142 |
7. The Game Model |
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demand of player -i
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demand 
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allocation process
Figure 7.3: The different QoS parameters of player i in repeated SSGs. The requirement is given by the QoS a player is trying to support. As part of its action within a game, a player selects its demand based on the estimated requirements of the opponent player, taken from the history of observations.
7.2.3Utility
The utility U i +0 of player i defines what a player i gains from its action ai ( n ) . This utility is a set-based function over the set Αi , and defined as follows. The definition takes into account all relevant aspects of what a CCHC actually requires, and whether or not the observed QoS is satisfying a particular QoS. In general, utility functions represent a “measure of satisfaction” (Goodman and Mandayam 2000:48) an individual player is experiencing. In the CCHC coexistence game, this level is determined by the QoS requirements in an overlapping QBSS.
7.2.3.1Utility as Function of Observed Share of Capacity and Observed Resource Allocation Interval
Based on the selection of an action, a player i obtains a utility that is dependent on two normalized utility terms UΘi +0 and U∆i +0 that represent the utility as a nonnegative real number with respect to the share of capacity and with respect to the observed resource allocation interval, respectively:
U i =U i |
( Θi |
,Θi |
,Θi |
) U i ( ∆i |
, ∆i |
), U i + . |
(7.11) |
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dem |
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Both utility terms can have values between 0 and 1, thus 0 ≤U i ≤1 . To maximize its utility, a player has to consider both, share of capacity as well as delays. The utility as it is defined in this section reflects all relevant characteristics of the CCHC requirements in terms of QoS. There are many approaches to reflect such characteristics with different functions. In this thesis, an approach based on rational functions is selected to simplify the analytical game analysis. However,
7.2 The Single Stage Game (SSG) Competition Model |
143 |
other utility functions may be applicable when modeling the preferences of a player in the CCHC coexistence game.
The first utility termUΘi , which is related to the share of capacity, is defined as
UΘi := (Θdemi ,Θobsi ,Θreqi )→
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if Θobsi ≥ Θreqi −Θtolerancei and 0 other wise. The observations and demands in this equation (and in the following Equation (7.13)) vary with n, whereas the require-
ment stays constant over the repeated SSGs.
The u and v parameters in Equation (7.12) define what is referred to as elasticity, i.e., tolerable deviation of the share of capacity, as
Θtolerancei |
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+u v −v |
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This parameter is indicated in Figure
parameters u,v + , u,v > 0 as isUΘi ( |
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first utility term UΘi ( Θdemi ,Θobsi ,Θreqi |
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ters u, v with a requirement Θreqi = 0.4
7.4. It is dependent on the two shaping
Θdemi ,Θobsi ,Θreqi ) . Figure 7.4 illustrates the and the influence of the shaping parame-
as example.
1
UΘi with variable u parameters
0
0
u=100 →
u=50 →
u=20 →
→
v=1
Shaping parameter u in UΘ determines the elasticity.
Θ
←tolerance
← Θreq= 0.4
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1 |
Θiobs=Θidem
1
UΘi with variable v parameters
0
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EDCF channel access. |
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Figurei 7.4: First utility term UΘi (Θdemi , Θobsi , Θreqi ) of a player i vs. observation Θobsi = Θdemi , with Θreq = 0.4 . Left: varying u, right: varying v. Maximum utility is observed if the require-
ment is reached. The share of capacity can be any value between 0 and 1.
144 |
7. The Game Model |
Note that UΘi ( Θdemi ,Θobsi ,Θreqi ) is a function of the demand as well as observation, violating a pure game theoretical approach. The utility a player observes
decreases with increasing demand for share of capacity. There is no obvious reason for the utility to decrease in case of high capacity demand, i.e., Θdemi →1.
However, demanding higher capacity for the highest priority access would result in poor EDCF results due to its lower priority.
To influence a player not to overload the channel by demanding all available resources, the v parameter shapes the function to decrease the resulting utility with increasing demand, for the benefit of the EDCF based medium access.
This is related to the demand-driven cost of resource in economic competition models, which determines a price to be paid when demanding an individual share of common good. The cost of resources is a result of the cumulative demand of all involved players for that particular resource. The phrase “price” suggests a monetary transaction, but it does not involve real money; it is an abstract model of cost.
The u-parameter allows modeling what is referred to as elasticity: depending on this parameter, a player may strictly demand what it requires, or it may be satisfied with observations that actually differ to a certain extent from the requirement. Both shaping parameters are assumed constant throughout the game, and known by all players.
The same shaping parameters as before, u and v are used for the definition of the second utility termU∆i , which is related to the resource allocation period, see Equation (7.13). The shaping parameters are now multiplied by 10 to consider
smaller |
∆ -parameters. Note that in typical situations, all ∆ = [ 0 ... 0.1] , whereas |
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if ∆obsi ≤ ∆reqi −∆tolerancei |
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7.2 The Single Stage Game (SSG) Competition Model |
145 |
The value of this parameter is given by the tolerated variation of the delay of resources allocations Ξi that was discussed in Section 7.2.2.1.
Figure 7.5 illustrates the second utility termU∆i ( ∆obsi , ∆reqi ) . As
inUΘi ( Θdemi ,Θobsi ,Θreqi ) , the v parameter models a decrease in utility with decreasing observation, i.e., U∆i →0 if ∆obsi →0 . This is motivated by the fact that
real-time applications, as well as HiperLAN/2 MAC frames integrated into the 802.11 protocol require constant time periods that should not be extended by unpredictable delays.
If ∆obsi > ∆reqi , QoS support is limited. In addition, too short resource allocation intervals are not useful as well, i.e., when ∆obsi ∆reqi . This is represented by the utility function discussed here. The utility function therefore decreases for small resource allocation intervals.
The v parameter in U∆i ( ∆obsi , ∆reqi ) has thus a different intention, although it is used the same way as inUΘi ( Θdemi ,Θobsi ,Θreqi ) . However, U∆i ( ∆obsi , ∆reqi ) does not depend on the demand ∆demi .
In the next chapters, where the utility defined here is extensively used for the analysis of the SSG and the discussion of behaviors and strategies, the u and v parameters are set to these common values:
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resource allocation. |
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allocation period. |
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Figure |
7.5: Second utility |
term U∆i (∆obsi , ∆reqi ) of a player i vs. observation ∆obsi , |
with ∆reqi |
= 0.04 . U∆i (∆obsi , ∆reqi |
) is not dependent on the demand. Left: varying u, right: |
varying v. This figure is shown for ∆obsi = 0 ... 0.1 , as the minimum number of resource allocation attempts per player and stage is 10.
146 |
7. The Game Model |
In the approach that is taken in this thesis it is assumed that these numbers are used by all players, and further they are assumed common knowledge among the players. In real life, individual players may have different preferences and therefore may implement autonomous shaping parameters, which vary from player to player. However, in this thesis the players attempt to establish a share of resources between networks instead of traffic classes with individual preferences in terms of the shaping of the utility functions. Therefore, u and v are equally used as defined in Equation (7.15).
As stated above, the observations and demands in Equation (7.12) and (7.13) vary with n, whereas the requirement stays constant over the repeated SSGs.
Figure 7.6 shows the resulting utility U i as a function of the observation for a
requirement of( Θreqi , ∆reqi ) =( 0.4,0.04 ) . It can be seen that there is one single observation that – if it is selected by the player i – maximizes the utility. In the
ideal case that is shown here, the player observes what it is demanding, i.e., there is no competition for resources.
Therefore, the optimal action ai * =( Θdemi |
*, ∆demi *) |
here is to demand the re- |
quirement ( Θreqi , ∆reqi ) =( 0.4,0.04 ) . Note that the utility U i is strictly concave in |
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( Θobsi , ∆obsi ) for all U i > 0 and concave in ( Θobsi , ∆obsi |
) for any utility U i .15 |
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The principle of a utility as a function of |
observation and demand is used to |
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inspire players not to overload the channel by selecting actions without taking into account the effect of their decisions on the opponent players. Accurately defined utility functions will lead to the selection of demands that allow colocated CCHCs to coexist. A utility function is a tool to encourage players to act in a way that optimizes the usage of shared resources. Interacting players are forced by the utility to use shared resources most efficiently. As part of the game approach, it is supposed that there is a common knowledge about the utility function, i.e., all players know the utility function as part of radio regulations or as part of standardized protocols. A fundamental assumption that is taken for the underlying game model is that the utility function including the shaping parameters of the utility function is common knowledge, i.e., a player knows the utility function, and “knows that its opponents know it, and knows that its opponents know that it knows, and so on ad infinitum” (Fudenberg and Tirole 1991:4). This is true if all CCHCs follow the game model. However, this does necessarily re-
15 A function f : → is concave |
if |
f( α x+(1- α) x') ≥ α f(x)+(1-α) f(x') , |
and strictly |
concave if |
f( α x+(1- α) x') >α f(x)+(1- α) f(x') |
for |
all x , x ' , and all α (0,1) |
(Osborne |
and Rubin- |
stein 1994:7). In particular, a linearly increasing function is concave. |
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