Chapter 8

THE SUPERFRAME AS SINGLE STAGE GAME

8.1Approximation of the QoS Observations of the

	Single Stage Game .............................................................	160
8.2	Nash Equilibrium Analysis...............................................	172
8.3	Pareto Efficiency Analysis, and Behaviors.......................	181

BEFORE the actual play of a Single Stage Game (SSG), players must take their actions for that particular stage. A player i takes its action based on

its own QoS requirements that are given by Θreqi , ∆reqi , with the consideration of the opponent player’s demands Θdem−i , ∆dem−i . As before, the index -i

refers to the opponent of a player i. Note, that the superscript “~” indicates the fact that the demands of any opponent player -i are not known to a player i, but estimated from the history of earlier stages of repeated SSGs. A player attempts to optimize its expected utility that depends on the QoS observations Θobsi , ∆obsi . For this reason, a player must be able to estimate the expected QoS observations, i.e., the outcome of a game in advance, while decision taking. It has to approximate the expected observations based on its own demands and the demands of the opponents. An analytical approximation of the SSG that can be used by a player for this purpose is developed in the next Section 8.1. In this section, after the approximation is explained in detail, it will be demonstrated that it is accurate enough to allow players to capture the statistical characteristics of the current stage of a game, i.e., an upcoming SSG. The analytical approximation of the SSG results in values for the observed share of capacity and for the observed resource allocation period. For the observed resource allocation period an upper bound is given. Such an upper bound of the resource allocation period is the relevant information that is required for the support of QoS.