## A Duality Model of TCP and Queue Management Algorithms (2002)

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Venue: | IEEE/ACM Trans. on Networking |

Citations: | 239 - 34 self |

### BibTeX

@ARTICLE{Low02aduality,

author = {Steven H. Low},

title = {A Duality Model of TCP and Queue Management Algorithms},

journal = {IEEE/ACM Trans. on Networking},

year = {2002},

volume = {11},

pages = {525--536}

}

### Years of Citing Articles

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### Abstract

We propose a duality model of congestion control and apply it to understand the equilibrium properties of TCP and active queue management schemes. Congestion control is the interaction of source rates with certain congestion measures at network links. The basic idea is to regard source rates as primal variables and congestion measures as dual variables, and congestion control as a distributed primal-dual algorithm carried out over the Internet to maximize aggregate utility subject to capacity constraints. The primal iteration is carried out by TCP algorithms such as Reno or Vegas, and the dual iteration is carried out by queue management such as DropTail, RED or REM. We present these algorithms and their generalizations, derive their utility functions, and study their interaction.

### Citations

2263 |
Congestion avoidance and control
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(Show Context)
Citation Context ...out by TCP, and the link algorithm is carried out by (active) queue management (AQM) schemes such as DropTail or RED [6]. Di#erent protocols use di#erent metrics to measure congestion, e.g., TCP Reno =-=[10, 24]-=-, and its variants, use loss probability as congestion measure, and TCP Vegas [4], it turns out, uses queueing delay as congestion measure [18]. Both are implicitly updated at the links and implicitly... |

2178 | Random early detection gateways for congestion avoidance
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(Show Context)
Citation Context ...urces that use that link. On the current Internet, the source algorithm is carried out by TCP, and the link algorithm is carried out by (active) queue management (AQM) schemes such as DropTail or RED =-=[6]-=-. Di#erent protocols use di#erent metrics to measure congestion, e.g., TCP Reno [10, 24], and its variants, use loss probability as congestion measure, and TCP Vegas [4], it turns out, uses queueing d... |

1366 |
Rate Control for Communication Networks: Shadow Prices, Proportional Fairness and Stability
- Kelly
- 1998
(Show Context)
Citation Context ... appear in [15]. + This work is supported by the Australian Research Council through grant A49930405, NSF through grant ANI0113425, and the Caltech Lee Center for Advanced Networking. 1 formulated in =-=[11]-=-. The objective is to maximize aggregate source utility subject to capacity constraints. We will interpret source rates as primal variables, congestion measures as dual variables, and TCP/AQM protocol... |

744 | Tsitsiklis, Neuro-Dynamic Programming - Bertsekas, N - 1996 |

687 | Charging and rate control for elastic traffic
- Kelly
- 1997
(Show Context)
Citation Context ...ions, and study their interaction. 1 Introduction Congestion control is a distributed algorithm to share network resources among competing sources. An optimal rate allocation problem is formulated in =-=[13]-=- where the goal is to choose source rates so as to maximize aggregate source utility subject to capacity constraints. This problem is solved using a penalty function approach in [15, 16, 10], and exte... |

528 | The macroscopic behavior of the TCP congestion avoidance algorithm
- Mathis, Semke, et al.
- 1997
(Show Context)
Citation Context ...D and REM. Let w s (t) be the window size. Let D s be the equilibrium round trip time (propagation plus equilibrium queueing delay), which we assume is constant, as customary in the literature, e.g., =-=[13, 20]-=-. Let x s (t) defined by x s (t) = w s (t)/D s be the source rate at time t. The time unit is on the order of several round trip times and source rate x s (t) should be interpreted as the average rate... |

514 | Optimization flow control, I. Basic algorithm and convergence
- Low, Lapsley
- 1999
(Show Context)
Citation Context ...to regard x(t) as primal variables, p(t) as dual variables, and (F, G) = (F s , G l , s # S, l # L) as a distributed primal-dual algorithm to solve the primal problem (9) and its Lagrangian dual (see =-=[16]-=-): min p#0 D(p) := # s max xs#0 (U s (x s ) - x s q s ) + # l p l c l (10) Hence, the dual variable is a precise measure of congestion in the network. The dual problem has an optimal solution since th... |

461 | End-to-End Window-based Congestion Control
- Mo, Walrand, et al.
- 1998
(Show Context)
Citation Context ... is to choose source rates so as to maximize aggregate source utility subject to capacity constraints. This problem is solved using a penalty function approach in [15, 16, 10], and extended in, e.g., =-=[26, 24, 18]-=-. It is solved using a duality approach in [21] leading to a basic algorithm whose convergence has been proved in an asynchronous environment. A practical implementation of this algorithm is studied i... |

411 | Tcp vegas: end to end congestion avoidance on a global internet
- Brakmo, Peterson
- 1995
(Show Context)
Citation Context ...chemes such as DropTail or RED [6]. Di#erent protocols use di#erent metrics to measure congestion, e.g., TCP Reno [10, 24], and its variants, use loss probability as congestion measure, and TCP Vegas =-=[4]-=-, it turns out, uses queueing delay as congestion measure [18]. Both are implicitly updated at the links and implicitly fed back to sources through end-to-end loss or delay, respectively. In this pape... |

384 | The performance of TCP/IP for networks with high bandwidth-delay products and random loss
- Lakshman, Madhow
- 1997
(Show Context)
Citation Context ...ry in the literature, e.g.,528 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 11, NO. 4, AUGUST 2003 TABLE I SUMMARY OF DUALITY MODEL OF TCP/AQM ALGORITHMS. NOTATIONS ARE EXPLAINED IN SECTIONS III AND IV =-=[14]-=-, [21]. Let defined by be the source rate at time . The time unit is on the order of several round-trip times and source rate should be interpreted as the average rate over this time scale. Dynamics s... |

306 | Resource pricing and the evolution of congestion control
- Gibbens, Kelly
- 1969
(Show Context)
Citation Context ...on. Under DropTail, a packet that arrives to a full bu#er is dropped. We do not know a convenient expression for the dynamics of marking probability. A model of loss rate that has been used, e.g., in =-=[7, 12]-=-, is that at a bu#erless queue, p(t + 1) = [1 - c/ # s x s (t)] + . This model is suitable for the penalty function approach to solving (9), but not the duality approach because of the feasibility con... |

269 | Bandwidth sharing: objectives and algorithms - Massoulié, Roberts - 1999 |

262 | Connections with multiple congested gateways in packet-switched networks part 1: one-way traffic
- Floyd
- 1991
(Show Context)
Citation Context ... or reduce equilibrium queueing delay. Remarks: 1. The relations (23) and (24) imply that Reno-1 and Reno-2 discriminate against sources with large D s , as well known in many previous studies, e.g., =-=[5, 6, 13, 20]-=-. Moreover, (23) for Reno-1 can be rewritten as x s = # 2 D s # 1 - q s q s # # 2 D s 1 # q s (29) when loss probability q s is small, a relation widely observed previously. Some authors, e.g., [9, 2]... |

220 | On designing improved controllers for AQM routers supporting TCP flows
- Hollot, Misra, et al.
(Show Context)
Citation Context ...linear in price r l (t): p l (t) = min{#r l (t), 1} (22) for some constant # > 0. The version with nonzero target queue length b l and linear marking probability is equivalent to the PI controller of =-=[9]-=-. Other proposed AQM's such as Adaptive Virtual Queue of [12] can also be modeled in the form of (2--3). The equations (17), (20), and (21) or (22) define the model (G, H) for REM. 3.2 Utility functio... |

218 | A control theoretic analysis of RED
- Hollot, Misra, et al.
- 2001
(Show Context)
Citation Context ...te vector)? Second, what are the dynamic properties, such as local stability around an equilibrium, and speed of convergence to a stable equilibrium? Local stability of Reno/RED is analyzed in, e.g., =-=[8, 14]-=-. In this paper, we present a model to understand the equilibrium properties. To this end, consider equilibria (x, p) of (1--3). The fixed point of (1) defines an implicit relation between equilibrium... |

205 | Active Queue Management
- Athuraliya, Low, et al.
- 2001
(Show Context)
Citation Context ...bles, and TCP/AQM protocols as distributed primal-dual algorithms to solve this optimization problem and its associated dual problem (Section 2). Di#erent protocols, such as Reno, Vegas, RED, and REM =-=[1]-=-, all solve the same prototypical problem with di#erent utility functions, and we derive these functions explicitly (Sections 3 and 4). Moreover all these protocols generate congestion measures (Lagra... |

179 | End-tc-end congestion control schemes: utility functions, random losses and ECN marks
- Kunniyur, Srikant
- 1995
(Show Context)
Citation Context ...on. Under DropTail, a packet that arrives to a full bu#er is dropped. We do not know a convenient expression for the dynamics of marking probability. A model of loss rate that has been used, e.g., in =-=[7, 12]-=-, is that at a bu#erless queue, p(t + 1) = [1 - c/ # s x s (t)] + . This model is suitable for the penalty function approach to solving (9), but not the duality approach because of the feasibility con... |

177 | Binomial Congestion Control Algorithms
- Bansal, Balakrishnan
- 2001
(Show Context)
Citation Context ...3, 20]. Moreover, (23) for Reno-1 can be rewritten as x s = # 2 D s # 1 - q s q s # # 2 D s 1 # q s (29) when loss probability q s is small, a relation widely observed previously. Some authors, e.g., =-=[9, 2]-=-, assume that Reno increases its window by 1 every round trip time deterministically. This corresponds to replacing (1 - q s (t)) by 1 in (15), which holds when the marking probability is small. This ... |

152 | Mathematical modelling of the internet
- Kelly
- 2000
(Show Context)
Citation Context ...erefore, Reno solves (3--4) with utility functions U s (x s ) = p 2 D s tan \Gamma1 ` x s D s p 2 ' (11) which is unique on the set of all possible equilibrium rates. This result is first obtained in =-=[14]-=-. We summarize. Theorem 1 The equilibrium rates of Reno/DropTail as modeled by (9) solve (3--4) with utility functions U s given by (11). Moreover the equilibrium rates are x s = D D s c; and the equi... |

139 | Internet congestion control
- Low, Paganini, et al.
- 2002
(Show Context)
Citation Context .... For Reno-2, using (26), F s in (16) can be rewritten in terms of the target rate x s (t) as x s (t + 1) = # x s (t) + q s (t) 2D s (x s (t) - x s (t)) # + 3. The approach taken here follows that in =-=[17]-=- where queue management mechanisms are modeled entirely by (G l , H l ). In contrast, the model in [15] includes the marking probability function as a part of F s , making utility function dependent o... |

131 |
TCP Vegas: end-to-end congestion avoidance on a global Internet
- Bramko, Peterson
- 1995
(Show Context)
Citation Context ...mes such as DropTail or RED [6]. Different protocols use different metrics to measure congestion, e.g., TCP Reno [10], [25] and its variants, use loss probability as congestion measure, and TCP Vegas =-=[4]-=-, it turns out, uses queueing delay as congestion measure [18]. Both are implicitly updated at the links and implicitly fed back to sources through end-to-end loss or delay, respectively. In this pape... |

117 | Understanding Vegas: A Duality Model
- Low, Peterson, et al.
- 2002
(Show Context)
Citation Context ...#erent metrics to measure congestion, e.g., TCP Reno [10, 24], and its variants, use loss probability as congestion measure, and TCP Vegas [4], it turns out, uses queueing delay as congestion measure =-=[18]-=-. Both are implicitly updated at the links and implicitly fed back to sources through end-to-end loss or delay, respectively. In this paper, we present a general model of congestion control and apply ... |

95 | Scalable laws for stable network congestion control
- PAGANINI, DOYLE, et al.
- 2001
(Show Context)
Citation Context ...e Associated with each source s is its transmission rate x s (t) at time t, in packets/sec. Associated with each link l is a scalar congestion measure p l (t) # 0 at time t. Following the notation of =-=[22]-=-, let y l (t) = # s R ls x s (t) be the aggregate source rate at link l and let q s (t) = # l R ls p l (t) be the end-to-end congestion measure for source s. In vector notation, we have ( T denotes tr... |

70 |
A Time-Scale Decomposition Approach to Adaptive Explicit Congestion Notification (ECN
- Kunniyur, Srikant
- 2002
(Show Context)
Citation Context ...probability m and m(t) = m(p(t)) should not be interpreted as the probability of a mark but rather the probability that the window is halved. 6 p(t). A model of loss rate that has been used, e.g., in =-=[9, 16, 17]-=-, is that at a bufferless queue, p(t + 1) = [1 \Gamma c= P s x s (t)] + . This model is suitable for the penalty function approach to solving (3--4), but not the duality approach because of the feasib... |

50 | Linear stability of TCP/RED and a scalable control
- Low, Paganini, et al.
- 2003
(Show Context)
Citation Context ...lim b D→0 ∗ l = lim p ∗ →1 b∗ l = 2bl To reduce equilibrium queue length b ∗ l ,alargeml (max p) and a small bl (max th) should be used. But this increases the slope ρ1 and compromises stability; see =-=[14]-=-. Hence, RED parameters can be tuned either to maintain stability or reduce equilibrium queueing delay. αlb ∗ l + y∗ l − cl = 0 (27) We know y∗ l ≤ cl. If y∗ l <cl, then (16) implies that b∗ l = 0; bu... |

36 | Linear and Nonlinear Programming, 2 nd Ed - Luenberger - 2005 |

34 | Comparison of TCP Reno and TCP Vegas via fluid approximation
- Bonald
- 1998
(Show Context)
Citation Context ... by noting that P s x s = c in equilibrium and using (10). Remarks: 1. The relation (10) implies that Reno discriminates against sources with large D s , as well known in many previous studies, e.g., =-=[7, 8, 19, 25, 5, 16]-=-. Moreover, (10) can be rewritten as x s = p 2 D s r 1 \Gamma p p p 2 D s 1 p p (13) when loss probability p is small, a relation widely observed previously. Some authors, e.g., [11, 3], assume that R... |

33 |
TCP/IP Illustrated: The Protocols, volume 1
- Stevens
- 1993
(Show Context)
Citation Context ...out by TCP, and the link algorithm is carried out by (active) queue management (AQM) schemes such as DropTail or RED [6]. Di#erent protocols use di#erent metrics to measure congestion, e.g., TCP Reno =-=[10, 24]-=-, and its variants, use loss probability as congestion measure, and TCP Vegas [4], it turns out, uses queueing delay as congestion measure [18]. Both are implicitly updated at the links and implicitly... |

33 |
Optimization flow control
- Athuraliya, Low
- 2000
(Show Context)
Citation Context ...It is solved using a duality approach in [21] leading to a basic algorithm whose convergence has been proved in an asynchronous environment. A practical implementation of this algorithm is studied in =-=[2]-=-. This set of work leads to abstract congestion control algorithms that can be regarded as distributed computations over a network to solve the optimal rate allocation problem. On the surface, the var... |

30 |
A global stability result in network flow control
- Paganini
- 2002
(Show Context)
Citation Context ...rties and have ignored the stability and dynamics of these protocols. The global stability of REM in the absence Fig. 5. Equilibrium rates under REM as Vegas sources varies of delay is established in =-=[22]-=- using a Lyapunov argufrom 0 to 200. N1 =200,D= 100 ms, c =20pkts/ms. ment. Local stability of Reno/RED has been studied in [8], [14]. It would be interesting to investigate deVegas receives much less... |

22 |
A duality model of TCP flow controls
- Low
- 2000
(Show Context)
Citation Context ...ess of congestion control as carrying out a distributed computation by sources and links over a network in real time to solve a global optimization problem # Partial and preliminary results appear in =-=[15]-=-. + This work is supported by the Australian Research Council through grant A49930405, NSF through grant ANI0113425, and the Caltech Lee Center for Advanced Networking. 1 formulated in [11]. The objec... |

22 |
End-to-End Congestion Control for the Internet: A Global Optimization Framework
- Golestani, Bhattacharyya
- 1998
(Show Context)
Citation Context ... is formulated in [13] where the goal is to choose source rates so as to maximize aggregate source utility subject to capacity constraints. This problem is solved using a penalty function approach in =-=[15, 16, 10]-=-, and extended in, e.g., [26, 24, 18]. It is solved using a duality approach in [21] leading to a basic algorithm whose convergence has been proved in an asynchronous environment. A practical implemen... |

14 |
Lakshman and Upamanyu Madhow. The performance of TCP/IP for networks with high bandwidth–delay products and random loss
- V
- 1997
(Show Context)
Citation Context ...D and REM. Let w s (t) be the window size. Let D s be the equilibrium round trip time (propagation plus equilibrium queueing delay), which we assume is constant, as customary in the literature, e.g., =-=[13, 20]-=-. Let x s (t) defined by x s (t) = w s (t)/D s be the source rate at time t. The time unit is on the order of several round trip times and source rate x s (t) should be interpreted as the average rate... |

14 | On the stability of optimization-based flow control
- Paganini
- 2001
(Show Context)
Citation Context ... too restrictive. We have only studied the equilibrium properties and have ignored the stability and dynamics of these protocols. The global stability of REM in the absence of delay is established in =-=[21]-=- using a Lyapunov argument. Local stability of Reno/RED has been studied in [8, 14]. It would be interesting to investigate delayed global stability of various TCP/AQM protocols. Here we derive the ut... |

13 |
Dynamics of tcp/aqm and a scalable control
- Low, Paganini, et al.
- 2002
(Show Context)
Citation Context ...te vector)? Second, what are the dynamic properties, such as local stability around an equilibrium, and speed of convergence to a stable equilibrium? Local stability of Reno/RED is analyzed in, e.g., =-=[8, 14]-=-. In this paper, we present a model to understand the equilibrium properties. To this end, consider equilibria (x, p) of (1--3). The fixed point of (1) defines an implicit relation between equilibrium... |

8 | Flow control via pricing: a feedback perspective - Paganini - 2000 |

8 |
Active queue management
- “REM
- 2001
(Show Context)
Citation Context ...nd TCP/AQM protocols as distributed primal-dual algorithms to solve this optimization problem and its associated dual problem (Section II). Different protocols, such as Reno, Vegas, RED, and REM [1], =-=[2]-=-, all solve the same prototypical problem with different utility functions, and we derive these functions explicitly (Sections III and IV). Moreover, all these protocols generate congestion measures (... |

7 |
and Venkat Anantharam. Charge-sensitive TCP and rate control in the Internet
- La
- 2000
(Show Context)
Citation Context ... is to choose source rates so as to maximize aggregate source utility subject to capacity constraints. This problem is solved using a penalty function approach in [15, 16, 10], and extended in, e.g., =-=[26, 24, 18]-=-. It is solved using a duality approach in [21] leading to a basic algorithm whose convergence has been proved in an asynchronous environment. A practical implementation of this algorithm is studied i... |

5 |
designing improved controllers for AQM routers supporting TCP flows
- “On
- 2001
(Show Context)
Citation Context ...ty that is a piecewise linear increasing function of : where (18) for some constant . The version with nonzero target queue length and linear marking probability is equivalent to the PI controller of =-=[10]-=-. Other proposed AQMs, such as Adaptive Virtual Queue of [13], can also be modeled in the form of (2) and (3). Equations (16), (19), and (20) or (21) define the model for REM. B. Utility Functions of ... |

5 |
Optimization flow control—Part I: Basic algorithm and convergence
- Low, Lapsley
- 1999
(Show Context)
Citation Context ...ms in both directions. We can start with general utility functions, e.g., tailored to our applications, and then derive TCP/AQM algorithms to maximize aggregate utility, as done in, e.g., [12], [13], =-=[17]-=-, [20], and [22]. Conversely, and historically, we can design TCP/AQM algorithms and then reverse-engineer the algorithms to determine the underlying utility functions they implicitly optimize and the... |

2 | Tables of series, products, and integrals. VEB Deutscher Verlag der Wissenschaften - Ryshik, Gradstein - 1963 |

2 |
Resourcepricingandthe evolution of congestion control
- Kelly
- 1999
(Show Context)
Citation Context ...n. Under DropTail, a packet that arrives to a full buffer is dropped. We do not know a convenient expression for the dynamics of marking probability. A model of loss rate that has been used, e.g., in =-=[7]-=-, [12], is that for a bufferless queue, p(t +1)=[1− c/ ∑ s xs(t)] + .This model is suitable for the penalty function approach to solving (8), but not the duality approach because of the feasibility co... |