Whereas the model is simple enough to allow players to estimate the outcomes in a game in advance, this model is also used for the equilibrium analysis of the SSG, see Section 8.2. This equilibrium analysis will show the characteristics of the competition scenario of two CCHCs operating with the same radio resources.

Based on this analysis, various behaviors are defined in Section 8.3, where the efficiency of equilibria in the SSG is discussed. These behaviors are the base for the strategies that are investigated in Chapter 9.

8.1Approximation of the QoS Observations of the Single Stage Game

8.1.1The Markov Chain P

In this section, a model for the game of two players that allows an analytical approximation of the expected observations as functions of the demands is presented:

The approximation is calculated by means of a Markov chain with five states, which will be explained in the following sections. Note that in the rest of this chapter, the dependency of some game parameters on the game stage n is not indicated, since it is the SSG that is analyzed in this chapter.

8.1.1.1Illustration and Transition Probabilities

In an SSG of two players, the calculation of the QoS observations is performed using the discrete-time Markov chain P illustrated in Figure 8.1 and defined by Equation (8.2). The Markov chain P is the model of the stochastic process, which approximates the SSG of two players. This Markov chain P is irreducible, as each state communicates with each other (Bertsekas and Gallager 1992).

It is assumed that the single stage game is stationary. The longer the duration of an SSG and the higher the number of allocation attempts per stage, the more stationary the process becomes. In a short SSG with relatively long and few resource allocations, the stationary distribution may depend on the starting state, or may even not exist. Therefore, a minimum of 10 resource allocations per player is required, i.e., ∆demi ,−i < 0.1. With this restriction, stationary of the SSG can be generally assumed. Further, it is assumed that none of the states is periodic.

P41

Figure 8.1: Discrete-time Markov chain P with five states to model the game of two players that attempt to allocate common resources. The default state in which the channel is idle or an EDCF frame exchange is ongoing is denoted as state 0. In case of high offered traffic, P43→ 0, P21→ 0 , as explained in Section 8.1.2.

The aperiodic characteristic of P is a necessary condition for the game analysis, and cannot be assumed in general. The influence of the assumption that P is aperiodic is discussed in Section 8.1.3, when discussing the delay of resource allocations during an SSG, see Equation (8.12).

8.1.1.2 Definition of Corresponding States and Transitions

The five states the SSG process, which is modeled by P, can be in are:

3: Player 2 successfully allocates resources with highest priority, player 1 does not attempt to allocate resources. This state is equivalent to state p1 that models the same situation for the opponent player 1.

4: Player 2 successfully allocates resources with highest priority, player 1 waits for the channel to get idle, to allocate resources immediately. This state is equivalent to state p2 that models the same situation for the opponent player 1.

Let

P {state (t +1)= i|state (t )= slot} = Pki , i,k = 0…4

be the transition probabilities of P.

The transition probabilities with Pki ≥ 0, i, k=0…4 are identified as:

P01: Probability that player 1 allocates resources while the channel is idle or allocated by low priority EDCF-TXOPs that allocate resources via contention.

P03: Probability that player 2 allocates resources while the channel is idle or allocated by low priority EDCF-TXOPs that allocate resources via contention. In a game with 2 players, P03=1-P01 .

P10: Probability that player 2 does not attempt to allocate resources during an ongoing resource allocation of player 1.

P12: Probability that player 2 does attempt to allocate resources during an ongoing resource allocation of player 1, thus, P12 =1-P10 .

P21: Probability that player 2 gives up its attempt to allocate resources before player 1 finishes its resource allocation.

P23: Probability that player 2 allocates resources right after player 1 finished its resource allocation, if it attempted to allocate resources during an ongoing allocation of player 1.

P30: Probability that player 1 does not attempt to allocate resources during resource allocation of player 2.

P34: Probability that player 1 does attempt to allocate resources during resource allocation of player 2, thus, P34=1-P30 .

P41: Probability that player 1 gives up its attempt to allocate resources before player 2 finishes its resource allocation.

P43: Probability that player 1 allocates resources right after player 2 finished its resource allocation, if it attempted to allocate resources during an ongoing allocation of player 1.

8.1.1.3Solution of P

The stationary distributions of the Markov chain P are defined by

lim P {state = i} =: pi , i = 0…4 ,

t →∞

and can be calculated to

p0 =1 − p1 − p2 − p3 − p4 ,

Here, P23→1 and P41→1, assuming that players tolerate delays of their resource allocation attempts, which occur when the opponent player allocates resources. It is assumed that a player never gives up its attempt to allocate resources when it waits for the opponent player to finish its resource allocation. This implies the simplification that a player does not attempt to allocate more than one resource during one single ongoing resource allocation by the opponent player. The model actually fails to represent an SSG in situations where the two players demand very dissimilar resource allocation periods. On the other hands, numerical results show that even in such unlikely configurations, the observations are still qualitatively represented by P.

8.1.1.4Collisions of Resource Allocation Attempts

The two players may attempt to allocate a resource at nearly the same point in time, thus, the first MPDUs that are transmitted during resource allocation by each player may collide. It is obvious that the probability of collision does increase with decreasing ∆i ,−i . The smaller the resource allocation period of any player, the higher the probability of medium access and collision. Further, the TXOPlimit that defines the maximum duration of a resource allocation through contention-based medium access, i.e., under the rules of the EDCF, affects the