated On-Off (MMOO)), whose amenability to per-flow analysis is typica ..."
Abstract - Cited by 3 (3 self) - Add to MetaCart
The practicality of the stochastic network calculus (SNC) is often questioned on grounds of potential looseness of its performance bounds. In this paper, it is uncovered that for bursty arrival processes (specifically Markov-Modulated On-Off (MMOO)), whose amenability to per-flow analysis is typically proclaimed as a highlight of SNC, the bounds can unfortunately be very loose (e.g., by several orders of magnitude off). In response to this uncovered weakness of SNC, the (Standard) per-flow bounds are herein improved by deriving a general sample-path bound, using martingale based techniques, which accommodates FIFO, SP, and EDF scheduling. The obtained (Martingale) bounds capture an extra exponential decay factor of O
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...t)’s are rather uncorrelated (see Talagrand [39]). Shroff and Schwartz [37] argued that the effective bandwidth theory yields reasonable bounds only for Poisson processes. Along the same lines, Ciucu =-=[9]-=- provided numerical evidence that SNC itself renders reasonably tight bounds for Poisson arrivals. III. QUEUEING MODEL This section introduces the queueing model and necessary SNC formalisms. For the ...

Exponential Supermartingales for Evaluating End-to-End Backlog Bounds

by Florin Ciucu , 2007
"... A common problem arising in network performance analysis with the stochastic network calculus is the evaluation of (min, +) convolutions. This paper presents a method to solve this problem by applying a maximal inequality to a suitable constructed supermartingale. For a network with D/M input, end-t ..."
Abstract - Cited by 2 (0 self) - Add to MetaCart
A common problem arising in network performance analysis with the stochastic network calculus is the evaluation of (min, +) convolutions. This paper presents a method to solve this problem by applying a maximal inequality to a suitable constructed supermartingale. For a network with D/M input, end-toend backlog bounds obtained with this method improve existing results at low utilizations. For the same network, it is shown that at utilizations smaller than a certain threshold, fluid-flow models may lead to inaccurate approximations of packetized models.
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... > e θx ) ≤ E[e θX0 ]e −θx , where e θXs is a supermartingale. This technique is reminiscent of the derivation of backlog bounds in GI/GI/1 queues, in a classic note by Kingman [10]. In previous work =-=[5]-=-, we have shown in a network calculus setting that Doob’s inequality leads to improved bounds than those obtained with Boole’s inequality, in the single-node case. In the multi-node case, multiple (mi...

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