R&I R 28/2008
Olav Østerbø and Paal E. Engelstad
WLAN QoS Analysis - A New
Modelling Approach for 802.11e
WLAN QoS Analysis - A New Modelling Approach for
802.11e
R&I Research Report
Title
R 28/2008
WLAN QoS Analysis - A New Modelling
Approach for 802.11e
Author(s)
ISBN / ISSN
Security group
Date
Olav Østerbø and Paal E. Engelstad
82-423-0617-6 / 1500-2616
OPEN
2008.11.08
Abstract
This document presents an analytical method to calculate the performance of 802.11
systems in general and the differentiation characteristics of 802.11e in particular. Like
similar analytical methods, it is based on a Markov model of the system. Our proposed
method is based on ensuring consistency between the macro-view and micro-view of
the system, while previous models are mainly focusing on the micro-view. With the new
method the system characteristics can be found by the macro-view approach, which
represents a great simplification compared to previous methods. Another simplification
is that the Laplace transform is used to study the delay in each Markov state, as well as
the overall access delay and queueing delay. Finally, the model is extended to also
include varying packet lengths and a more detailed description of the AIFS
differentiation of 802.11e.
Keywords
802.11, 802.11e, Markov, Bianchi, QoS, Performance, Virtual Collisions, Laplace, access
delay, Queueing delay.
Telenor R&I R 28/2008
WLAN QoS Analysis - A New Modelling Approach for
802.11e
© Telenor ASA 2008.08.08
All rights reserved. No part of this publication may be reproduced or utilized in
any form or by any means, electronic or mechanical, including photocopying,
recording, or by any information storage and retrieval system, without
permission in writing from the publisher.
Telenor R&I R 28/2008
WLAN QoS Analysis - A New Modelling Approach for
802.11e
Preface
This document presents the work of the WLAN QoS Analysis project that was
carried out in 2008. This activity is part of the Wireless Broadband project at
Telenor R&I.
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802.11e
Contents
1
Introduction .......................................................... 1
2
Combining micro and macro view to obtain
throughput and system parameters ...................... 5
3
Virtual collisions and the mean slot length ........... 7
1.1
1.2
1.3
1.4
3.1
3.2
3.2.1
3.2.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
4
4.1
4.2
4.3
4.4
Communication scenarios addressed by the model ....................... 1
Our original model ................................................................... 3
The need for improving the model .............................................. 3
An outline of the document ....................................................... 4
What are virtual collisions?........................................................ 7
How virtual collisions affect model parameters ............................. 8
The collision probability ............................................................ 8
The duration of a slot on the radio channel (the “Slot Length”) ....... 8
The probability of a transmission on the radio channel .................. 9
The probability of a busy time slot ........................................... 10
The probability of a successful transmission............................... 10
Mean duration of a busy time slot without virtual collisions .......... 11
Mean duration of a busy time slot with virtual collisions ............... 12
Mean duration of a time slot with a successful packet ................. 14
Mean duration of a time slot with a frame collision without
virtual collisions..................................................................... 15
Mean duration of a time slot with a frame collision with virtual
collisions .............................................................................. 15
The slot length distribution for the RTS/CTS case both with and
without virtual collisions (using the Laplace transform) ............... 15
Using the Laplace transform to obtain the
access delay distribution..................................... 17
Description of the delay components ........................................ 17
The Laplace transform of the access delay ................................. 18
Specification of the countdown delay ........................................ 20
Delay distributions during countdown and freeze periods ............. 23
5
The queueing delay ............................................. 25
6
Concluding remarks ............................................ 27
6.1
6.2
Summary ............................................................................. 27
Issues for further work ........................................................... 27
References .................................................................. 29
7
7.1
7.2
Appendix ............................................................. 31
Sums ................................................................................... 31
Elements of Laplace inversion .................................................. 31
Significance for Telenor .............................................. 35
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WLAN QoS Analysis - A New Modelling Approach for
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1 Introduction
1.1 Communication scenarios addressed by the model
This document presents an analytical model for IEEE 802.11 WLAN
communication. The model covers communication both with and without
802.11e (i.e. with or without traffic differentiation/QoS.) As WLAN
communication without 802.11e is basically a subset and a simplified version of
our model, we will focus mainly on 802.11e communication in the document.
Figure 1 illustrates a typical scenario addressed by the model. In the figure, the
“AP” refers to the Access Point (AP) and each “STA” refers to a WLAN Station
(STA). A number of STAs contend for the wireless medium to send traffic to the
AP.
This is a pure uplink scenario, where all traffic is sent from the STAs to the AP.
However, the model might as well be used for other communication patterns,
including the pure downlink scenario where all communication goes from the AP
to the STAs.
STA 8
STA 1
A
STA 7
STA 2
STA 6
STA 3
STA 4
Figure 1
STA 5
A typical WLAN communication scenario addressed by the model
The 802.11e extensions allow for differentiation between four different traffic
classes, which are referred to as Access Categories (ACs). Each AC is allocated
a separate queue on the station. The ACs are numbered so that the lowest
priority AC is referred to as “AC 0” and the highest priority AC is referred to as
“AC 3”.
Figure 1 does not indicate the offered traffic load per AC. A typical scenario is
that each STA is sending traffic of all four ACs and – for simplicity – is trying to
send an equal amount of traffic of each of these ACs. To further simplify the
scenario, one might assume that the packet lengths do not differ between the
different ACs. (The latter might not be very realistic, because normally the
voice traffic that is sent at the highest priority AC is normally of shorter packet
length than the packet length of the best-effort or background data traffic sent
at the lowest ACs). In this scenario, there will be 24 actively transmitting
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queues in the scenario depicted in the figure, i.e. eight actively transmitting
stations, each with four actively transmitting queues. In this scenario, the four
actively transmitting queues on a station are contending internally for accessing
the channel, before the queue that “wins” the internal contention is trying to
transmit on the radio channel, contending with transmissions from the other
STAs. The first round of internal contention between the queues on a STA is
referred to as “virtual collisions” (VCs), and a model addressing this scenario is
referred to as a “model with virtual collisions”. Virtual collisions will be
explained in further detail later in this document.
Another simple traffic scenario in Figure 1 is typically that each STA is sending
traffic of only one AC each – and for simplicity – one might assume that each
station tries to transmit an equal amount of traffic independent of which AC it is
sending. To further simplify the scenario, one might assume that the packet
lengths do not differ between the different ACs. For example, in Figure 1 STA 1
and STA 2 send traffic of AC 0, STA 3 and STA 4 send traffic of AC 1, STA 5 and
STA 6 send traffic of AC 2, and STA 7 and STA 8 send traffic of AC 3. In this
scenario, there is a total of only eight actively transmitting queues in the
system (i.e. two queues of each AC). Furthermore, there is no internal
contention between the queues on a station, because only one queue is actively
transmitting traffic here. Consequently, there are no Virtual Collisions in the
system. A model addressing this scenario is referred to as a “model without
Virtual Collisions”.
In Figure 2 we observe some simulation results illustrating the typical
performance characteristics according to these types of uplink scenarios. In this
specific case, 802.11b is used as the PHY layer and there are five actively
transmitting stations, all trying to transmit an equal amount of traffic for each
of the four ACs to the AP. The curves show the total amount of throughput per
access category of successfully transmitted packets (sometimes also referred to
as “goodput”).
4000
Throughput per AC [Kb/s]
3500
3000
2500
2000
1500
1000
500
0
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Traffic generated per AC [Kb/s]
Figure 2
AC[3]: Simulations
AC[2]: Simulations
AC[0]: Simulations
Input = Output
AC[1]: Simulations
A performance study of the differentiation characteristics of 802.11e
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1.2 Our original model
We developed a first version of the model in the period 2004 – 2006. At that
time there were few efforts modelling the differentiation characteristics of
802.11e, as the original Bianchi model considered plain 802.11 with only one
traffic class.
Furthermore, most efforts considered only the saturation throughput, which is
found as the horizontal curves on the right side of Figure 2. However, in the
leftmost part of the figure, the traffic load is so low that the system is totally
unsaturated and nearly all traffic that is trying to be transmitted is eventually
being successfully transmitted. (However, a frame might have to be
retransmitted after a collision has been detected, in order to finally be
considered as successfully transmitted.) Since nearly all traffic is eventually
transmitted with success, the traffic curves follow the 45o input=output line in
the leftmost part of the figure. By having a model that takes both saturation
and non-saturation conditions into account, the model is able to predict the
performance over all possible traffic loads. Especially when the differentiation
characteristics of 802.11e are considered, it is fruitful to have a model that
gives the full picture of how the performance changes with a changing traffic
load, as illustrated in Figure 2.
The differentiation characteristics between the two highest priority ACs in
Figure 2 are primarily given by the fact that these two ACs have different
minimum contention windows (and thus a different mean number of postbackoff slots). However, the fact that the two lowest priority ACs face full
starvation at saturation conditions, is mainly attributed to the AIFS
differentiation. Furthermore, since these two ACs have the same contention
window (CW) settings, the difference between these two curves observed in the
region between non-saturation and starvation is a direct consequence of the
difference in AIFS differentiation. Another contribution of our previous model
was a simple – and fairly accurate - model of the AIFS differentiation.
The transmission of a frame (or “packet”) consists of two main delay
components. First, the frame waits in the transmission queue, and this delay
component is referred to as the queueing delay. When at the head of the
queue, the frame is being attempted to be transmitted. This is referred to as
the “access delay”. In fact, since the frame ends with a post-backoff, the
service time of the frame is the sum of the access delay and the duration of the
post-backoff.
In our previous model, we calculated both access delay, service time and
queueing delay. Using the z-transform to calculate the delay, we were able to
calculate all moments of the delay distribution and thus, in principle, the full
delay distribution of both access delay, service time and queueing delay.
1.3 The need for improving the model
To the best of our knowledge, all models – including Bianchi’s original model for
802.11 saturation conditions and our own model for 802.11e differentiation and
non-saturation conditions – were taking a micro-view approach to deriving the
system characteristics. However, after taking a macro-view check of the model,
we observed that inconsistencies between a micro-view approach and a macroview approach could be observed under some conditions. That potential
inconsistency needed to be eliminated.
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As a consequence, one of the main goals of the work presented in this
document was to ensure consistency of the model.
Furthermore, there was a need to simplify the model. This was another
objective of the work presented in this document. It was assumed that a
macro-view approach of the model could lead to a simpler method to derive the
system characteristics. Thus, by ensuring consistency, there would also be
potentials for simplifications.
Inconsistency occurred especially in a non-slotted delay view, i.e. where each
state of the Markov diagram could have a delay distribution not equal to all the
other Markov states. Thus, by solving the aforementioned inconsistency,
arbitrary delay characteristics would be an integral part of the model. The ztransform, which we used in our previous work, is particularly suitable for
slotted systems and not these kinds of arbitrary delay characteristics. Therefore
there was a need to use the Laplace transform instead, which is more
appropriate in continuous-time systems. Thus, in addition to finding a simpler
macro-view method, we introduced the Laplace transform as another way of
meeting the objective of simplifying the model.
Finally, we wanted to extend the model to cover more system aspects. To
obtain this goal we had to find a more detailed description of the AIFS delay.
Since our original model did not take differing packet lengths appropriately into
account when calculating the slot length on the radio channel, we also had to
consider different packet lengths of different ACs.
1.4 An outline of the document
Chapter 2 presents a model that provides consistency between the micro-view
and the macro-view. In Chapter 3, the slot length on the radio channel is
calculated with differing packet lengths. It is found both for model variants with
virtual collisions and model variants without virtual collisions. Then, the access
delay and the queueing delay are found in Chapter 4 and Chapter 5, and a
detailed model for AIFS differentiation is used when finding the access delay in
Chapter 4. The Laplace transform is used when finding the delay characteristics
in Chapter 4 and Chapter 5. Finally, conclusions are drawn in Chapter 6.
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2 Combining micro and macro view to
obtain throughput and system
parameters
Performance modelling of IEEE 802.11 DCF and 802.11e EDCF is usually based
on an imbedded Markov chain to obtain the transmission probability. Hence, an
important part of the analysis is to find the steady state probabilities of the
Markov chain, from which the transmission probability is derived [1]-[14]. With
the imbedded Markov chain, the distribution of the delay in each state is
absent. Instead, the state probabilities are found by counting the number of
states that are visited as part of a successful transmission. This method is
possible, because it is assumed that the system spends an equal amount of
time in each Markov state (i.e. it is assumed that the model is “slotted”).
At first sight this imbedded approach seems simple, because the state
probabilities can be found without calculating the access delay. However, for
the non-saturated analysis the mean service time (and thus also the mean
access delay) must be found to determine the utilization factor (i.e. the
probability of having an empty access queue). Thus, a delay analysis must be
carried out anyhow.
In this paper we will instead take a new, non-imbedded approach and start with
doing the delay analysis. Our new non-imbedded method is considerably easier
that the traditional imbedded method, because the system performance can be
found without finding the steady state probabilities of the Markov chain. In
addition, the new approach is quite intuitive.
Our new approach is based on ensuring consistency between a macroscopic and
microscopic viewpoint. Consistency is ensured by equating two different
expressions for the throughput:
•
The first expression is obtained by a traditional microscopically slot level
view, i.e. the throughput is expressed as the fraction of time for which
the channel is used to successfully transmit frames.
•
The second way of expressing the throughput is by macroscopic level
quantities as the offered traffic load and the mean access delay.
In the following we consider a particular access queue for an Access Category
(AC) in a WLAN operating under the EDCA mechanism. We assume that the
arrival of frames follows a Poisson process with intensity λi , which we for
simplicity assume to be equal for all class i AC queues. To keep full generality
we assume that there is a total of N different classes indexed i = 0,1,.., N − 1
and that the number of queues for class i is ni , i.e. the total number of access
queues equals K =
N −1
∑n
i =0
i
. (If each STA have all classes present, i.e. ni = n is
the number of STAs in the WLAN and hence
K = Nn .)
Below we obtain two expressions for the throughput for a particular AC queue.
The first one is obtained by a microscopic view of the channel in terms of slots
where the throughput is expressed in the average payload information in a slot
divided by the average duration of a slot time. The microscopic expression is
found in the original model of Bianchi [1] and can be written:
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si =
τ i (1 − p i ) E[ DiMSDU ]
σ
(1)
where σ is the mean slot length, E[ DiMSDU ] is the mean time required to
transmit the data payload of the DATA frame, τ i is the transmission probability
and pi , is the collision probability for AC i .
On the other hand, by applying a macroscopic approach, we may consider this
particular queue as a single server with mean service time E[ DiSat ] , i.e. the
mean time it takes to transmit a frame including the post-backoff period. With
the arrival rate λi the queue has the load (utilization factor) ρ i = min[1, λi E[ DiSat ]] .
The “carried load”, obtained by subtracting those frames that are lost by
expiration of the retry limit Li , is then (1 − piLi +1 ) ρ i . However, on average only
E[ DiMSDU ]
part of the transmissions carries useful full traffic. Hence the
E[ DiSat ]
normalized throughput si may be written as:
si = (1 − piL +1 ) ρ i
i
E[ DiMSDU ]
E[ DiSat ]
(2)
By (1) and (2) we obtain the fundamental τ -equation:
(1 − p iLi +1 )
σ
τi =
ρi
1 − pi
E[ DiSat ]
(3)
This can be written separately as:
(1 − p iLi +1 )
τi =
λiσ for λi E[ DiSat ] < 1 and
1 − pi
(4)
(1 − p iLi +1 ) σ
for λi E[ DiSat ] ≥ 1
1 − p i E[ DiSat ]
(5)
τi =
The two equations above give the important and basic relation between the
transmission probability and the collision probability. They show that we have
two different modes of operation:
•
Non-saturation (when λi E[ DiSat ] < 1 ) and
•
Saturation (when λi E[ DiSat ] ≥ 1 )
This modelling approach is fundamentally different from other approaches using
Markov modelling. Moreover, it is more general and only depends on the mean
slot length and the mean access delay.
Hence, to specify the model, both the mean slot length
σ
and the mean
service time E[ D ] must be expressed in terms of system parameters. This is
done in the next two chapters of this paper.
Sat
i
Finally, observe that the saturation case coincides with a Bianchi type model
E[ Sl ]
1
=
(equation (5)) only if
where E[ N iSat ] is the mean number of
Sat
E[ N i ] E[ DiSat ]
Markov-states visited for a frame.
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3 Virtual collisions and the mean slot length
3.1 What are virtual collisions?
IEEE 802.11e introduces a mechanism called “virtual collisions”, which occurs
between queues on the same station. If two or more queues at the station are
trying to transmit in the same time slot (of the radio channel), only the highest
priority queue of these queues is allowed to transmit (Figure 3). The other
queues go into backoff, just as if the transmission of the frame in this queue
had been involved in a real collision on the radio channel. The highest priority
queue of the queues that wants to transmit in a time slot goes only into backoff
if the frame it transmits is involved in a collision on the radio channel.
Figure 3
Internal transmission conflicts between the different queues at a
station are taken care of by the Virtual Collision Handler
With “vanilla” 802.11 (i.e. 802.11 without 802.11e), there are naturally no
virtual collisions. With 802.11e, on the other hand, there might be scenarios
both without and with virtual collisions. In a scenario where each station is
actively transmitting traffic of only one AC at the time, there will be
differentiation between the ACs on the radio channel, but no virtual collisions
occurring between queues on the stations. On the other hand, in a scenario
where one or more stations are actively transmitting traffic of more than one
AC simultaneously, there will be virtual collisions in the system. In this case,
the virtual collisions need to be taken into account in the model.
When discussing 802.11e in the following, we will consider two extreme
scenarios. In the first scenario, referred to as “with virtual collisions”, there are
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n stations in total, each queue transmitting traffic of all N ACs simultaneously.
In the other scenario, referred to as “without virtual collisions”, there are n0
stations transmitting only traffic of AC 0, n1 stations transmitting only traffic of
AC 1, and so forth, and finally nN-1 stations transmitting only traffic of AC N - 1.
If in the latter case we assume that ni = n for all i, there is a total of n*N
stations in the system. Then, both scenarios are quite similar since there is a
total of n * N actively transmitting queues in both. In this case, the biggest
difference between the two scenarios is whether or not virtual collisions occur
within each of the n groups of N queues in the system.
3.2 How virtual collisions affect model parameters
3.2.1 The collision probability
Whether virtual collisions are part of the model or not affects the definition of
the collision probability. Without virtual collisions, the collision probability can
be written [11]:
N −1
pi =
(1 − τ j )
∏
j =0
nj
(Without virtual collisions)
(1 − τ i )
(6)
Otherwise, if there are virtual collisions, the collision probability can instead be
written [11]:
N −1
pi =
∏ (1 − τ
j
)
nj
j =0
i
(With virtual collisions)
∏ (1 − τ j )
(7)
j =0
3.2.2 The duration of a slot on the radio channel (the “Slot
Length”)
The mean duration of a slot on the radio channel, σ, can be written as the sum
σ = σe + σc + σs, where:
•
σe is the probability that the slot is empty, (1 − pb ) , times its
duration, Te,
•
σc is the probability that the slot contains a packet collision, times
c
the duration of the collision D , and
•
σs is the probability that the slot contains a successfully
s
transmitted packet times its duration D .
If all packets are of the same length, the slot length can generally be written:
σ = (1 − p b )Te + ( p b − p s ) D c + p s D s
(8)
In this case, the presence of virtual collisions only affects the collision
parameter of the model. However, if packets are of different lengths, the slot
length is affected. The way to calculate the mean slot length then depends on
whether or not there are virtual collisions.
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3.3 The probability of a transmission on the radio
channel
Before estimating the slot duration with or without the existence of virtual
collisions, and with or without varying packet lengths, we need to determine
the probability of a transmission on the radio channel. This probability will be
used later in this chapter.
We define τ~i as the probability that a station transmits on the radio channel. If
there are no virtual collisions in the system, τ~i is defined as:
τ~i = τ i .
(Without virtual collisions)
(9)
Otherwise, if there are virtual collisions, τ~i is defined as:
N −1
τ~i = τ i ∏ (1 − τ j ) ,
(With virtual collisions)
(10)
j = i +1
where
∏
is defined so that
N −1
∏ (1 − τ j ) = (1 − τ N
j N
= −2
Let
−1
N −1
N −1
(1 − τ j ) = 1 , ∏ (1 − τ j ) = (1 − τ N −1 ) ,
∏
j = N −1
j=N
)(1 − τ N − 2 ) and so forth.
τ~ STA denote the probability that a STA will transmit on the radio channel.
τ~ STA can be written as:
N −1
τ~ STA = 1 − ∏ (1 − τ j ) .
(11)
j =0
This equation holds whether there are virtual collisions or not. Without virtual
collisions, this relation is trivial, since obviously the probability that no queues
N −1
at the STA is transmitting can be written
∏ (1 − τ
j
) . In the following it will be
j =0
shown that the relation is also valid if there are virtual collisions. First note that
with virtual collisions, transmission attempts on a node are mutually exclusive,
so that the probability that an STA will transmit on the radio channel,
STA
=
be written as the sum τ~
N −1
N −1
N −1
i =0
i =0
j =i +1
τ~ STA , can
∑τ~i = ∑τ i ∏ (1 − τ j ) . Thus, if one is able to
show that the relation
N −1
N −1
N −1
i =0
j = i +1
j =0
∑τ i ∏ (1 − τ j ) = 1 − ∏ (1 − τ j )
(12)
holds for all N>1, it proves the validity of the equation above, also when there
are virtual collisions.
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Relation (12) is shown by induction: For N=1, we have
0
0
i =0
j = i +1
∑τ i ∏ (1 − τ j ) = τ 0
and
1−1
1 − ∏ (1 − τ j ) = τ 0 , so the relation is obviously valid for N=1. If we assume that
j =0
the relation is valid for N. Then we have for N + 1:
N
N
⎡ N −1
N −1
⎤
⎡
N −1
⎤
N
i =0
j = i +1
⎣ i =0
j = i +1
⎦
⎣
j =0
⎦
j =0
∑τ i ∏ (1 − τ j ) = (1 − τ N )⎢∑τ i ∏ (1 − τ j )⎥ + τ N = (1 − τ N )⎢1 − ∏ (1 − τ j )⎥ + τ N = 1 − ∏ (1 − τ j )
Since the relation is also valid for N + 1, the relation is proved by induction.
Q.E.D.
3.4 The probability of a busy time slot
During countdown periods the access queue in consideration does not try to
transmit, which means that the corresponding probabilities must be calculated
as a system without that particular queue present. Similarly, during AIFSN
“freeze periods” all the lower priority STAs are blocked and the probabilities
must be calculated as if all these STAs were removed from the system. To write
the probabilities effectively, we introduce the following vector notation where
( n 0 ,..., n N −1 ) denotes a system with n i class i queues ( i = 0,..., N − 1 ):
Let pb denote the probability of a busy time slot. Note that p e = (1 − p b ) is the
probability that no STA is transmitting, so that with n different queues of each
AC i, we have (1 − pb ) = (1 − τ~ STA ) n . Hence:
N −1
p e = p e (n 0 ,..., n N −1 ) = ∏ (1 − τ l ) n
(13)
p b = p b (n 0 ,..., n N −1 ) p e = 1 − p e
(14)
l
l =0
These relations are valid whether there are virtual collisions or not.
3.5 The probability of a successful transmission
The probability of a successful transmission for a class j attempt is
p s , j = p s , j ( n 0 ,..., n N −1 ) = n j τ j (1 − p j ) ; j = 0,..., N − 1
(15)
and further the probability of a successful transmission
N −1
p s = p s (n 0 ,..., n N −1 ) = ∑ n j τ j (1 − p j )
(16)
j =0
where p j is the collision probability for a class j attempt and is given by either
(6) for a system without virtual collision handling or (7) for a system with
virtual collision handling.
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3.6 Mean duration of a busy time slot without virtual
collisions
In order to study the mean duration on the radio channel of a transmission
attempt, σc + σs, we first introduce an ordering function, F(j) = i, i,j in {0, 1, …,
N-1} that reorders the indices access categories so that DF(j) ≤ DF(j+1). This
means that by using the F(j) instead of i as the index of the AC, all the ACF(j) are
ordered so that the average transmission length (i.e. transmission overhead
and packet length) is non-increasing as the index j is incremented.
Let us assume that in a given time slot on the radio channel, there is one or
many frames attempted to be transmitted. The mean duration of this slot can
be found by dividing the transmission attempts into N - 1 of mutually exclusive
events:
0.
Amongst the transmission attempts in a time slot on the radio
channel, there is at least one frame of ACF(0), i.e. at least one frame of
the shortest packet duration.
1.
•
Then, the duration of the time slot is then DF(0), i.e. since DF(j) ≥
DF(j+1).
•
The probability of this event is [1 − (1 − τ F ( 0 ) )
nF ( 0 )
].
Amongst the transmission attempts in a time slot on the radio
channel, there are no frames of ACF(0), but at least one frame of
ACF(1).
2.
•
The duration of the time slot is then DF(1), i.e. since DF(j) ≥ DF(j+1).
•
The probability of this event is (1 − τ F ( 0 ) )
nF ( 0 )
[1 − (1 − τ F (1) )
n F (1 )
].
Amongst the transmission attempts in a time slot on the radio
channel, there are no frames of ACF(0) and ACF(1), but at least one
frame of ACF(2), i.e. the duration of the time slot is then E(DF(2)).
•
The duration of the time slot is then DF(2), i.e. since DF(j) ≥ DF(j+1).
•
The probability of this event is
(1 − τ F ( 0 ) )
...
nF (0)
(1 − τ F (1) )
n F (1 )
[1 − (1 − τ F ( 2 ) )
nF ( 2)
].
(and so forth)
N-1. Amongst the transmission attempts in a time slot on the radio
channel, there are no frames of ACF(0)…ACF(N-2), but at least one frame
of ACF(N-1),
•
The duration of the time slot is then DF(N-1).
•
The probability of this event is
(1 − τ F ( 0 ) )
nF (0)
(1 − τ F (1) )
n F (1)
...(1 − τ F (1) )
n F ( N −1)
[1 − (1 − τ F ( N −1) )
nF ( 2)
].
0
If we define
∏ ( f ( j )) = 0 , we can write the probabilities of an event e (being
j =0
any of the events 0, 1, …, N - 1 described above) as:
e
∏ (1 − τ
F ( k −1)
)
n F ( k −1)
[1 − (1 − τ F ( e ) )
nF (e)
]
(17)
k =0
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and the mean delay of the transmission attempts is thus:
N −1
σc + σs =
j
∑ D ∏ (1 − τ
j =0
F ( j)
F ( k −1)
)
n F ( k −1)
[1 − (1 − τ F ( j ) )
nF ( j )
(18)
]
k =0
It is easy to show that the mean delay of the time slot with one or many
transmission attempts can be rewritten as:
σc + σs = D F ( 0 ) − (1 − pb ) D F ( N −1) +
N −2
∑[D
j =0
j
F ( j +1)
− DF ( j ) ]∏ (1 − τ F ( k −1) )
n F ( k −1)
(19)
k =0
N −1
where the definition pb = 1 − ∏ (1 − τ k ) nk is used.
k =0
If we set all the packet lengths as equal, we have DF(j) = DF(j+1) = D for all j ,
and the mean delay of the time slot with one or many transmission attempts
resolves to:
σc + σs = p b D
3.7 Mean duration of a busy time slot with virtual
collisions
When considering virtual collisions, let us assume that there are n STAs, each
with N-1 actively transmitting queues, i.e. when considering virtual collisions it
is natural to assume that ni = n for all i ∈ {0, 1, …, N - 1}.
On a station, there are N mutually exclusive transmission events during a time
slot on the radio channel. These N events comprise the N-1 events that a frame
of AC i i ∈ {0, 1, …, N - 1}, is transmitted, occurring at the probability τ~i , and
the event that no frame is transmitted. The latter event occurs at the
probability
N −1
N −1
i =0
j =0
1 − ∑ τ~i = ∏ (1 − τ j ) = 1 − τ~ STA
(20)
according to Section 3.2.
Let us first reuse the ordering function, F(j) = i, i,j in {0, 1, …, N - 1}, from
Section 3.2. F(j) reorders the indices of access categories so that DF(j) ≥ DF(j+1).
N −1
Note that
∑τ~
i= j
F (i )
is the probability that a station transmits a frame with a
duration DF(j) or longer, but no frames of durations DF(0), …, DF(j-1). Thus,
N −1
(1 − (1 − ∑τ~F (i ) ) n ) is the probability that at least one station transmits a packet
i= j
with a duration DF(j) or longer.
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DERIVATION ALTERNATIVE 1:
N −1
Note that
(1 − (1 − ∑τ~F (i ) ) n ) is the probability that at least one station transmits
i= j
N −1
a packet with a duration DF(j). Hence,
N −1
(1 − (1 − ∑τ~F ( i ) ) n ) − (1 − (1 −
∑τ~
i= j
i = j +1
F (i )
) n ) is
the probability that at least one station transmits a packet with a duration DF(j)
or longer minus the probability that at least one station transmits a packet with
a duration DF(j+1) or longer, under the condition that j < N - 1. In other words,
N −1
(1 − (1 − ∑τ~F ( i ) ) n ) − (1 − (1 −
i= j
N −1
∑τ~F (i ) )n ) = (1 −
i = j +1
N −1
N −1
i = j +1
i= j
∑τ~F (i ) )n − (1 − ∑τ~F (i ) )n
is the
probability that at least one station transmits a frame of duration DF(j), but no
station transmits frames of longer durations. For j = N - 1, on the other hand,
(1 − (1 − τ~F ( N −1) ) n ) is the probability that at least one station transmits of frame
duration DF(N-1) or longer, since there are no frames of longer duration anyway.
Hence, the mean delay of the busy slot can be written:
σc + σs = D F ( N −1) (1 − (1 − τ~F ( N −1) ) n ) +
N −2
∑D
j =0
F ( j)
N −1
N −1
⎤
⎡
n
n
~
~
⎢(1 − ∑ τ F (i ) ) − (1 − ∑ τ F (i ) ) ⎥
i= j
i = j +1
⎦
⎣
(21)
If we set all the packet lengths as equal, we have Di = Di+1= E(D) for all i =0,
…, N - 2, then the mean delay of the busy time slot resolves to:
σc + σs = DF ( N −1) − DF (0) (1 −
N −1
N −1
N −1
i =0
i =0
i =0
∑τ~F (i) ) n = DF ( N −1) − DF (0) ∏ (1 −τ i ) n = D(1 − ∏ (1 −τ i ) n )
With ni = n, the definition of pb is:
N −1
pb = 1 − ∏ (1 − τ i ) n
(22)
i =0
Using this definition, the mean delay of a busy time slot, given that the
transmission duration of all ACs is equal to D, can be written:
σc + σs = Dpb
(23)
DERIVATION ALTERNATIVE 2:
The delay of a packet of the AC with the shortest frame duration, ACF(0), is DF(0).
Therefore, the mean delay of a busy time slot is minimum equal to
N −1
N −1
i =0
i =0
D F ( 0 ) (1 − (1 − ∑ τ~F ( i ) ) n ) , since (1 − (1 − ∑τ~F ( i ) ) n ) is the probability that there is a
frame transmitted on the channel. We use this measure as our first
approximation. However, the other ACs, AC F(1) … AC F(N-1) have all a frame
duration that is at least equal to DF(1), and the probability that there are any
such frames on the channel is (1 − (1 −
N −1
∑τ~
i =1
[
F (i )
) n ) . We compensate for this in our
]
first approximation by adding the term D F (1) − D F ( 0 ) (1 − (1 −
N −1
∑τ~
i =1
F (i )
) n ) Thus, as
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our improved approximation, the mean delay of a busy time slot is minimum
equal to D F ( 0 ) (1 − (1 −
∑τ~F (i ) ) n ) + [DF (1) − DF (0) ](1 − (1 − ∑τ~F (i ) ) n ) .
N −1
N −1
i =0
i =1
Compensating for this in the same way leads to the new approximation
N −1
N −1
N −1
D (1 − (1 − τ~ ) n ) + [D − D ](1 − (1 − τ~ ) n ) + [D − D ](1 − (1 − τ~ ) n ) Finally, when
F ( 0)
∑
i =0
F (i )
F (1)
∑
F ( 0)
i =1
F (i )
F ( 2)
F (1)
∑
i =1
F (1)
we have compensated for the assumption that the delay of the ACF(N-2), DF(N-2)
equals the delay of the highest ACF(N-1), DF(N-1), we have finally derived an exact
expression for the mean delay of a busy time slot:
σc + σs = D F ( 0 ) (1 − (1 −
N −1
N −1
∑τ~
i =0
F (i )
[
]
N −1
) n ) + ∑ D F ( j ) − D F ( j −1) (1 − (1 − ∑ τ~F (i ) ) n )
j =1
(24)
i= j
We observe that this is another way of writing the same expression found for
“derivation alternative 1” above.
With ni = n, the definition of pb is:
N −1
pb = 1 − ∏ (1 − τ i ) n
(25)
i =0
Using this definition, the mean delay of a busy time slot can be written:
σc + σs = D F ( 0 ) p b +
∑ [D
N −1
j =1
F ( j)
]
N −1
− D F ( j −1) (1 − (1 − ∑ τ~F (i ) ) n )
(26)
i= j
As before, if we set all the packet lengths as equal, we have Di = Di+1 = D for
all i = 0,…, N - 2, then the mean delay of the busy time slot resolves to:
σc + σs = Dpb
(27)
3.8 Mean duration of a time slot with a successful packet
In the radio slots where there is only one STA performing a transmission
attempt, the transmission is successful. The duration of the radio slot is
determined by the length of the successfully transmitted frame. The probability
of the occurrence of a successfully transmitted frame is a sum of N-1 mutually
exclusive events, where the transmitted frame is of ACi , i = 0, …, N - 1.
The mean duration of a slot on the radio channel that contains a successful
packet is therefore:
N −1
σs =
∑D nτ
i =0
i
i i
(1 − pi )
(28)
Note that the probability (1 - pi) is the probability that no other STAs are
transmitting and that no higher ACi on the same STA are trying to transmit. In
other words, if there are virtual collisions in the model, this is accommodated
for by (1 - pi) in the expression above. Hence, the expression is valid whether
or not there are virtual collisions in the system. (However, the expression for pi
depends on whether there are virtual collisions or not, i.e. expressed by either
(6) or (7) above. Furthermore, when considering virtual collisions it is natural to
assume that ni = n for all i ∈ {0, 1, …, N - 1} in the expression, as pointed out
earlier).
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If we set all the packet lengths as equal, we have DF(j) = DF(j+1) = D for all j,
and the mean delay of a time slot with a successfully transmitted packet
resolves to:
N −1
σs = D
∑p
i =0
i ,s
=Dp s
(29)
3.9 Mean duration of a time slot with a frame collision
without virtual collisions
In the radio slots where more than one frame is attempted to be transmitted, a
collision will occur.
The mean delay of a collision is the mean delay of a busy slot (i.e. a slot
containing one or more transmission attempts) minus the mean delay of a slot
containing exactly one transmission attempt:
N −1
N −1
N −1
σc = D p + [ D − D ] (1 − τ ) n [1 − (1 − τ ) n ] − D n τ (1 − p )
∑ j −1) j ∏ k
∑ F ( j) j j
0 b
i
j
k
j =1
i
(30)
j =0
k = j +1
This expression is derived by subtracting equation (28) from (18) above.
If we set all the packet lengths as equal, we have Di = Dj+1 = D for all j =0, …,
N - 2, and the mean delay of a time slot with a collision resolves to:
σc = D ( p b − p s )
(31)
3.10
Mean duration of a time slot with a frame
collision with virtual collisions
When considering the virtual collisions, the mean delay of a collision is the
mean delay of a busy slot (calculated with virtual collisions) minus the mean
delay of a slot containing exactly one transmission attempt (also calculated with
virtual collisions):
σc = D p + [D − D
∑ F ( j ) F ( j −1) ](1 − (1 − ∑τ~F (i ) )n ) − ∑ DF ( j ) n jτ j (1 − p j )
F (0) b
N −1
N −1
N −1
j =1
i= j
j =0
(32)
This expression is derived by subtracting equation (28) from (21) above.
If we set all the packet lengths as equal, we have Di = Dj+1 = D for all j =0, …,
N - 2, and the mean delay of a time slot with a collision resolves to:
σc = D ( p b − p s ) .
3.11
The slot length distribution for the RTS/CTS case
both with and without virtual collisions (using the
Laplace transform)
When considering the slot length above we have assumed that all data traffic
are sent with the basic medium access mechanism, comprising a two-way
handshake of a DATA frame and the return of an ACK (acknowledgment) frame.
However, there is also an alternative access mechanism where the DATA and
ACK exchange is preceded by an exchange of an RTS (request-to-send) and a
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CTS (clear-to-send) frame. The latter mechanism results in a four-way
handshake.
In the DATA/ACK case considered above, the duration of a slot with a
successfully transmitted packet, as well as the duration of a slot with a packet
collision, are given by the packet lengths. However, with the RTS/CTS
mechanism, all collisions (at least in theory) are resolved during the RTS/CTS
exchange. Since this exchange has a fixed, predefined duration, the durations
of the collisions are not dependent on packet lengths. In this section, we will
consider this case.
N −1
Recall first that the idle/busy channel probabilities are pe = ∏ (1 − τ l ) nl and
l =0
pb = 1 − pe and that the probability of a successful transmission for a class j
N −1
attempt is p s , j = n jτ j (1 − p j ) for j = 0,..., N − 1 . Recall also that p s = ∑ p s ,i is the
j =0
probability of a successful transmission where p j is the collision probability for
a class j attempt and is given by either (6) for a system without virtual
collision handling or (7) for a system with virtual collision handling.
For the RTS/CTS case the duration of a collision is constant and equal for all
ACs, i.e. we have DiC = D C ; i = 0,..., N − 1 . With this assumption the Laplace
transform of the duration of the generic slot length Σ( s ) will be the weighted
sum:
N −1
Σ( s ) = (1 − pb )e − sT + ∑ p s ,i e
e
i =0
− sD S
+ ( pb − p s )e − sD
C
(33)
and the corresponding mean slot length is easily found by differentiation:
N −1
σ = (1 − pb )T e + ∑ ps ,i DiS + ( pb − ps ) D C .
i =0
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4 Using the Laplace transform to obtain the
access delay distribution
Except for a few earlier works (especially [13] but also [12] and [15]) where
the z-transform has been used to obtain the distribution of channel access
delay, the main focus has been on deriving expressions for the mean access
delay. However, to be able to obtain higher layer delay due to queueing, we
need to analyse the distribution of the access delay. This queueing delay will
contribute to the overall delay and thus be part of the corresponding QoS over
the WLAN. Since we propose a non-slotted model for analysing the 802.11e
EDCA mechanism, we use the Laplace transform rather than the z-transform as
the tool to describe the delay distribution. Another aspect is also that the
Laplace transform is more commonly used in queueing analysis, and hence
many results are given in terms of Laplace rather than z-transforms.
4.1 Description of the delay components
We let DiSat denote the channel access delay for a successful transmission for
AC i . In the following we shall find an expression for the Laplace transform
DiSat ( s ) = E[ s − sDi ] of the access delay.
Sat
For AC i we define the following (discrete) stochastic variables:
•
the number of retries (collisions) before obtaining a successful
C
Ni
transmission
•
CD
N i, j
the (starting) value of the backoff counter for the j ’th retry
attempt; j = 0,1,..., Li where Li is the retry limit
By assuming a fixed collision probability for each transmission attempt;
follows that
C
P ( N i = j ) = (1 − p i ) p i
j
pi ,
it
; j = 0,1,..., Li is a (truncated) geometrical
distribution, and we also define
C
P ( N i = Li + 1) = P ( drop ) = pi
Li +1
as the probability that a
frame is dropped due to expiration of the retry limit. For each retry attempt j
the backoff counter is chosen from a uniform distribution over [0, Wi , j − 1] , i.e.,
CD
P( N i, j = l ) =
1
Wi , j
; l = 0,1,..., Wi , j − 1 , where Wi , j is the corresponding window size.
Similarly, we also define the corresponding (stochastic) delay components:
•
DiS the delay for a successful transmission which is assumed to be
constant,
•
DiC, j the delay for the j ’th (unsuccessful transmission) retry which is
assumed to be constant equal to DiC ; independent of the actual retry
number,
•
DiCD
, j ,k the delay in connection with the k ’th countdown decrement cycle
for the j ’th retry also including periods where the countdown process is
frozen due to AIFS differentiation.
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Hence we have:
DiSat = DiS +
CD
N iC N i , j −1
N iC −1
∑ DiC, j + ∑
j =0
DiCD
∑
, j ,k
k =0
j =0
(35)
for a transmission attempt that is successful, i.e. that is not dropped due to the
retry limit and
Sat
i
D
Li
= ∑D
j =0
C
i, j
CD
Li N i , j −1
+∑
∑ DiCD, j ,k
(36)
j =0 k =0
for a transmission attempt that is dropped due to expiration of the retry limit.
In these expressions j = 0 in the last (double) sums represents the delay due to
post-backoff, which is part of the access delay (Figure 4).
Idle
-Idle state
-Collision
-Successful
1 − ρi
-Countdown
pi
states
1 / Wi , 0
1 − pi
ρ i (1 − p i )
Post-backoff
DS
ρ i pi
D 0C
D i,0,0
D i , 0 ,1
D i , 0 ,W
Di ,1, 0
Di ,1,1
Di ,1,W
i,0
D i , 0 ,W
−2
i ,0
−1
1 / W i ,1
1 − pi
1’st retry
DS
pi
D 1C
Di ,1,W
i ,1 − 2
i , 1 −1
1 / Wi , j
1 − pi
j’st retry
DS
Di , j ,0
pi
D Cj
Di , j ,1
Di , j ,W
Di , j ,W
i , j −2
i , j −1
1 / W i , j +1
1 / W i , Li
1 − pi
Li’st retry
DS
pi
Drop
Figure 4
Di , Li , 0
Di , Li ,1
Di , L ,W
1
i , Li
−2
Di , L ,W
i
i , Li
−1
Countdown diagram with the different states
4.2 The Laplace transform of the access delay
We now assume that the stochastic variables are all independent and take
DiS ( s ) = s − sDi as the Laplace transform for a successful transmission and
S
further DiC, j identically distributed for j = 0,1,..., Li with the Laplace transform
DiC ( s ) = e − sDi . When first entering the window countdown procedure for each
C
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retry attempt, i.e. when k = 0 , one has to perform a full AIFS deferring period
before sensing if the channel is idle or not. This is not the case for the other
states k = 1,..., Wi , j − 1 where the countdown occurs if the channel is idle. Hence
AIFS
we have DiCD
and an ordinary
, j , 0 as the sum of an AIFS deferring period Di
CD
countdown period DiCD
, . For the other countdown periods Di , j , k we assume that
they are all identically distributed with Laplace transforms DiCD ( s ) = E[ s
k = 1,..., Wi , j − 1 ; and further the Laplace transform for D
E[ s
] = DiAIFS ( s ) DiCD ( s ) ; j = 0,1,..., Li .
By conditioning on
E[e −sD
Sat
i
E[e −sD
Sat
i
] for
is
CD
i , j ,0
− sDiCD
, j,0
− sDiCD
, j ,k
C
Ni
C
Ni
CD
= j; N i ,s = l s ,
and
C
Ni = j
CD
N i ,s = l s
s = 0,1,..., j we may write:
(
) ∏ (D
CD
i
(s)
) ∏ (D
CD
i
( s)
s = 0,1,..., j ] = DiS ( s ) DiAIFS ( s ) DiC ( s )
CD
= L i + 1; N i , s = l s ,
(
s = 0,1,..., Li ] = DiAIFS ( s ) DiC ( s )
j
j
Li +1
s =0
Li
s =0
)
ls
)
ls
for j = 0,1,..., Li and
.
Finally by un-conditioning we obtain:
DiSat ( s) = E[ s − sDi ]
Sat
(
Li
)
(
AIFS
= (1 − pi ) DiS ( s )∑ pi DiAIFS ( s ) DiC ( s ) Dilevel
( s ) DiC ( s)
, j ( s ) + p i Di
j
)
j =0
Li +1
Dilevel
, Li ( s )
(37)
where we have defined the product
j
stage
Dilevel
( s ) with Distage
( s) =
, j ( s ) = ∏ Di ,l
,l
l =0
(
)
Wi , l
1 1 − DiCD ( s )
Wi ,l 1 − DiCD ( s )
(38)
The Laplace transform DiSat (s) is the key for the further analysis.
After differentiating, we find the mean channel access delay as:
⎛
⎞ R1
p
E[ DiSat ] = (1 − p L +1 )⎜⎜ DiS + i ( DiC + E[ DiAIFS ]) ⎟⎟ + i E[ DiCD ]
1 − pi
⎝
⎠ 2
i
i
(39)
and R i is given by the sum:
1
Li
Ri1 = ∑ pij (Wij − 1)
(40)
j =0
By performing the summation for the case m i ≤ Li the following explicit
expression for Ri1 is obtained:
m +1
L +1
⎛ 1 − (2 pi ) m +1
⎞ 1 − pi L +1
p
− pi
⎟−
Ri1 = Wi 0 ⎜
+ 2m i
⎜ 1− 2 p
⎟ 1− p
1 − pi
i
i
⎝
⎠
i
i
i
i
(41)
i
Higher order moments of the access delay Di
Sat
are possible to obtain.
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4.3 Specification of the countdown delay
To complete the description of the access delay we also need to specify the
countdown delay DiCD . This modelling will be a very important part of the
analysis, since the AIFSN differentiation must be included in this delay. To do
that we introduce a so-called “AIFSN freeze counter” for each window
countdown state representing the slots where a STA is blocked due to AIFSN
differentiation; i.e. these slots are only available for higher priority STAs.
Hence, for a STA from a particular AC to be able to decrease the backoff
counter by one, Ai consecutive slots have to be idle (for the higher priority
STAs), and then also the next slot must be idle. Hence, we decrease the AIFS
freeze counter by one for each idle slot (for higher priority STAs). If some of the
above mentioned slots are taken, the freeze counter is reset to its initial value
Ai and the algorithm is repeated, see Figure 5.
For an ordinary countdown period (not the first one after a retry) the delay is
the slot length T e if the channel is sensed idle with probability g i . However, if
the channel is sensed busy with the delay of a transmission attempt DiTr with
probability 1 − g i , the STA enters an AIFS deferring period with delay DiAIFS .
D iAIFS
D iCD
1− gi
gi
Te
Figure 5
D iTr
Model for the backoff window countdown period
Then after each AIFS deferring period DiAIFS , with probability g i the channel is
idle for a duration equal to the length of a slot length T e . Otherwise (i.e. with
probability 1 − g i ), the channel is busy or a duration DiTr corresponding to the
length of a transmission attempt, either successful or not. In the latter case,
the freeze counter is reset and a new AIFS freeze countdown period is entered.
The Laplace transform of the countdown delay DiCD (s) is therefore obtained as a
geometrical sum of contributions, and may be written as:
DiCD ( s ) =
g i e − sT
1 − (1 − g i ) DiTr ( s ) DiAIFS ( s )
e
(42)
To obtain the delay for the AIFSN freeze countdown period, we consider the
delay in a system consisting of Ai stages where the time spent in each stage
k = (1,..., Ai ) is either the slot length T e , with probability hi ,k , or DiTr,k , the delay
corresponding to the duration of a transmission attempt, either successful or
not, with probability 1 − hi ,k , and the freeze counter is reset and enters the first
level, see Figure 6. During the AIFSN freeze period all the lower priority STAs
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are blocked and do not contribute to the corresponding “idle” channel
probabilities hi ,k . Therefore the probabilities hi ,k have to be different form g i
above (and hence these “idle” channel probabilities will depend on the stage
k ).
We denote Di ,k the time it takes to reach level k + 1 for the first time (without
including the service time of that level). Recursively, we then obtain the
following relations between the Laplace transforms Di ,k ( s ) :
Di , k ( s ) =
hi ,k e − sT Di ,k −1 ( s )
e
1 − (1 − hi ,k ) DiTr,k ( s ) Di ,k −1 ( s )
for k = 1,..., Ai
(43)
where we define Di , 0 ( s ) = 1 .
D i ,1
1 − h i ,1
h i ,1
Te
D iTr,1
D i,2
1 − hi ,2
hi , 2
Te
D iTr, 2
D iAIFS = D i , A
i
hi , A
i
1 − hi , A
i
Te
Figure 6
D iTr, Ai
Model for the delay of an AIFSN freeze period
By taking k = Ai as a starting point and then by using (43) recursively, we
obtain the Laplace transform of the AIFSN freeze period DiAIFS (s ) as a (finite)
continued fraction on the form:
hi , A e −sT
hi , A −1e − sT ...
e
i
e
i
DiAIFS ( s) = Di , A ( s ) =
1 − (1 − hi , A ) D
i
i
1 − (1 − hi , A ) D ( s)
i
Tr
i , Ai
Tr
i , Ai −1
− sTe
i , Ai −1
( s )...
h
e
(44)
....
1 − (1 − hi , A −1 ) DiTr, A −1 ( s)...
i
i
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(If Ai = 0 for some values of i , the AIFSN freeze period is set to zero, and
hence we may write Di , Ai ( s) = 0 for those cases.)
Although (44) gives an explicit expression for the Laplace transform DiAIFS (s ) the
recursion (43) is more useful and gives an effective algorithm to calculate the
transform numerically. Moreover, it is also effective to obtain the moments of
the delay. Differentiating the relation (43) gives:
E[ Di ,l ] =
E[ Di ,l −1 ] + Gi ,l
hi ,l
where
(45)
Gi ,l = hi ,l T e + (1 − hi ,l ) E[ DiTr,l ] for l = 1,..., Ai .
(46)
Solving the recursion (45) gives:
E[ Di ,k ] =
1
(Gi,1 + hi,1Gi,2 + ... + hi,1hi,2 ....hi,k −1Gi,k ) for k = 1,..., Ai
hi ,1hi , 2 ...hi ,k
(47)
Where the expression for Gi ,l is given by (46).
Similarly for the second order moment we find:
E[ Di ,l −1 ] + Gi2,l
2
E[ Di ,l ] =
2
hi ,l
where
2
(48)
2
Gi2,l = hi ,l T e + (1 − hi ,l ) E[ DiTr,l ] − 2(hi ,l T e + (1 − hi ,l ) E[ DiTr,l ]) 2 +
(49)
2hi ,l (1 − hi ,l ) E[ Di ,l ]2 + 2hi ,l (2(hi ,l T e + (1 − hi ,l ) E[ DiTr,l ]) − T e ) E[ Di ,l ]
for l = 1,..., Ai . As above we find the following expression for the second order
moment:
E[ Di ,k ] =
2
(
1
Gi2,1 + hi ,1Gi2, 2 + ... + hi ,1hi , 2 ....hi ,k −1Gi2,k
hi ,1hi , 2 ...hi ,k
) for
(50)
k = 1,..., Ai
where Gi2,l is given by (49) (and (46) and (47) give expressions for E[ Di ,k ] ).
Finally by differentiating (42) we obtain the following expressions for the first
two moments of the (window) countdown delay:
E[ DiCD ] = T e +
1 − gi
( E[ DiTr ] + E[ Di , Ai ])
gi
(51)
where expression for E[ Di , Ai ] is given by (47) and
2
2
E[ DiCD ] = T e +
⎛ 1 − gi
2⎜⎜
⎝ gi
(
)
2
2
(1 − g i )
E[ DiTr ] + 2 E[ DiTr ]E[ Di , A ] + E[ DiTr ] + 2T e ( E[ DiTr ] + E[ Di , A ]) +
gi
i
2
⎞
⎟⎟ ( E[ DiTr ] + E[ Di , A ]) 2
⎠
i
(52)
i
where expressions for E[ Di , Ai ] and E[ Di , A 2 ] are given by (47) and (50) above
i
(and we define E[ Di , A ] = E[ Di , A ] = 0 if Ai = 0 ).
2
i
i
In the following we shall assume that all the freeze countdown delays DiTr,k are
equally distributed according to DiTrFr (which may be different from the delay
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DiTr since during the AIFSN freeze periods only the higher priority STAs are
active) and also that “freeze idle channel” probabilities hi ,k = hi are all equal for
k = 1,..., Ai . With these assumptions the summations in (47) yield the summation
of geometrical series, and we obtain:
E[ DiAIFS ] = E[ Di , Ai ] =
1 − hi
hi
Ai
Ai
( E[ DiTrFr ] +
hi
T e)
1 − hi
(53)
and hence:
E[ DiCD ] = T e +
1 − gi
gi
Ai
⎛
⎞
⎜ E[ DiTr ] + 1 − hAi ( E[ DiTrFr ] + hi T e ) ⎟
i
⎜
⎟
1 − hi
hi
⎝
⎠
(54)
(Observe that (54) also applies for the ACs where Ai = 0 since the last term
including Ai vanishes.) The second order moment is given by (52) where we
find the following explicit expression for the second order moment of the AIFSN
freeze delay period:
AIFS 2
i
E[ D
] = E[ Di , Ai
2
A
⎛ 1 − hi Ai
1 − hi i
hi
TrFr 2
e2
⎜
E
D
T
]=
(
[
]
+
)
+
2
i
A
⎜ h Ai
1 − hi
hi i
⎝ i
2
⎞
⎟ ( E[ DiTrFr ] + hi T e ) 2
⎟
1 − hi
⎠
A (1 − h ) 1 − h i
h
Te
− 2( i Ai i − Ai −i1 )( E[ DiTrFr ] + i T e )
1 − hi
(1 − hi )
hi
hi
(55)
A
To complete the description above we must specify the “media idle”
probabilities g i and hi ,k , as well as the Laplace transforms of the corresponding
duration of a transmission attempt DiTr (s) and DiTr, k ( s ) .
4.4 Delay distributions during countdown and freeze
periods
To find the Laplace transform D Tr (s) for the duration of a transmission attempt,
we limit ourselves to the case where the duration of colliding transmissions are
all equal i.e. we take DiC = D C ; i = 0,..., N − 1 . With this assumption the Laplace
transform will be a weighted sum:
N −1
p s ,l
l =0
pb
D Tr ( s ) = D Tr ( s; n 0 ,..., n N −1 ) = ∑
e − sDl + (1 −
S
p s − sD C
)e
pb
(56)
The corresponding first two moments are easily found by differentiation:
N −1
p s ,l
l =0
pb
E[ D Tr ] = D1Tr ( n 0 ,..., n N −1 ) = ∑
N −1
p s ,l
l =0
pb
E[ D Tr ] = D 2 Tr ( n 0 ,..., n N −1 ) = ∑
2
DlS + (1 −
2
ps
) D C and
pb
(57)
ps
2
)D C
pb
(58)
DlS + (1 −
The corresponding results valid during the countdown period of a class i
transmission are found by removing that particular queue, i.e. by letting
n i → n i − 1 . Hence we have for i = 0,..., N − 1 :
g i = pe (n0 ,..., ni − 1,..., n N −1 ) ,
(59)
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DiTr ( s) = DTr ( s; n0 ,..., ni − 1,..., nN −1 ) ,
(60)
E[ DiTr ] = D1Tr (n 0 ,..., ni − 1,..., n N −1 ) and
(61)
2
(62)
E[ DiTr ] = D 2 Tr ( n 0 ,..., n i − 1,..., n N −1 )
For the AIFSN differentiation we assume that the parameters are ordered so
that A0 ≤ A1 ≤ .... ≤ AN −2 ≤ AN −1 = 0 , i.e. the queues with priority l have AIFSN
parameter AN −l ; l = 1,..., N . We consider only the cases where Ai > 0 . (If Ai = 0
there will be no freeze countdown for that class.) For each “freeze counter”
value k we let lk be the largest integer such that k > AN −l , i.e. lk = max {l k > AN −l }.
l =1,..., N
Then all the queues with lower priority than lk are blocked due to AIFSN
differentiation, and hence all these low priority queues must be removed from
the calculations. We find for i = 0,..., N − 1 ; k = 1,..., Ai :
hi , k = pe (0,...,0, n N −l ,..., n N −1 ) ,
(63)
DiTr,k ( s ) = D Tr ( s;0,...,0, n N −l ,..., n N −1 ) ,
(64)
E[ DiTr,k ] = D1Tr (0,...,0, n N −l ,..., n N −1 ) and
(65)
k
k
k
2
(66)
E[ DiTr,k ] = D 2Tr (0,...,0, nN −lk ,..., nN −1 )
For the approximation where we assume that all the freeze countdown delays
DiTr,k are equally distributed according to DiTrFr and where the “freeze idle
channel” probabilities hi ,k = hi are all equal, we take the value for k = Ai in the
expression above, i.e. hi = hi , Ai , DiTrFr ( s ) = DiTr, A ( s ) , E[ DiTrFr ] = E[ DiTr, A ] and
i
TrFr 2
i
E[ D
Tr 2
i , Ai
] = E[ D
i
].
Observe that the derivation above covers both the RTS/CTS case and also the
case where all packet lengths are equal.
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5 The queueing delay
The access delay only gives part of the delay for frames competing for
transmission over the shared 802.11e radio network. For each STA there is a
transmission queue - and hence a waiting time - associated to each AC. This
waiting time has to be added to the access delay to get the performance of the
WLAN. Seen as a whole, the STAs and the radio channel may be viewed as a
single server where several queues are attached and where the EDCF regulates
the usage of the server. Approximately, however, when seen from a particular
queue, the system acts as a single server queue. The service time equals the
time duration from a frame is at the front of the queueing device until it has
completed the transmission over the radio channel also have completed the
post backoff. Thus, the service time is equal to DiSat , which is given by (35) or
(36), and with the Laplace transform DiSat ( s ) = E[ s − sDi ] by (37).
Sat
In this view, the interaction between the different STAs is incorporated in the
collision probabilities, as well as in the parameters describing the countdown
delay. Hence with this approximation and the assumption of Poisson arrival
processes, we may describe the queueing delay by an M/G/1 model. If we
denote the waiting time in the access queue as Wi Sat , then the PollaczekKhinchin formula [16] gives the Laplace transform Wi Sat ( s ) = E[ s − sWi ] as:
Sat
Wi Sat ( s) =
s(1 − ρ i )
s − λi (1 − DiSat ( s))
(67)
and further the mean waiting time E[Wi Sat ] is given as:
2
E[Wi Sat ] =
λi E[ DiSat ]
2(1 − ρ i )
(68)
Hence, the first order moment of the waiting time Wi Sat requires the second
order moment of DiSat as well as the load ρ i = λi E[ DiSat ] . After some algebra we
obtain by differentiating (37) twice:
⎛ 2
2
2
2 ⎞
p
E[ DiSat ] = 1 − p Li i +1 ⎜⎜ DiS + i ( DiC + 2 DiC E[ DiAIFS ] + E[ DiAIFS ]) ⎟⎟ +
1 − pi
⎝
⎠
(
)
⎛
⎞ p
p
2( DiC + E[ DiAIFS ]) 1 − ( Li + 1) p Li i + Li p iLi +1 ⎜⎜ DiS + i ( DiC + E[ DiAIFS ]) ⎟⎟ i +
1 − pi
⎝
⎠ 1 − pi
(
( D + E[ D
C
i
AIFS
i
(E[ D ]) ⎛⎜⎜ R3
CD
i
4
i
2
⎝
)
⎡⎛
⎤
⎞
p
]) ⎢⎜⎜ DiS + i ( DiC + E[ DiAIFS ]) ⎟⎟ Ri1 − p Li i +1Ri2 + ( DiC + E[ DiAIFS ]) Ri3 ⎥ +
1 − pi
⎠
⎣⎢⎝
⎦⎥
(
+
)
(69)
1
2 R
Ri5 − Ri3 ⎞
⎟ + E[ DiCD ] i
2 ⎟⎠
2
where the sum Ri1 is given by (41) and the other sums Ri2 , …, Ri5 are defined
by:
Li
Ri2 = ∑ (Wij − 1) ,
(70)
j =0
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Li
Ri3 = ∑ jpij (Wij − 1) ,
(71)
j =1
Li
Ri4 = ∑ pij (Wij − 1)(Wij − 2) and
(72)
j =0
Li
j −1
j =1
s =0
Ri5 = ∑ pij (Wij − 1)∑ Wis ;
(73)
where the two first moments of DiAIFS and countdown delay DiCD are given in
section 4.3. Moreover, explicit expressions for the sums Ri2 , …, Ri5 are given in
the Appendix.
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6 Concluding remarks
6.1 Summary
This document presents an analytical method to calculate the performance of
802.11 systems in general, and the differentiation characteristics of 802.11e in
particular. Like similar analytical methods, it is based on a Markov model of the
system.
The main contribution of this document is the proposal of a new method for
finding the system performance. Our proposed method is ensuring consistency
between the macro-view and micro-view of the system. Previous methods, on
the contrary, are mainly focusing on the micro-view, and inconsistency between
these two views could therefore be observed when the distribution of the time
the system spends in a Markov state could differ between two different states.
With the new method the system characteristics can be found by using the
macro-view approach, instead of using the micro-view approach. The macroview approach represents a great simplification compared to previous
methods, because the state distributions of the Markov chain do not have to be
resolved. Instead, the system characteristics are found directly by studying the
time the system spends in each of the Markov states, and the state
distributions do not need to be resolved.
Furthermore, we have also provided extensions compared to our previous
method. The first extension is to allow for different packet lengths for different
access categories, in contrast to our previous model, where we assumed that all
packets of all access categories were of the same length. The packet length
primarily influences the mean duration of the slot length on the radio channel.
The slot length was found both for a system with virtual collisions and for a
system without virtual collisions. The second extension is model for the AIFS
differentiation of 802.11e that is more physically detailed than that in our
previous work.
Since the new method accounts for an arbitrary delay of each Markov state, the
Laplace transform is more appropriate than the z-transform. While our previous
work used the z-transform to study the delay characteristics of the Markov
chain and of the system in general, this document analyses the delay with the
Laplace transform. The access delay and the queueing delay are found by the
Laplace transform.
6.2 Issues for further work
The scope of our work was to provide consistency, simplifications and model
extensions. However, we went a step further and started simulation work to
fine-tune the model in accordance with the simulation tool. As this can be quite
a time-consuming task, we started the work more than a month before the
deadline of this document. Unfortunately, after three weeks of work a disc
crash occurred. Our work was lost, and the simulation results could not be
presented in this report. However, carrying out this task would be a fruitful
issue for further work, since a fine-tuned model might be used to calculate the
performance of the WLAN systems of Telenor with higher accuracy.
Telenor R&I R 28/2008 - 27
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The model can also be used to gain further insight into WLAN technology and
performance. For example, it could prove useful to do a comparison study
between different AIFS mechanisms. Since the high priority traffic, such as
voice, typically consists of short packets, while low priority best-effort traffic
typically consists of long packets one should also study the effect that the
packet length has on the differentiation features of the system.
It might also be valuable for Telenor to use the model to explore more realistic
scenarios, including actual services, different traffic patterns, asymmetry in the
patterns etc. The performance of TCP over WLAN, including the layer
interactions between WLAN differentiation and TCP congestion control, is also
an issue that deserves further exploration.
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References
[1]
Bianchi, G., "Performance Analysis of the IEEE 802.11 Distributed
Coordination Function", IEEE J-SAC Vol. 18 N. 3, Mar. 2000, pp. 535-547.
[2]
Ziouva, E. and Antonakopoulos, T., "CSMA/CA performance under high
traffic conditions: throughput and delay analysis", Computer
Communications, vol. 25, pp. 313-321, Feb. 2002.
[3]
Xiao, Y., "Performance analysis of IEEE 802.11e EDCF under saturation
conditions", Proceedings of ICC, Paris, France, June 2004.
[4]
Malone, D.W., Duffy, K. and Leith, D.J., "Modelling the 802.11 Distributed
Coordination Function with Heterogeneous Load", Proceedings of Rawnet
2005, Riva Del Garda, Italy, April 2005.
[5]
G. Bianchi and I. Tirinello, “Remarks on IEEE 802.11 DCF Performance
Analysis”, IEEE Comm Lett., vol. 9, no. 8, Aug. 2005.
[6]
Y. Xiao, “Performance analysis of priority schemes for IEEE 802.11 and
IEEE 802.11e wireless LANs”, IEEE Trans. Wirel. Comm., vol. 9, no. 4,
July 2005.
[7]
P. E. Engelstad and O. Østerbø, “Analysis of QoS in WLAN”, Telectronikk ,
vol 1, 2005.
[8]
Engelstad, P.E., Østerbø O.N., "Non-Saturation and Saturation Analysis of
IEEE 802.11e EDCA with Starvation Prediction", Proceedings of the Eighth
ACM International Symposium on Modeling, Analysis & Simulation of
Wireless and Mobile Systems (ACM MSWiM 2005), Montreal, Canada, Oct.
10-13, 2005.
[9]
Engelstad, P.E., Østerbø O.N., "An Analytical Model of the Virtual Collision
Handler of 802.11e", Proceedings of the Eighth ACM International
Symposium on Modeling, Analysis & Simulation of Wireless and Mobile
Systems (ACM MSWiM 2005), Montreal, Canada, Oct. 10-13, 2005.
[10] Engelstad, P.E., Østerbø O.N., "Differentiation of Downlink 802.11e EDCA
Traffic in the Virtual Collision Handler", Proceedings of the 30th Annual
IEEE Conf. on Local Computer Networks (LNC ’05), WLN, Sydney,
Australia, Nov. 15-17, 2005.
[11] Engelstad, P.E., Østerbø O.N., "Delay and Throughput Analysis of IEEE
802.11e EDCA with AIFS Differentiation under Varying Traffic Loads",
Proceedings of the 30th Annual IEEE Conf. on Local Computer Networks
(LNC ’05), WLN, Sydney, Australia, Nov. 15-17, 2005.
[12] Engelstad, P.E., Østerbø O.N., "Queueing Delay Analysis of 802.11e
EDCA", Proceedings of The Third Annual Conference on Wireless On
demand Network Systems and Services (WONS 2006), Les Menuires,
France, Jan. 18-20, 2006.
[13] Engelstad, P.E., Østerbø O.N., "The Delay Distribution of IEEE 802.11e
EDCA", Proceedings of the 25th IEEE International Performance
Computing and Communications Conference (IPCCC'06), Phoenix,
Arizona, April 10 - 12, 2006.
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WLAN QoS Analysis - A New Modelling Approach for
802.11e
[14] Engelstad, P.E., Østerbø O.N., "Closed-form Solution of the Bianchi Model
for 802.11 DCF and IEEE 802.11e EDCA", Proceedings of the 15th IST
Mobile & Wireless Communication Summit (ISTsummit’06), Myconos,
Greece (Hellas), June 4-8, 2006.
[15] Engelstad, P.E., Østerbø O.N., "Analysis of the Total Delay of IEEE
802.11e EDCA and 802.11 DCF", Proceedings of IEEE International
Conference on Communication (ICC'2006), Istanbul, June 11-15, 2006.
[16] Kleinrock, L., “Queueing Systems,Vol. 1”, John Wiley, 1975.
[17] Abate J.,Whitt W., Numerical Inversion of Laplace Transforms of
Probability Distributions, ORSA Journal on Computing, Vol. 7, No.1, Winter
1995, pp 38-43.
[18] Abate, J., Valko P. P.,Multi-precision Laplace transform inversion, Int. J.
Numer. Meth. Engng. 2004; 60, pp 979–993.
[19] Wong R. 1989. Asymptotic Approximations of Integrals, Academic Press,
Inc. San Diego, CA.
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7 Appendix
7.1 Sums
By performing the summation for the case mi ≤ Li we obtain the following
explicit expressions for the sums Ri2 ,…, Ri5 :
(
)
Ri2 = Wi 0 2 m ( 2 + Li − mi ) − 1 − ( Li + 1)
i
(
(74)
)
⎛ 2 p 1 − ( mi + 1)(2 pi ) m + mi (2 pi ) m +1
+
Ri3 = Wi 0 ⎜ i
⎜
(1 − 2 pi )2
⎝
2
mi
(
pi (mi + 1) pi
i
mi
− mi pi
mi +1
i
− (Li + 1) pi + Li pi
Li
Li +1
(1 − pi )2
)⎞⎟ + p (1 − (L + 1) p
⎟
⎠
i
i
Li
i
(1 − pi )2
+ Li pi
Li +1
)
mi +1
L +1
mi +1
− pi i ⎞⎟
2 ⎛ 1 − ( 4 pi )
mi pi
Ri4 = Wi 0 ⎜
+
4
⎜ 1− 4p
⎟−
1 − pi
i
⎝
⎠
(76)
m +1
L +1
L +1
⎛ 1 − (2 pi ) mi +1
1 − pi i
p i − pi i ⎞⎟
3Wi 0 ⎜⎜
+ 2 mi i
+
2
⎟
1 − pi
1 − pi
⎝ 1 − 2 pi
⎠
mi +1
⎛ 1 − (2 pi ) mi +1 1 − pi mi +1 ⎞
1 − (2 pi ) mi +1 ⎞
2 ⎛ 1 − ( 4 pi )
⎟+
⎟ − Wi 0 ⎜
Ri5 = Wi 0 ⎜⎜
−
−
⎟
⎜ 1− 2p
1 − 2 pi ⎠
1 − pi ⎟⎠
i
⎝ 1 − 4 pi
⎝
(
) (
m
L
L +1
m +1
L +1
⎛
p p i − (Li − mi + 1) pi i + ( Li − mi ) pi i
pi i − pi i ⎞⎟
mi
Wi 0 2 Wi 0 − 1 ⎜⎜ 2 mi i i
+
−
2
1
⎟
1 − pi
(1 − pi )2
⎝
⎠
(
)
mi
(75)
)
(77)
The sum
S kN =
N
∑ j ( j − 1)...( j − k ) =
j = k +1
( N + 1) N ( N − 1)...( N − k )
for N ≥ k + 1
( k + 2)
Proof by induction:
Ok for N = k + 1 .
Assume hypothesis ok for N ≥ k + 1 then
S kN +1 = S kN + ( N + 1) N ( N − 1)...( N + 1 − k ) =
( N + 1) N ( N − 1)...( N − k )
+ ( N + 1) N ( N − 1)...( N + 1 − k )
(k + 2)
⎛ N − k ⎞ ( N + 2)( N + 1) N ( N − 1)...( N + 1 − k )
= ( N + 1) N ( N − 1)...( N + 1 − k )⎜
+ 1⎟ =
(k + 2)
⎝ k +2
⎠
i.e. hypothesis ok for N + 1 .
7.2 Elements of Laplace inversion
Suppose we are working with a stochastic variable W where the Laplace
~
transform W ( s) = E[e − sW ] is known. Then the probability density function (PDF)
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and distribution function (DF) may be written by the inversion integral as
follows:
w(t ) =
~
1
e stW ( s )ds and
2πi ∫γ
W (t ) = P{W ≤ t} =
(78)
1 e st ~
W ( s )ds
2πi ∫γ s
(79)
{
}
where the integration line is parallel with the imaginary axis γ = s s =a + iy and
where a > 0 is a constant and y ∈ (−∞, ∞) . Numerical inversion of Laplace transforms is generally difficult, but several methods are available in the literature
[17], [18].
It is possible to obtain an accurate approximation of the inversion of Laplace
transforms by applying Large Deviation (LD) theory. Normally this is done by
writing the inversion integral above on the following form:
w(t ) =
1
e f ( s ,t ) ds
2πi ∫γ
W (t ) = 1 +
(80)
1 e f ( s ,t )
ds where
2πi ∫γ s
~
f ( s, t ) = st + log(W ( s))
(81)
(82)
We have changed the integration in the last integral to a line to the left of the
origin, i.e. γ = s s =a + iy where a < 0 . (By doing this we have to pick up the
{
}
residue for s = 0 ).
Approximations of the inversion integrals may be found by considering asymptotic expansions for large t by applying the method of Steepest Descent or
Saddle Point method. (See for instance [19] for a thorough description of the
method.) This method will work well for the integral (80) but will not give
accurate approximation for (81) due to the pole at s = 0 .
By applying the ordinary saddle point method for the integral (80) one finds the
following approximation
w LD (t ) =
1
2πh(r (t ))
e f ( r ( t ),t )
(83)
where r (t ) is the saddle point which is the value when f ( s, t ) attains its
minimum as function of s , i.e. is the solution of the equation
~
∂f ( s, t ) W ′( s )
= ~
+ t = 0 and further
∂s
W (s)
2
~
~
∂ 2 f ( s, t ) W ′′( s) ⎛ W ′( s) ⎞
= ~
−⎜ ~ ⎟ .
h( s ) =
∂s 2
W ( s) ⎜⎝ W ( s) ⎟⎠
(84)
(85)
It turns out that the “ordinary” Saddle Point method fails to approximate the
integral (81) when the root r (t ) is close to the pole at s = 0 . It is, however,
possible to extract the pole and obtain an Uniform Asymptotic Approximation
(UAA) that is uniform with respect to t and also yields in the area where r (t ) is
close to zero. The corresponding asymptotic expansion is given by:
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W UAA (t ) = 1 −
⎧⎪
sign (r (t )) ⎫⎪
1
1
erfc (− sign (r (t )) − 2 f (r (t ), t ) ) + e f ( r (t ),t ) ⎨
−
⎬
2
⎪⎩ r (t ) 2πh(r (t )) 2 − πf (r (t ), t ) ⎪⎭
(86)
The main effort (or complexity) by applying the Saddle Point methods is to
locate the saddle point r (t ) , and the equation has to be solved for each value of
t.
The accuracy of the UAA method is impressive and gives results very close to
the exact convolutions for a very broad range of parameters also including
cases where the asymptotic is not exactly fulfilled.
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WLAN QoS Analysis - A New Modelling Approach for
802.11e
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802.11e
Significance for Telenor
The work carried out in the WLAN QoS Analysis project and the presented
model that was developed in this project give a profound understanding of the
detailed working of 802.11 WLAN and of the QoS features of 802.11e. This
insight will be useful in future research projects, but also to answer questions
regarding Telenor’s daily WLAN operations.
Furthermore, by using the model we are able to predict the performance of
WLAN without having to do time consuming simulations. While simulations
normally are done in days and weeks, calculations from the model are done in
seconds. Therefore, the model will hopefully be useful for future WLAN projects.
Finally, the model contributes to enhancing Telenor R&I’s visibility and impact,
which is part of Telenor R&I’s key performance indicators. As an example, our
previous model that has now been improved as described in this document, has
been cited by more than 86 different other works, i.e. in journal articles,
conference papers, standardization documents, theses, etc. This means that
our model is highly visible abroad and that our work is appreciated by persons
and entities outside Telenor.
Telenor R&I R 28/2008 - 35