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RateBased Equilibria in Random Access Wireless Networks

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Collision channel - If two users transmit at the same slot, their data ... We show that the above function is continuous and unimodal. 14. Equilibrium Analysis ... – PowerPoint PPT presentation

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Title: RateBased Equilibria in Random Access Wireless Networks


1
Rate-Based Equilibria in Random Access Wireless
Networks
  • Ishai Menache
  • Joint work with Nahum Shimkin
  • QCG Meeting, December 07

2
Wireless Medium Access Games
  • Traffic types
  • User preferences (QoS) (e.g, throughput, delay,
    power)
  • Physical channel characteristics
  • Reception model
  • Fading
  • Information structure
  • Adjustable parameters and protocol constraints

3
Our Model
  • A multiple access network shared by a finite
    number of users i1,,n who transmit to a
    common base-station.
  • Time is slotted.
  • Collision channel - If two users transmit at the
    same slot, their data gets lost.

4
Fading Channels
  • At the beginning of every time slot k, each user
    obtains a Channel State Information (CSI) signal
    representing its channel quality.
  • For each user, is a sequence of i.i.d.
    random variables.
  • and are independent for every
    i,j.
  • The useful data rate per slot (i.e., in case of
    no collision) is increasing with the CSI signal.

5
Fading Channels
  • Example The continuous CSI case

6
User Model
  • Users always have packets to send.
  • Each user is characterized by an average
    throughput requirement (bits/slot)
  • The objective
  • Note Transmission rate is proportional to power
    consumption

7
User Strategy
  • In this work we focus on stationary transmission
    strategies.
  • Thus, a transmission strategy for user i is a
    mapping which assigns a transmission
    probability for each possible CSI.

8
User Strategy
  • Lemma The best response stationary strategy is
    such that there exists a signal threshold
    above which the user always transmits, and below
    which it never transmits.

9
Nash Equilibrium
  • Basic assumption users are self-interested and
    do not coordinate their strategies.
  • A multi-strategy is a Nash
    equilibrium if

ConcaveIncreasing
10
Equilibrium Analysis
  • Lemma A multi-strategy is
    a Nash equilibrium if and only if

11
Equilibrium Analysis

12
Equilibrium Analysis
  • Theorem There are two Nash equilibrium points or
    none for the game (the case of a single
    equilibrium point is non-generic).
  • Proof sketch

at equilibrium
13
Equilibrium Analysis
  • Proof sketch (cont.) Define the scalar function
  • We show that the above function is continuous
    and unimodal.

14
Equilibrium Analysis
  • Hence, when an equilibrium exists, there are two
    equilibria, one uniformly better than the other.
  • We show that at the best equilibrium.
  • Moreover, the better equilibrium coincides with
    the social optimum with respect to the latter
    criterion.

15
Best Response Dynamics
  • Best Response
  • Theorem Starting from an empty configuration,
    asynchronous best response dynamics
    asymptotically converge to the best equilibrium.
  • Theorem These dynamics are resilient to multiple
    user arrivals or multiple departures.

Can be measured
16
Summary
  • There are two Nash equilibria or none for the
    non-cooperative game.
  • One equilibrium is strictly better than the other
    for all users.
  • A simple distributed mechanism converges to the
    best equilibrium.

17
Extensions
  • Examine other reception models (e.g.,
    multi-packet reception)
  • Additional performance criteria
  • Incorporating delay
  • Allow multiple power levels
  • Convergence analysis, including identification
    and prevention of worse equilibria.
  • Non-stationary, history dependent policies
  • Non-cooperative dynamic games
  • Does the system benefit from the use of more
    complex strategies?
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