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## Stochastic service curve and delay bound analysis: a single node case (2013)

Venue: | Computer Science from University of Kaiserslautern |

Citations: | 2 - 0 self |

### Citations

1007 | An architecture for differentiated services
- Blake, Black, et al.
- 1998
(Show Context)
Citation Context ...regate. We call (26) the “aggregate behavior” of the node, which is in consistence with the definition of the aggregate per-hop behavior [17] under the Differentiated Services (DiffServ) architecture =-=[1]-=-. With (26) and following the same proof of Theorem 1, the following result can be verified. Lemma 3. Consider a work-conserving system with constant capacity C, shared by a traversing flow f and a cr... |

408 |
Network Calculus: A Theory of Deterministic Queuing Systems for the Internet, volume LNCS 2050. Springer-Verlag, revised version 4, may 10, 2004 edition
- Boudec, Thiran
- 2001
(Show Context)
Citation Context ...g two tracks — deterministic and stochastic. Deterministic network calculus, coined by [10], has been extensively studied since its introduction in early 1990s, and is nicely covered by two books [5] =-=[27]-=-. Stochastic network calculus is the probabilistic extension or generalization of deterministic network calculus. The development of stochastic network calculus (SNC) began also in early 1990s. Early ... |

386 |
Queueing Systems, Volume 1: Theory
- Kleinrock
- 1975
(Show Context)
Citation Context ...ority system, for which, the classic queueing theory has exact result for the delay expectation of the low priority traffic. Note that, given the delay CCDF P{D ≥ τ}, the average delay is obtained as =-=[24]-=- E[D] = ∫ ∞ 0 P{D ≥ τ}dτ. 9Note that in real computer networks the time is discrete. For this reason, we have also assumed discrete time at the beginning. Nevertheless, this paper does not specify the... |

294 |
Performance Guarantees in Communication Networks
- Chang
- 2000
(Show Context)
Citation Context ...along two tracks — deterministic and stochastic. Deterministic network calculus, coined by [10], has been extensively studied since its introduction in early 1990s, and is nicely covered by two books =-=[5]-=- [27]. Stochastic network calculus is the probabilistic extension or generalization of deterministic network calculus. The development of stochastic network calculus (SNC) began also in early 1990s. E... |

154 | Notes on effective bandwidth,”
- Kelly
- 1996
(Show Context)
Citation Context ...node to the flow and the v.b.c SAC characterization of the flow are known, a delay bound can be readily obtained from the existing snetcal results. In the literature, while many results (e.g. [19][22]=-=[23]-=-) may be exploited to find the v.b.c SAC characterization of a flow, there are very few for SSC analysis. The difficulties are inherent in the SSC definition. Suppose S(t) is the service process provi... |

154 |
On Computing Per-Session Performance Bounds in High-Speed Multi-Hop Computer Networks.
- Kurose
- 1992
(Show Context)
Citation Context ... is the probabilistic extension or generalization of deterministic network calculus. The development of stochastic network calculus (SNC) began also in early 1990s. Early representative works include =-=[26]-=-[33][4] for traffic modeling, and [28] for server modeling. The book [5] also covers the theory of effective bandwidth, a first approach to SNC. However, due to challenges specific to stochastic netwo... |

150 | Performance and stability of communication networks via robust exponential bounds.
- Yaron, Sidi
- 1993
(Show Context)
Citation Context ...the probabilistic extension or generalization of deterministic network calculus. The development of stochastic network calculus (SNC) began also in early 1990s. Early representative works include [26]=-=[33]-=-[4] for traffic modeling, and [28] for server modeling. The book [5] also covers the theory of effective bandwidth, a first approach to SNC. However, due to challenges specific to stochastic networks,... |

115 | Stochastic Network Calculus
- Jiang, Liu
- 2008
(Show Context)
Citation Context ...ective bandwidth, a first approach to SNC. However, due to challenges specific to stochastic networks, it is recently that crucial network calculus properties have been proved for SNC, e.g. [3][8][29]=-=[16]-=-[31][22][12]. A selection of recent results can be found in the book [21]. In addition, three surveys/overviews are available [32][13][20]. In SNC, stochastic service curve is the fundamental concept ... |

106 |
queue length, and delay of deterministic and stochastic queueing networks
- Stability
- 1994
(Show Context)
Citation Context ...probabilistic extension or generalization of deterministic network calculus. The development of stochastic network calculus (SNC) began also in early 1990s. Early representative works include [26][33]=-=[4]-=- for traffic modeling, and [28] for server modeling. The book [5] also covers the theory of effective bandwidth, a first approach to SNC. However, due to challenges specific to stochastic networks, it... |

66 | An end-to-end probabilistic network calculus with moment generating functions,”
- Fidler
- 2006
(Show Context)
Citation Context ...idth, a first approach to SNC. However, due to challenges specific to stochastic networks, it is recently that crucial network calculus properties have been proved for SNC, e.g. [3][8][29][16][31][22]=-=[12]-=-. A selection of recent results can be found in the book [21]. In addition, three surveys/overviews are available [32][13][20]. In SNC, stochastic service curve is the fundamental concept for server m... |

61 | A network calculus with effective bandwidth,”
- Li, Burchard, et al.
- 2007
(Show Context)
Citation Context ... effective bandwidth, a first approach to SNC. However, due to challenges specific to stochastic networks, it is recently that crucial network calculus properties have been proved for SNC, e.g. [3][8]=-=[29]-=-[16][31][22][12]. A selection of recent results can be found in the book [21]. In addition, three surveys/overviews are available [32][13][20]. In SNC, stochastic service curve is the fundamental conc... |

51 | A network service curve approach for the stochastic analysis of networks
- Ciucu, Burchard, et al.
- 2005
(Show Context)
Citation Context ... of effective bandwidth, a first approach to SNC. However, due to challenges specific to stochastic networks, it is recently that crucial network calculus properties have been proved for SNC, e.g. [3]=-=[8]-=-[29][16][31][22][12]. A selection of recent results can be found in the book [21]. In addition, three surveys/overviews are available [32][13][20]. In SNC, stochastic service curve is the fundamental ... |

49 |
Communication Networking: An Analytical Approach,
- Kumar, Manjunath, et al.
- 2004
(Show Context)
Citation Context ...m, the challenge already exists. A. A Pitfall When the node has constant capacity C (in bps), the following equation has sometimes been wrongly believed in the network calculus literature (see, e.g., =-=[25]-=-), A∗(t) = inf 0≤s≤t {A(s) + C · (t− s)}. (11) Or in other words, it is wrongly believed, for the constant capacity node: S(s, t) = C · (t− s). (12) To give a counterexample, let’s consider a single p... |

42 | Performance bounds in communication networks with variable-rate links. - Lee - 1995 |

35 | A min-plus calculus for end-to-end statistical service guarantees.
- Burchard, Liebeherr, et al.
- 2006
(Show Context)
Citation Context ...ory of effective bandwidth, a first approach to SNC. However, due to challenges specific to stochastic networks, it is recently that crucial network calculus properties have been proved for SNC, e.g. =-=[3]-=-[8][29][16][31][22][12]. A selection of recent results can be found in the book [21]. In addition, three surveys/overviews are available [32][13][20]. In SNC, stochastic service curve is the fundament... |

35 | Quality of service management in integrated services networks.
- Cruz
- 1996
(Show Context)
Citation Context ...y be mapped [21]. Definition 1. A flow is said to have a v.b.c (virtual backlog centric) stochastic arrival curve α(t) with bounding function F̄ , if its arrival process A(t) satisfies, for any t ≥ 0 =-=[11]-=-[34][22], P{A(t)−A⊗ α(t) > x} ≤ F̄ (x) (6) where α(t) is non-negative non-decreasing on t, and F̄ (x) non-negative non-increasing on x. In Definition 1, if F̄ (0) = 0, implying F̄ (x) = 0 for all x ≥ ... |

34 | Analysis of Generalized Stochastically Bounded Bursty Traffic for Communication Networks.
- Yin, Jiang, et al.
- 2002
(Show Context)
Citation Context ... mapped [21]. Definition 1. A flow is said to have a v.b.c (virtual backlog centric) stochastic arrival curve α(t) with bounding function F̄ , if its arrival process A(t) satisfies, for any t ≥ 0 [11]=-=[34]-=-[22], P{A(t)−A⊗ α(t) > x} ≤ F̄ (x) (6) where α(t) is non-negative non-decreasing on t, and F̄ (x) non-negative non-increasing on x. In Definition 1, if F̄ (0) = 0, implying F̄ (x) = 0 for all x ≥ 0 or... |

29 | Delay bounds for a network of Guaranteed Rate servers with FIFO aggregation.
- Jiang
- 2002
(Show Context)
Citation Context ...eterministic) service curve of the server [27]. (ii) Within any backlog period of length t, the amount of service provided by a constant rate server with capacity C is lower bounded by (C · t− Lmax)+ =-=[14]-=-. Fundamentally, the following inequality can be proved [14]: A∗(t) ≥ A(t)⊗ (C · t− Lmax(t))+. (13) where Lmax(t) ≡ max{l1, l2, l(t)} with l(t) denoting the length of the most recent packet that arriv... |

27 |
A survey of deterministic and stochastic service curve models in the network calculus,”
- Fidler
- 2010
(Show Context)
Citation Context ...work calculus properties have been proved for SNC, e.g. [3][8][29][16][31][22][12]. A selection of recent results can be found in the book [21]. In addition, three surveys/overviews are available [32]=-=[13]-=-[20]. In SNC, stochastic service curve is the fundamental concept for server modeling. If some flows and servers are independent, it is expected that tighter analytical bounds can be obtained by makin... |

22 | Network calculus delay bounds in queueing networks with exact solutions
- Ciucu
- 2007
(Show Context)
Citation Context ...plicable to any type of traffic that has v.b.c stochastic arrival curve, which covers a wide range of traffic types [22]. For the examples, delay bound analysis of M/M/1 using snetcal can be found in =-=[6]-=-[18]. However, the technique used in this paper has fundamental difference from the techniques used in [6][18]. Particularly in [6][18], the analysis directly works on the arrival process and the serv... |

21 | Relationship between guaranteed rate server and latency rate server
- Jiang
- 2003
(Show Context)
Citation Context ...lowing result is crucial, which establishes a link between the stochastic service curve model and the virtual time function. For deterministic network calculus, a similar relationship can be found in =-=[15]-=- (Lemma 2). The proof is long and is included in the appendix. Lemma 1. Consider a flow f served by a system. For any time t > 0 and R > 0, the following relationship holds (for any sample path of the... |

20 | Perspectives on network calculus - No free lunch, but still good value.
- Ciucu, Schmitt
- 2012
(Show Context)
Citation Context ...tant capacity node provides a stochastic service curve C ·t with bounding function F̄L max(t)(x). Unfortunately, Lmax(t) is non-decreasing with t, implying that F̄Lmax(t)(x) may approach 1 as t grows =-=[9]-=-3. Consequently, using F̄Lmax(t) as a bounding function is meaningless. The problem becomes even more challenging when there is cross traffic. First, in order to apply the leftover service property to... |

14 | On superlinear scaling of network delays
- Burchard, Liebeherr, et al.
- 2011
(Show Context)
Citation Context ...ary 2 and Corollary 3 will lead to Theorem 6 by ignoring the packetization effect. In the general sense of taking packetization effect into stochastic service curve and delay bound analysis, the work =-=[2]-=- is most related. However, the obtained results in [2] are mostly functions of ∫∞ x F̄ l(y)dy, while in our results, they are related directly to F̄ l. In addition, how to make use of independence inf... |

14 | A survey of envelope processes and their applications in quality of service provisioning
- Mao, Panwar
- 2006
(Show Context)
Citation Context ... network calculus properties have been proved for SNC, e.g. [3][8][29][16][31][22][12]. A selection of recent results can be found in the book [21]. In addition, three surveys/overviews are available =-=[32]-=-[13][20]. In SNC, stochastic service curve is the fundamental concept for server modeling. If some flows and servers are independent, it is expected that tighter analytical bounds can be obtained by m... |

13 | Per-domain packet scale rate guarantee for Expedited Forwarding.
- Jiang
- 2003
(Show Context)
Citation Context ...between the two flows, and (26) is only concerned about the aggregate. We call (26) the “aggregate behavior” of the node, which is in consistence with the definition of the aggregate per-hop behavior =-=[17]-=- under the Differentiated Services (DiffServ) architecture [1]. With (26) and following the same proof of Theorem 1, the following result can be verified. Lemma 3. Consider a work-conserving system wi... |

12 | Delay bounds in communication networks with heavy-tailed and self-similar traffic
- Liebeherr, Burchard, et al.
- 2012
(Show Context)
Citation Context ...(C − θ) · t, for some θ ≥ 0, with a resultant bounding function related to ∫∞ x F̄ l(y)dy, by exploiting an approach used in SNC in dealing with maximal random processes [21]. Recently, the effort in =-=[30]-=- shows that, without compromising the service 3An exception is when all packets have the same length L or their lengths are upper-bounded by L. In this case, for any t, F̄Lmax(t)(x) ≤ 1 for all x ≤ L;... |

9 |
Fundamental Calculus on Generalized Stochastically Bounded Bursty Traffic for Communication Networks.
- Jiang, Yin, et al.
- 2009
(Show Context)
Citation Context ...the node to the flow and the v.b.c SAC characterization of the flow are known, a delay bound can be readily obtained from the existing snetcal results. In the literature, while many results (e.g. [19]=-=[22]-=-[23]) may be exploited to find the v.b.c SAC characterization of a flow, there are very few for SSC analysis. The difficulties are inherent in the SSC definition. Suppose S(t) is the service process p... |

9 | A calculus for stochastic QoS analysis
- Liu, Tham, et al.
(Show Context)
Citation Context ...ve bandwidth, a first approach to SNC. However, due to challenges specific to stochastic networks, it is recently that crucial network calculus properties have been proved for SNC, e.g. [3][8][29][16]=-=[31]-=-[22][12]. A selection of recent results can be found in the book [21]. In addition, three surveys/overviews are available [32][13][20]. In SNC, stochastic service curve is the fundamental concept for ... |

8 |
A calculus for network delay, part I and part II
- Cruz
- 1991
(Show Context)
Citation Context ...ervice processes using some bounds and perform analysis based on such bounds. Network calculus has developed along two tracks — deterministic and stochastic. Deterministic network calculus, coined by =-=[10]-=-, has been extensively studied since its introduction in early 1990s, and is nicely covered by two books [5] [27]. Stochastic network calculus is the probabilistic extension or generalization of deter... |

8 | Network calculus and queueing theory: Two sides of one coin
- Jiang
- 2009
(Show Context)
Citation Context .... The difficulties are inherent in the SSC definition. Suppose S(t) is the service process provided by the node to the flow. The following equation, called the min-plus convolution queueing principle =-=[18]-=-, holds [21]: A∗(t) = inf 0≤s≤t {A(s) + S(s, t)} (10) where S(t) denotes the service process provided to the flow and S(s, t) ≡ S(t)− S(s). Essentially, SSC defines a way to characterize the service p... |

6 | Scaling Properties in the Stochastic Network Calculus
- Ciucu
- 2007
(Show Context)
Citation Context ...e characterization of the system taking into consideration the packetization effect. For delay bound analysis of M/M/1/priority using snetcal, the same delay expectation bound as (64) may be found in =-=[7]-=-. However, beside the fundamental difference in the used analytical technique, the bound in [7] is derived under some additional conditions/assumptions, e.g., preemptive priority and ignoring the pack... |

4 | A note on applying stochastic network calculus
- Jiang
- 2010
(Show Context)
Citation Context ... by the node to the flow and the v.b.c SAC characterization of the flow are known, a delay bound can be readily obtained from the existing snetcal results. In the literature, while many results (e.g. =-=[19]-=-[22][23]) may be exploited to find the v.b.c SAC characterization of a flow, there are very few for SSC analysis. The difficulties are inherent in the SSC definition. Suppose S(t) is the service proce... |

4 | Stochastic network calculus for performance analysis of Internet networks—an overview and outlook,”
- Jiang
- 2012
(Show Context)
Citation Context ... calculus properties have been proved for SNC, e.g. [3][8][29][16][31][22][12]. A selection of recent results can be found in the book [21]. In addition, three surveys/overviews are available [32][13]=-=[20]-=-. In SNC, stochastic service curve is the fundamental concept for server modeling. If some flows and servers are independent, it is expected that tighter analytical bounds can be obtained by making us... |