## On Deterministic Traffic Regulation and Service Guarantees: A Systematic Approach by Filtering (1997)

Venue: | IEEE Transactions on Information Theory |

Citations: | 62 - 4 self |

### BibTeX

@ARTICLE{Chang97ondeterministic,

author = {Cheng-shang Chang},

title = {On Deterministic Traffic Regulation and Service Guarantees: A Systematic Approach by Filtering},

journal = {IEEE Transactions on Information Theory},

year = {1997},

volume = {44},

pages = {1096--1107}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper, we develop a filtering theory for deterministic traffic regulation and service guarantees under the (min; +)-algebra. We show that traffic regulators that generate f-upper constrained outputs can be implemented optimally by a linear time invariant filter with the impulse response f under the (min; +)-algebra, where f is the subadditive closure defined in the paper. Analogous to the classical filtering theory, there is an associate calculus, including feedback, concatenation, "filter bank summation" and performance bounds. The calculus is also applicable to the recently developed concept of service curves that can be used for deriving deterministic service guarantees. Our filtering approach not only yields easier proofs for more general results than those in the literature, but also allows us to design traffic regulators via systematic methods such as concatenation, filter bank summation, linear system realization, and FIR-IIR realization. We illustrate the use of ...

### Citations

3267 | Convex analysis
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Citation Context ... \Gammad 2 )( )] i : (38) Note that ~ g is known as the concave envelope of g?O d\Gammad 1 \Gammad 2 and ~ g is the smallest concave function that is not less than g ?O d\Gammad 1 \Gammad 2 (see e.g. =-=[20]-=-). We first show such an allocation guarantees the overall service curve g ? O d . Note that the overall service curve is ~ g ? O d1 ? ~ g ? O d2 = ~ g ? O d1 +d2 : Thus, the overall service curve is ... |

1673 | A generalized processor sharing approach to flow control in integrated services networks ⎯ the single node case
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- 1993
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Citation Context ...ks, we propose in this paper a general filtering approach for the traffic regulation problem. Our approach is also applicable to the concept of service curve recently developed by Parekh and Gallager =-=[19]-=-, Cruz [12] and Sariowan [21]. Our approach is built upon filtering under the (min; +)-algebra. In such an algebra, one replaces the usual addition operator by the min operator and the usual multiplic... |

722 |
Inequalities: Theory of Majorization and Its Applications
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- 1979
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Citation Context ...e in the family of sequences F 1 = ff : f 2 F 0 ; f(s) s f(t) t ; 8sstg: (21) Proof. Note that for all f 2 F 1 , \Gammaf is star-shaped, i.e., \Gammaf (t)=t is increasing in t and f(0) = 0 (see e.g., =-=[18]-=- pp. 453). Since a star-shaped function is superadditive, f 2 F 1 is subadditive. This can also be seen directly from f(s) + f(t \Gamma s)ss t f(t) + t \Gamma s t f(t) = f(t); 8sst: 13 Thus, f is suba... |

557 |
A calculus for network delay, part I: Network elements in isolation
- Cruz
- 1991
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Citation Context ...tworks aim to serve integrated traffic, such as voice, video, fax, and so forth. To control interaction among traffic generated by different sources, traffic regulation seems inevitable. In the paper =-=[10]-=-, Cruz proposed the following deterministic traffic characterization. For an increasing sequence A j fA(t); t = 0; 1; 2; : : :g (with A(0) = 0), it is f-upper constrained for some function f if A(t 2 ... |

300 | A calculus for network delay, part II: Network analysis - Cruz - 1991 |

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- Baccelli, Cohen, et al.
- 1992
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Citation Context ...2]. Results along this line is reported in [9]. (ii) Transfer functions under the (min; +)-algebra: one may define the so called z-transform under the (min; +)-algebra by F (z) = min t [f(t)+zt] (see =-=[3]-=-). This definition corresponds to the Legendre transform (the convex transform) [20] in the continuous time setting. (iii) Time varying filtering: time varying filtering is related to stochastic exten... |

219 | New Directions in Communications (or Which Way to the Information Ap - Turner - 1986 |

210 | Quality of service guarantees in virtual circuit switched networks
- Cruz
- 1995
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Citation Context ...ose in this paper a general filtering approach for the traffic regulation problem. Our approach is also applicable to the concept of service curve recently developed by Parekh and Gallager [19], Cruz =-=[12]-=- and Sariowan [21]. Our approach is built upon filtering under the (min; +)-algebra. In such an algebra, one replaces the usual addition operator by the min operator and the usual multiplication opera... |

172 | queue length, and delay of deterministic and stochastic queueing networks[J
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- 1994
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Citation Context ...ration in [3]. Remark 2.3 We note that over all f 2 F (with h 2 F fixed), there is a minimum solution of the equation h = h ? f . The minimum solution, denoted by h, is the minimum envelope of h (see =-=[6]-=-), i.e.,sh(t) = sup s0 [h(t + s) \Gamma h(s)]: (7) To see this, note from (7) that h(s) +sh(t \Gamma s)sh(t) for all sst (with the equality when s = t). Thus, h ?sh = h. On the other hand, if f is a s... |

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Citation Context ...luding feedback, concatenation, "filter bank summation", and performance bounds, is also applicable to f - servers. We illustrate the use of the theory by considering a window flow control p=-=roblem in [2]-=-. We also show that maximal regulators, in conjunction with the maximum delay guarantee, guarantee shifted-subadditive service curves in [21]. Based on this, we provide a couple of rules for service c... |

22 | Service guarantees for window flow control
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- 1996
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Citation Context ...on of servers. We note that the importance of the role of the (min; +)-algebra in deterministic traffic 2 regulation and service guarantees is also recognized by Agrawal and Rajan [2], Cruz and Okino =-=[15] and Le Bo-=-udec [4]. In particular, a "division" operator in the (min; +)-algebra is defined in [2, 15]. Such an operator provides a simple representation of performance bounds and output burstiness. E... |

22 | Subadditive processes, in École d’Été de Probabilités de SaintFlour V - Kingman - 1975 |

22 |
An optimal flow control scheme that regulates the burstiness of traffic subject to delay constraints
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Citation Context ...(oe; ae) regulator (leaky bucket) and the FIFO queue. Both only generate traffic that satisfies a linear constraint (f is a linear function) in (1). For f(t) = oe + aet, Ananthram and Konstantopoulos =-=[1, 17]-=- recently showed that the leaky bucket is the best causal traffic regulator to generate such a constraint in terms of maximizing the total number of departures at any moment in time (an equivalent sta... |

19 |
A Service-Curve Approach to Performance Guarantees in Integrated Service Networks
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- 1996
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Citation Context ...a general filtering approach for the traffic regulation problem. Our approach is also applicable to the concept of service curve recently developed by Parekh and Gallager [19], Cruz [12] and Sariowan =-=[21]-=-. Our approach is built upon filtering under the (min; +)-algebra. In such an algebra, one replaces the usual addition operator by the min operator and the usual multiplication operator by the additio... |

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- 1996
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Citation Context ...the deterministic filtering theory developed here. It is expected such an extension would be closely related to the exponential transformation from the (min; +)-algebra to the usual algebra (see e.g. =-=[7]-=- and references therein). Cruz [14] also noted that time varying filtering could be used for transient analysis of service guarantees. (iv) Constrained filtering: In [17], Ananthram and Konstantopoulo... |

10 |
A filtering theory for deterministic traffic regulation
- Chang
- 1997
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Citation Context ...ulator) For f 2 F 0 , the construction in (12) is called the maximalsf-regulator (for the input A). We note that the definition in Definition 3.4 is a generalization of that in our conference version =-=[8]-=- (see also [21, 2]), where the sequence f is required to be subadditive. Analogous to the classical filtering theory, we may view the maximal f-regulator as a linear time invariant filter with the imp... |

8 |
Optimality and interchangeability of leaky buckets in tandem
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- 1995
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Citation Context ...(oe; ae) regulator (leaky bucket) and the FIFO queue. Both only generate traffic that satisfies a linear constraint (f is a linear function) in (1). For f(t) = oe + aet, Ananthram and Konstantopoulos =-=[1, 17]-=- recently showed that the leaky bucket is the best causal traffic regulator to generate such a constraint in terms of maximizing the total number of departures at any moment in time (an equivalent sta... |

7 |
Boudec, "Network calculus made easy
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Citation Context ...ote that the importance of the role of the (min; +)-algebra in deterministic traffic 2 regulation and service guarantees is also recognized by Agrawal and Rajan [2], Cruz and Okino [15] and Le Boudec =-=[4]. In parti-=-cular, a "division" operator in the (min; +)-algebra is defined in [2, 15]. Such an operator provides a simple representation of performance bounds and output burstiness. Extensions of such ... |

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Citation Context ...ow control problem, Cruz and Okino [15] considered a more detailed model than the one in Agrawal and Rajan [2]. They obtained a tighter bound than the one in this paper. See also Le Boudec and Thiran =-=[5]-=-. The paper is organized as follows: in Section 2, we review the (min; +)-algebra and develop the concept and the properties of subadditive closure. We then use these properties to show equivalent sta... |

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Service Displines For Guaranteed Performance Service
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Citation Context ... Served (FCFS) policy and the static priority policy, or more complicated deadline scheduling polices such as the Generalized Processor Sharing (GPS) and the Earliest Deadline First policy (EDF) (see =-=[10, 6, 19, 23]-=-). We note that deadline scheduling policies usually provide a larger admission set, but they are at the cost of design complexity, e.g. time stamping and sorting. Corollary 5.8 shows that the best se... |

5 | Matrix extensions of the filtering theory for deterministic traffic regulation and service guarantees
- Chang
- 1998
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Citation Context ...ator in the (min; +)-algebra is defined in [2, 15]. Such an operator provides a simple representation of performance bounds and output burstiness. Extensions of such an operator can be found in Chang =-=[9]-=- and Cruz [13]. For the window flow control problem, Cruz and Okino [15] considered a more detailed model than the one in Agrawal and Rajan [2]. They obtained a tighter bound than the one in this pape... |

3 |
personal communication
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- 1997
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Citation Context ... developed here. It is expected such an extension would be closely related to the exponential transformation from the (min; +)-algebra to the usual algebra (see e.g. [7] and references therein). Cruz =-=[14]-=- also noted that time varying filtering could be used for transient analysis of service guarantees. (iv) Constrained filtering: In [17], Ananthram and Konstantopoulos considered traffic regulation pro... |