1 / 42

TCP Modeling

Explore essential TCP modeling concepts and various models including stochastic processes. Understand control and network system models. Study TCP interactions with Active Queue Management algorithms like RED.

kimberlycarver
Download Presentation

TCP Modeling

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. TCP Modeling CMPT 765/408: Computer Networks Simon Fraser University Cheng-Hsin Hsu cha16@cs.sfu.ca

  2. Agenda • Motivation for mathematical TCP modeling • Essentials of TCP modeling • Gallery of TCP models • Periodic model • Detailed packet loss model • Stochastic model with general loss process • Control system model • Network system model • Summary

  3. Stochastic Model with General Loss Process • Previous models assume i.i.d. packet loss process with loss probability • Not true in the Internet -- bursty errors and depends on the window size. • Need a general loss process • Incorporate a general loss process to the detailed packet-loss model • Ignore inessential details to ease the burden of analysis

  4. Dynamics of the TCP Transmission Rate

  5. Simple Model of TCP Transmission Rate • Consider packet losses as events • : instantaneous transmission rate • : interval between events • Write , where • : multiplicative decrease factor (=1/2) • : additive increase factor (= ) • Assume has the correlation function:

  6. Expected Sending Rate • Altman et al. show the model has a stationary solution: • given: • (loss process) is ergodic stationary

  7. Expected Value of • Expected sending rate: , where • is the loss event frequency

  8. The Second Moment of

  9. Average Sending Rate • Using Palm probability, Altman et al. provide the average TCP sending rate as: • Observation: average sending rate is a function of loss frequency, loss interval correlation functions, linear increase factor (thus RTT), and multiplicative decrease factor

  10. Average Loss Probability • Define • : average loss probability • : the number of transmitted packet • : the number of loss events • We have

  11. Rewrite the Average Sending Rate • Multiple these two equations: • We have

  12. Rewrite the Average Sending Rate (cont.) • Define a normalized correlation function: • We have

  13. Final Average Sending Rate • We write • Observation: average transmission rate is inversely related to • The round-trip time • The square root of the loss probability

  14. Receiver Rate Limitation • Receiver places a packet receiving rate limit M, such that • Same as limiting window size to • This results in a nonlinear model, the explicit expressions for and are hard to derive (if all possible)

  15. Receiver Rate Limitation (cont.) • Instead, upper and lower bounds are given if the sending rate is limited by receiver window • E.g., the lower bound (of the sending rate) converges to: as

  16. Stochastic model with general loss process • Consider a general loss process -- e.g., i.I.d. random losses, Markovian arrival loss process, etc. • Losses are modeled as events • Result follows the inverse square-root p law

  17. Agenda • Motivation for mathematical TCP modeling • Essentials of TCP modeling • Gallery of TCP models • Periodic model • Detailed packet loss model • Stochastic model with general loss process • Control system model • Network system model • Summary

  18. Control System Model • Previous model assumes that losses occur because of insufficient resources • Active Queue Management (AQM) techniques are proposed to cope with congestion problem • A router intentionally drops packets when it detects congestions • Random Early Detection (RED) is a key AQM proposal Some slides are based on the online notes at http://www.cse.cuhk.edu.hk/~cslui/CSC5480/stochastic_tcp_notes.ps.gz

  19. Active Queue Management Algorithm -- RED • Packet drop function is a function of the average queue length at that router 1 Drop probability p pmax tmin tmax Average queue length x Modified from: Fluid-based Analysis of a Network of AQM Routers Supporting TCP Flows with an Application to RED -- ppt file at http://dna-wsl.cs.columbia.edu/pubsdb/citation/presentationfile/27/sigcomm2000.ppt

  20. Key Features of Control System Model • Study the interaction of TCP with AQM (e.g., RED) • Model data traffic as fluid • Model packet losses as Poisson process • Derive a set of differential equations to describe the AQM policy and queue length

  21. Model a Single Congested Router • Consider a bottleneck router with transmission capacity C • The packet drop functionis denoted by p(x) • The queueing length at time t is q(t) • Let N TCP flows (labeled as Ni, where i=1,2,…,N) pass through this bottleneck router

  22. Model RTT • Wi(t): window size of flow i at time t • Ri(t): RTT of flow i at time t • RTT is modeled (assumed) as: • is a fixed propagation delay • models the queueing delay

  23. Model Sending Rate and Packet Losses • Bi(t): instantaneous throughput (sending rate) of flow i at time t • Follows the fluid model • Assume the number of packet losses is describe by a Poisson process {Ni(t)} with rate

  24. Model Window Size • Window size is modeled by the Poisson Counter Driven Stochastic Differential equations • AIMD behavior of TCP • Take expectation, we have:

  25. Packet Drop/Mark Round Trip Delay (t) Revisit Loss Rate AQM Router B(t) p(t) Sender Receiver Loss Rate as seen by Sender: l(t) = B(t-t)*p(t-t) Copied from: Fluid-based Analysis of a Network of AQM Routers Supporting TCP Flows with an Application to RED -- ppt file at http://dna-wsl.cs.columbia.edu/pubsdb/citation/presentationfile/27/sigcomm2000.ppt

  26. Revisit Loss Rate (cont.) • Let x(t) be the total traffic load at the bottleneck router • Recall: • Write loss rate as:

  27. Final Model • Finally, we have N differential equations

  28. Control System Model • Capture the relationship between window size and packet drop function p(.) used by AQM • Can be used to design better AQM • The original paper also analyzes the interaction between queue length and window size • The original paper generalizes the single bottleneck case to complete networks

  29. Agenda • Motivation for mathematical TCP modeling • Essentials of TCP modeling • Gallery of TCP models • Periodic model • Detailed packet loss model • Stochastic model with general loss process • Control system model • Network system model • Summary

  30. Network System Model • Consider a collection of TCP flows for optimal network bandwidth allocation • Optimization-based approach: formulate the bandwidth allocation problem as nonlinear programming problems • The formulation is useful to various communication networks (not only to IP)

  31. System Overview • Assume the network contains l links, each of them has capacity Cl (in bps) • Each TCP flow using a route (path) r, r can be written as a list of links; let set R be the collection of all routes • Matrix A is defined as Air =1 if route r uses link l; Air =0, otherwise

  32. System Model • Models the rates as differential equations: where • : route transmission rate • : feedback information regarding the link condition

  33. System Model (cont.) • : a function of RTT, known as the gain of the differential equation system • : willingness-to-pay, describes how aggressive the rate control algorithm is • Capture AIMD feature

  34. Objective Functions • Individual route: • Network Obj. Function:

  35. Optimal Solution • Kelly et al. show there is a unique solution that is the optimal set of transmission rates (maximize the obj. function) • is solved by differentiating the obj. fcn. w.r.t. all , and set them to be zero. • Observation: it is a distributed algorithm!

  36. Why Centralized Algorithm is Bad? • Not scalable • Solving nonlinear programming problems is not computational trivial • Consider the scale of the Internet, we have too many routes! • Even we mange to build a super centralized point, the (injected) control messages would interfere with the actual data

  37. Uniqueness and Stability • Hence, U(x(t)) is strictly increasing with t unless • is the unique maximum and is stable about the optimum point

  38. Rate of Convergence • Linearize the system around the optimum solution using: • Write these equation into a vector: • Where X, W, P’ are diagonal matrices with entries

  39. Rate of Convergence (cont.) • The smallest eigenvalue of determine the convergence rate • Close to zero, then the convergence takes a long time • Large, the system will return to the optimal performance rather quickly

  40. Proportional Fairness • is proportionally fair if is feasible and for any other feasible vector • If the utility function is of the form Ur(xr)=wr log xr, the optimal allocation satisfies proportional fairness • Proportional fairness states a connection achieves a sending rate in proportion to the number of network links that it requires.

  41. Network System Model • Formulate resource allocation problems • Exists a unique and stable optimum solution • Convergence rate can be derived by computing eigenvalues • Achieves proportional fairness

  42. Agenda • Motivation for mathematical TCP modeling • Essentials of TCP modeling • Gallery of TCP models • Periodic model • Detailed packet loss model • Stochastic model with general loss process • Control system model • Network system model • Summary

More Related