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Model of TCP/AQM & Market Mechanism for Resource Allocation

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### Model of TCP/AQM & Market Mechanism for Resource Allocation

Resource Allocation

Problems

- How do multiple users share the network?
- Are the system resources used efficiently?
- Are the current TCP/AQM good enough?
- Do all users get a fair amount of bandwidth?

Resource Allocation

Outline

- Part 1 –Resource Allocation
- Modeling the optimization problem of network
- Solutions to the relaxations of the problem
- Part 2–Duality Model
- Different approach to solve the optimization problem
- Part 3– TCP/AQM
- A model to analyze TCP/AQM
- Deriving TCP/AQM algorithms
- Part 4– Fairness
- Max-Min and Proportional

Resource Allocation

Network Utility Maximization (NUM)

- Maximum sum of the utilization of each user subject to the rates on each link is less than its capacity
- U be unknown by network
- User point of view and network point of view of NUM

Resource Allocation

NUM (cont.)

- xr : user r’s transfer rate
- Ur(xr) : utilization of the user as function of user’s rate
- Utility is additive, so utility of all users is the sum of the utility of each r
- Cj: capacity of resource (router) j
- A: routing matrix
- Ajr = 1 if resource j is on route r
- Ajr = 0 otherwise

Resource Allocation

USER view of NUM

Utility Maximization for user r is USERr(Ur;λr) :

max utility of route r minus price per unit time on route r

User place a “bid” wr with “price per flow”λr

“Goods” xr = bid /price per unit flow = wr / λr

“bid” and “goods” are measured per unit time

Resource Allocation

Nets view of NUM

- Net knows bid (vector w) for each r
- Network’s max problem NETWORK (A,C,w) is
- Maximize sum over all routes of each route’s “bid” multiplied by the “log of flow”
- As if network max logarithmic utility function, but with bids (wr, rR) chosen by users

Resource Allocation

Decomposition

- There exist vectors , w and x such that
- wr = rxr for r R
- wr solves USERr (Ur; r)
- x solves NETWORK (A, C; w)
- By decomposing the SYS into USER and NET, we solve SYS.

Resource Allocation

Solving NETWORK (A,C,w)

- Lagrangian problem for network where µ is a vector of shadow prices
- derivative of L:
- Set derivative to 0 to find optimal solution

Resource Allocation

Solving NETWORK (A,C,w) cont…

- optimal flow is price per time divided by the sum of the “shadow prices” on each resource (router) j on that route r

Resource Allocation

Primal Algorithm

Rates vary gradually plus be decentralized

pj(x) is price charged by resource j per unit flow through it when the flow is x

Resource Allocation

Primal Algorithm (cont.)

- Resource j generates feedback price j(t)
- Price j sent to each user r whose route passes through resource j
- Multiplicative decrease in xr at rate proportional to stream of feedback price received
- Linear increase in xr at rate proportional to wr
- By adjusting xr(t), net tries to equalize cost of flow with its target value wr

Resource Allocation

Primal Algorithm (cont.)

solves the maximization of

- x* that maximizes V(x) is the unique and stable equilibrium of the primal algorithm
- Interpretation of V(x)? How close x* is to the real solution of NETWORK (A,C,w) ?

Resource Allocation

Dual Algorithm

shadow prices vary gradually and rates be functions of shadow prices

xr(t) is the rate according to price per unit flow (demand)

Prices adjust according to supply and demand

qj(µ) is the flow through resource j which generates µ (supply)

Resource Allocation

User Adaptation

- Primal and Dual algorithms assume that wr chosen are fixed variables
- wr can be varied in the algorithms
- User r can monitor xr(t) and vary wr(t) in order to optimize USER problem

Resource Allocation

Outline

- Part 1 –Resource Allocation
- Modeling the optimization problem of network
- Solution to the relaxations of the problem
- Part 2–Duality Model (covered already)
- Different approach to solve the optimization problem
- Part 3– TCP/AQM
- A model to analyze TCP/AQM
- Deriving TCP/AQM algorithms
- Part 4– Fairness
- Max-Min and Proportional

Resource Allocation

Outline

- Part 1 –Resource Allocation
- Modeling the optimization problem of network
- Solution to the relaxations of the problem
- Part 2–Duality Model
- Different approach to solve the optimization problem
- Part 3– TCP/AQM
- A primal-dual model to analyze TCP/AQM
- Deriving TCP/AQM algorithms
- Part 4– Fairness
- Max-Min and Proportional

Resource Allocation

TCP & AQM

pl(t)

xi(t)

A link algorithm

Update congestion measure and send it back to the sources

Carried out by AQM

A source algorithm:

Adjust sending rate based on congestion

Implemented in TCP

Resource Allocation

Primal-Dual Model

Each source s updates rate

Each link l updates congestion measure

Resource Allocation

Primal-Dual and TCP/AQM

- Primal and Dual

- Distributed solution

- Distributed primal dual algorithm to solve the problem

- TCP/AQM

Reno, Vegas

- TCP iterates on rates (windows) - F

- AQM iterates on congestion measures – G,H

- With different utility functions

DropTail, RED, REM

Resource Allocation

F, G, H in primal-dual model

- Derivation
- Derive (F, G, H) from protocol
- Derive U, which the protocol implicitly optimizes

- Performance
- equilibrium
- throughput, loss, queue length, delay
- fairness
- friendliness

Resource Allocation

Active Queue Management (AQM)

- Idea: provide congestion information by probabilistically dropping or marking packets

DropTail, RED, REM

- RED: Random Early Detection
- REM: Random Exponential Marking

Resource Allocation

Random Exponential Marking (REM)

- Price adjusted to match rate and clear buffer
- Marking probability exponential in `price’

H:

REM

G:

Resource Allocation

Transmission Control Protocol (TCP)

- Idea: adjust source rate (window size) based on the congestion indication

Reno, Vegas

- Reno
- Vegas

Resource Allocation

Reno/RED equilibrium properties

- Single link with capacity c shared by Reno users with RTT Ds

- Rates

- Marking Probability

- Queue length

Resource Allocation

Summary

- Duality model interprets protocols as distributed primal-dual algorithms over internet
- Reverse engineering: start with a given protocol
- Equilibrium properties - throughput, delay, loss, fairness

Resource Allocation

Outline

- Part 1 –Resource Allocation
- Modeling the optimization problem of network
- Solution to the relaxations of the problem
- Part 2–Duality Model
- Different approach to solve the optimization problem
- Part 3– TCP/AQM
- A model to analyze TCP/AQM
- Deriving TCP/AQM algorithms
- Part 4– Fairness
- Max-Min and Proportional

Resource Allocation

Fairness

- What is fairness
- Proportional fairness
- Generalized proportional fairness
- Max-Min fairness
- Proportional fairness algorithm

Resource Allocation

Network Optimization and Fairness

Resource Allocation

Max-Min Fairness

Resource Allocation

Max-Min Fairness (cont.)

Resource Allocation

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