Download presentation
Presentation is loading. Please wait.
1
Network Routing Problem r Input: m network topology, link metrics, and traffic matrix r Output: m set of routes to carry traffic A B C D E S1S1 R1R1 R3R3 R2R2 S3S3 S2S2
2
Network Routing: Classical Approach r Routing as optimization problem m e.g., minimum total delay in network m focus on global network performance (social optimal) m performance of individual user not important r Centralized or distributed algorithms m e.g., link state or distance vector r Passive users m users are oblivious to routing decisions
3
Network Routing: Game-Theoretic Approach r Routing as game between users m users determine route m decision based solely on individual performance (selfish routing) m strongly dependent on other users’ decisions r Non-cooperative game (non-zero sum) m users compete for network resources r Equilibrium point of operation m Nash equilibrium point (NEP)
4
Selfish Network Routing r Advantages m no need of centralized control or global agreement on routing algorithm m individual user’s performance considered m greater adaptability changes in user demands or changes in network conditions r Disadvantages m multiple equilibria (eq. selection problem) m convergence to equilibrium m no network-wide optimality at equilibrium cost of “selfish routing” m user’s must have detailed knowledge of network
5
Applications of Game Theory to Network Routing r Competitive routing in multiuser communication networks A. Orda, R. Rom and N. Shimkin IEEE/ACM Transactions on Networking, 1 (5) 1993 r How bad is selfish routing? T. Roughgarden and E. Tardos Journal of the ACM, 49 (2) 2002 r Selfish routing with atomic players T. Roughgarden ACM/SIAM Symp. on Discrete Algorithms (SODA) 2005
6
Simple Model: Network of Parallel Links r set of users share a set of parallel links r each user has fixed demand (data rate) r users decide how to split demand across links m minimize individual cost r link has a load dependent cost (e.g., delay) A S1S1 B R1R1 S2S2 S3S3 S4S4 R2R2 R3R3 R4R4
7
r set of parallel links: r set of users: r each user has a fixed demand (data rate): r user splits its demand across links flow of user i on link l : flow configuration of user i : r system flow configuration: r feasible configurations m satisfy nonnegative and demand constraints Network of Parallel Links
8
User’s Cost Function Cost function of user i : m cost depends on flow configuration of all users r Assumptions on cost function m sum of user-link cost function: m can be infinite m convex in m when finite, continuously differentiable in m at least one user with infinite cost can change its flow configuration to have finite cost aggregate demand must be less than aggregate link capacity Very mild conditions on cost function
9
The Game r Users individually decide their flow configuration m goal is to minimize its own cost r Nash Equilibrium Point (NEP) m system flow configuration such that no user can reduce their cost by changing its flow allocation is a NEP if for all i, the following holds: No user can reduce their cost by rerouting their own flow
10
The Issues r Existence of NEP m is at least one NEP guaranteed to always exist r Uniqueness of NEP m under which conditions (if any) do we have a single NEP r Convergence to (and stability of) NEP m play dynamics that lead to a NEP r System properties at the NEP m e.g., how does users divide allocate their flows
11
Existence of NEP r N-person convex game [Rosen65] m joint strategy set is convex, closed and bounded m each player’s payoff function is convex in their own strategy m existence of NEP proven by Katutani fixed point theorem r Can also show using Kuhn-Tucker conditions m necessary and sufficient for system flow configuration to be a NEP
12
Uniqueness of NEP r Uniqueness of NEP only under a type of cost functions (type-A functions) cost function has two parameters: user’s i and aggregate of all others m monotonically increasing in each parameter m still very general (e.g., M/M/1 delay function) r Proof by contradiction using Kuhn-Tucker conditions System has a single operating point
13
System Properties at NEP r Assume all users share same type-A cost function m but users can have different demands r Monotonicity of link usage m user with higher demand uses more of each and every link used m a user with higher demand uses more links r Higher capacity links receive more users m does not hold in general, only under yet another type of cost function (which still captures M/M/1) Intuitive but assuring properties
14
r Simple case study m two-users sharing two parallel links r Dynamical model: Elementary Stepwise System m Users take turns in updating their flow configuration measure load on links, adjust its flow to minimize cost flow of user i on link l at step n Dynamical System A S1S1 B R1R1 S2S2 R2R2
15
Convergence to NEP r Let denote unique NEP of game r Initialize system with any feasible flow configuration: f(0) r Convergence to NEP guaranteed r Framework used in proof not aplicable in general m limited to two link, two user structure
16
General Topology r Users decide how to split their demands over possible paths m users know network topology (directed graph) A B C D E S1S1 R1R1 R3R3 R2R2 S3S3 S2S2
17
Existence and Uniqueness of NEP r Existence of NEP m same argument as before (N-person convex game) r No unique NEP for type-A cost functions m shown by counterexample r Uniqueness shown only under very strict conditions for cost function m not very interesting networking scenarios Analysis of general network in this modeling framework is much harder
18
The “Price of Anarchy” r Equilibria of non-cooperative games usually inefficient m e.g., prisoner’s dilemma m Pareto optimal usually not a NEP r Quantify inefficiency in terms of a global objective m “price of anarchy” (coordination versus competition) Price of Anarchy of a Game objective function value at NEP optimal objective function value = m if multiple NEP exists, take sup (or inf) over NEP set
19
Cost of Selfish Routing r How does total cost compare? m flow allocation at a NEP m optimal flow allocation r Total cost of flow configuration: m where is load dependent link cost function m e.g., link delay
20
Example (1/4) r flow configuration cost r optimal flow allocation r can be realized with A S1S1 B R1R1 S2S2 R2R2 r 1 = 0.5 r 2 = 0.5
21
Example (2/4) r But this is not NEP… Cost of a flow configuration to user i A S1S1 B R1R1 S2S2 R2R2 r 1 = 0.5 r 2 = 0.5 r By rerouting traffic user 1 (or 2) can reduce its cost: lower cost!
22
higher cost Example (3/4) r NEP given by m link 1 is a dominant strategy (link 2 never used) Cost to user i at NEP r Total cost of NEP configuration A S1S1 B R1R1 S2S2 R2R2 r 1 = 0.5 r 2 = 0.5 higher cost!
23
Example (4/4) r Optimal cost: r NEP cost: r Price of Anarchy: A S1S1 B R1R1 S2S2 R2R2 r 1 = 0.5 r 2 = 0.5 Thm:[Roughgarden/Tardos00] POA of selfish routing w/affine cost functions is at most 4/3 m for any network topology and traffic matrix!
24
Another example (non-linear cost)… r NEP: both users only use link 1 m cost is 1 r Optimal: 1-ε for link 1 and ε for link 2 ε depends on d, but is small for large d m cost ≈ 0 r Price of anarchy can be arbitrarily large goes to infinity as d goes to infinity A S1S1 B R1R1 S2S2 R2R2 r 1 = 0.5 r 2 = 0.5
25
So how bad is selfish routing? r It depends... m cost functions, network topology, traffic matrix, user demands, etc. r In reality, not so bad m achieves close to optimal cost in Internet-like environments (simulation study) r Another positive (and nice) result: Thm:[Roughgarden/Tardos00] selfish routing is no worst than the optimal routing of twice as much traffic m for any cost function, network topology and traffic matrix!
26
Title
27
Congestion Control Problem r Input: m network topology, routes, link characteristics, traffic matrix r Output: m set of data rates to be used A B C D E S1S1 R1R1 R3R3 R2R2 S3S3 S2S2
28
Congestion Control: Classical Approach r Congestion control as optimization problem m match user’s demand to network capacity and achieve some fairness among users m focus on global network performance (social optimal) m performance of individual user not important r Centralized or distributed algorithms m e.g., TCP, max-min fairness r Passive users m users are oblivious to congestion decisions
29
Congestion Control: Game- Theoretic Approach r Congestion control as game between users m users determine their own data rates m decision based solely on individual performance r Non-cooperative game (non-zero sum) m users compete for network resources r Equilibrium point of operation m Nash equilibrium point (NEP) Key Assumption: A higher sending rate do not necessarily yields better performance for user
30
Routing Games vs Congestion Control Games r Routing games m users determine network routes m multi-path routing and traffic splitting is possible m users’ data rates are given and must be routed r Congestion games m users determine their data rate m network routes are given (single path)
31
Applications of Game Theory to Congestion Control r Making greed work in networks: a game-theoretic analysis of switch service disciplines S. Shenker IEEE/ACM Transactions on Networking, 3 (6) 1995 r An evolutionary game-theoretic approach to congestion control D. Menasché, D. Figueiredo, E. de Souza e Silva Performance Evaluation, 62 (1-4) 2005
32
Simple Model: Single Bottleneck Link S1S1 R1R1 S2S2 S3S3 S4S4 R2R2 R3R3 R4R4 r set of users share a bottleneck link r users decide their data rates m maximize individual performance r user’s performance depends on link load m e.g., quality of service provided by link
33
Single Bottleneck Link r Users determine sending rate: r Link modeled as M/M/1 queue m unit capacity m packet scheduling policy r Scheduling policy induces average queue length for each user : avg. queue length of user i r User’s utility function m strictly increasing in m strictly decreasing in m convex and derivable everywhere
34
Scheduling Policy r Determined by system operator r Allocation function m scheduling policy P induces an avg. queue length for each user given all user’s data rate r FIFO example r Must satisfy some constraints m aggregate average queue size same as M/M/1 r Allocation function can be realized by different service disciplines
35
Fair Share Allocation r Allocate service capacity fairly among user’s demand m user’s requesting less obtain higher priority r Implemented through a priority queueing algorithm Assume : r 1 < … < r N User Priority Level ABCD 1 r1r1 --- 2 r1r1 r 2 - r 1 -- 3 r1r1 r 3 - r 2 - 4 r1r1 r 2 - r 1 r 3 - r 2 r 4 - r 3 fraction of traffic gets lower priority
36
MAC: Set of Monotonic Allocation Functions r Consider a set of possible allocation functions m : increases, increases m : increases, does not decrease m r Includes all typical service disciplines m FIFO, LIFO, PS, fair share allocation at for all with r k r o k
37
The Problem Investigated r Relationship between NEP and service disciplines (MAC functions) r Which service disciplines yield good NEP? r Properties of NEP of a given MAC m efficiency m fairness m convergence to equilibrium m user protection System designer can select service discipline that yields good equilibrium
38
Efficiency of NEP r Efficiency in terms of Pareto optimal m no global objective function of system outcome r Pareto optimal outcome: m no other outcome is preferred by all users Thm:[Shenker95] There is no allocation function in MAC such that every NEP is Pareto optimal r Under some additional constraints fair share is always efficient m constrained users’ utility function m symmetric rate vector
39
Uniqueness of NEP r Allocation functions can induce multiple NEP m undesirable since users cannot coordinate Thm:[Shenker95] Fair share mechanism always has a unique NEP Fair share is the only allocation function that always yields a unique NEP
40
Convergence to Equilibrium r Dynamics through a generalized hill climbing algorithm m users eliminate strategies that always perform worst m system converges to a reduced set of strategies r Different from best-response dynamics Thm:[Shenker95] With the fair share mechanism, all generalized hill climbing algorithm converges to the NEP m Convergence is also fast (superlinear) and stable
41
Title
42
Application to Multimedia Traffic r Users share common bottleneck link r User’s choose data rate to be sent by source m only few data rates available r Utility given by “perceived” quality S1S1 R1R1 R2R2 R3R3 R4R4 Investigate dynamics and convergence using evolutionary game theory
43
Why Evolutionary Game Theory r Model how users change their strategy r Users are not perfect: stochastic dynamics, myopic, etc r Which NEP will be achieved (if more than one exists) r Efficiency of selected NEP Evolutionary Game Theory
44
Entities of Model and Interactions Users (strategy set) QoS Model (E-model or other) Link model (M/M/1/k or other) choice of strategies causes impact on performance metric feeds yields perceived quality to
45
Two-layer Markovian Model users’ actions 0, 3 1, 2 2, 1 3, 0 link perf. layer 1 layer 2 QoS of each user QoS of each user QoS of each user QoS of each user
46
States and Users Utility : number of users selecting strategy l in state : number of data rates available to users : state : utility function of strategy l in state r No constraints on users’ utility function m should be defined for every state
47
Transition Matrix r Transitions determined by QoS in each state m rate of change proportional to gain m transitions can reduce QoS (users make errors) m Markov chain is ergodic
48
Main Problem Investigated r System in steady state r Users make no mistakes Assume: States that have non-negligible steady state probability States that correspond to NEP What is the relationship?
49
Proposition 1 r under the condition that this state is contained in a quasi- absorbing set If a state has non- negligible steady state probability This state is also a NEP
50
Proposition 2 r proof via simple counter-example This state also has non-negligible steady state probability If a state is a NEP
51
Summary of Results r States with non-negligible SS probability are NEP m correspond to “stable” states r Some NEP are not stable m system dynamics cannot converge on them r Still possible to have multiple stable NEP m not clear where system will converge m state with highest probability?
52
Title
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.