TCP Modeling

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TCP Modeling

• CMPT 765/408 Computer Networks
• Simon Fraser University
• Cheng-Hsin Hsu
• cha16_at_cs.sfu.ca

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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

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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

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Dynamics of the TCP Transmission Rate
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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

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Expected Sending Rate

• Altman et al. show the model has a stationary
  solution
• given
• (loss process) is ergodic stationary

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Expected Value of

• Expected sending rate
• , where
• is the loss event frequency

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The Second Moment of

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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

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Average Loss Probability

• Define
• average loss probability
• the number of transmitted packet
• the number of loss events
• We have

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Rewrite the Average Sending Rate

• Multiple these two equations
• We have

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Rewrite the Average Sending Rate (cont.)

• Define a normalized correlation function
• We have

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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

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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)

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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

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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

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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

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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/stochast
ic_tcp_notes.ps.gz

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Active Queue Management Algorithm -- RED

• Packet drop function is a function of the
  average queue length at that router

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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/pre
sentationfile/27/sigcomm2000.ppt
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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

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Model a Single Congested Router

• Consider a bottleneck router with transmission
  capacity C
• The packet drop function is denoted by p(x)
• The queueing length at time t is q(t)
• Let N TCP flows (labeled as Ni, where i1,2,,N)
  pass through this bottleneck router

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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

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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

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Model Window Size

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

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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/pre
sentationfile/27/sigcomm2000.ppt
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Revisit Loss Rate (cont.)

• Let x(t) be the total traffic load at the
  bottleneck router
• Recall
• Write loss rate as

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Final Model

• Finally, we have N differential equations

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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

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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

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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)

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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

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System Model

• Models the rates as differential equations
• 
  where
• route transmission rate
• feedback information regarding the link
  condition

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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

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Objective Functions

• Individual route
• Network Obj. Function

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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!

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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

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Uniqueness and Stability

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

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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

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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

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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.

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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

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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