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Achieving Network Optima Using Stackelberg Routing Strategies. Yannis A. Korilis, Member, IEEE Aurel A. Lazar, Fellow, IEEE & Ariel Orda, Member IEEE IEEE/ACM transactions on networking, vol. 5, No. 1, February 1997 Sanjeev Kohli EE 228A. Presentation Outline.

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achieving network optima using stackelberg routing strategies

Achieving Network Optima Using Stackelberg Routing Strategies

Yannis A. Korilis, Member, IEEE

Aurel A. Lazar, Fellow, IEEE

&

Ariel Orda, Member IEEE

IEEE/ACM transactions on networking, vol. 5, No. 1, February 1997

Sanjeev Kohli

EE 228A

presentation outline

Presentation Outline

Introduction to non cooperative networks

Overview of approach

Model and Problem Formulation

Non cooperative User & Manager

Single Follower Stackelberg Routing game

Multi Follower Stackelberg Routing game

Issues

non cooperative networks1

Non-cooperative Networks

Users take control decisions individually to max own performance

non cooperative networks2

Non-cooperative Networks

Users take control decisions individually to max own performance

Similar to non cooperative games

non cooperative networks3

Non-cooperative Networks

Users take control decisions individually to max own performance

Similar to non cooperative games

Operating points of such networks are determined by Nash equilibria

non cooperative networks4

Non-cooperative Networks

Users take control decisions individually to max own performance

Similar to non cooperative games

Operating points of such networks are determined by Nash equilibria

Nash Equilibria – Unilateral deviation doesn’t help any user

non cooperative networks5

Non-cooperative Networks

Users take control decisions individually to max own performance

Similar to non cooperative games

Operating points of such networks are determined by Nash equilibria

Nash Equilibria – Unilateral deviation doesn’t help any user

Inefficient, leads to sub optimal performance

non cooperative networks6

Non-cooperative Networks

Users take control decisions individually to max own performance

Similar to non cooperative games

Operating points of such networks are determined by Nash equilibria

Nash Equilibria – Unilateral deviation doesn’t help any user

Inefficient, leads to sub optimal performance

Better solution needed !

network manager1

Network Manager

Architects the n/w to achieve efficient equilibria

network manager2

Network Manager

Architects the n/w to achieve efficient equilibria

Run time phase

network manager3

Network Manager

Architects the n/w to achieve efficient equilibria

Run time phase

Awareness of users behavior

network manager4

Network Manager

Architects the n/w to achieve efficient equilibria

Run time phase

Awareness of users behavior

Aims to improve overall system performance through maximally efficient strategies

network manager5

Network Manager

Architects the n/w to achieve efficient equilibria

Run time phase

Awareness of users behavior

Aims to improve overall system performance through maximally efficient strategies

Maximally efficient strategy

Optimizes overall performance

network manager6

Network Manager

Architects the n/w to achieve efficient equilibria

Run time phase

Awareness of users behavior

Aims to improve overall system performance through maximally efficient strategies

Maximally efficient strategy

Optimizes overall performance

Individual users are well off at this operating point [Pareto Efficient]

presentation outline1

Presentation Outline

Introduction to non cooperative networks

Overview of approach

Model and Problem Formulation

Non cooperative User & Manager

Single Follower Stackelberg Routing game

Multi Follower Stackelberg Routing game

Issues

overview of this approach1

Overview of this approach

Total flow: Flow of users + Flow of manager

overview of this approach2

Overview of this approach

Total flow: Flow of users + Flow of manager

Example of manager’s flow

Traffic generated by signaling/control mechanism

overview of this approach3

Overview of this approach

Total flow: Flow of users + Flow of manager

Example of manager’s flow

Traffic generated by signaling/control mechanism

Users traffic that belongs to virtual networks

overview of this approach4

Overview of this approach

Total flow: Flow of users + Flow of manager

Example of manager’s flow

Traffic generated by signaling/control mechanism

Users traffic that belongs to virtual networks

Manager optimizes system performance by controlling its portion of flow

overview of this approach5

Overview of this approach

Total flow: Flow of users + Flow of manager

Example of manager’s flow

Traffic generated by signaling/control mechanism

User traffic that belongs to virtual networks

Manager optimizes system performance by controlling its portion of flow

Investigates manager’s role using routing as a control paradigm

non cooperative routing scenario

Non Cooperative Routing Scenario

IPv4/IPv6 allow source routing

User determines the path its flow follows from source- destination

goal of manager

Goal of Manager

Capability of Manager

Optimize overall network performance according to some system wide efficiency criterion

  • It is aware of non cooperative behavior of users and performs its routing based on this information
central idea1

Central Idea

Manager can predict user responses to its routing strategies

central idea2

Central Idea

Manager can predict user responses to its routing strategies

Allows manager to choose a strategy that leads of optimal operating point

central idea3

Central Idea

Manager can predict user responses to its routing strategies

Allows manager to choose a strategy that leads of optimal operating point

Example of Leader-Follower Game [Stackelberg]

slide30

MAN

VP’s k

VP’s k

VP’s k

Org1

Org n

Org2

User 1

User p

User 2

User 3

need to derive

Need to derive

A necessary and sufficient condition that guarantees that the manager can enforce an equilibrium that coincides with the network optimum

Above condition requires –

Manager’s flow Control > Threshold

need to derive1

Need to derive

A necessary and sufficient condition that guarantees that the manager can enforce an equilibrium that coincides with the network optimum

Above condition requires –

Manager’s flow Control > Threshold

If the above criterion is met, we can show that the maximally efficient strategy of manager is unique and we will specify its structure explicitly

presentation outline2

Presentation Outline

Introduction to non cooperative networks

Overview of approach

Model and Problem Formulation

Non cooperative User & Manager

Single Follower Stackelberg Routing game

Multi Follower Stackelberg Routing game

Issues

model and problem formulation

1

2

Source

Destination

L

Model and Problem Formulation

User set I = {1,…..,I}

Communication Links L= {1,.....,L}

model and problem formulation contd

Model and Problem Formulation (contd)

Manager is referred at user 0

I0= I U {0}

cl = capacity of link l

c = (c1,….cL) : capacity configuration

C = lL cl : total capacity of the system of parallel links

c1 >= c2 >= …. >= cL

Each i  I0 has a throughput demand ri > 0

r1 >= r2 >= …. >= rI r = iI ri

R = r + r0

Demand is less than capacity of links  R < C

model and problem formulation contd1

Model and Problem Formulation (contd)

User i  I0 splits its demand ri over the set of parallel links to send its flow

Expected flow of user i on link l is fli

Routing strategy of user i fi = (f1i,….fLi)

Strategy space of user i 

Fi = {fi IRL : 0 <= fli <= cl, l  L; lL fli = ri}

Routing strategy profile f = {f0, f1,….,fI)

System strategy space  F = iIoFi

model and problem formulation contd2

Model and Problem Formulation (contd)

Cost function quantifying GoS of user i’s flow is

Ji : F  IR i  I0

Cost of user i under strategy profile f is Ji(f)

Ji(f) = lL fliTl(fl); Tl(fl) is the average delay on link l, depends only on the total flow fl = iIofli on that link

Tl(fl) = (cl - fl)-1, fl < cl

= , fl >= cl

Total cost J(f) = iIoJi(f) = lL fl/ (cl - fl)

Higher cost  lower GoS provided to the user’s flow, higher average delay

model and problem formulation contd3

Model and Problem Formulation (contd)

is a convex function of (f1, …, fL)

 a unique link flow configuration exists – min cost

(f1*,….fL*) ;

Above is solution to classical routing opt problem, routing of all flow (users+manager) is centrally controlled; referred to as network optimum.

kuhn tucker optimality conditions

Kuhn – Tucker Optimality conditions

(f1*,….fL*) is the network optimum if and only if there exists a Lagrange Multiplier , such that for every link lL

presentation outline3

Presentation Outline

Introduction to non cooperative networks

Overview of approach

Model and Problem Formulation

Non cooperative User & Manager

Single Follower Stackelberg Routing game

Multi Follower Stackelberg Routing game

Issues

non cooperative users1

Non cooperative users

Each user tries to find a routing strategy fi Fi that minimizes its cost Ji (average time delay)

non cooperative users2

Non cooperative users

Each user tries to find a routing strategy fi Fi that minimizes its cost Ji (average time delay)

This minimization depends on strategies of the manager and other users, described by strategy profile

f-i = (f0 , f1 ,… fi-1, fi+1 ,… fI )

non cooperative users3

Non cooperative users

Each user tries to find a routing strategy fi Fi that minimizes its cost Ji (average time delay)

This minimization depends on strategies of the manager and other users, described by strategy profile

f-i = (f0 , f1 ,… fi-1, fi+1 ,… fI )

Routing strategy of manger is FIXED  f0

non cooperative users4

Non cooperative users

Each user tries to find a routing strategy fi Fi that minimizes its cost Ji (average time delay)

This minimization depends on strategies of the manager and other users, described by strategy profile

f-i = (f0 , f1 ,… fi-1, fi+1 ,… fI )

Routing strategy of manger is FIXED  f0

Each user adjusts its strategy to other users actions

non cooperative users5

Non cooperative users

Each user tries to find a routing strategy fi Fi that minimizes its cost Ji (average time delay)

This minimization depends on strategies of the manager and other users, described by strategy profile

f-i = (f0 , f1 ,… fi-1, fi+1 ,… fI )

Routing strategy of manger is FIXED  f0

Each user adjusts its strategy to other users actions

Can be modeled as a non cooperative game, any operating point is Nash Equilibrium; dependent on f0 !

non cooperative users6

Non cooperative users

Each user tries to find a routing strategy fi Fi that minimizes its cost Ji (average time delay)

This minimization depends on strategies of the manager and other users, described by strategy profile

f-i = (f0 , f1 ,… fi-1, fi+1 ,… fI )

Routing strategy of manger is FIXED  f0

Each user adjusts its strategy to other users actions

Can be modeled as a non cooperative game, any operating point is Nash Equilibrium; dependent on f0 !

From users view point, manager reduces capacity on each link l by fl0 , the system reduces to a set of parallel links with capacity configuration c – f0 has a unique Nash Equilibrium

f0  f-0 ……. N0(f0)

non cooperative users7

Non cooperative users

For a given strategy profile f-i of other users in I0, the cost of i, Ji(f) = lL fliTl(fl), is a convex fn of its strategy fi , hence the following min problem has a unique solution

kuhn tucker optimality conditions1

Kuhn – Tucker Optimality conditions

fi is the optimal response of user i if and only if there exists a (Lagrange Multiplier) , such that for every link lL, we have

non cooperative users8

Non cooperative users

f-0  F-0 is a Nash Equilibrium of the self optimizing users induced by strategy f0 of the manger.

The function N0 : F0 F-0 that assigns the induced equilibrium of the user routing game (to each strategy of the manger) is called the Nash Mapping. It is continuous.

role of the manager

Role of the Manager

It has knowledge of non cooperative behavior of users; determines the Nash Equilibrium N0(f0) induced by any routing strategy it f0 chosen by him

Acts as Stackelberg leader, that imposes its strategy on the self optimizing users that behave as followers

Aims to optimize the overall network performance, plays a social rather than selfish role

To find f0 such that if f-0 = N0(f0), then iIofli = fl* for all l

Thisf0 is called maximally efficient strategy of manager

It is Pareto efficient !

outline of results

Outline of Results

In case of a single user, the manager can always enforce network optimum; its MES is specified explicitly

In case of any no of users, the manager can enforce the network optimum iff its demand is higher that some threshold r0, in which case the MES is specified explicitly

r0 is feasible if total demand of users plus r0 is less than C

It is easy for manager to optimize heavily loaded networks as r0is small

As the no of user increases, threshold increases i.e. harder for manager to enforce network optimum

The higher the difference in throughput demands of any two users, the easier it is for manager to enforce network optimum

slide53

Network optimum: (f1*,….fL*)

Flow on link l, fl* is decreasing in link no l L

There exists some link L*, such that fl* > 0 for l <= L* and fl* = 0 for l > L* ; L* is determined by (from [1] & [2]),

where

and G1=0, GL+1=ln=1cn = C

cl >= cl+1 Gl <= Gl+1

slide55

Best reply fi of user i  I0 to the strategies of manager and other users, described by f-i, can be determined as network optimum for a system of parallel links with capacity configuration (c1i,…, cLi)

Assuming cli >= cl+1i , l=1,…,L-1

the flow fliis decreasing in the link no l L

There exists some link Li, such that fli> 0 for l <= Li and fli= 0 for l > Li ; The threshold Li is determined by

slide56

Best reply fi of user i to strategy profile f-i of the other users in I0 is given by

Best reply doesn’t depend on detailed description of f-i but only on residual capacity cli seen by user on every link l L

In practice, residual capacity info can be acquired by measuring the link delays using an appropriate estimation technique

presentation outline4

Presentation Outline

Introduction to non cooperative networks

Overview of approach

Model and Problem Formulation

Non cooperative User & Manager

Single Follower Stackelberg Routing game

Multi Follower Stackelberg Routing game

Issues

single follower stackelberg routing game1

Single Follower Stackelberg Routing Game

In this game, there exists a MES of the manager then it is unique and is given by

single follower stackelberg routing game2

Single Follower Stackelberg Routing Game

  • The best reply f1 of the follower is
  • Therefore, {1,…,L1} is the set of links over which the follower sends its flow when manager implements f0.
  • For manager: Send flow fl* on every link l that will not receive any flow from the follower

Split the rest of its flow among the links that will receive user flow proportional to their capacities

presentation outline5

Presentation Outline

Introduction to non cooperative networks

Overview of approach

Model and Problem Formulation

Non cooperative User & Manager

Single Follower Stackelberg Routing game

Multi Follower Stackelberg Routing game

Issues

multi follower stackelberg routing game1

Multi Follower Stackelberg Routing Game

  • An arbitrary number I of self optimizing users share the system of parallel links
multi follower stackelberg routing game2

Multi Follower Stackelberg Routing Game

  • An arbitrary number I of self optimizing users share the system of parallel links
  • Maximally Efficient Strategy of manager (if it exists) and the corresponding Nash Equilibrium of non cooperative users is:
multi follower stackelberg routing game3

Multi Follower Stackelberg Routing Game

  • Equilibrium strategy fi of user i I is described by
  • If a MES exists, then the induced Nash equilibrium of the followers has precisely the same structure with the best reply follower in the single follower case
remarks m f stackelberg routing game1

Remarks - M F Stackelberg Routing Game

  • {1,…., Li} is the set of links that receive flow from follower i  I
remarks m f stackelberg routing game2

Remarks - M F Stackelberg Routing Game

  • {1,…., Li} is the set of links that receive flow from follower i  I
  • Il is the set of followers that send flow on link l. Since H1 = 0 < ri, i  I, all users send flow on link 1  I1 = I
remarks m f stackelberg routing game3

Remarks - M F Stackelberg Routing Game

  • {1,…., Li} is the set of links that receive flow from follower i  I
  • Il is the set of followers that send flow on link l. Since H1 = 0 < ri, i  I, all users send flow on link 1  I1 = I
  • For f0 to be admissible, fl0 >= 0, for all l  L
remarks m f stackelberg routing game4

Remarks - M F Stackelberg Routing Game

  • {1,…., Li} is the set of links that receive flow from follower i  I
  • Il is the set of followers that send flow on link l. Since H1 = 0 < ri, i  I, all users send flow on link 1  I1 = I
  • For f0 to be admissible, fl0 >= 0, for all l  L
  • If fl0 < 0  fl-10 < 0
remarks m f stackelberg routing game5

Remarks - M F Stackelberg Routing Game

  • {1,…., Li} is the set of links that receive flow from follower i  I
  • Il is the set of followers that send flow on link l. Since H1 = 0 < ri, i  I, all users send flow on link 1  I1 = I
  • For f0 to be admissible, fl0 >= 0, for all l  L
  • If fl0 < 0  fl-10 < 0
  • Admissible condition reduces to f10 >= 0
remarks m f stackelberg routing game6

Remarks - M F Stackelberg Routing Game

  • {1,…., Li} is the set of links that receive flow from follower i  I
  • Il is the set of followers that send flow on link l. Since H1 = 0 < ri, i  I, all users send flow on link 1  I1 = I
  • For f0 to be admissible, fl0 >= 0, for all l  L
  • If fl0 < 0  fl-10 < 0
  • Admissible condition reduces to f10 >= 0
  • f10 is an increasing function of the throughput demand r0 of leader, r0 [0, C - r] ………. [3]
theorem

Theorem

  • There exists some r0, with 0 < r0 < C – r, such that the leader in multi follower Stackelberg routing game can enforce the network optimum if and only if its throughput demand r0 satisfies r0 < r0 < C – r. The maximally efficient strategy of leader is given by
presentation outline6

Presentation Outline

Introduction to non cooperative networks

Overview of approach

Model and Problem Formulation

Non cooperative User & Manager

Single Follower Stackelberg Routing game

Multi Follower Stackelberg Routing game

Issues

properties of leader threshold r 01

Properties of Leader Threshold r0

  • r0 of the leader is a unique solution of the equation

“f10(r0) = 0” in r0 [0, C - r]

properties of leader threshold r 02

Properties of Leader Threshold r0

  • r0 of the leader is a unique solution of the equation

“f10(r0) = 0” in r0 [0, C - r]

  • When r C, r0 0 i.e. in heavily loaded networks, controlling a small portion of flow can drive the system into the network optimum
properties of leader threshold r 03

Properties of Leader Threshold r0

  • r0 of the leader is a unique solution of the equation

“f10(r0) = 0” in r0 [0, C - r]

  • When r C, r0 0 i.e. in heavily loaded networks, controlling a small portion of flow can drive the system into the network optimum
  • With throughput demand r fixed, the leader threshold r0 increases with increase in no of users.
properties of leader threshold r 04

Properties of Leader Threshold r0

  • r0 of the leader is a unique solution of the equation

“f10(r0) = 0” in r0 [0, C - r]

  • When r C, r0 0 i.e. in heavily loaded networks, controlling a small portion of flow can drive the system into the network optimum
  • With throughput demand r fixed, the leader threshold r0 increases with increase in no of users.
  • Leader threshold r0 decreases with increase in difference in user demands
slide80

Set of Parallel links with capacity conf (12,7,5,3,2,1), shared by I identical followers with total demand r

slide81

Set of Parallel links with capacity conf (12,7,5,3,2,1), shared by 100 identical self optimizing users with total demand r and the manager

r0= r0

slide82

Set of Parallel links with capacity conf (12,7,5,3,2,1), shared by 100 identical self optimizing users with total demand r and the manager

r0= r0

slide83

Set of Parallel links with capacity conf (12,7,5,3,2,1), shared by 100 identical self optimizing users with total demand r and the manager

r0= r0

slide84

Set of Parallel links with capacity conf (12,7,5,3,2,1), shared by 100 identical self optimizing users with total demand r and the manager

r0= r0

scalability1

Scalability

  • To determine maximally efficient strategy, manager needs throughput demand ri of every user.
scalability2

Scalability

  • To determine maximally efficient strategy, manager needs throughput demand ri of every user.
  • In many networks, user declare average rate ri during negotiation phase
scalability3

Scalability

  • To determine maximally efficient strategy, manager needs throughput demand ri of every user.
  • In many networks, user declare average rate ri during negotiation phase
  • Alternatively, the manager can estimate average rates by monitoring the behavior of users
scalability4

Scalability

  • To determine maximally efficient strategy, manager needs throughput demand ri of every user.
  • In many networks, user declare average rate ri during negotiation phase
  • Alternatively, the manager can estimate average rates by monitoring the behavior of users
  • Manager can adjust its strategy to maximally efficient one whenever a user departs or a new one joins the network
scalability5

Scalability

  • To determine maximally efficient strategy, manager needs throughput demand ri of every user.
  • In many networks, user declare average rate ri during negotiation phase
  • Alternatively, the manager can estimate average rates by monitoring the behavior of users
  • Manager can adjust its strategy to maximally efficient one whenever a user departs or a new one joins the network
  • User not necessarily mean a single user, it can be a group of users joining the network as an organization. It also reduces threshold r0
references

References

[1] A. Orda, R. Rom, and N. Shimkin, “Competitive routing in multi- user communication networks,” IEEE/ACM Trans. Networking, vol. 1, pp. 510-521, Oct. 1993.

[2] Y.A. Korilis, A.A. Lazar, and A. Orda, “Capacity allocation under non cooperative routing,” IEEE Trans. Automat. Contr.

[3] Y.A. Korilis, A.A. Lazar, and A. Orda, “Achieving network optima using Stackelberg routing strategies,” Center for Telecommunications Research, Columbia University, NY, CTR Tech. Rep. 384-94-31, 1994.