Information Theory for Mobile Ad-Hoc Networks (ITMANET): The FLoWS Project

1. Information Theory for Mobile Ad-Hoc Networks (ITMANET): The FLoWS Project Competitive Scheduling in Wireless Networks with Correlated Channel State Ozan Candogan, Ishai Menache, Asu Ozdaglar, Pablo Parrilo

2. Competitive Scheduling in Wireless Collision Channels withCorrelated Channel State (Candogan, Menache, Ozdaglar, Parrilo) FLOWS ACHIEVEMENT STATUS QUO IMPACT p1 p1 Central component p2 p2 p3 BR(p-3)‏ +1 0 Local components (buildings)‏ +1 -1, -1 4, 0 0 0, 4 -1, -1 A similar game with 2 users and a single state. Our problem is more complex, with continuous action spaces and multiple fading states NEXT-PHASE GOALS NEW INSIGHTS • Existing work on scheduling focuses on hard to analyze centralized schemes with single performance objective • Competitive scheduling models allow the flexibility to incorporate different user objectives, but focus mainly on users with independent channel models • Correlated channels are more realistic extensions of the current model as they incorporate joint fading effects • Aggregate utility maximization problem is non-convex and difficult to solve. Instead, a simple distributed framework is suggested. • Robust system design in the presence of non-cooperative users is achieved and efficiency loss due to selfishness of users is bounded. Achievement: • Convergent dynamics and equilibria characterization for competitive scheduling in collision channels • Bounds on price of stability (efficiency losses due to selfishness) • Equilibrium paradoxes: Bad-quality states can be used more frequent than good states. How it works: • Best response dynamics (or fictitious play) and potential game property imply convergence • Characterization of the social optimum in order to find efficiency losses due to selfishness. • Bounds on price of stability is achieved by bounding the change in aggregate utility when a Nash equilibrium is obtained by perturbing the social optimum Assumptions and limitations: • Perfect correlation across channel state processes of users • Symmetric-rate assumption for a more tractable analysis Combine tools from optimization and game theory • Selfish utility maximization framework for collision channels • Use tools from optimization theory to analyze wireless network games • Potential games and convergence algorithms Principle: When user 3 updates its strategy through best-response, the potential function increases after each update • Extend the results to partial channel state-correlation (local & central fading components)‏ • Additional channel models (e.g., CDMA)‏ • Convergence of dynamics with asynchronous updates and with limited information • Extending the results to general networks models which also include routing Robust scheduling and power allocation in wireless networks with selfish users

3. Motivation Centralized resource allocation in wireless networks is usually Hard to implement and sustain not robust to selfishness of users Hence, there is an increasing interest in distributed resource allocation in wireless networks under the presence of self-interested users. One of the most distinctive features of wireless networks is the time-varying nature of the channel quality, an effect known as fading. Users may base their transmission decision on the current channel quality. This decision model naturally leads to a non-cooperative game, since each user’s decision affects other users through the commonly shared channel.

4. Motivation • Unlike existing work in the area which assumes independent state-processes for different users, we consider the case where the channel state is correlated across users. • This correlated-fading model allows to capture global network elements that affect all mobiles.

5. Goals and Relevance • Goals: • Study the equilibria of the competitive scheduling game • Establish tight bounds on efficiency loss due to noncooperative behavior of the users • Suggest network dynamics that converge to desired equilibrium points. • Relevant examples: • Satellite networks • Any uplink wireless network in which global noise effects are possible.

6. The Model • We consider a finite set of mobiles transmitting to a common base-station. • Time is slotted, and the channel quality is revealed to the user prior to each transmission. • Each user may base its transmission decision on the current channel state. • In the present work, we assume perfect correlation, meaning that all users observe the same channel state, yet may obtain different rates for a given state. User 1, p1 User 2, p2 User 3, p3 Due to perfect correlation, all users observe the same fading state at a given time slot • Let be the state space. We denote by the rate that user m will obtain in state j, provided that there are no simultaneous transmissions • Monotonicity assumption: If j<i then for every user m.

7. The Model • Underlying fading process is assumed to be stationary-ergodic. We denote by the state-state probability for channel state i. • Reception Model: We assume a collision channel: Simultaneous transmissions are lost. A single transmission is always successful. • Users are assumed to adopt stationary strategies, : policy of user m, where : transmission probability of user m in state j

8. The Noncooperative Game • Performance measures: Average power and average throughput • Average power: • Average throughput: • Each user is interested in maximizing its throughput with minimal power investment. We capture this tradeoff via the following utility where is a positive constant. • Each user is interested in maximizing the above utility subject to an individual power constraint, i.e., • Nash equilibrium:

9. Centrally Optimal Scheduler • Central optimization problem • The central optimization problem is non-convex and hard to characterize. • Partial characterization: There exists an optimal solution of the central optimization problem where users utilize threshold policies - meaning that for each user m there exist at most a single state j so that

10. Equilibrium Characterization • A Nash equilibrium point always exists. • There can be infinitely many Nash equilibria. • For the case where , best equilibrium coincides with the social optimum (i.e., price of stability is one). • Price of anarchy (upper bound on the performance ratio between the social optimum and the worst Nash equilibrium) can be arbitrary large. • Bad quality states might be utilized more frequently than good states, where utilization is defined as the overall transmission probability at a given state

11. Convergence to Equilibrium • Additional assumption: Assume that rates are user independent, i.e., • Potential games are games in which a change in a single user’s payoff is equivalent to a change in a system’s potential function. An important property of such games is the convergence of sequential best-response dynamics. • Thus, under the above assumption, our game convergences to an equilibrium in case that users update their strategies in a sequential manner. • Convergence properties without the above assumption are under current study.

12. Future Work Further characterization of equilibrium points. Explicit bounds on price of stability/anarchy for special cases of interest. Partial state-correlation models. Asynchronous best response dynamics and their properties Multi-hop networks. Additional reception models (such as CDMA based networks).‏