Congestion Control. Topics

Size: px
Start display at page:

Download "Congestion Control. Topics"

Transcription

1 Congestion Control Topics Congestion control what & why? Current congestion control algorithms TCP and UDP Ideal congestion control Resource allocation Distributed algorithms Relation current algorithms and resource allocation Broad implications 6.976/ESD.937 1

2 Congestion Control: What and Why? Internet is used by many independent users Resources (link capacity) are finite If every user sends data at very high rate, it will cause congestion packets will be dropped, i.e. unreliable transmission seemingly high utilization of resource may be actually very low! If every user sends data at very low rate, resource will not be well-utilized Users need to send data at the correct rate so that Resources are well-utilized Users get reliable data transfer Resources should be shared fairly This is the primary goal of congestion control 6.976/ESD.937 2

3 Congestion Control Main question: How to implement congestion control in the network With the help of only distributed protocols at the users is it even possible? if not, what is the best possible thing to do? An example: Sender link Receiver ( unknown capacity for sender/receiver ) 6.976/ESD.937 3

4 Congestion Control If senders get acknowledgment of receipt of packets They know if packets are dropped or not Based on this, senders can infer the following If packets are not dropped sender is sending at rate lower than the capacity If packets are dropped sender is sending at rate higher than capacity Use drops of packets as signal of congestion Change rate as a reaction to packet drop so as to achieve fair-share Congestion control is essentially feedback-control 6.976/ESD.937 4

5 Congestion Control Mechanism To perform congestion control, we need two basic protocols Algorithm I: Drop detection ask receiver to acknowledge receipt of packets if a sent packet is not acknowledged fast enough the packet is assumed to be dropped Algorithm II: Rate-control if packets are not dropped, i.e. no congestion and hence increase rate by certain amount else there is congestion and hence decrease rate by certain amount Algorithms I and II are key ideas behind current congestion control protocols 6.976/ESD.937 5

6 Congestion Control Recall, Internet has layered architecture Congestion control is essentially required for reliable transmission at fair-rate with high-resource utilization Implemented in Network Layer The congestion control protocol is also called transport protocol Transmission Control Protocol (TCP) is a popular protocol provides reliable transmission when all users exhibit good-citizen behavior but has higher delay (cost of reliability) User Data-gram Protocol (UDP) is another protocol unreliable and user selfish protocol but has lower delay useful for media communication 6.976/ESD.937 6

7 TCP Brief History TCP (some version) has been around since ARPANET Older versions(s) required users to send probes to network to detect level of congestion led network to always operate under congestion often led to congestion collapse Popular well-documented example Congestion collapse in October 1986 Rate between Lawrence-Berkeley Lab and UCB went down from 32 kbps 40 bytes! 6.976/ESD.937 7

8 TCP Post-mortem of congestion collapse Older TCP sent more and more packets without confirming their delivery violation of packet conservation principle was key reason for failure Did not account for inhomogeneity in network bandwidth Lack of rate-control Current TCP Packet conservation: inject new packet when old packet wave has reached destination Slow-start: search for capacity starting from zero Rate-control: control rate via packet drop feedback, and be good user 6.976/ESD.937 8

9 TCP: Current Algorithm Algorithm has two components Sender algorithm and receiver algorithm Sender algorithm: Parameters: STHRESH: threshold for slow-start W max : maximum window-size CW: current window size Initially: number the data to be sent 1, 2,...,N /ESD.937 9

10 TCP: Current Algorithm Slow-start: Set STHRESH = W max /2 Start with CW= 1, packet sent P= 0 Each time, send packets from P to P+CW When a packet P+1 is acknowledged, set P P+1 For every acknowledgment, increase CW to CW+1 till CW < STHRESH or Packet loss is detected if packet loss, then set STHRESH STHRESH/2 and CW CW/2 Continue above until CW=STHRESH go to congestion-avoidance phase If P=N ever, transmission is over! 6.976/ESD

11 TCP: Current Algorithm Congestion-avoidance For acknowledgement of P+1, CW CW + 1 [CW] If packet-loss then CW is decreased different versions have different ways of dealing with the window size decrease one version: set STHRESH CW/2 CW 1 go to slow-start phase 6.976/ESD

12 TCP: Current Algorithm Receiver algorithm: For every received packet, send acknowledgment With request for the next packet # required if packets 1-10 and 13 are received, then send request for packet number 11 (and not 14) Packet-loss detection: Essentially, packet is dropped when acknowledgment is not received Specific protocol if ack. not received within TIME-OUT, or 3 consecutive requests sent for the same packet # by receiver 6.976/ESD

13 TCP: An Example Suppose, CW = 10 at some-time T= packets 1 CW:( ) 1 CW:( ) T=4 T=6 ACK 1 ACK 10 CW = 11 T= packets ACK T=14 T=20 TIME-OUT packets Sender Receiver 6.976/ESD

14 TCP: Performance The main performance metric Throughput of TCP That is, what is the net equilibrium rate of users as a fraction of the total capacity Evaluation of TCP throughput Model: TCP dynamics effect of slow-start vs. congestion-avoidance Characterization of equilibrium of TCP First, we ll consider a model for TCP 6.976/ESD

15 TCP: Performance We ll consider a very simple situation single link and single user ignore extra complications Given this model, we will identify Key system dynamics affecting its performance Specifically, we ll try to compare effect of Slow-start vs. Congestion control And, use this insight to evaluate performance precisely 6.976/ESD

16 TCP: Performance Model: Single user accessing link of capacity C T be round-trip time B be the butter size at the link Maximum window size W max = ct + B STHRESH = W max /2 = ct+b 2 User wants to transmit infinite data Goal: Evaluate average rate of transmission 6.976/ESD

17 TCP: Performance TCP dynamics Periodic between slow-start and congestion avoidance slow-start congestion-avoidance slow-start t = 0 Tss Tss + Tca Nss Nca Rate = N ss + Nca Tss + Tca Next, we evalute Nss, Nca, Tss and Tca To obtain rate of TCP 6.976/ESD

18 TCP: Performance We first study the Slow-start phase Specifically, evolution of window size and queue-size Slow-start phase has cyclic behavior Divide time into cycles where window size doubles in each cycle: t = 0: first packet sent; W(0)= 1 t = T: packet ACK; W(T)= 2 t = 2T: both packets ACK; W(2T)= 3. each mini-cycle has length =T Window-size evolution: W(nT) 2 n /ESD

19 TCP: Performance Next, we study queue-length evolution In slow-start phase using window-evoluation Again, queue-size has the same cyclic behavior Queue-size at the beginning of cycle 0 During nth cycle, 2 n 2 packets are sent total packets sent (until nth cycle) = 2 n n = 2 n+1 total packets acknowledged by end of nth cycle W(nT) = 2 n 2 Max-queue length in nth cycle, Q(nT) 2 n n 2 2 n 2 W(nT)/ /ESD

20 TCP: Performance In slow-start phase, buffer-overflow happens if Q is larger than B But window can be at most STHRESH Now, Q = W 2 which is at most STHRESH 2 = B+cT 4 ( ) Hence, for overflow B+cT 4 > B That is, ct > 3B B < ct/3 Equivalently, no overflow when B ct/ /ESD

21 TCP: Performance First, we consider no overflow situation That is, B ct/3 When there is no overflow, we have the following Tss: time for slow-start W(Tss) B+cT 2 2 T ss /T B+cT ( 2 ) B+cT Tss = T log 2 2 Nss: # of packets transmitted = # of packets acknowledge = window-size = B+cT /ESD

22 TCP: Performance Now, iff overflow happens in slow-start, i.e. B < c T/3 There are two slow-start phases: Phase 1: at the overflow Q= B = W/2; but detection happens after T time, i.e., window has doubled W = 2 B at Tss 1 (time to overflow) 2 T ss 1 /T 2 B T ss 1 = T log 2 2B Nss 1 = 2B Set STHRESH = 2B 2 = B 6.976/ESD

23 TCP: Performance Phase 2: after overflow Now, the overflow does not happen as STHRESH is low enough! W = B at the end of this phase 2 T ss 2 /T B T ss 2 T logb Nss 2 = B In summary, when overflow happens Tss = Tss 1 + Tss 2 = 2T log B + T Nss = 3B And when overflow does not happen ( ) B+cT Tss = T log 2 2 Nss = B+cT /ESD

24 TCP: Performance Now, we study Congestion-avoidance phase Start of window-size W 0 = W max /2 = ct+ B 2 if B ct 3 B if B < ct 3 Next, we study evolution of window-size with time t = W(t) Let a(t) = # of acknowledgments received till time t Then, change in window-size is dw dt = dw da da dt { C if W is large, server is busy Rate change of a = W/T if W is smaller da That is, dt = min{w/t,c} Also: dw da = 1 (by definition of algorithm) W 6.976/ESD

25 TCP: Performance Hence, dw dt = { 1/T if W < ct c/w if W ct Congestion avoidance will end when W = W max Two-phases Phase 1: W 0 W < ct Phase 2: ct W W max First, we consider Phase /ESD

26 TCP: Performance Phase 1: W 0 W < ct Tca 1 = T (ct W 0 ), since dw dt = 1/T N ca1 = a(t ca1 ) = T ca1 0 da(t), where Tca1 0 da(t) = = = Tca1 0 Tca1 0 Tca1 0 Tca1 da(t) dt dt W(t) T dt W 0 + t/t dt T W 0 T ca1 = 0 T [ = W 0 (ct W 0 ) + T2 ca 1 2T ct W 0 2 ] 6.976/ESD

27 TCP: Performance Phase 2: W ct; dw dt = c W(t) W(t)dW = cdt W 2 (t) W 2 (0) = 2ct (where W(0) = ct) W 2 (t) = 2ct + c 2 T 2 Phase ends when W(T ca2 ) = W max W 2 max = 2cT ca2 + c 2 T 2 2cT ca2 = W 2 max c 2 T 2 T ca2 = W2 max c 2 T 2 2c N ca2 = ct ca2 because, node is running at capacity c 6.976/ESD

28 TCP: Performance In summary, in congestion avoidance phase Tca 1 = T (ct W 0 ), [ N ca1 = W 0 (ct W 0 ) 1 + ct W ] 0 2 T ca2 = W2 max c2 T 2 2c, and N ca2 = ct ca2 Let, compare the contribution of slow-start and congestion avoidance phases When, B = ct, for large c We ll find that congestion avoidance dominates Only concentrate on TCP-dynamics for congestion-avoidance We ll use this insight to find throughput for simple model 6.976/ESD

29 TCP: Performance We consider only congestion-avoidance dynamics Single-link Many-sources Modeling T r : RTT of user r W r (t): window size of user r at time t q r (t): fraction of packet list at time t for user r x r (t) = W r(t) : rate of user r at time t T a r (t) = acknowledgment until time t for user r Dynamics Acknowledgment: increase window by Drop: decrease window by βw r (t) 1 W r (t) 6.976/ESD

30 TCP: Performance To study throughput of TCP We need to study evolution of W r (t) Let q r (t) fraction of packets are dropped at time t for user r Then, the rate at which packets are dropped at t for user r x r (t T r ) q r (t) That is, drop rate at time t for user r is x r (t T r )q r (t) Acknowledgment rate at t: x r (t T r )(1 q r (t)) Then, W r (t) changes as follows: dw r (t) dt = (1 q r (t)) [ ] 1 W r (t) x r(t T r ) [ ] q r (t) βw r (t) x r (t T r ) Since, x r (t) = W r (t)/t r dx r (t) dt = (1 q r(t))x r (t T r ) T r X r (t) q r (t)βt r x r (t)x r (t T r ) 6.976/ESD

31 TCP: Performance To obtain long-term effective throughput We evaluate the equilibrium point: dx r(t) = 0 dt Ignoring the delay Tr in equation usually, can not ignored we ll study when is this justified This gives us the following: In equilibrium That is, 0 = (1 ˆq r) T 1 βˆx 2 rt rˆq r ˆx r = 1 T r 1 ˆq r βˆq r 6.976/ESD

32 TCP: Performance The analysis has following main message Throughput is mainly affected by congestion avoidance phase, and not by slow-start Qualitatively, throughput is inversely proportional to RTT (T r ) square root inversely proportional to drop-probability 6.976/ESD

33 TCP: Performance Questions: We assumed deterministic dynamics how valid is it? The Law-of-Large-Numbers or Fluid-models provide justification We ignored the effect of other users implicit in q r ( ) What happens when many links? naturally, hard to quantify exact relation however, its useful to ask the following basic question: what do we really what to achieve from TCP and has it achieved that? next, we address this basic question 6.976/ESD

34 Resource Allocation Suppose a piece of cake is to be shared between two people in a fair manner: How should we do it? Well, divide it into half each, assuming both care for size only. What if one cares only for cherry but other cares about actual cake? Division scheme should care about utility of cake to the people. Known literature of cake-cutting algorithm has inspired a lot of interesting work in Game Theory and Algorithms 6.976/ESD

35 Resource Allocation Consider a single link with capacity C R users want to use it and let f r, 1 r R be rate at which user r wants to send data If f f R C allocate demanded rate to each user But, if f f R > C, then we need an allocation mechanism to allocate rates x 1,...,x R to all users such that 0 x r f r ; 1 r R R x r C r=1 and, allocation should maximize the overall system utility 6.976/ESD

36 Resource Allocation Let u r (x r ) be utility of rate x r to user r Then, allocation (x r ) should be solution of R max u r (y r ) r=1 subject to 0 y r f r ; 1 r R R y r C Now, a bad user can have u r very high can t rely on users utility for fair allocation Question: what should be u r ( ) so that allocation is fair? r= /ESD

37 Fair Allocation Let s consider a simple example C = 10 ; f 1 = 4, f 2 = 20. x 1 = 4 ; x 2 = 6 makes sense because allocation is as much equal as possible Max-min fair: (x r ) is max-min fair iff For any (r, r ); x r > x r only if x r = f r In the above example, x 1 = 3 ; x 2 = 1 is not max-min fair because x 2 > x 2 but x 1 f 1. Philosophy of max-min fair: At the fair allocation, the only way to become richer is to make a poor, poorer /ESD

38 Max-Min Fair Allocation Network resource allocation Question: Consider network with L links with capacities C 1,...,C L Sources 1,...,R with demands f 1,...f r. Let S denote set of all sources. Source r sends data from node s r to d r using some links (according to routing algorithm) Rates (x r ) are feasible if data transmission, according to the rates, satisfies link capacities constraints that is, no link is over-subscribed What about fair allocation? Similar definition generalizes /ESD

39 Max-Min Fair Allocation Definition (max-min fair): Question: Let (x r ) be max-min fair for a given capacitated network only if (1) (x r ) is feasible (2) for any other feasible (y r ), if y s > x s for some source s, then there exists source p s.t. x p x s and y p < x p How to find such an allocation? Next, we will see a simple centralized algorithm. Later we will consider distributed algorithms and their relationship to current TCP! 6.976/ESD

40 Max-Min Fair Allocation (1) Let n l be number of sources in S that use link l. For each l s.t. c l 0, define fair share f l as: g l = c l n l (2) Define z r = min g l, which is the min rate over links that source r is using. l r (3) Define z min = min r z r. (4) Let R = {r : z r = z min } (5) For all r R; the max-min rate x r = min(z r,f r ) (6) Set S S\ R ; c l c l x r ; r R ;l r n l n l r R 1 {l r} (7) Repeat (1) (6) until S is empty /ESD

41 Max-Min Fair Allocation Algorithm Not distributed Not implementable Fairness There can be other fairness criteria Next, we will see A range of fairness criteria max-min fair as one of them Study distributed algorithm for allocation based on them /ESD

42 α-fair Allocation Let utility of user r be u r (x r ) = x 1 α r r w r 1 α r α r > 0 ; α r 1 w r log x r α r = 1. Also, f r =, i.e. everyone wants maximal rate. Fair allocation is solution to optimization problem subject to max (x r ) ur (x r ) x r c l ; l L l r x r 0 Next, we consider special examples of the above class of fair allocation /ESD

43 Examples (I) Minimum delay fair α r = 2 ; r x 1 α r r u r (x r ) = w r 1 α r = w r x r (II) Proportional fair α r = 1 ; r (III) Max-min fair u r (x r ) = +w r log x r α r = α ; r ;α An example 6.976/ESD

44 Resource Allocation Consider routing matrix M = [M re ] R L M rl = 1 if data of user r passes through link l y = Mx ; x = (x r ) rate vector Then, resource allocation problem (RAC) max ur (x r ) (x r ) subject to x r 0 ; y = Mx C Instead of solving RAC, we will first solve (RAC1) max u r (x r ) x 0 r l L 0 Here, constraint y = Mx C is absent s: l s x s Instead, f l ( ) is penalty (price) function s.t. y 0 f l(x)dx as y non-decreasing, continuum, non-negative f l (y)dy /ESD

45 RAC1 Note that, for all choice of (α r ) u r ( ) is strictly concave, increasing Hence, the new utility function of RAC1 V (x) = u r (x r ) r l is strictly concave s: l s x s Proof [pg. 24, [SRIKANT]] 0 f l (y)dy We also assume that u r (x r ) as x r /ESD

46 RAC1 For maximizing strictly concave function V ( ) V (x) as x 0 V (x) + as x There is a unique solution of RAC1 that lies in the interior of set x 0. Hence, optimal solution must satisfy dv dx r = 0 r u r(x r ) f l ( s ) = 0 r l:l r s:l sx Solution of these equations leads to optimal route But, difficult to solve in decentralized manner 6.976/ESD

47 Decentralize Solution for RAC1 Each node can compute traffic through outgoing link and corresponding price at time t; i.e. y l (t) = s:l s x s(t) ; p l (t) = f l (y)l(t)) price on a route = sum of prices on its links, i.e. q r (t) = l:l r p l (t) or q = Mp Then, optimality condition is u r(x r ) q r = 0 A natural gradient algorithm is ẋ r (t) = k r (x r )(u r(x r ) q r ), with k r ( ) non-negative, non-decreasing continuous function /ESD

48 RAC1 Then: The gradient algorithm converges to optimal solution starting from any initial condition Proof [pg. 26, [SRIKANT]] Implication: To reach modified resource allocation, simple radiant algorithm based on prices and utility function is sufficient Question: What prices/penalty lead to correct algorithm? well, penalty function are approximations for capacity constraints: f l (x) as x C l actual utilities are never exact Next, some exact penalty functions 6.976/ESD

49 Exact Penalty Function An explanation of penalty functions can be given in terms of dual variables [explained later]. Based on this idea, it is possible to introduce adaptive penalty function ( yl ) Bl Define, f l (y l, c l ) = ; c virtual capacity c l We will adapt c l during the course of algorithm as follows: d c l (A) dt = α l(c l y l ) + c l And, primal algorithm ẋ r = k r (x r )(u r(x r ) q r ) ; q r = l:l r p l = l:l r f l (y l1, c l ). The above combination provides a primal algorithm with adaptive price-function 6.976/ESD

50 Exact Penalty Function Then: If routing matrix M is full rank, then algorithm solves RAC, exactly. Proof [pg. 30, [SRIKANT]] Implications: It is possible to change rates and price function adaptively to obtain solution of Resource allocation problem The price has natural optimization interpretation in terms of dual variable 6.976/ESD

51 RAC: Dual RAC: subject to max x 0 ur (x r ) Mx c. Lagrangian of RAC: x 0 ; λ 0 L(x;λ) = r u r (x r ) λ T (Mx c). Dual D(λ) = where, y l = s:l s x s. max x 0;Mx c = max x 0;Mx c [ ] u r (x r ) λ T (Mx c) r [ u r (x r ) r l λ l y l ] + l λ l c l, 6.976/ESD

52 Dual and It s Properties Dual optimal By strong-duality: inf D(λ) = RAC λ 0 inf D(λ) λ 0 For given λ 0: maximizing x is s.t. for all r L = 0 u r(x r ) x r x r l:l r λ l = 0 Further, by strong duality, for the optimizing (λ, x) pair λ l (y l c l ) = 0. Based on above optimality conditions, a simple gradient algorithm: [pg. 28, [SRIKANT]] Set x r = u 1 r ( l:l r λ l), λ l = h l (λ l )(y l c l ) + λ l /ESD

53 RAC: Dual Then: The gradient descent algorithm described for dual of RAC converges to optimal solution if the routing matrix M has full rank Proof [pg. 29, [SRIKANT]] Implication: By adjusting prices (λ l = p l ) and rational use behavior, desired rate-allocation can be achieved What if both prices and rate are changed simultaneously? 6.976/ESD

54 RAC: Primal Dual Primal-dual algorithm Primal algorithm at sources ẋ r = k r (x r )(u r(x r ) q r ), Dual algorithm at links ṗ l = k l (p l )(y l c l ) + p l. Then: The algorithm converges to the solution of RAC, starting from initial condition Proof [pg. 28, [SRIKANT]] 6.976/ESD

55 Relation to Current Algorithms Primal algorithm ẋ r = k r (x r )(u r(x r ) q r ) Changing rate based on feedback Recall, TCP changes window-size w r (t) be windowsize T r be RTT Then, rate x r (t) = w r(t) T r Let q r (t) be drop probability Then, TCP-Reno increases window if no drop (prb. 1 q r (t)) decreases window if drop (prb. q r (t)) 6.976/ESD

56 TCP versus Primal Algorithm Specifically, TCP dynamics ẋ r = 1 q r(t)x r (t T r ) T 2 r x r (t) Let x r (t T r ) x r (t); then βq r (t)x r (t)x r (t T r ). ẋ r = 1 q r(t) βx 2 T r(t)q r 2 r (t). If q r (t) small, 1 q r (t) 1; then ẋ r = 1 βx 2 T r(t)q r 2 r (t) [ = βx 2 r(t) 1 βt 2 r x 2 r(t) q r(t) = k r (x r (t)) [u r(x r (t)) q r (t)] 1 Then, u r (t) = βtr 2 x r (t) That is: TCP weighted delay fair allocation! ] 6.976/ESD

57 What About Dual? TCP corresponds to primal algorithm What about dual algorithm? q r ( ): corresponds to drop probability Queue management algorithm corresponds to dual algorithm Popular queue-management algorithm Random Early Detection (RED) Mark (or drop) packet with probability proportional to queue-size If butter-size is B, then probability of marking packet when queue-size is Q is Q B. Let y be arrival rate 6.976/ESD

58 Dual = Active Queue Management Now, queue-dynamics Then, marking (drop) probability Q = (y c) + Q p = Q B ṗ = α(y c)+ p This has exactly the same dynamics as prices or Lagrange multiplier in dual or primal-dual algorithm Note that q r l:l r p l ; because if drops at links are independent then (1 q r ) = (1 p l ) 1 p l, l:l r l:l r when p l is small /ESD

59 Next We assumed that RCA algorithm or TCP has immediate feedback. In practice, feedback is always delayed Questions: (1) How does delayed feedback affect performance? (2) If delay is very large: feedback is useless. Then how large can delay be so as the network can still operate? Next, we will see how ideas from control theory can be useful. Professor Mitter s lectures 6.976/ESD

60 Other Issues If everything works well, then TCP and RED seem good algorithms Question: What if uses do not cooperate? Malicious users can lead to undesirable network performance. Next, we study network security /ESD

Resource Allocation and Pricing. R. Srikant University of Illinois

Resource Allocation and Pricing. R. Srikant University of Illinois Resource Allocation and Pricing R. Srikant University of Illinois References The Mathematics of Internet Congestion Control, Birkhauser, 2004. Pricing: Kelly Distributed Resource Allocation: Kelly, Mauloo

More information

Routing. Topics: 6.976/ESD.937 1

Routing. Topics: 6.976/ESD.937 1 Routing Topics: Definition Architecture for routing data plane algorithm Current routing algorithm control plane algorithm Optimal routing algorithm known algorithms and implementation issues new solution

More information

Motivation. Lecture 2 Topics from Optimization and Duality. network utility maximization (NUM) problem:

Motivation. Lecture 2 Topics from Optimization and Duality. network utility maximization (NUM) problem: CDS270 Maryam Fazel Lecture 2 Topics from Optimization and Duality Motivation network utility maximization (NUM) problem: consider a network with S sources (users), each sending one flow at rate x s, through

More information

Network Optimization and Control

Network Optimization and Control Foundations and Trends R in Networking Vol. 2, No. 3 (2007) 271 379 c 2008 S. Shakkottai and R. Srikant DOI: 10.1561/1300000007 Network Optimization and Control Srinivas Shakkottai 1 and R. Srikant 2 1

More information

Rate Control in Communication Networks

Rate Control in Communication Networks From Models to Algorithms Department of Computer Science & Engineering The Chinese University of Hong Kong February 29, 2008 Outline Preliminaries 1 Preliminaries Convex Optimization TCP Congestion Control

More information

cs/ee/ids 143 Communication Networks

cs/ee/ids 143 Communication Networks cs/ee/ids 143 Communication Networks Chapter 4 Transport Text: Walrand & Parakh, 2010 Steven Low CMS, EE, Caltech Agenda Internetworking n Routing across LANs, layer2-layer3 n DHCP n NAT Transport layer

More information

Topics in Data Mining Fall Bruno Ribeiro

Topics in Data Mining Fall Bruno Ribeiro Network Utility Maximization Topics in Data Mining Fall 2015 Bruno Ribeiro 2015 Bruno Ribeiro Data Mining for Smar t Cities Need congestion control 2 Supply and Demand (A Dating Website [China]) Males

More information

Internet Congestion Control: Equilibrium and Dynamics

Internet Congestion Control: Equilibrium and Dynamics Internet Congestion Control: Equilibrium and Dynamics A. Kevin Tang Cornell University ISS Seminar, Princeton University, February 21, 2008 Networks and Corresponding Theories Power networks (Maxwell Theory)

More information

TCP over Cognitive Radio Channels

TCP over Cognitive Radio Channels 1/43 TCP over Cognitive Radio Channels Sudheer Poojary Department of ECE, Indian Institute of Science, Bangalore IEEE-IISc I-YES seminar 19 May 2016 2/43 Acknowledgments The work presented here was done

More information

Dynamic resource sharing

Dynamic resource sharing J. Virtamo 38.34 Teletraffic Theory / Dynamic resource sharing and balanced fairness Dynamic resource sharing In previous lectures we have studied different notions of fair resource sharing. Our focus

More information

An Optimal Index Policy for the Multi-Armed Bandit Problem with Re-Initializing Bandits

An Optimal Index Policy for the Multi-Armed Bandit Problem with Re-Initializing Bandits An Optimal Index Policy for the Multi-Armed Bandit Problem with Re-Initializing Bandits Peter Jacko YEQT III November 20, 2009 Basque Center for Applied Mathematics (BCAM), Bilbao, Spain Example: Congestion

More information

A Mathematical Model of the Skype VoIP Congestion Control Algorithm

A Mathematical Model of the Skype VoIP Congestion Control Algorithm A Mathematical Model of the Skype VoIP Congestion Control Algorithm Luca De Cicco, S. Mascolo, V. Palmisano Dipartimento di Elettrotecnica ed Elettronica, Politecnico di Bari 47th IEEE Conference on Decision

More information

Optimization and Stability of TCP/IP with Delay-Sensitive Utility Functions

Optimization and Stability of TCP/IP with Delay-Sensitive Utility Functions Optimization and Stability of TCP/IP with Delay-Sensitive Utility Functions Thesis by John Pongsajapan In Partial Fulfillment of the Requirements for the Degree of Master of Science California Institute

More information

Solving Dual Problems

Solving Dual Problems Lecture 20 Solving Dual Problems We consider a constrained problem where, in addition to the constraint set X, there are also inequality and linear equality constraints. Specifically the minimization problem

More information

Wireless Internet Exercises

Wireless Internet Exercises Wireless Internet Exercises Prof. Alessandro Redondi 2018-05-28 1 WLAN 1.1 Exercise 1 A Wi-Fi network has the following features: Physical layer transmission rate: 54 Mbps MAC layer header: 28 bytes MAC

More information

Reliable Data Transport: Sliding Windows

Reliable Data Transport: Sliding Windows Reliable Data Transport: Sliding Windows 6.02 Fall 2013 Lecture 23 Exclusive! A Brief History of the Internet guest lecture by Prof. Hari Balakrishnan Wenesday December 4, 2013, usual 6.02 lecture time

More information

Utility, Fairness and Rate Allocation

Utility, Fairness and Rate Allocation Utility, Fairness and Rate Allocation Laila Daniel and Krishnan Narayanan 11th March 2013 Outline of the talk A rate allocation example Fairness criteria and their formulation as utilities Convex optimization

More information

THE Internet is increasingly being used in the conduct of

THE Internet is increasingly being used in the conduct of 94 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 14, NO. 1, FEBRUARY 2006 Global Stability Conditions for Rate Control With Arbitrary Communication Delays Priya Ranjan, Member, IEEE, Richard J. La, Member,

More information

CSE 123: Computer Networks

CSE 123: Computer Networks CSE 123: Computer Networks Total points: 40 Homework 1 - Solutions Out: 10/4, Due: 10/11 Solutions 1. Two-dimensional parity Given below is a series of 7 7-bit items of data, with an additional bit each

More information

Singular perturbation analysis of an additive increase multiplicative decrease control algorithm under time-varying buffering delays.

Singular perturbation analysis of an additive increase multiplicative decrease control algorithm under time-varying buffering delays. Singular perturbation analysis of an additive increase multiplicative decrease control algorithm under time-varying buffering delays. V. Guffens 1 and G. Bastin 2 Intelligent Systems and Networks Research

More information

A Globally Stable Adaptive Congestion Control Scheme for Internet-Style Networks with Delay 1

A Globally Stable Adaptive Congestion Control Scheme for Internet-Style Networks with Delay 1 A Globally Stable Adaptive ongestion ontrol Scheme for Internet-Style Networks with Delay Tansu Alpcan 2 and Tamer Başar 2 (alpcan, tbasar)@control.csl.uiuc.edu Abstract In this paper, we develop, analyze

More information

Understanding TCP Vegas: A Duality Model

Understanding TCP Vegas: A Duality Model Understanding TCP Vegas: A Duality Model Steven Low Departments of CS and EE, Caltech, USA slow@caltech.edu Larry Peterson Limin Wang Department of CS, Princeton University, USA {llp,lmwang}@cs.princeton.edu

More information

These are special traffic patterns that create more stress on a switch

These are special traffic patterns that create more stress on a switch Myths about Microbursts What are Microbursts? Microbursts are traffic patterns where traffic arrives in small bursts. While almost all network traffic is bursty to some extent, storage traffic usually

More information

Microeconomic Algorithms for Flow Control in Virtual Circuit Networks (Subset in Infocom 1989)

Microeconomic Algorithms for Flow Control in Virtual Circuit Networks (Subset in Infocom 1989) Microeconomic Algorithms for Flow Control in Virtual Circuit Networks (Subset in Infocom 1989) September 13th, 1995 Donald Ferguson*,** Christos Nikolaou* Yechiam Yemini** *IBM T.J. Watson Research Center

More information

Min Congestion Control for High- Speed Heterogeneous Networks. JetMax: Scalable Max-Min

Min Congestion Control for High- Speed Heterogeneous Networks. JetMax: Scalable Max-Min JetMax: Scalable Max-Min Min Congestion Control for High- Speed Heterogeneous Networks Yueping Zhang Joint work with Derek Leonard and Dmitri Loguinov Internet Research Lab Department of Computer Science

More information

9. Dual decomposition and dual algorithms

9. Dual decomposition and dual algorithms EE 546, Univ of Washington, Spring 2016 9. Dual decomposition and dual algorithms dual gradient ascent example: network rate control dual decomposition and the proximal gradient method examples with simple

More information

Analysis of Scalable TCP in the presence of Markovian Losses

Analysis of Scalable TCP in the presence of Markovian Losses Analysis of Scalable TCP in the presence of Markovian Losses E Altman K E Avrachenkov A A Kherani BJ Prabhu INRIA Sophia Antipolis 06902 Sophia Antipolis, France Email:altman,kavratchenkov,alam,bprabhu}@sophiainriafr

More information

Congestion Control In The Internet Part 1: Theory. JY Le Boudec 2018

Congestion Control In The Internet Part 1: Theory. JY Le Boudec 2018 Congestion Control In The Internet Part 1: Theory JY Le Boudec 2018 1 Contents 1. What is the problem; congestion collapse 2. Efficiency versus Fairness 3. Definitions of fairness 4. Additive Increase

More information

Information in Aloha Networks

Information in Aloha Networks Achieving Proportional Fairness using Local Information in Aloha Networks Koushik Kar, Saswati Sarkar, Leandros Tassiulas Abstract We address the problem of attaining proportionally fair rates using Aloha

More information

Fairness comparison of FAST TCP and TCP Vegas

Fairness comparison of FAST TCP and TCP Vegas Fairness comparison of FAST TCP and TCP Vegas Lachlan L. H. Andrew, Liansheng Tan, Tony Cui, and Moshe Zukerman ARC Special Research Centre for Ultra-Broadband Information Networks (CUBIN), an affiliated

More information

384Y Project June 5, Stability of Congestion Control Algorithms Using Control Theory with an application to XCP

384Y Project June 5, Stability of Congestion Control Algorithms Using Control Theory with an application to XCP 384Y Project June 5, 00 Stability of Congestion Control Algorithms Using Control Theory with an application to XCP . Introduction During recent years, a lot of work has been done towards the theoretical

More information

communication networks

communication networks Positive matrices associated with synchronised communication networks Abraham Berman Department of Mathematics Robert Shorten Hamilton Institute Douglas Leith Hamilton Instiute The Technion NUI Maynooth

More information

COMP9334: Capacity Planning of Computer Systems and Networks

COMP9334: Capacity Planning of Computer Systems and Networks COMP9334: Capacity Planning of Computer Systems and Networks Week 2: Operational analysis Lecturer: Prof. Sanjay Jha NETWORKS RESEARCH GROUP, CSE, UNSW Operational analysis Operational: Collect performance

More information

This Dissertation. entitled. Event-triggered distributed algorithms for network optimization. Pu Wan

This Dissertation. entitled. Event-triggered distributed algorithms for network optimization. Pu Wan This Dissertation entitled Event-triggered distributed algorithms for network optimization typeset with nddiss2ε v3.0 (2005/07/27) on November 30, 2009 for Pu Wan This L A TEX2ε classfile conforms to the

More information

Impact of Cross Traffic Burstiness on the Packet-scale Paradigm An Extended Analysis

Impact of Cross Traffic Burstiness on the Packet-scale Paradigm An Extended Analysis Impact of ross Traffic Burstiness on the Packet-scale Paradigm An Extended Analysis Rebecca Lovewell and Jasleen Kaur Technical Report # TR11-007 Department of omputer Science University of North arolina

More information

NICTA Short Course. Network Analysis. Vijay Sivaraman. Day 1 Queueing Systems and Markov Chains. Network Analysis, 2008s2 1-1

NICTA Short Course. Network Analysis. Vijay Sivaraman. Day 1 Queueing Systems and Markov Chains. Network Analysis, 2008s2 1-1 NICTA Short Course Network Analysis Vijay Sivaraman Day 1 Queueing Systems and Markov Chains Network Analysis, 2008s2 1-1 Outline Why a short course on mathematical analysis? Limited current course offering

More information

Processor Sharing Flows in the Internet

Processor Sharing Flows in the Internet STANFORD HPNG TECHNICAL REPORT TR4-HPNG4 Processor Sharing Flows in the Internet Nandita Dukkipati, Nick McKeown Computer Systems Laboratory Stanford University Stanford, CA 9434-93, USA nanditad, nickm

More information

Modelling an Isolated Compound TCP Connection

Modelling an Isolated Compound TCP Connection Modelling an Isolated Compound TCP Connection Alberto Blanc and Denis Collange Orange Labs 905 rue Albert Einstein Sophia Antipolis, France {Email: alberto.blanc,denis.collange}@orange-ftgroup.com Konstantin

More information

Congestion Control. Need to understand: What is congestion? How do we prevent or manage it?

Congestion Control. Need to understand: What is congestion? How do we prevent or manage it? Congestion Control Phenomenon: when too much traffic enters into system, performance degrades excessive traffic can cause congestion Problem: regulate traffic influx such that congestion does not occur

More information

Channel Allocation Using Pricing in Satellite Networks

Channel Allocation Using Pricing in Satellite Networks Channel Allocation Using Pricing in Satellite Networks Jun Sun and Eytan Modiano Laboratory for Information and Decision Systems Massachusetts Institute of Technology {junsun, modiano}@mitedu Abstract

More information

PIQI-RCP: Design and Analysis of Rate-Based Explicit Congestion Control

PIQI-RCP: Design and Analysis of Rate-Based Explicit Congestion Control PIQI-RCP: Design and Analysis of Rate-Based Explicit Congestion Control Saurabh Jain Joint work with Dr. Dmitri Loguinov June 21, 2007 1 Agenda Introduction Analysis of RCP QI-RCP PIQI-RCP Comparison Wrap

More information

Understanding TCP Vegas: A Duality Model

Understanding TCP Vegas: A Duality Model Understanding TCP Vegas: A Duality Model STEVEN H. LOW Caltech, Pasadena, California AND LARRY L. PETERSON AND LIMIN WANG Princeton University, Princeton, New Jersey Abstract. We view congestion control

More information

Markovian Model of Internetworking Flow Control

Markovian Model of Internetworking Flow Control Информационные процессы, Том 2, 2, 2002, стр. 149 154. c 2002 Bogoiavlenskaia. KALASHNIKOV MEMORIAL SEMINAR Markovian Model of Internetworking Flow Control O. Bogoiavlenskaia Petrozavodsk State University

More information

Methodology for Computer Science Research Lecture 4: Mathematical Modeling

Methodology for Computer Science Research Lecture 4: Mathematical Modeling Methodology for Computer Science Research Andrey Lukyanenko Department of Computer Science and Engineering Aalto University, School of Science and Technology andrey.lukyanenko@tkk.fi Definitions and Goals

More information

On the Resource Utilization and Traffic Distribution of Multipath Transmission Control

On the Resource Utilization and Traffic Distribution of Multipath Transmission Control On the Resource Utilization and Traffic Distribution of Multipath Transmission Control Bo Jiang 1, Yan Cai, Don Towsley 1 1 {bjiang, towsley}@cs.umass.edu ycai@ecs.umass.edu University of Massachusetts,

More information

Congestion Control 1: The Chiu-Jain Model

Congestion Control 1: The Chiu-Jain Model Mathematical Modelling for Computer Networks- Part I Spring 2013 (Period 4) Congestion Control 1: The Chiu-Jain Model Lecturers: Laila Daniel and Krishnan Narayanan Date:11th March 2013 Abstract This lesson

More information

Fairness and Optimal Stochastic Control for Heterogeneous Networks

Fairness and Optimal Stochastic Control for Heterogeneous Networks λ 91 λ 93 Fairness and Optimal Stochastic Control for Heterogeneous Networks sensor network wired network wireless 9 8 7 6 5 λ 48 λ 42 4 3 0 1 2 λ n R n U n Michael J. Neely (USC) Eytan Modiano (MIT) Chih-Ping

More information

On the Resource Utilization and Traffic Distribution of Multipath. Transmission Control

On the Resource Utilization and Traffic Distribution of Multipath. Transmission Control On the Resource Utilization and Traffic Distribution of Multipath Transmission Control UMass Computer Science Technical Report UM-CS-2011-005 Bo Jiang 1, Yan Cai 2, Don Towsley 1 1 {bjiang, towsley}@cs.umass.edu

More information

Distributed Systems Principles and Paradigms. Chapter 06: Synchronization

Distributed Systems Principles and Paradigms. Chapter 06: Synchronization Distributed Systems Principles and Paradigms Maarten van Steen VU Amsterdam, Dept. Computer Science Room R4.20, steen@cs.vu.nl Chapter 06: Synchronization Version: November 16, 2009 2 / 39 Contents Chapter

More information

Analytic Performance Evaluation of the RED Algorithm

Analytic Performance Evaluation of the RED Algorithm Prof. Dr. P. Tran-Gia Analytic Performance Evaluation of the RED Algorithm Stefan Köhler, Michael Menth, Norbert Vicari TCP Model RED Model TCP over RED Results TCP > Reliable transmission > Closed loop

More information

Strong Performance Guarantees for Asynchronous Buffered Crossbar Schedulers

Strong Performance Guarantees for Asynchronous Buffered Crossbar Schedulers Strong Performance Guarantees for Asynchronous Buffered Crossbar Schedulers Jonathan Turner Washington University jon.turner@wustl.edu January 30, 2008 Abstract Crossbar-based switches are commonly used

More information

Alternative Decompositions for Distributed Maximization of Network Utility: Framework and Applications

Alternative Decompositions for Distributed Maximization of Network Utility: Framework and Applications Alternative Decompositions for Distributed Maximization of Network Utility: Framework and Applications Daniel P. Palomar Hong Kong University of Science and Technology (HKUST) ELEC5470 - Convex Optimization

More information

MASTER THESIS. Development and Testing of Index Policies in Internet Routers

MASTER THESIS. Development and Testing of Index Policies in Internet Routers Universidad del País Vasco / Euskal Herriko Unibertsitatea MASTER THESIS Development and Testing of Index Policies in Internet Routers Author: Josu Doncel Advisor: Peter Jacko Urtzi Ayesta Leioa, September

More information

Rate adaptation, Congestion Control and Fairness: A Tutorial. JEAN-YVES LE BOUDEC Ecole Polytechnique Fédérale de Lausanne (EPFL)

Rate adaptation, Congestion Control and Fairness: A Tutorial. JEAN-YVES LE BOUDEC Ecole Polytechnique Fédérale de Lausanne (EPFL) Rate adaptation, Congestion Control and Fairness: A Tutorial JEAN-YVES LE BOUDEC Ecole Polytechnique Fédérale de Lausanne (EPFL) December 2000 2 Contents 31 Congestion Control for Best Effort: Theory 1

More information

TRANSMISSION STRATEGIES FOR SINGLE-DESTINATION WIRELESS NETWORKS

TRANSMISSION STRATEGIES FOR SINGLE-DESTINATION WIRELESS NETWORKS The 20 Military Communications Conference - Track - Waveforms and Signal Processing TRANSMISSION STRATEGIES FOR SINGLE-DESTINATION WIRELESS NETWORKS Gam D. Nguyen, Jeffrey E. Wieselthier 2, Sastry Kompella,

More information

Distributed Systems Principles and Paradigms

Distributed Systems Principles and Paradigms Distributed Systems Principles and Paradigms Chapter 6 (version April 7, 28) Maarten van Steen Vrije Universiteit Amsterdam, Faculty of Science Dept. Mathematics and Computer Science Room R4.2. Tel: (2)

More information

A Generalized FAST TCP Scheme

A Generalized FAST TCP Scheme A Generalized FAST TCP Scheme Cao Yuan a, Liansheng Tan a,b, Lachlan L. H. Andrew c, Wei Zhang a, Moshe Zukerman d,, a Department of Computer Science, Central China Normal University, Wuhan 430079, P.R.

More information

Modeling Impact of Delay Spikes on TCP Performance on a Low Bandwidth Link

Modeling Impact of Delay Spikes on TCP Performance on a Low Bandwidth Link Modeling Impact of Delay Spikes on TCP Performance on a Low Bandwidth Link Pasi Lassila and Pirkko Kuusela Networking Laboratory Helsinki University of Technology (HUT) Espoo, Finland Email: {Pasi.Lassila,Pirkko.Kuusela

More information

A Simple Model for the Window Size Evolution of TCP Coupled with MAC and PHY Layers

A Simple Model for the Window Size Evolution of TCP Coupled with MAC and PHY Layers A Simple Model for the Window Size Evolution of TCP Coupled with and PHY Layers George Papageorgiou, John S. Baras Institute for Systems Research, University of Maryland, College Par, MD 20742 Email: gpapag,

More information

Distributed Approaches for Proportional and Max-Min Fairness in Random Access Ad Hoc Networks

Distributed Approaches for Proportional and Max-Min Fairness in Random Access Ad Hoc Networks Distributed Approaches for Proportional and Max-Min Fairness in Random Access Ad Hoc Networks Xin Wang, Koushik Kar Department of Electrical, Computer and Systems Engineering, Rensselaer Polytechnic Institute,

More information

Node-based Distributed Optimal Control of Wireless Networks

Node-based Distributed Optimal Control of Wireless Networks Node-based Distributed Optimal Control of Wireless Networks CISS March 2006 Edmund M. Yeh Department of Electrical Engineering Yale University Joint work with Yufang Xi Main Results Unified framework for

More information

Performance Analysis of Priority Queueing Schemes in Internet Routers

Performance Analysis of Priority Queueing Schemes in Internet Routers Conference on Information Sciences and Systems, The Johns Hopkins University, March 8, Performance Analysis of Priority Queueing Schemes in Internet Routers Ashvin Lakshmikantha Coordinated Science Lab

More information

Giuseppe Bianchi, Ilenia Tinnirello

Giuseppe Bianchi, Ilenia Tinnirello Capacity of WLAN Networs Summary Per-node throughput in case of: Full connected networs each node sees all the others Generic networ topology not all nodes are visible Performance Analysis of single-hop

More information

Controlo Switched Systems: Mixing Logic with Differential Equations. João P. Hespanha. University of California at Santa Barbara.

Controlo Switched Systems: Mixing Logic with Differential Equations. João P. Hespanha. University of California at Santa Barbara. Controlo 00 5 th Portuguese Conference on Automatic Control University of Aveiro,, September 5-7, 5 00 Switched Systems: Mixing Logic with Differential Equations João P. Hespanha University of California

More information

Stochastic Hybrid Systems: Applications to Communication Networks

Stochastic Hybrid Systems: Applications to Communication Networks research supported by NSF Stochastic Hybrid Systems: Applications to Communication Networks João P. Hespanha Center for Control Engineering and Computation University of California at Santa Barbara Talk

More information

A Time-Scale Decomposition Approach to Adaptive Explicit Congestion Notification (ECN) Marking

A Time-Scale Decomposition Approach to Adaptive Explicit Congestion Notification (ECN) Marking University of Pennsylvania ScholarlyCommons Departmental Papers (ESE) Department of Electrical & Systems Engineering June 2002 A Time-Scale Decomposition Approach to Adaptive Explicit Congestion Notification

More information

Fair Scheduling in Input-Queued Switches under Inadmissible Traffic

Fair Scheduling in Input-Queued Switches under Inadmissible Traffic Fair Scheduling in Input-Queued Switches under Inadmissible Traffic Neha Kumar, Rong Pan, Devavrat Shah Departments of EE & CS Stanford University {nehak, rong, devavrat@stanford.edu Abstract In recent

More information

On Selfish Behavior in CSMA/CA Networks

On Selfish Behavior in CSMA/CA Networks On Selfish Behavior in CSMA/CA Networks Mario Čagalj1 Saurabh Ganeriwal 2 Imad Aad 1 Jean-Pierre Hubaux 1 1 LCA-IC-EPFL 2 NESL-EE-UCLA March 17, 2005 - IEEE Infocom 2005 - Introduction CSMA/CA is the most

More information

Agreement. Today. l Coordination and agreement in group communication. l Consensus

Agreement. Today. l Coordination and agreement in group communication. l Consensus Agreement Today l Coordination and agreement in group communication l Consensus Events and process states " A distributed system a collection P of N singlethreaded processes w/o shared memory Each process

More information

Lecture on Sensor Networks

Lecture on Sensor Networks Lecture on Sensor Networks Cyclic Historical Redundancy Development Copyright (c) 2008 Dr. Thomas Haenselmann (University of Mannheim, Germany). Permission is granted to copy, distribute and/or modify

More information

Switched Systems: Mixing Logic with Differential Equations

Switched Systems: Mixing Logic with Differential Equations research supported by NSF Switched Systems: Mixing Logic with Differential Equations João P. Hespanha Center for Control Dynamical Systems and Computation Outline Logic-based switched systems framework

More information

A Signal Processing Approach to the Analysis of Chemical Networking Protocols

A Signal Processing Approach to the Analysis of Chemical Networking Protocols Laurea Specialistica 19 July 2010 A Signal Processing Approach to the Analysis of Chemical Networking Protocols Author Supervisors Prof. Marco Luise Prof. Filippo Giannetti (University of Pisa) (University

More information

Efficient Mechanism Design

Efficient Mechanism Design Efficient Mechanism Design Bandwidth Allocation in Computer Network Presenter: Hao MA Game Theory Course Presentation April 1st, 2014 Efficient Mechanism Design Efficient Mechanism Design focus on the

More information

CHAPTER 4. Networks of queues. 1. Open networks Suppose that we have a network of queues as given in Figure 4.1. Arrivals

CHAPTER 4. Networks of queues. 1. Open networks Suppose that we have a network of queues as given in Figure 4.1. Arrivals CHAPTER 4 Networks of queues. Open networks Suppose that we have a network of queues as given in Figure 4.. Arrivals Figure 4.. An open network can occur from outside of the network to any subset of nodes.

More information

A New Technique for Link Utilization Estimation

A New Technique for Link Utilization Estimation A New Technique for Link Utilization Estimation in Packet Data Networks using SNMP Variables S. Amarnath and Anurag Kumar* Dept. of Electrical Communication Engineering Indian Institute of Science, Bangalore

More information

MS&E 246: Lecture 17 Network routing. Ramesh Johari

MS&E 246: Lecture 17 Network routing. Ramesh Johari MS&E 246: Lecture 17 Network routing Ramesh Johari Network routing Basic definitions Wardrop equilibrium Braess paradox Implications Network routing N users travel across a network Transportation Internet

More information

1.225 Transportation Flow Systems Quiz (December 17, 2001; Duration: 3 hours)

1.225 Transportation Flow Systems Quiz (December 17, 2001; Duration: 3 hours) 1.225 Transportation Flow Systems Quiz (December 17, 2001; Duration: 3 hours) Student Name: Alias: Instructions: 1. This exam is open-book 2. No cooperation is permitted 3. Please write down your name

More information

Lagrange duality. The Lagrangian. We consider an optimization program of the form

Lagrange duality. The Lagrangian. We consider an optimization program of the form Lagrange duality Another way to arrive at the KKT conditions, and one which gives us some insight on solving constrained optimization problems, is through the Lagrange dual. The dual is a maximization

More information

Getting Connected. Chapter 2, Part 2. Networking CS 3470, Section 1 Sarah Diesburg

Getting Connected. Chapter 2, Part 2. Networking CS 3470, Section 1 Sarah Diesburg Getting Connected Chapter 2, Part 2 Networking CS 3470, Section 1 Sarah Diesburg 1 Five Problems Encoding/decoding Framing Error Detection Error Correction Media Access 2 Five Problems Encoding/decoding

More information

Modelling TCP with a Discrete Time Markov Chain

Modelling TCP with a Discrete Time Markov Chain Modelling TCP with a Discrete Time Markov Chain José L Gil Motorola josegil@motorola.com ABSTRACT TCP is the most widely used transport protocol in the Internet. The end-to-end performance of most Internet

More information

Distributed Systems Fundamentals

Distributed Systems Fundamentals February 17, 2000 ECS 251 Winter 2000 Page 1 Distributed Systems Fundamentals 1. Distributed system? a. What is it? b. Why use it? 2. System Architectures a. minicomputer mode b. workstation model c. processor

More information

M C. Analytical Methods for Network Congestion Control. Steven H. Low Engineering & Applied Science, Caltech

M C. Analytical Methods for Network Congestion Control. Steven H. Low Engineering & Applied Science, Caltech Analytical Methods for Network Congestion Control Steven H. Low Engineering & Applied Science, Caltech SYNTHESIS LECTURES ON COMMUNICATION NETWORKS & M C Morgan & claypool publishers ABSTRACT The congestion

More information

14. Duality. ˆ Upper and lower bounds. ˆ General duality. ˆ Constraint qualifications. ˆ Counterexample. ˆ Complementary slackness.

14. Duality. ˆ Upper and lower bounds. ˆ General duality. ˆ Constraint qualifications. ˆ Counterexample. ˆ Complementary slackness. CS/ECE/ISyE 524 Introduction to Optimization Spring 2016 17 14. Duality ˆ Upper and lower bounds ˆ General duality ˆ Constraint qualifications ˆ Counterexample ˆ Complementary slackness ˆ Examples ˆ Sensitivity

More information

UNIVERSITY OF YORK. MSc Examinations 2004 MATHEMATICS Networks. Time Allowed: 3 hours.

UNIVERSITY OF YORK. MSc Examinations 2004 MATHEMATICS Networks. Time Allowed: 3 hours. UNIVERSITY OF YORK MSc Examinations 2004 MATHEMATICS Networks Time Allowed: 3 hours. Answer 4 questions. Standard calculators will be provided but should be unnecessary. 1 Turn over 2 continued on next

More information

Link Models for Packet Switching

Link Models for Packet Switching Link Models for Packet Switching To begin our study of the performance of communications networks, we will study a model of a single link in a message switched network. The important feature of this model

More information

Computer Networks ( Classroom Practice Booklet Solutions)

Computer Networks ( Classroom Practice Booklet Solutions) Computer Networks ( Classroom Practice Booklet Solutions). Concept Of Layering 0. Ans: (b) Sol: Data Link Layer is responsible for decoding bit stream into frames. 0. Ans: (c) Sol: Network Layer has the

More information

CS276 Homework 1: ns-2

CS276 Homework 1: ns-2 CS276 Homework 1: ns-2 Erik Peterson October 28, 2006 1 Part 1 - Fairness between TCP variants 1.1 Method After learning ns-2, I wrote a script (Listing 3) that runs a simulation of one or two tcp flows

More information

Efficient Network-wide Available Bandwidth Estimation through Active Learning and Belief Propagation

Efficient Network-wide Available Bandwidth Estimation through Active Learning and Belief Propagation Efficient Network-wide Available Bandwidth Estimation through Active Learning and Belief Propagation mark.coates@mcgill.ca McGill University Department of Electrical and Computer Engineering Montreal,

More information

MS&E 246: Lecture 18 Network routing. Ramesh Johari

MS&E 246: Lecture 18 Network routing. Ramesh Johari MS&E 246: Lecture 18 Network routing Ramesh Johari Network routing Last lecture: a model where N is finite Now: assume N is very large Formally: Represent the set of users as a continuous interval, [0,

More information

Continuous-Model Communication Complexity with Application in Distributed Resource Allocation in Wireless Ad hoc Networks

Continuous-Model Communication Complexity with Application in Distributed Resource Allocation in Wireless Ad hoc Networks Continuous-Model Communication Complexity with Application in Distributed Resource Allocation in Wireless Ad hoc Networks Husheng Li 1 and Huaiyu Dai 2 1 Department of Electrical Engineering and Computer

More information

Bounded Delay for Weighted Round Robin with Burst Crediting

Bounded Delay for Weighted Round Robin with Burst Crediting Bounded Delay for Weighted Round Robin with Burst Crediting Sponsor: Sprint Kert Mezger David W. Petr Technical Report TISL-0230-08 Telecommunications and Information Sciences Laboratory Department of

More information

TCP is Competitive Against a Limited AdversaryThis research was supported in part by grants from NSERC and CITO.

TCP is Competitive Against a Limited AdversaryThis research was supported in part by grants from NSERC and CITO. TCP is Competitive Against a Limited AdversaryThis research was supported in part by grants from NSERC and CITO. Jeff Edmonds Computer Science Department York University, Toronto, Canada jeff@cs.yorku.ca

More information

Cooperative HARQ with Poisson Interference and Opportunistic Routing

Cooperative HARQ with Poisson Interference and Opportunistic Routing Cooperative HARQ with Poisson Interference and Opportunistic Routing Amogh Rajanna & Mostafa Kaveh Department of Electrical and Computer Engineering University of Minnesota, Minneapolis, MN USA. Outline

More information

Distributed Systems. Time, Clocks, and Ordering of Events

Distributed Systems. Time, Clocks, and Ordering of Events Distributed Systems Time, Clocks, and Ordering of Events Björn Franke University of Edinburgh 2016/2017 Today Last lecture: Basic Algorithms Today: Time, clocks, NTP Ref: CDK Causality, ordering, logical

More information

cs/ee/ids 143 Communication Networks

cs/ee/ids 143 Communication Networks cs/ee/ids 143 Communication Networks Chapter 5 Routing Text: Walrand & Parakh, 2010 Steven Low CMS, EE, Caltech Warning These notes are not self-contained, probably not understandable, unless you also

More information

Queueing Theory I Summary! Little s Law! Queueing System Notation! Stationary Analysis of Elementary Queueing Systems " M/M/1 " M/M/m " M/M/1/K "

Queueing Theory I Summary! Little s Law! Queueing System Notation! Stationary Analysis of Elementary Queueing Systems  M/M/1  M/M/m  M/M/1/K Queueing Theory I Summary Little s Law Queueing System Notation Stationary Analysis of Elementary Queueing Systems " M/M/1 " M/M/m " M/M/1/K " Little s Law a(t): the process that counts the number of arrivals

More information

10-725/36-725: Convex Optimization Spring Lecture 21: April 6

10-725/36-725: Convex Optimization Spring Lecture 21: April 6 10-725/36-725: Conve Optimization Spring 2015 Lecturer: Ryan Tibshirani Lecture 21: April 6 Scribes: Chiqun Zhang, Hanqi Cheng, Waleed Ammar Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer:

More information

Value-Aware Resource Allocation for Service Guarantees in Networks

Value-Aware Resource Allocation for Service Guarantees in Networks Value-Aware Resource Allocation for Service Guarantees in Networks Parimal Parag, Student Member, IEEE, Sankalp Sah, Srinivas Shakkottai, Member, IEEE, Jean-Francois Chamberland, Senior Member, IEEE Abstract

More information

On the Flow-level Dynamics of a Packet-switched Network

On the Flow-level Dynamics of a Packet-switched Network On the Flow-level Dynamics of a Packet-switched Network Ciamac Moallemi Graduate School of Business Columbia University ciamac@gsb.columbia.edu Devavrat Shah LIDS, EECS Massachusetts Institute of Technology

More information

WiFi MAC Models David Malone

WiFi MAC Models David Malone WiFi MAC Models David Malone November 26, MACSI Hamilton Institute, NUIM, Ireland Talk outline Introducing the 82.11 CSMA/CA MAC. Finite load 82.11 model and its predictions. Issues with standard 82.11,

More information