164	8. The Superframe as Single Stage Game

collision probability as well: the larger the TXOPlimit, the larger may be the duration of EDCF-TXOPs allocated by any station in the environment. With larger durations of EDCF-TXOPs, a player has to wait longer times until it can start its own resource allocation. While waiting for the channel to become idle, the opponent player may also decide to attempt an allocation once the channel will become idle again. As a result, the two players will observe a collision in this case. The TXOPlimit defines a granularity in medium access and must be kept small to achieve a small probability of collision. Note that in this thesis a single TXOPlimit is assumed that is valid across all QBSSs.

However, it must not be a poll frame that collides. According to the CCHC concept and the 802.11e draft standard (IEEE 802.11 WG, 2002a), it is up to the station that allocates resources, which MPDU it selects for initial transmission after the PIFS duration. Usually a CCHC detects a collision from the missing response of the station the MPDU was directed to. How much capacity is lost when a collision occurs is therefore dependent on the length of the first transmitted MPDU within a resource allocation, i.e., within the TXOP. Here, it is assumed that the duration of the first frame is very small compared to the duration of a resource allocation by a player. Such allocations typically last for a couple of milliseconds, whereas the duration of a single first MPDU typically is smaller than100 µs .

It is not the intention of this thesis to discuss collision resolution protocols. As part of the CCHC coexistence problem of two players, a simple rule to solve collisions is used: in the case when two players detect a collision of their allocation attempts, the one which allocated a resource the last time before the collision refrains from repeating its attempt. The other player immediately transmits another MPDU in order to start its resource allocation before any EDCF-TXOP can start. In case a player’s resource allocation attempt is unsuccessful because of a collision with a starting EDCF-TXOP, the player can repeat its attempt immediately, whereas the station that transmitted the colliding MPDU has to backoff according to the EDCF protocol.

The simple rule to solve collisions, as well as the relatively short duration of MPDUs that actually collide, allow ignoring collisions in the model of the SSG. No specific state in the Markov chain P represents the event of a collision.

8.1.2Transition Probabilities Expressed with the QoS Demands

In this section, the QoS demands are used to determine the transition probabilities of P. The transition probability that player 1 allocates resources while the

8.1 Approximation of the QoS Observations of the Single Stage Game

165

channel is idle or allocated by low priority EDCF-TXOPs via contention is approximated as

P	L1	, L1, L2 > 0 .	(8.3)
01	L1 + L2

During an SSG, the more TXOPs, L1, player 1 attempts to allocate compared to the number of all high priority TXOPs, L1+L2 , the higher the probability of resource allocation of this player 1. With P03=1-P01 , the probability of resource allocation of player 2 can be calculated similarly.

It is further approximated that the transition probability that player 2 attempts to allocate resources during an ongoing allocation of player 1, is given by

				d1							, D2	−d 2 > d1					d1
P		D		2	−d			2				P	min 1,								,	(8.4)
																2
12												12			D		−d		2
					1							else
and equivalently,
					d		2				, D1	−d1 > d 2				d 2
P				1		−d			1			P	min 1,							.		(8.5)
															1
34			D									34					−d	1
						1						else			D

These transition probabilities are declared in a piecewise way. The probability P12 is either d1 /D2 −d 2 or approximated to 1, as expressed by Equation (8.4). The probability that player 1 decides to attempt a resource allocation while player 2 is allocating resources, depends on the ratio between the duration of this allocation d1 and the duration of the time interval between two consecutive demanded resource allocations of player 2, i.e., D2 −d 2 . In the case that the time interval between two consecutive demanded resource allocations of player 2, given by D2 −d 2 , is smaller than the duration d1 of a resource allocation by player 1, the player 2 will attempt to allocate resources immediately after the ongoing resource allocation, with probability 1. The same is valid for the reverse situation when player 2 allocates resources with duration d 2 , as is stated in Equation (8.5), where P34 is equivalently defined.

The parameters Li , Di , d i determine the times of the resource allocations of player i. They are introduced in Section 7.2.1 and illustrated in Figure 7.1. These parameters can also be determined from the demanded QoS parameters:

	166			8. The Superframe as Single Stage Game
		1
	Li =		, ∆demi		> 0;
		∆demi
	Di = SFDUR ∆demi			;	(8.6)
	d i = SFDUR Θdemi			∆demi .

The operator is neglected in the following for the sake of analytical tractability of the model. Any nonnegative real value L > 0 is allowed, as already explained in Section 7.2.1. With the QoS demands as given in Equation (8.6), the transition probabilities of P are

∆dem2

1,2

P

=

,

∆

> 0 ,

∆dem2 +∆1dem

01

dem

∆1

Θ1

∆2

> 0, Θ2

P

= min 1,

dem

dem

,

<1,

2

12

2

dem

dem

∆dem

1 −Θdem

∆dem2

Θdem2

1

1

P

= min 1,

,

∆

dem

> 0, Θ

dem

<1 ,

1

1

34

∆dem

1 −Θdem

with 0 ≤ P01, P12 , P34 ≤1 . Most relevant for the CCHC coexistence problem are situations where the overall throughput demands of all involved players are high, i.e., ∑i Θdemi →1 . In such situations, the transition probabilities P12 and P34 can be simplified to

	P12	→1,	(8.7)
	P34 →1.
	For this reason, P21→0 and P43→0	in situations of high offered traffic. The
	equilibrium analysis in Section 8.2 will make use of this simplification.

8.1.3Average State Durations Expressed with the QoS Demands

The average state durations T0 ,T1,T2 ,T3 ,T4 are further required to calculate the QoS observations from the stationary distributions of P.

The average duration of the model P being in the idle state, T0 , is approximated to

8.1 Approximation of the QoS Observations of					the Single Stage Game					167
	T0 min (D2 −d 2 , D1−d1 ),
which can be expressed using the QoS demands of Equation (8.6) as
T	SFDUR min	(	∆1	1-Θ1		, ∆2	1-Θ2		,	(8.8)
0			dem	(	dem )	dem	(	dem ))

with SFDUR being the duration of an SSG, given in ms. This is understood as follows. If both players attempt to allocate resources periodically, the idle times between the resource allocations of a player i is denoted as Di −d i . In general, the player that requires shorter periods determines the average T0 of the SSG. This is represented by Equation (8.8). The value of T0 can be simplified to

T0→0

(8.9)

for situations where the overall throughput demands of all involved players are relatively high, i.e., ∑i Θdemi →1 . In this case, it is very probable that the conten- tion-based channel access through EDCF cannot allocate any resources due to its low priority in medium access. Therefore, if ∑i Θdemi →1 , resources are nearly always allocated by one of the two players; the channel is busy most of the time.

The mean state duration is given by

TMean =p0T0+p1T1+p2T2+p3T3+p4T4 , and can be expressed by

T	Mean	p T + p d1 + p		3	d 2	,
		0 0	1
because the duration of	the process P being in state p1 is determined by the
duration of a resource allocation of			player 1, d1 , if			the opponent player 2 does

not decide to attempt resources during this allocation. In addition, if the opponent player decides to attempt a resource allocation during this allocation, the process changes to state p2 . The duration of the process P consecutively being in the states p1 and p2 is again determined by the duration of a resource allocation of player 1, d1 . Therefore, it can be approximated that

p1T1+p2T2 ≈ p1 d1 ,

and equivalently,

p3T3+p4T4 ≈ p3 d 2 .

The mean state duration TMean can now be expressed by using the QoS demands of Equation (8.6) as

168	8. The Superframe as Single Stage Game
TMean SFDUR	(1-Θdem2 ))+ p1 Θ1dem ∆1dem + p3 Θdem2 ∆dem2 ),
(p0 min (∆1dem (1-Θ1dem ), ∆dem2

where p0 , p1 , and p3 are given through the solution of P, see Section 8.1.1.3.

With this definition of the mean state durationTMean , the observed throughput of the players is given by

	1	1	1		p1
	Θobs	= SFDUR ∆dem	Θdem			,
					TMean		(8.10)
	Θobs2	= SFDUR ∆dem2	Θdem2		p3
						.
					TMean

Assuming high offered traffic, with Equation (8.7) the throughput observation of player i is calculated as

∑i Θdemi →1 Θobsi =		Θi	∆i
		dem	dem		, i , −i Ν ={1, 2} . (8.11)
	Θi			∆−i
		∆i	+Θ−i
	dem	dem	dem	dem

In any other case, Θobsi =Θdemi . The maximum resource allocation period a player may observe due to delayed allocations during an SSG is calculated as

∆obsi =	∆demi	+	∆dem−i Θdem−i	+TXOPlimit	, i, −i Ν ={1, 2} ,
	demanded allocation interval		unwanted increase of	allocation interval (delay)

where the unwanted maximum increase of resource allocation intervals is dependent on the demand of the opponent player as well as the maximum duration of the EDCF-TXOPs. The latter is defined by the TXOPlimit. This TXOPlimit is neglected in the following as it was defined to be relatively small compared to

the typical	duration	of resource allocations of the two players
( TXOPlimit	∆demi Θdemi	with i Ν ={1, 2} ), and further because of the lower

priority in medium access through EDCF. Thus, the maximum observed resource allocation period is given by

∆oi bs = ∆demi +∆dem−i Θdem−i , i , −i Ν ={1, 2} .

(8.12)

All calculations in the following are performed using this upper bound of ∆obsi instead of any mean or expected value. Such an expected value can be derived from P as well (see Hettich 2001), but is not a realistic approximation, mainly due to the fact that depending on the ∆reqi ,−i -parameters, the process that is modeled by P may become periodic. The stationary distributions that are calculated from P are only true approximations if the process that is modeled by P, i.e., here the