Loading in 2 Seconds...

Achieving Network Optima Using Stackelberg Routing Strategies

Loading in 2 Seconds...

- By
**trudy** - Follow User

- 95 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about 'Achieving Network Optima Using Stackelberg Routing Strategies' - trudy

**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.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

### Achieving Network Optima Using Stackelberg Routing Strategies

### Presentation Outline

### Non-cooperative Networks

### Non-cooperative Networks

### Non-cooperative Networks

### Non-cooperative Networks

### Non-cooperative Networks

### Non-cooperative Networks

### Network Manager

### Network Manager

### Network Manager

### Network Manager

### Network Manager

### Presentation Outline

### Overview of this approach

### Overview of this approach

### Overview of this approach

### Overview of this approach

### Overview of this approach

### Non Cooperative Routing Scenario

### Goal of Manager

### Central Idea

### Central Idea

### Central Idea

### Need to derive

### Need to derive

### Presentation Outline

### Model and Problem Formulation

### Model and Problem Formulation (contd)

### Model and Problem Formulation (contd)

### Model and Problem Formulation (contd)

### Model and Problem Formulation (contd)

### Kuhn – Tucker Optimality conditions

### Presentation Outline

### Non cooperative users

### Non cooperative users

### Non cooperative users

### Non cooperative users

### Non cooperative users

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

### Non cooperative users

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

### Non cooperative users

### Kuhn – Tucker Optimality conditions

### Non cooperative users

### Role of the Manager

### Outline of Results

### Presentation Outline

### Single Follower Stackelberg Routing Game

### Single Follower Stackelberg Routing Game

### Presentation Outline

### Multi Follower Stackelberg Routing Game

### Multi Follower Stackelberg Routing Game

### Multi Follower Stackelberg Routing Game

### Remarks - M F Stackelberg Routing Game

### Remarks - M F Stackelberg Routing Game

### Remarks - M F Stackelberg Routing Game

### Remarks - M F Stackelberg Routing Game

### Remarks - M F Stackelberg Routing Game

### Remarks - M F Stackelberg Routing Game

### Theorem

### Presentation Outline

### Properties of Leader Threshold r0

### Properties of Leader Threshold r0

### Properties of Leader Threshold r0

### Properties of Leader Threshold r0

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

### 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

### 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

### 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

### 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

### Scalability

### Scalability

### Scalability

### Scalability

### Scalability

### References

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

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

Users take control decisions individually to max own performance

Users take control decisions individually to max own performance

Similar to non cooperative games

Users take control decisions individually to max own performance

Similar to non cooperative games

Operating points of such networks are determined by Nash equilibria

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

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

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 !

Architects the n/w to achieve efficient equilibria

Architects the n/w to achieve efficient equilibria

Run time phase

Awareness of users behavior

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

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

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]

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

Total flow: Flow of users + Flow of manager

Total flow: Flow of users + Flow of manager

Example of manager’s flow

Traffic generated by signaling/control mechanism

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

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

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

IPv4/IPv6 allow source routing

User determines the path its flow follows from source- destination

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

Manager can predict user responses to its routing strategies

Manager can predict user responses to its routing strategies

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

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]

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

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

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

2

Source

Destination

L

User set I = {1,…..,I}

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

Manager is referred at user 0

I0= I U {0}

cl = capacity of link l

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

C = lL 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 = iI ri

R = r + r0

Demand is less than capacity of links R < C

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; lL fli = ri}

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

System strategy space F = iIoFi

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) = lL fliTl(fl); Tl(fl) is the average delay on link l, depends only on the total flow fl = iIofli on that link

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

= , fl >= cl

Total cost J(f) = iIoJi(f) = lL fl/ (cl - fl)

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

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.

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

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

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

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 )

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 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

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

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 !

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

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)

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

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

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.

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 iIofli = fl* for all l

Thisf0 is called maximally efficient strategy of manager

It is Pareto efficient !

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

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

Using Lagrange Multiplier’s equations, we get,

Network Optimum is given by [2]

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

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

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

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

- 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

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

- An arbitrary number I of self optimizing users share the system of parallel links

- 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:

- 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

- {1,…., Li} is the set of links that receive flow from follower i I

- {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

- {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

- {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

- {1,…., Li} is the set of links that receive flow from follower i I
- For f0 to be admissible, fl0 >= 0, for all l L
- If fl0 < 0 fl-10 < 0
- Admissible condition reduces to f10 >= 0

- {1,…., Li} is the set of links that receive flow from follower i 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]

- 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

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

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

“f10(r0) = 0” in r0 [0, C - r]

- 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

- 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.

- 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

r0= r0

r0= r0

r0= r0

r0= r0

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

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

- 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

- 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

- 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

[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.

Download Presentation

Connecting to Server..