- •Introduction
- •Increasing Demand for Wireless QoS
- •Technical Approach
- •Outline
- •The Indoor Radio Channel
- •Time Variations of Channel Characteristics
- •Orthogonal Frequency Division Multiplexing
- •The 5 GHz Band
- •Interference Calculation
- •Error Probability Analysis
- •Results and Discussion
- •IEEE 802.11
- •IEEE 802.11 Reference Model
- •IEEE 802.11 Architecture and Services
- •Architecture
- •Services
- •802.11a Frame Format
- •Medium Access Control
- •Distributed Coordination Function
- •Collision Avoidance
- •Post-Backoff
- •Recovery Procedure and Retransmissions
- •Fragmentation
- •Hidden Stations and RTS/CTS
- •Synchronization and Beacons
- •Point Coordination Function
- •Contention Free Period and Superframes
- •QoS Support with PCF
- •The 802.11 Standards
- •IEEE 802.11
- •IEEE 802.11a
- •IEEE 802.11b
- •IEEE 802.11c
- •IEEE 802.11d
- •IEEE 802.11e
- •IEEE 802.11f
- •IEEE 802.11g
- •IEEE 802.11h
- •IEEE 802.11i
- •Overview and Introduction
- •Naming Conventions
- •Enhancements of the Legacy 802.11 MAC Protocol
- •Transmission Opportunity
- •Beacon Protection
- •Direct Link
- •Fragmentation
- •Traffic Differentiation, Access Categories, and Priorities
- •EDCF Parameter Sets per AC
- •Minimum Contention Window as Parameter per Access Category
- •Maximum TXOP Duration as Parameter per Access Category
- •Collisions of Frames
- •Other EDCF Parameters per AC that are not Part of 802.11e
- •Retry Counters as Parameter per Access Category
- •Persistence Factor as Parameter per Access Category
- •Traffic Streams
- •Default EDCF Parameter Set per Draft 4.0, Table 20.1
- •Hybrid Coordination Function, Controlled Channel Access
- •Controlled Access Period
- •Improved Efficiency
- •Throughput Improvement: Contention Free Bursts
- •Throughput Improvement: Block Acknowledgement
- •Delay Improvement: Controlled Contention
- •Maximum Achievable Throughput
- •System Saturation Throughput
- •Modifications of Bianchi’s Legacy 802.11 Model
- •Throughput Evaluation for Different EDCF Parameter Sets
- •Lower Priority AC Saturation Throughput
- •Higher Priority AC Saturation Throughput
- •Share of Capacity per Access Category
- •Calculation of Access Priorities from the EDCF Parameters
- •Markov Chain Analysis
- •The Priority Vector
- •Results and Discussion
- •QoS Support with EDCF Contending with Legacy DCF
- •1 EDCF Backoff Entity Against 1 DCF Station
- •Discussion
- •Summary
- •1 EDCF Backoff Entity Against 8 DCF Stations
- •Discussion
- •Summary
- •8 EDCF Backoff Entities Against 8 DCF Stations
- •Discussion
- •Summary
- •Contention Free Bursts
- •Contention Free Bursts and Link Adaptation
- •Simulation Scenario: two Overlapping QBSSs
- •Throughput Results with CFBs
- •Throughput Results with Static PHY mode 1
- •Delay Results with CFBs
- •Conclusion
- •Radio Resource Capture
- •Radio Resource Capture by Hidden Stations
- •Solution
- •Mutual Synchronization across QBSSs and Slotting
- •Evaluation
- •Simulation Results and Discussion
- •Conclusion
- •Prioritized Channel Access in Coexistence Scenarios
- •Saturation Throughput in Coexistence Scenarios
- •MSDU Delivery Delay in Coexistence Scenarios
- •Scenario
- •Simulation Results and Discussion
- •Conclusions about the HCF Controlled Channel Access
- •Summary and Conclusion
- •ETSI BRAN HiperLAN/2
- •Reference Model (Service Model)
- •System Architecture
- •Medium Access Control
- •Interworking Control of ETSI BRAN HiperLAN/2 and IEEE 802.11
- •CCHC Medium Access Control
- •CCHC Scenario
- •CCHC and Legacy 802.11
- •CCHC Working Principle
- •CCHC Frame Structure
- •Requirements for QoS Support
- •Coexistence Control of ETSI BRAN HiperLAN/2 and IEEE 802.11
- •Conventional Solutions to Support Coexistence of WLANs
- •Coexistence as a Game Problem
- •The Game Model
- •Overview
- •The Single Stage Game (SSG) Competition Model
- •The Superframe as SSG
- •Action, Action Space A, Requirements vs. Demands
- •Abstract Representation of QoS
- •Utility
- •Preference and Behavior
- •Payoff, Response and Equilibrium
- •The Multi Stage Game (MSG) Competition Model
- •Estimating the Demands of the Opponent Player
- •Description of the Estimation Method
- •Evaluation
- •Application and Improvements
- •Concluding Remark
- •The Superframe as Single Stage Game
- •The Markov Chain P
- •Illustration and Transition Probabilities
- •Definition of Corresponding States and Transitions
- •Solution of P
- •Collisions of Resource Allocation Attempts
- •Transition Probabilities Expressed with the QoS Demands
- •Average State Durations Expressed with the QoS Demands
- •Result
- •Evaluation
- •Conclusion
- •Definition and Objective of the Nash Equilibrium
- •Bargaining Domain
- •Core Behaviors
- •Available Behaviors
- •Strategies in MSGs
- •Payoff Calculation in the MSGs, Discounting and Patience
- •Static Strategies
- •Definition of Static Resource Allocation Strategies
- •Experimental Results
- •Scenario
- •Discussion
- •Persistent Behavior
- •Rational Behavior
- •Cooperative Behavior
- •Conclusion
- •Dynamic Strategies
- •Cooperation and Punishment
- •Condition for Cooperation
- •Experimental Results
- •Conclusion
- •Conclusions
- •Problem and Selected Method
- •Summary of Results
- •Contributions of this Thesis
- •Further Development and Motivation
- •IEEE 802.11a/e Simulation Tool “WARP2”
- •Model of Offered Traffic and Requirements
- •Table of Symbols
- •List of Figures
- •List of Tables
- •Abbreviations
- •Bibliography
8.2 NashTP PT Equilibrium Analysis |
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throughputs,observed |
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throughput demand of player 1, |
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Figure 8.2: Resulting observed QoS parameters of an SSG for two interacting players via
Θ1 , calculated with P, and simulated. Left: observed share of capacity, right: observed
dem 1 2
resource allocation interval. In this figure, ∆dem < ∆dem .
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Figure 8.3: Resulting observed QoS parameters of an SSG for two interacting players via
Θ1 , calculated with P, and simulated. Left: observed share of capacity, right: observed
dem 1 2
resource allocation interval. In this figure, ∆dem = ∆dem .
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Θ1 |
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Figure 8.4: Resulting observed QoS parameters of an SSG for two interacting players via
Θ1 , calculated with P, and simulated. Left: observed share of capacity, right: observed
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resource allocation interval. In this figure, ∆dem > ∆dem .
8.2.1Definition and Objective of the Nash Equilibrium
The Nash equilibrium is defined in the following Definition 8.1.
174 |
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8. The Superframe as Single Stage Game |
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Definition 8.1 (Osborne and Rubinstein 1994) A Nash equi- |
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librium of |
actions |
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Stage Game is a vector of |
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tions a*=( ai*,a−i* ) Α, with |
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i N |
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The Nash equilibrium of actions in the SSG is characterized by the fact that for each player i, the particular action ai* maximizes V i(a)=V i(ai ,a−i* ) in Ai for a given a−i* . When operating in Nash equilibrium, no player can profitably deviate from its choice of action, thus the action taken by any player will remain stable as long as QoS requirements do not change. Given the action of the other players, no player has reason to take an action than the one that leads to the Nash equilibrium, as each player’s action is an optimal, payoff maximizing response to the opponent player’s action. In general, either none, or a unique, or multiple Nash equilibria may exist in a strategic game.
In a course of an MSG, if a Nash equilibrium exists in the SSG, players that adjust their demands rationally in the sense that they attempt to maximize their payoffs, will end up in an action profile that is a Nash equilibrium. If there is no Nash equilibrium in the SSG, then rational players keep adjusting their demands continuously without meeting a stable point of operation. If multiple Nash equilibria exist in the SSG, then it is not clear to which one the players are adjusting their demands throughout the course of an MSG.
8.2.2Existence of the Nash Equilibrium in the SSG of Coexisting CCHCs
The SSG as it was defined in the previous Chapter 7 allows an uncountable number of actions per player. Such a game is referred to as infinite game (Fudenberg 1991). The analytical approximation of Section 8.1 works with a continuum of actions, whereas a simulation is performed with a large finite set of actions. During simulation, the size of the action set was chosen such that sensitivities to the precision of the discrete grid of actions are negligible. Therefore, the existence of a Nash equilibrium is verified by using a theorem that is known from the theory of infinite games, i.e., games that allow an infinite number of actions per player. The theorem uses an attribute of the payoff, which is referred to as quasi-concavity. Quasi-concavity of a payoff is defined in the following Definition 8.2. The Definition 8.3 recalls another necessary attribute, the continuity of the payoff.
8.2 NashTP PT Equilibrium Analysis |
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Definition 8.2 |
For any two actions of a player i, |
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Definition 8.3 |
(Debreu 1959:15) Let the payoff V i( a ) be a |
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y'' =V i( a '') it follows that y''→ y' . The func- |
tion is continuous in Α if it is continuous at every point of Α .
With these two definitions, it is possible to formulate a sufficient condition for the existence of at least one Nash equilibrium in the SSG. The following Theorem 8.1 for the existence of a Nash equilibrium in an infinite game is taken from Fudenberg (1991).
Theorem 8.1 (Fudenberg 1991:34) Consider a strategic game of whose action spaces Ai are nonempty compact convex subsets of an Euclidean space, i Ν ={1, 2} . If all
payoffs |
V i (ai ,a-i ) are continuous in Α and quasi-concave |
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in Ai , |
there exists |
a pure |
action Nash equilib- |
rium a* =( ai*, a−i* ) in |
Α =× |
Ai , i, −i Ν . |
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i Ν |
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The theorem is for example proven in Debreu (1952). Note that in the SSG competition model mixed actions are not defined, players take pure actions only (see Section 7.2). Nash equilibria can still exist in the SSG if the conditions of this theorem are not satisfied, as these conditions are sufficient but not necessary.
Proposition 8.1 postulates the existence of a Nash equilibrium in the SSG of two coexisting CCHCs. This proposition is proven underneath.
Proposition 8.1 In the Single Stage Game of two coexisting CCHCs exists a Nash equilibrium in Α =×i Ν Ai .
176 |
8. The Superframe as Single Stage Game |
Proof: By definition, the action spaces Ai , i Ν ={1, 2} , are each nonempty, compact, and convex sets, and they are subsets of an Euclidean space, as it was stated in the definition of the action spaces in Section 7.2.2.2. The outcome of an SSG for a player i is the payoff V i , which is given by the observed utility U i , as defined in Equation (7.16), page 148. The ob-
served utility is given by the product U i =UΘi U∆i of the two
utility terms UΘi ( Θdemi ,Θobsi ,Θreqi ) and U∆i ( ∆obsi , ∆reqi ) , see
Equation (7.11) on page 142. The two utility terms are continuous functions of QoS parameters by definition. Equation (8.14), page 169, defines a continuous function for the QoS parameters, thus, the payoff V i is evidently continuous in Α .
Since the two utility terms UΘi ,U∆i are each concave functions
of the observed QoS parameters for any shaping parameters |
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u,v |
+ , with u,v >0 (see Section 7.2.3.1), it remains to be |
shown that the functions for the observed QoS parameters,
the |
share |
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capacity Θobsi , and |
resource allocation inter- |
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val ∆obsi , which are given by Equation (8.14), are concave |
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action. |
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+∆dem−i Θdem−i |
is a concave function of ∆demi |
, as it in- |
creases linearly with increasing demanded resource allocation
interval ∆demi |
. The observed resource allocation interval ∆obsi |
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player i, i.e., |
Θdemi |
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capacity Θobsi |
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→0 then Θobsi =Θdemi |
, see Equation (8.14). Thus, it is a |
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concave function of |
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for small demands. |
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The function |
Θobsi =Θdemi |
is called curve for smaller demands |
in the following. For small demands, this curve is not dependent on the demanded resource allocation interval ∆demi .
It can be concluded that Θobsi is a concave function in Ai for small demands of share of capacity, i.e., if Θdemi →0 .
8.2 NashTP PT Equilibrium Analysis |
177 |
For large demands, i.e., if Θdemi →1, the observed share of capacity is defined by Equation (8.14) to
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This function is called the curve for larger demands in the following. This function is a concave function of Θdemi and a
concave function of ∆demi . It can be concluded that Θobsi is a concave function in Ai for large demands of share of capac-
ity, i.e., if Θdemi →1.
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The two curves intersect in the interval of interest, 0 < Θ0i <1 ,
if ( Θdem−i ∆dem−i /∆demi ) <1 . At this intersection point, if it exists in the interval 0 < Θ0i <1 , the gradient of the curve for a larger
demand is required to be equal or smaller than the gradient of the curve for smaller demand, which is 1, in order to achieve
that Θobsi is a concave function of Θdemi and ∆demi . Since the gradient of the curve for a larger demand at point Θobsi is
given by
178 |
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8. The Superframe as Single Stage Game |
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Θobsi is a concave function of Θdemi |
for any demand. It can be |
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concluded that Θobsi |
is a concave function in |
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■ |
The Proposition 8.1 implies the existence of at least one Nash equilibrium, which may not be unique. The uniqueness of Nash Equilibria in the SSG is discussed in the next section.
8.2.3Calculation of Nash Equilibria in the SSG of Coexisting CCHCs
A necessary condition for a to be a Nash equilibrium |
a* =(ai*,a−i* ) Αwith |
the property that for every player i Ν , the action ai* |
is preferred to any other |
action ai , is given by |
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(8.15) |
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requirement |
demand |
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8.2 NashTP PT Equilibrium Analysis |
179 |
In the SSG, the utilities U i are concave functions in Ai for any player i, therefore Equation (8.15) is a sufficient condition for a Nash equilibrium (Mangold et al. 2001h).
In a game of two players, Equation (8.15) describes a system of four equations with the four unknown variables ∆demi ,Θdemi ,Θdem−i , ∆dem−i , given the requirements per player. In order to achieve differentiability, when calculating a Nash equilibrium, the utility functions have to be considered piecewisely. From Equation (8.15), the resulting vectors of demands in Nash equilibrium can be calculated numerically, if the requirements of all players and the shaping parameters u, v of the utility function are known. In general, the resulting vectors of demands in a Nash equilibrium can be given in closed form as function of all parameters of the game (resulting vectors of demands in a Nash equilibrium are given by the action profile that is selected when playing the Nash equilibrium). Because of the complexity of the general term of the Nash equilibrium, this is not explicitly shown here.
Depending on the profile of requirements, and given the shaping parameters of the utility function, Nash equilibria can be calculated with Equation (8.15). As an example for the calculation, in the following the Nash equilibrium is calculated for one example.
Let
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and 2, both players select their |
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as defined in Equation (7.8), with i Ν . Equation (8.15) is used to calculate the Nash equilibria with the following set of equations, where the utility functions are shaped as stated in Equation (7.15).
∂ |
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180 |
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8. The Superframe as Single Stage Game |
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This system of differential equations can be solved numerically for any specific requirement profile. With this solution, the resulting demands in Nash equilibrium are
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and with Equations (7.12)-(7.16), the resulting payoffs in Nash equilibrium are
V( a * ) = (0.817,1.000).
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players achieve a high payoff |
in this example. However, player 1 suffers |
from |
the competition and the |
allocation process, and its payoff is |
V1( a * ) = 0.817 . Therefore, it cannot completely achieve its QoS requirements in Nash equilibrium. It depends on the services the player 1 attempts to support if it is satisfied with this degradation or not. The Nash equilibrium is achieved either through complete knowledge in one SSG or through adjustment in repeated SSGs, when both players play rational. Playing rational means playing the best response action, i.e., attempting to optimize the payoff by taking the action that results in the highest payoff per SSG.
In general, multiple Nash equilibria can exist in the SSG, which may be the case for some QoS requirements the players attempt to support. Depending on the profile of requirements of the players, the Nash equilibrium in the SSG of two coexisting CCHCs is not always unique. A game with multiple Nash equilibria is given for example in games where both players have very high QoS requirements, such as
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In this game, typically Θobsi |
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number of action profiles. This is a result of the shape of the utility function used in all games: for any shaping parameter, the utility function is concave, but not strictly concave.
8.3 ParetoTP PT Efficiency Analysis, and Behaviors |
181 |
8.3Pareto17 Efficiency Analysis, and Behaviors
In a SSG, Nash equilibria are stable and predictable points of operation. Players that take rational actions will automatically adjust into a Nash equilibrium, because it has been shown that at least one Nash equilibrium exists in the SSG. If the Nash equilibrium is unique, then the respective action profile can be predicted as point of operation, as a result of rational behavior. However, there may in general exist action profiles in the SSG that can lead to higher payoffs than what is achieved in Nash equilibrium. If such profiles do not exist, the Nash equilibrium is referred to as Pareto efficient, or equivalently, Pareto optimal. If such a profile exists, the Nash equilibrium is not Pareto efficient. In this case, payoffs may be optimized through actions taken by all players. This is discussed in the following.
The Definition 8.4 highlights the concept of Pareto efficiency, which is also referred to as Pareto optimality.
Definition 8.4 |
(Shubik 1981:347) Let the payoff V i( a ) be |
a function from |
Α to , and consider an action profile rep- |
resented by point a =( ai ,a−i ) Α, with i Ν ; Ν being the set of players of a game. The action profile (the point) is said
to be Pareto efficient (or Pareto optimal) if there is no other action profile a ' =( ai ',a−i ') Α such that V i( a ') ≥V i( a ) for all i Ν and V i( a ') >V i( a ) for at least one i Ν .
Definition 8.4 indicates that there can be one or more Pareto efficient action profiles in the SSG. A unique Nash equilibrium in the SSG that is Pareto efficient is a desirable scenario in the sense that in this case there is no other action profile that allows the usage of radio resources more efficiently, i.e., with larger payoffs for all involved players. It depends on the requirements of the players in the CCHC coexistence game if a Nash equilibrium is Pareto efficient. In general, Nash equilibria are less probable to be Pareto efficient in games where all players require more resources than available, i.e., with high offered traffic.
In addition, in case the Nash equilibrium is not Pareto efficient, a Pareto efficient action profile is reached if all players deviate from the Nash equilibrium point. If one player alone deviates, it achieves a lower payoff than before, according to the definition of a Nash equilibrium. Thus, when deviating from action profiles that are Nash equilibria and not Pareto efficient towards action profiles that are not
17 Vilfredo Pareto (1848-1923), Italian economist.