Telenor R&D/ UniK
1331 Fornebu, Norway paal.engelstad@telenor.com
Abstract — A number of analytical models have been proposed to describe the priority schemes of the Enhanced Distributed
Channel Access (EDCA) mechanism of the IEEE 802.11e standard. EDCA provides a class-based differentiated Quality of
Service (QoS) to IEEE 802.11 WLANs. Many have used a multiple number of nodes to study the differentiation behaviour of the model. However, in many real-life usage scenarios Internet traffic is often asymmetric with much downlink traffic from the access point to the stations and little traffic in the reverse direction. Hence, most of the overall traffic differentiation will happen in the Virtual Collision Handler (VCH) of the access point. If the access point uses EDCA, it should know the characteristics of the VCH to be able to control the differentiation of the downlink traffic. The main contribution of this paper opposed to other works is that it demonstrates how a generic channel model of 802.11e can be modified to predict the performance behaviour of the VCH with a remarkably good accuracy. In doing so, we first introduce virtual collision handling into the generic model. We also show how to derive a closed-form solution for the performance. We observe good match between the analytical model and simulations.
Keywords-802.11e, Virtual Collision Handler, Performance
Analysis, EDCA, Non-Saturation, AIFS, Starvation.
I. I NTRODUCTION
IEEE 802.11 WLAN [1] is the most widely used technology for wireless access to wired Local Area Network
(LAN) infrastructures and to the Internet.
So far, the majority of works that do analytical performance evaluations and empirical simulations of 802.11 seem to focus on the uplink traffic problem. They present results with a number of wireless stations contending for the channel, and with fairly equal shares of traffic transmitted from each station.
However, the situation where the majority of traffic is downlink traffic from the access point is of higher practical interest. In daily life, the station is often a client that retrieves a large amount of information from the wired network (e.g. video streaming from a server on the Internet). In other words, traffic patterns are normally asymmetric, with little uplink traffic from the stations, but a large amount of downlink traffic from the access point.
In order to analyse the effects of asymmetric traffic patterns, this paper assumes for simplicity the extreme situation where all traffic is downlink traffic from the access point - in
Telenor R&D
1331 Fornebu, Norway olav-norvald.osterbo@telenor.com
contrast to most other relevant works. Here, we explore how the new IEEE 802.11 amendment [2], which provides 802.11 with mechanisms for Quality of Service (QoS), works in a downlink scenario. When 802.11e is being used, it is of utmost importance to ensure QoS and appropriate differentiation of the downlink traffic.
This paper focuses on situations where the mandatory
Enhanced Distributed Channel Access (EDCA) of 802.11e is being used. EDCA works as an extension to the Distributed
Coordination Function (DCF) of legacy 802.11. EDCA enhances DCF by permitting four different Access Categories
(ACs) at each station and a transmission queue associated with each AC. Each AC at a station has a conceptual module responsible for channel access for each AC and in this paper the module is referred to as a ”backoff instance”.
A key point is that the different backoff instances in a station do not access the channel completely independently, due to the virtual collision handling between the queues in the station. If two or more backoff instances on the same station try to access the channel in the same timeslot, the station attempts to transmit the frame of the highest priority AC, while the lower priority frames will enter backoff.
When EDCA is being used, a QoS-enabled Access Point
(QAP) will be interested in ensuring appropriate QoS of all traffic it is transmitting. Not only is it important to ensure appropriate QoS on the wireless channel. The QAP must also somehow control the performance of the Virtual Collision
Handler (VCH), which performs the virtual collision handling internally in the node. With the analytical model presented in this paper, the QAP can predict how the VCH responds to different traffic patterns and different parameter settings of each AC.
Needless to say, being able to predict and control the performance of the VCH can be useful not only for a QAP, but also for any QoS-enabled station (QSTA) using EDCA. As such, the results presented in this paper can be useful for any
QSTA, although this paper primarily focuses on the VCH of a
QAP.
Since we assume for simplicity the extreme situation where all traffic is downlink traffic from the QAP, the VCH is always free to use the wireless channel, and the QAP will not experience collision from any other station. This actually means that all traffic contention will occur in the VCH. The

http://folk.uio.no/paalee/
VCH represents a “virtual” traffic channel, and we can use analytical models that incorporate virtual collision handling to derive the performance of the VCH.
The main purpose of this paper is to show how the model presented extensively in [3] can be used to model the performance behaviour of the VCH of e.g. a QAP. This model is capable of predicting the performance not only in the saturated case, but in the whole range from a non-saturated medium to a fully saturated channel. The reader is encouraged to consult [3] for more details and explanations of the model used in this paper. Virtual collision handling is first incorporated into the aforementioned model. This is necessary to later be able to predict the behaviour of the VCH.
Furthermore, the paper shows how a closed-form solution for the performance of the VCH can be derived when the channel is saturated.
The remaining part of the paper is organized as follows:
The next section presents the analytical model presented in [3] and demonstrates how the modified model can be used to predict the behaviour of a VCH (e.g. on a QAP). Then, Section
III shows how this adaptation of the model influences on the expression of the throughput. In Section IV a closed-form solution of the model is found for saturation conditions and in
Section V expressions for the mean delay are found – also in a closed form. Our findings are then validated against simulations in Section VI, and concluding remarks follow in the last section.
A.
II. MODEL
The Markov Model i
A backoff instance is associated with each AC[i], where
N
and where N is the number of different ACs
(i.e. normally four). The state of the backoff instance can be described by a stochastic Markov process. The Markov chain for this transmission process is illustrated in Figure 1.
The backoff instance makes a transmission attempt when it is in any of the states ( i , j , 0 ) , j = 0 , 1 ,..., L i number of unsuccessful transmission attempts. The transmission is successful at a probability 1 − p i
. At probability p i
, on the contrary, the backoff instance experiences a collision and adjusts the contention window, W i , j
. It selects a backoff slot at uniform probability and moves to the corresponding state in the row below, i.e. to the ( j + 1 ) -th stage.
If the packet has not been successfully transmitted after L i
+ 1 attempts, where L i
denotes the retry limit of the retry counter, the packet is dropped.
Figure 1. Markov Chain (both saturation and non-saturation)
If b i , 0 , k , e and b i , j , k denote the state distributions of the
Markov chain, we see that b i , j , 0
= p i j b i , 0 , 0
, due to the probability, p i
, for experiencing a collision. Since transmission attempts only occur in these states, the transmission probability, τ i
, can be written:
τ i
=
L i
j = 0 b i , j , 0
= b i , 0 , 0
1
1
−
− p i
L i p i
+ 1
. (2) k
The utilization factor, ρ i
, shown in the Markov chain represents the probability that there is a packet waiting in the transmission queue of the backoff instance of AC at the time a transmission is completed successfully (or a packet is dropped). Now, the backoff instance selects a backoff interval
at random within the minimum contention window,
W i , 0
, and goes into post-backoff. If the queue is non-empty, the postbackoff is started by entering the state ( i , 0 , k ) .
W i , j
starts at the minimum value, CW i , min
, and is doubled until it reaches its maximum value, CW i , max
, in the m i
-th backoff stage. After this, W i , j remains unchanged. Hence,
W i , j
=
2 m i
W i
2
, 0 j W i , 0
= CW i , max j j = 0 , 1 ,...., m i
= m i
,...., L i
− 1
(1)
On the contrary, if the queue is empty, at a probability 1 − ρ i
, the post-backoff is started by entering one of the states ( i , 0 , k , e ) . If it receives a packet in the transmission queue while in any of these states, at a probability q i
∗
, it moves to a corresponding state in the second row with a packet waiting for transmission. Otherwise, it remains in the first row with no packets waiting for transmission. http://www.unik.no/personer/paalee

If no packet arrives in the queue, it finally ends up in the state ( i , 0 , 0 , e ) . Now, the backoff instance has completed postbackoff and is only waiting for a packet to arrive in the queue.
If it receives a packet during a timeslot at a probability , it does a ”listen-before-talk” channel sensing and moves to a new state in the second row, since a packet is now ready to be sent.
If the backoff instance senses the channel busy, at a probability p i
, it performs a new backoff. Otherwise, it moves to state
( i , 0 , 0 , e ) to do a transmission attempt. can simply predict from the downlink traffic load that it pours into the transmission queues, whether any AC will face starvation when the traffic is handled by the VCH.
A complete description of the system can be found by solving the above set of equations (one equation per AC[ i ]). In the following, we will therefore find ways to express b i , 0 , 0
in terms of the other parameters of the system.
By looking at regularities in the Markov chain and by undertaking the normalization, it is possible to show that [3]:
While in all backoff states (i.e. all states for which k ≥ 1 ) p i
∗ is the probability that the backoff instance remains in the state between two transmission slots. The model can be implemented with a number of settings for this parameter:
1 b i , 0 , 0
= j
L i ∑
= 0




1 +
1 −
1 p i
∗
W i , j ∑
− k k = 0
W i , j
W i , j
− k



 p i j +
1 − q i
ρ i
1 − ( 1 −
W i , 0 q q
∗ i
∗ i
)
W i , 0
( 1 +
( W i , 0
2 ( 1
−
−
1 ) p i q i
) p i )
. (5)
0
By performing the summations in Eq. (5) above and by assuming m i
≤ L i
, we may rewrite Eq. (2) as: p i
∗ = (Bianchi [4]) (3a) or or p i
∗ = p i
∗ p i
=

 min 

1 , p i
+
A i


1 −
N
i =
− 1
0
( 1
1 − τ i
− τ i
)






(Xiao [5]) (3b)
(3c)
τ
1 i
W i , 0
=
( 1 −
2 ( 1 −
2 p p i
* i
*
)
)
(
( 1 − p i
+
)( 1 −
2 (
( 2
1 − p i
) p i
* m i
) +
)( 1 −
( 1 −
2 p i
(
1
1
−
− p p i
L i i
+ 1
)
1 − q i
ρ i
1 − ( 1 −
W i , 0 q i
∗ q i
∗
) W i , 0
( 1 +
2 p i
)( 1 −
)( 2 p i
L i p i
+ 1
) m i
)
( W i , 0
− 1 ) q i
2 ( 1 − p i
) p i )
( 1 − p i
L i
− m i
+ 1 )
)
+
(6) where
 
Here p i
is the important collision probability, which will be affected by the virtual collision handling. Expressions for p i will be derived in the following subsections.
A i
= AIFSN [ i ] − min
{
AIFSN [ 0 ], L , AIFSN [ N − 1 ]
}
. (4)
Eq. (3a) describes a situation where the backoff counter is decremented in each slot, even when there is a transmission or collision on the channel. This reduces the presented model to one similar to that of Bianchi [4], however, still extended to cover both saturation and non-saturation conditions and to incorporate finite retry counters.
According to Eq. (3b), on the contrary, the backoff counter is not decremented when there is a transmission or collision on the channel. When the backoff instance senses a slot is busy, at the probability p i
, it remains in the same backoff state. This reduces the presented model to that of Xiao [5], except that the latter is restricted to saturation conditions. for
[3].
For the non-saturation solution, where ρ i
< 1 , expressions
i
, q i
and q i
∗ are also needed. These are all described in
B. Collision Probabilities with Virtual Collision Handling
According to the original model in [3], the probability of unsuccessful transmission, p i
, from one specific backoff instance is given when at least one of the other backoff instances does transmit in the same slot. However, lower priority ACs on the same QSTA will never cause a collision, due to the virtual collision handling. This can be generalized by the expression [3]:
Finally, Eq. (3c) models the blocking of the countdown process, like in Xiao’s model [5]. However, differentiation based on Arbitration Inter-Frame Spaces (AIFSs) is incorporated into the countdown blocking probability p i
∗ , as described in [3]. When a backoff instance senses that the channel is idle after a packet transmission, it normally waits a guard time during which it is not allowed to transmit packets or do backoff countdown. The guard time of AC[i] consists of a
Short Interframe Space (SIFS) and AIFSN[i] number of additional time slots. The AIFSN[ i ] parameters appears in Eq.
(4). p i
= 1 −
N c
− 1
= 0 c i
= 0
( 1 −
( 1 −
τ c
τ c
) n c
)
. channel access in each priority class i , and N denotes the total number of classes. Note that here we have defined that AC[ N-
1] is of the highest priority and AC[0] is of the lowest.
C. Collision Probabilities within the Virtual Collision
Handler (VCH)
(7)
Under conditions when p i
∗ = 1
as a result of Eq. (3c), starvation occurs and AC[ i ] will not be able to access the channel. Hence, in cases where Eq. (3c) is applicable, the QAP
One may use the multi-node channel model to study the behaviour of the Virtual Collision Handler (VCH). Here the

VCH represents the channel, while there are only N (typically four) queues contending for access, i.e. one queue per AC i .
Hence, one may model the throughput of the VCH by setting n i
= 1 for all i . In this case, p i takes the form: where T e and T s are the real-time duration of an empty slot and of a slot containing a successfully transmitted packet, respectively. [The real time duration of a slot containing two or more colliding packets T c
, is not used in our special case, due p i
= 1 −
N − 1
∏ c = i + 1
( 1 − τ c
) to Eq. (12).] B denotes the nominal data bit-rate (e.g. 11 Mbps for 802.11b [6]), and T i , MSDU
denotes the average real-time
Here, we note that the highest priority class, AC[ N1], will correctly have p
N − 1
= 0 , which means that it is never blocked required transmitting the MSDU part of a data packet at this rate. and never experiences a collision when it tries to access the channel for transmission.
In the saturation channel situation all the utilization factors ρ i
, equal unity. This cancels the last term in Eq. (5) and
III. THROUGHPUT makes the corresponding model independent of the
With the generic model for n i
nodes, the probability p i , s
, that a packet from any of the backoff instances of class i is transmitted successfully in a time slot, is p i , s
= n i
τ i
( 1 − p i
) [3]. parameters q i and
Eq. (5) then yields: q i
* . The corresponding basic relation in
However, when examining the behaviour of the VCH within one station, we set n i
= 1 for all i , as we also did in Eq. (8).
Thus, p i , s
= τ i
( 1 − p i
)
τ
1 i
=
( 1 −
2 ( 1 −
2 p i
* p i
*
)
)
+
W i , 0
(
( 1 − p i
)( 1 − (
. (9) 2
( 1
2
− p i
) p i
* m i
)
)( 1
+
−
( 1
2
− p i
2 p i
)( 1 −
)( 2 p i
L i p i
+ 1
)
) m i
( 1 − p i
L i
− m i
+ 1 )
)
(14)
Then we can find the probability p s that any time slot contains a successful transmission, i.e. that a packet from any class is transmitted successfully in a time slot: p s
=
N − 1
i = 0 p i , s
. (10)
We present a recursive solution method where Eq. (14) first is resolved with respect to the highest priority AC, i.e. AC[ N-
1 ]. This result is then used to solve Eq. (14) for the secondhighest priority AC AC[ N-2 ]. Recursively these results are then used to solve Eq. (14) for AC[ N-i ] for i = 1 , 2 ,...
In the following, we demonstrate how this works.
Finally, we let p b
denote the probability that the channel is busy. It is busy when at least one backoff instance transmits during a slot time: p b
= 1 −
N i
− 1
= 0
( 1 − τ i
)
Since all traffic of the highest priority AC, AC[ N-1 ], have strict priority in the VCH without chance of colliding with the other traffic , we can always set:
. (11) and p
N p *
N
− 1
− 1
=
=
0
0
(15)
. (16)
By comparing Eq. (9) and Eq. (10) with Eq. (11), we observe that: p b
= p s p *
N − 1 in Eqs. in the special n i
= 1 case we are considering here. Eq. (12) implies that the probability of observing a collision on the channel ( p b
− p s
) is zero. This result corresponds to the reality.
Since all collisions are handled by the VCH, colliding packets are either transmitted successfully or pushed to backoff.
Collisions will not block transmissions in the VCH.
The throughput of class , S i
, can be written as the average real-time duration of successfully transmitted packets by the average real-time duration of a contention slot that follows the special time scale of our model. By inserting Eq.
(9), Eq. (11) and Eq. (12) into the generic throughput expression in [3], we get:
S i
=
T s
τ i
( 1 −
− ( T s p i
) T i , MSDU
B
− T e
)
N i =
− 1
0
( 1 − τ i
)
When inserting these values into Eq. (14) for the highest priority AC, AC[ N-1 ], the transmission probability of this AC is obtained in closed-form:
τ
N − 1
=
2
W
N − 1 , 0
+ 1
. (17)
For the lower priority ACs, the collision probabilities are written according to Eq. (8): p
N − 2
= 1 − ( 1 − τ
N − 1
) = τ
N − 1
, p
N − 3
= 1 − ( 1 − τ
N − 2
)( 1 − τ
N − 1
) = τ
N − 2
+ τ
N − 1 p
N − 4
= ...
− τ
N − 1
τ
N − 2
, (18)

The main obstacle for solving the model recursively is if the transmission probability of a higher priority AC is dependent on that of a lower priority AC, leading to circular dependencies between the N equations to be solved [i.e. Eq. (14)]. p i
*
The only parameter that may contain such a dependency is
. If countdown blocking is not taken into account in the model, there is no problem, since if countdown blocking is used, p i
* p i
* = 0 i
= p i
(Eq. (3a)). Similarly,
(Eq. (3b)), and there is no such dependency problems.
However, if AIFS differentiation is incorporated into the model, circular dependencies occur. The reason is that Eq. (3c) let p i
*
depend on the transmission probabilities of all other
ACs, through the term:
V. EXRESSION FOR THE MEAN
DELAY
The mean saturation delay, D i
SAT , can be found by inserting Eq. (12) into the expression for the delay derived in
[3]:
D i
SAT = T s
1
(
− p i
L i
+ 1
)
+


T e
+ T s ( 1 − p i
* p i
* )


L i
j = 0 p i j
( W ij
2
− 1 ) . (21)
(We have also set
T c
* = 0 in the original expression in [3], since collisions are instant in the VCH and do not consume additional time.) m i
By performing the summation above in Eq (21) for the case
≤ L i
we obtain:
N i
=
− 1
0
( 1 − τ i
) j
L i
= 0 p i j
( W ij
2
− 1 )
= W i 0




1 − ( 2 p i
1 − 2
) p i m i
+ 1
+ 2 m i p i m i
+ 1
1 −
− p p i i
L i
+ 1




−
1 −
1 − p i
L i p i
+ 1 . (22)
In this case, we have to make a simpler approximation for i to be able to derive with a closed-form expression. To obtain this we approximate:
The mean non-saturation delay D i
NON − SAT , can now be calculated by subtracting the post-backoff from the mean saturation delay [3]: p i
∗ ≈ max
(
0 , 1 − p i
(
1 + A i
) )
, (19) where A i was defined in Eq. (4). The original expression in Eq.
(3c) is based on the fact that the countdown process of an AC can potentially be blocked in the VCH by traffic from all other
ACs. The simpler approximation above, on the other hand, neglects the blocking of traffic from lower priority ACs, which is normally considerably smaller compared to that of the higher priority ACs. (In fact, validations in Section V below, show that this approximation is good in all scenarios that are validated.)
As an example, we show how to use this method with the recommended parameter settings of 802.11e (See [2] or scenario 1 in Table I later in this paper for the actual values).
Now, AIFSN [ N − 2 ] − min( AIFSN [ 0 ]...
AIFSN [ N − 1 ]) = 0 , and thus p i
* = p
2
= τ
3
. As a result, Eq. (14) gives the following expression for the transmission probability, τ
2
: between D i
NON
D i
NON − SAT
− SAT
Eq. (16) and get:
= D i
SAT −



T e
+ T s
Generally, the mean delay, and D i
SAT
D i
NON − SAT
:
D
T s
≤ D
N − 1
≤ T s i
≤ D i
+ T e
( 1 − p i
* p i
* ) 

 W i 0
2
− 1 (23)
, is expected to be somewhere
≤ D i
SAT
. (24)
For the highest priority AC, AC[N-1], we use Eq. (15) and
W i 0
2
− 1 . (25)
The delays for the lower priority ACs are calculated similarly, obtained as shown in the previous section. p p i
*
VI. VALIDATIONS
τ
1
2
=
( 1 − 2 τ
2 ( 1 − τ
3
)
3
)
+
W
2 , 0
( 1 −
2 ( 1 − 2 τ
( 2 τ
3
)( 1 −
3
) m i
τ
3
L i
)
+ 1 )
+
W
2 , 0
( 2 τ
2 ( 1 −
3
) m i
τ
( 1 −
3
)( 1 − τ
τ
3
L i
− m i
+ 1 )
L
3 i
+ 1 ) one calculated
A closed form expression for the saturation throughput can finally be found by inserting results into Eq. (13). The delay can also be found. This will be explored in further detail in the next section. p i
∗ = 1
(20)
A . Setup
If starvation occurs for lower priority ACs, this means that
as a result of Eq. (19). Otherwise, one uses Eq.(18) and Eq. (19) in combination with Eq.(14) to find the transmission probabilities also for the two lowest priority
ACs.
We compare numerical computations of the model with simulations. The TKN implementation of 802.11e for ns-2 [7] is used for the ns-2 simulations.
We selected 802.11b with the mandatory long preamble [6] without the optional RTS/CTS mechanism and with the short retry limit. Poisson distributed traffic consisting of 1024 bytes packets were sent at the maximum transmission rate, i.e. at 11
Mbps. The corresponding values for T e
, T s
, T c and T
MDSDU
are calculated in [3]. For simplicity, we assumed that the QAP generated the same amount of downlink traffic for each of the four ACs.

numerical saturation solution according to Eq. (3c). We see that the closed form solution does not introduce significant errors.
TABLE I. S ELECTED S CENARIOS .
Scenario:
Config.
CWmin of AC[3] CWmax
(AC_VO) AIFSN
1 2
4 8
8 16
2 2
3
4
8
2
4
4
8
2
Config.
CWmin 8 16 16 16 32 of AC[2] CWmax 16 32 32 32 64
(AC_VI) AIFSN 2 2 2 2 2
Config.
CWmin 16 32 16 32 32 of AC[1] CWmax 1024 2048 1024 2048 2048
(AC_BE) AIFSN 3 3 3 3 3
5
4
8
2
Config.
CWmin 16 32 16 32 32 of AC[0] CWmax 1024 2048 1024 2048 2048
(AC_BK) AIFSN 7 7 7 7 7
Figure 2. Simulation setup to validate numerical results of downlink traffic.
We considered the node topology depicted in Figure 1. The
QAP implements a VCH and uses four transmission queues.
The QSTAs were not actively initiating traffic. Their role was only to acknowledge all MAC frames that the QAP successfully transmit on the channel. This corresponds to the downlink scenarios presented in the frequently cited paper by
Mangold et al. [8], except that we considered 802.11b instead of 802.11a.
We selected a number of scenarios with different 802.11e access parameters. The selected scenarios are summarized in
Table I. Scenario 1 corresponds to the recommended (default) parameter settings for 802.11e, and is thus the major reference scenario. Other scenarios are variations of this. Scenario 2, for example, is a variation of scenario 1, where the CWmin and
CWmax values are the double of those of Scenario 1.
TABLE II. A NALYTICAL C LOSED F ORM S OLUTION C OMPARED WITH
S IMULATION R ESULTS .
Scenario
1
2
3
4
5
AC
AC[3]
AC[2]
AC[1]
AC[0]
AC[3]
AC[2]
AC[1]
AC[0]
AC[3]
AC[2]
AC[1]
AC[0]
AC[3]
AC[2]
AC[1]
AC[0]
AC[3]
AC[2]
AC[1]
AC[0]
Saturation Throughput
Simulations
5239
850
58
0
4298
1381
341
0
5687
388
64
0
5683
419
36
0
5895
200
40
0
Closed
Form
5246
828
21
0
4516
1223
245
0
5587
445
49
0
5609
446
24
0
5806
232
34
0
Numerical
5246
828
21
0
4516
1223
245
0
5587
445
49
0
5609
446
24
0
5806
232
34
0
Xiao
(p*=p)
5035
795
142
129
4177
1130
369
330
5306
422
191
171
5459
434
98
93
5636
225
115
101
Bianchi
(p*=0)
4620
1136
191
171
3873
1319
441
394
4955
635
275
240
5157
661
145
135
5466
358
178
83
To illustrate how well the starvation features represented by
Eq. (3c) and Eq. (19), we have also added the saturation solution according to Xiao’s and Bianchi’s models, replacing
Eq. (19) with Eq. (3b) and Eq. (3a), respectively. The AIFS differentiation according to Eq. (3a) or Eq. (19) keeps the lowpriority ACs, AC[1] and AC[0], relatively close to our simulation results. The results of Xiao and Bianchi, on the contrary, are not appropriate when AIFS differentiation is used.
6000
5000
4000
3000
2000
1000
0
0 2000
AC[3]: Simulations
AC[3]: Numerical
4000 6000
Traffic generated per AC [Kb/s]
AC[2]: Simulations
AC[2]: Numerical
AC[1]: Simulations
AC[1]: Numerical
8000 10000
AC[0]: Simulations
AC[0]: Numerical
Figure 3. Comparison between closed-form saturation solution and simulation results with the default 802.11e parameter settings (Scenario 1).
B. Closed form Saturation Solution
The selected scenarios were calculated analytically, using the recursive closed-form solution method described above, and compared with simulation results. To get saturation results form the simulations, we let the QAP generate 10 Mbps of traffic per AC.
Results are listed in Table II. We observe that there is a fairly good match between the analytical closed-form solution and our simulation results. To illustrate the accuracy of the approximation in Eq. (19) works, we have included the
We also show the comparisons of scenario 1 and scenario 2 in Figure 3 and Figure 4, respectively. The linearly increasing part of the line comes from the fact that there cannot be more traffic transmitted than traffic generated. We observe that there is a fairly good match between the closed-form solution and the simulation results, except in the intermediate state between the non-saturated and saturated conditions for the lower priority
ACs. In this area, one has to rely on numerical calculations of

the model. How well these calculations match with simulations will be explored in the following.
5000 results on a larger scale (up till 20000 Kbps per AC) to illustrate the remarkably good accuracy between model and simulation results in the saturation part of the figure.
5000
4000
4000
3000
3000
2000
2000
1000
1000
0
0
AC[3]: Simulations
AC[3]: Closed Form
5000 10000 15000
Traffic generated per AC [Kb/s]
AC[2]: Simulations
AC[2]: Closed Form
AC[1]: Simulations
AC[1]: Closed Form
20000
AC[0]: Simulations
AC[0]: Closed Form
Figure 4. Comparison between closed-form saturation solution and simulation results (Scenario 2).
0
0 5000
AC[3]: Simulations
AC[3]: Numerical
10000 15000
Traffic generated per AC [Kb/s]
AC[2]: Simulations
AC[2]: Numerical
AC[1]: Simulations
AC[1]: Numerical
20000
AC[0]: Simulations
AC[0]: Numerical
Figure 6. Comparison between analytical results and simulation results
(Scenario 2).
C. Solution
The non-saturation part of the solution is difficult to solve in a closed form. Here, we had to do numerical calculations in
Mathematica .
Figure 5 compares numerical calculations of the analytical model with the actual simulation results. We observe that our analytical model of the VCH, which describes the performance on the full range from a non-saturated (finite queues) to a saturated (infinite queues) system, gives a good match when compared with simulations.
6000
5000
4000
3000
2000
1000
6000
5000
4000
0
0 1000
AC[3]: Simulations
AC[2]: Numerical
2000 3000 4000
Traffic generated per AC [Kb/s]
AC[3]: Simulations
AC[1]: Numerical
AC[1]: Simulations
AC[1]: Numerical
5000
AC[0]: Simulations
AC[0]: Numerical
6000
3000 Figure 7. Comparison between analytical results and simulation results on a small scale (Scenario 1).
2000
1000
0
0 2000
AC[3]: Simulations
AC[3]: Numerical
4000 6000
Traffic generated per AC [Kb/s]
AC[2]: Simulations
AC[2]: Numerical
AC[1]: Simulations
AC[1]: Numerical
8000 10000
AC[0]: Simulations
AC[0]: Numerical
Figure 5. Comparison between analytical results and simulation results using
Recommended 802.11e parameter settings (Scenario 1).
In Figure 6 we repeat the validations using different values for the contention window. Here we have doubled all minimum and maximum contention windows compared to the recommended values given in Table I. We have also shown the
However, there are ranges of Figure 5 and Figure 6 where there are noticeable discrepancies between the curves. For the recommended parameter setting results in Figure 6, this range is expanded and shown on a smaller scale in Figure 7. Here, we observe that the model – probably the AIFS-approximation - is a little too rough on the lowest priority AC, AC[0]. Due to the fact that AC[0] and partly also AC[1] are underestimated here, the model incorrectly gives a throughput of AC[3] that exceeds the 1-to-1 linear line. This would mean that AC[3] transmits more traffic than is generated, which is obviously not correct. It is indeed possible to do some improvements of the model in this region, although one must keep in mind that the model is approximate, and a complete match might be difficult to find without adding considerable complexity to the model.

VII. CONCLUSIONS
This paper shows how virtual collision handling can be incorporated into an analytical model that covers the full range from a non-saturated to a fully saturated channel.
Using a model that encompasses virtual collision handling, we demonstrate that it is also possible to describe the behaviour of a Virtual Collision Handler internally on a node, such as on an Access Point. The Virtual Collision Handler is treated as a
"virtual" channel. Moreover, we also show how closed-form saturation solutions of the performance behaviour of the Virtual
Collision Handler can be found.
Given this model, an Access Point that uses EDCA for massive downlink traffic is able to predict the levels of QoS that the data traffic it is transmitting will obtain by its own
Virtual Collision Handler. In this way it is to a larger extent in control of the QoS of the traffic it is sending. (Needless to say, any station – whether it is an access point or not – may benefit from predicting the behaviour of the Virtual Collision Handler, although we anticipate that the model will be mostly appreciated by the Access Points.)
The model is calculated numerically and validated against simulations, using 802.11b and variations of the default parameter settings for 802.11e. We observed that the closedform saturation solution gives a good match with simulation results when the channel is either close to the saturated condition or close to the non-saturation condition.
In the intermediate area between saturation and nonsaturation, the analytical model has to be solved numerically in order to obtain sufficient accuracy. The numerical results of the
Virtual Collision Handler correspond well with simulation results in the whole region from a fully non-saturated to a completely saturated medium.
[7] Wietholter, S. and Hoene, C., "Design and verification of an IEEE
802.11e EDCF simulation model in ns-2.26", Technische Universitet at
Berlin, Tech. Rep. TKN-03-019, November 2003.
[8] Mangold, S., Choi, S., Hiertz, G., Klein, O. and Walke, B., "Analysis of
IEEE 802.11e for QoS support in wireless LANs", IEEE Wireless
Comm, Dec. 2003, pp. 40-50.
A CKNOWLEDGMENT
We would like to thank Bjørn Selvig for help with the development of the simulation tool used for the validations.
R EFERENCES
[1] IEEE 802.11 WG, "Part 11: Wireless LAN Medium Access Control
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[2] IEEE 802.11 WG, "Draft Supplement to Part 11: Wireless Medium
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