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Multiple Access Protocols for Mobile Communications: GPRS, UMTS and Beyond
Alex Brand, Hamid Aghvami
Copyright

2002 John Wiley & Sons Ltd
ISBNs: 0-471-49877-7 (Hardback); 0-470-84622-4 (Electronic)
5
MODELS FOR THE PHYSICAL
LAYER AND FOR USER TRAFFIC
GENERATION
In general, sophisticated communication systems are designed according to the concept
of layering, often adhering to the OSI layering approach. The designer of a certain layer
can then consider other layers as black boxes, which provide certain services defined in
terms of functional relations between their respective inputs and outputs. She does not
have to worry about the details of implementation of the next lower layer, but must only
be aware of the services it provides. Conversely, she has to be sure that the layer being
designed will cater for the services required by the next higher layer.
As we want to investigate the performance of multiple access protocols, we are inter-
ested in the MAC sub-layer, which will make use of the services provided by the physical
layer. We must therefore assess the performance of the latter, or indeed, establish the rele-
vant functional relations. Several options on how to model physical layer performance
will be discussed and the models chosen for the performance assessment of a few multiple
access protocols presented in Chapters 7 to 9 outlined in the following.
Regarding the relationship between the MAC sub-layer and higher (sub-)layers, of major
concern here is the traffic coming from the latter, which has to be handled by the MAC
making the best possible use of the available physical link(s). Where exactly (in terms of
layers) this traffic is generated depends very much on the service considered, but is not
of interest here. What is relevant is only the quantity and the temporal characteristics of
this traffic as seen by the MAC sub-layer (and hence as delivered by the RLC sub-layer).
For this purpose, traffic models are defined, which will then be used for the performance
assessment of the MAC solutions investigated in later chapters. These include a model


for packet-voice as an example of real-time packet data traffic, and models for Web
browsing and email transfer as examples for non-real-time traffic. Furthermore, some
aspects relating to video traffic are discussed.
5.1 How to Account for the Physical Layer?
5.1.1 What to Account For and How?
To carry traffic across the air interface, the MAC layer will use the services provided by
the physical layer. The fundamental question that arises is: under what conditions can
222
5 MODELS FOR THE PHYSICAL LAYER AND FOR USER TRAFFIC GENERATION
the physical layer be expected to deliver this traffic successfully to the peer MAC entity?
This may not only depend on the input from the MAC layer (e.g. the number of bursts
or packets to be carried at any one time), but also on conditions not directly under the
influence of the MAC layer, e.g. the current state of the radio channel. As far as the
latter is concerned, one would expect the physical layer designer to include means that
allow provision of the required degree of reliability, such as appropriate FEC protection
in combination with interleaving to combat the effects of fast fading.
Since increased reliability comes at the cost of reduced capacity, certain reliability
problems will almost always prevail at the physical layer of a mobile communications
system due to the typically adverse propagation conditions experienced on radio channels.
It is possible to include these effects in the functional relations to be established by
appropriate statistical modelling. However, for MAC layer performance optimisation of
primary concern are the direct interdependencies between MAC and physical layer, and
the main focus will be on modelling these. All the same, one has to be aware that the
physical layer may fail to deliver information over the air irrespective of the behaviour
of the MAC layer.
The physical layer model established in the following will predominantly be used
for investigating MD PRMA performance on a hybrid CDMA/TDMA air interface. The
fundamental functional relation of interest in this case is the error performance of the
physical layer as a function of the number of bursts or packets carried in a time-slot. The
error performance may also depend on particular spreading codes selected by the terminals

(in particular, code collisions will affect error performance) and on the distribution of the
power levels, at which the different bursts or packets are received by the base station.
The latter is actually something that can only partially be controlled by the system, and
is significantly affected by propagation characteristics.
There are two fundamental approaches to establish these functional relations:
• through use of appropriate mathematical approximations of the error performance; or
• through detailed assessment of physical layer performance, often via simulation.
5.1.2 Using Approximations for Error Performance Assessment
It would be nice if the physical layer performance could be approximated with reasonable
accuracy by a set of formulae readily available from literature, preferably parameterised
in a manner that permits different operating conditions to be investigated easily. MAC
layer investigations could then be carried out without having to undertake detailed phys-
ical layer investigations first. Such approximations of the error performance for CDMA
systems, such as the well known standard Gaussian approximation (SGA), have indeed
been discussed widely in the literature and will be considered in the next section.
The simplest way to detect direct-sequence CDMA signals is to use a simple correlation
receiver or matched filter, which detects a single path of the wanted signal. MAI is treated
as noise. Multipath propagation does normally not result in undesired signal distortion,
since the correlation receiver can lock onto and resolve paths individually
1
. Unfortunately,
1
A path can be resolved by a Rake receiver, if its temporal separation from other paths is at least equal to
the chip duration T
c
.
5.1 HOW TO ACCOUNT FOR THE PHYSICAL LAYER?
223
replicas of the signal to be detected arriving through other paths will manifest themselves
as self-interference, which will affect the error performance. To improve performance,

Rake receivers are commonly included in CDMA system design concepts (e.g. Refer-
ence [12]). In a Rake configuration, several correlation receivers lock each onto a different
path, and the individual signals are combined, which allows a path diversity gain to be
achieved. Whether simple matched filters or Rake receivers are implemented, MAI is the
main limiting factor to the error performance. Thus, to reduce errors further, the level of
MAI has to be reduced, for instance through methods such as interference cancellation or
joint detection (see below), or through the use of antenna arrays at the base station.
It is possible to invest arbitrary effort in error performance approximation, to account
for the effect of multipath propagation, use of Rake receivers, and even antenna
arrays [242,243]. However, the potential benefits of using such approximations in
the context considered here do not justify the added complexity involved. Instead,
when assessing the impact of MAI on MD PRMA performance, a standard Gaussian
approximation for simple correlation receivers will be used, multipath propagation will be
ignored, and it will be assumed that power fluctuations are compensated by power control.
However, the impact of power control errors on error performance will be studied. This
‘standard Gaussian model’, which is described in Section 5.2, can be used on its own
to establish the error performance of the physical layer, while ignoring code-assignment
matters. Alternatively, as outlined below, it can be combined with a ‘code-time-slot model’
to account for code assignment and potential code collisions.
5.1.3 Modelling the UTRA TD/CDMA Physical Layer
A very important issue in CDMA systems is power control. In order to avoid capacity
degradation due to the near-far effect, the power radiated by the different users on the
uplink should be controlled tightly, such that each user’s signal is received by the base
station at a power level which correspond as closely as possible to a certain reference
power level. Unfortunately, due to the fast power fluctuations caused by fast fading, it
is impossible to control the power perfectly. Tight power control is particularly difficult
to achieve in a hybrid CDMA/TDMA system, since closed-loop power control cannot
be fast enough. In fact, Baier argues that the introduction of a TDMA component in a
CDMA system with single-user detection (whether this be a single correlation receiver or
a Rake receiver) will be virtually impossible for exactly this reason [109] and that multi-

user detection should be used instead. In his research group, physical layer solutions were
developed for hybrid CDMA/TDMA systems which incorporate joint detection (JD) of all
signals transmitted in a time-slot, such that the near-far problem can be resolved without
requiring tight power control (e.g. Reference [13]). Such an approach was also adopted
for the UTRA TD/CDMA mode.
One could argue that with the fast and accurate open-loop power control possible
in TDD configurations with alternating up- and downlink slots (see Section 6.3), joint
or multi-user detection would not be required. However, TDD with alternating slots is
limited to small cells, and multi-user detection schemes would still be required in all other
cell types. Furthermore, according to Reference [86], the accuracy of open-loop power
control is in general not very good due to terminal hardware limitations
2
.
2
For completeness, it is reported that a Japanese company proposed a wideband CDMA system with a TDMA
element and without mandatory multi-user detection in the early phases of the UTRA standardisation, but also
224
5 MODELS FOR THE PHYSICAL LAYER AND FOR USER TRAFFIC GENERATION
Unfortunately, convenient approximations of the error performance of TD/CDMA with
JD do not exist yet, since the detection algorithms used are rather complex. An alternative
would be to establish the physical layer performance through simulation. The snapshots
produced with such simulations, however, are only valid for very specific scenarios,
thus not allowing for easy generalisation. It is possible to overcome this limitation, but
at the expense of complex interfacing between physical layer simulations and higher
layer simulations. These interfacing issues have actually resulted in a string of dedicated
publications (e.g. references [227–229]). To adopt such approaches, it is necessary to
process the results obtained during physical layer simulations in a particular manner.
In TD/CDMA, to perform joint detection, the receiver must be able to estimate reliably
the channels of all users transmitting in the same time-slot. This requires inclusion of
training sequences in the burst format, and limits the number of users that can simultane-

ously access a time-slot and the number of spreading codes available in this slot (note that
a single user may transmit on more than one code in a particular time-slot). As a general
rule, the fewer the number of users, the more codes are available, but the relationship is
not straightforward [90]. Here it is assumed that every user is allocated only one code.
This results in a fixed number of codes per time-slot, and thus in a rectangular grid of
code-time-slots representing a TDMA frame, as for instance shown in Figure 3.13, each
being able to carry a burst or a packet.
With the above considerations on the problems of assessing physical layer performance
in mind, the simplest possible model is adopted for TD/CDMA in Section 5.3, namely that
of the perfect-collision channel. This is probably also the most commonly used approach
for MAC investigations. In this model, if only one user accesses a particular code-time-
slot, its burst is assumed to be transmitted successfully, but if more than one user accesses
that slot, a collision occurs and all bursts involved in this collision are assumed to be
corrupted. This model is very basic; in particular it does not account for MAI. However,
since JD will at least partially eliminate the dominant source of MAI, namely intracell
interference, it is a reasonable approximation, provided that:
• strong FEC coding is used; and
• the number of code-slots provided per time-slot and the reuse factor are chosen such
that the intercell interference level is tolerable even in case of fully loaded cells (i.e.
the system is blocking limited).
The major drawback of this ‘code-time-slot model’ is that individual code-slots in a
time-slot are considered to be mutually orthogonal, and it is assumed that even excessive
intracell interference created by contending users in a particular time-slot will not affect
users holding a reservation in the same slot. Unfortunately, JD cannot completely remove
intracell interference, particularly not that of contending users. To model at least quali-
tatively the impact of non-orthogonality on the protocol operation, the models presented
in Sections 5.2 and 5.3 will be combined in Section 5.4. Collisions on individual code-
time-slots will again result in the erasure of the bursts involved in the collision, but on
top of that, the error performance of all bursts in a particular time-slot will depend on the
total level of interference in that time-slot.

pointed at the power control problem. Presumably, also the hybrid CDMA/TDMA candidate systems submitted
in early phases of the GSM standardisation process (see Reference [3]) did not mandate multi-user detection.
5.2 ACCOUNTING FOR MAI GENERATED BY RANDOM CODES
225
5.1.4 On Capture and Required Accuracy of Physical Layer
Modelling
The possibility of the receiver capturing one of several colliding bursts is ignored in
the following. The qualitative behaviour of the protocols to be investigated would not
be affected by capture. Quantitatively, protocol performance would improve without any
particular precautions being required other than modifying update values in the case
of backlog-based access control as outlined in Reference [131] (see also Sections 3.5
and 4.11).
On a more general note, it must be repeated that it would be beyond the scope of
this book, which is concerned with multiple access protocols, to assess the physical layer
performance in various environments in great detail. As long as it is made certain that
the suggested protocol can cope conceptually with all possible effects affecting physical
layer performance (whether for the better such as capture, or for the worse such as some
residual errors), it is justifiable to focus on those effects that have a fundamental impact on
protocol operation. These effects are collisions of bursts or packets and, where a CDMA
component is considered, the impact of multiple access interference. Correspondingly,
while protocol multiplexing efficiency and access delay performance will be investigated,
with respect to the limitations of the physical layer models used, it would be unwise to
quote any spectral efficiency figures.
5.2 Accounting for MAI Generated by Random Codes
5.2.1 On Gaussian Approximations for Error Performance
Assessment
In CDMA systems, MAI is usually generated by a large number of users. Applying the
Central Limit Theorem (CLT), one would therefore expect its distribution to be Gaussian.
If this were the case, and if the variance were known, the approximate bit error rate
P

e
could be calculated using the error function. Pursley proposed to do exactly this in
1977. Furthermore, expanding on a paper from 1976 [244], he provided expressions for
the variance in direct-sequence CDMA (DS/CDMA) systems, with random coding and
BPSK modulation, as a function of the spreading factor or processing gain X, the number
of simultaneous users K and additive white Gaussian noise [245]. This approximation is
now commonly referred to as the standard Gaussian approximation (SGA) [246].
The CLT in its strictest form states that the sum of a sequence of n zero-mean inde-
pendent and identically distributed (i.i.d.) random variables with finite variance σ
2
will
converge to a Gaussian random variable as n grows large. Using random spreading
sequences and assuming perfect power control, such that the power level received from
each mobile user at the base station is the same, the CLT in its strictest form can indeed
be applied, since each user looks statistically the same to the base station. We would
therefore expect the SGA to deliver accurate results when the number of simultaneous
users K is large, but might have to expect accuracy problems for small K. Indeed,
Morrow and Lehnert found that for small K, when the phases and delays of the inter-
fering signals are random, the MAI cannot be accurately modelled as a Gaussian random
variable. Thus, SGA delivers only reliable P
e
values when K is large [246]. Particularly
226
5 MODELS FOR THE PHYSICAL LAYER AND FOR USER TRAFFIC GENERATION
inaccurate (overly optimistic) results are obtained for large values of the spreading factor
X combined with small values of K.
The shortcomings of SGA can be overcome by an improved Gaussian approxima-
tion (IGA) proposed by Morrow and Lehnert in Reference [246]. In Reference [137], the
same authors demonstrate how to reduce the computational complexity of IGA. However,
even this simplified approach entails complex calculations to determine P

e
.IfafixedX
and perfect power control are considered, and only the MAI to the wanted user created by
K − 1 other users needs to be accounted for to assess P
e
,suchP
e
(K) values can be calcu-
lated once for the desired range of K and then simply looked up when required. Courtesy
of the authors of Reference [247], Perle and Rechberger, such results were available to
us for the investigations reported in References [136] and [31]. In Reference [247], Perle
and Rechberger propose an approach to extend IGA for unequal power levels, which
requires establishing first the power level distribution, and then involves similar calcu-
lations as in Reference [137]. Again, this approach is quite complex, and would require
separate calculations for every scenario that results in a different power level distribution.
Furthermore, the approach would need to be further extended to account for interference
created by users dwelling outside the test cell considered.
The interested reader is referred to a fairly recent letter by Morrow, which provides a
good summary on the issues discussed above and the relevant results reported in Refer-
ence [248]. As a matter of fact, we would have welcomed this letter to appear some years
earlier. The letter reports successful endeavours by Holtzmann to simplify IGA greatly for
equal power reception without compromising too much on accuracy, and provides further
simplifications to this approach. According to the letter, this simplified IGA (SIGA) can
also be extended easily to the unequal power case. Such an extended SIGA could have
been attractive for our investigations. However, it is perfectly justifiable to use SGA for
our purposes for the following reasons.
• We are mostly interested in small X, in which case the difference between SGA and
IGA is not so large (see Reference [246]).
• To maximise the normalised throughput, quite strong FEC coding is applied. In this
case, the values of K for which SGA underestimates P

e
will result in a packet success
rate of 1 anyway, whether SGA or IGA is used [31].
• SGA can easily be extended to unequal power reception. While this violates the i.i.d.
requirement of the strongest form of the CLT, a weaker form can be invoked, which
requires that at least the variances of the individual contributors should be of the
same order
3
. Unequal power reception of intracell interferers (which should dominate
intercell interferers) will be due to power control errors. If these errors are small,
the variances should indeed be of the same order. If they become too large, problems
with limited accuracy of SGA will be overshadowed anyway by severe degradation of
system performance to the point where the system being considered becomes useless.
• Finally, physical layer models in investigations dedicated to the MAC layer will
always be subject to some simplifications (e.g. we are ignoring multipath fading).
3
Actually, according to Reference [249], the weakest form of the CLT requires that every single contributor
shall only make an insignificant contribution to the sum of contributions.
5.2 ACCOUNTING FOR MAI GENERATED BY RANDOM CODES
227
Small accuracy problems within the framework of the simplified model are negligible
compared to the inaccuracies caused by these simplifications.
In the following, the physical layer performance is assessed in terms of the bit error
rate or BER and of the packet success probabilities. SGA is used to evaluate the BER.
5.2.2 The Standard Gaussian Approximation
The standard Gaussian approximation for DS/CDMA systems is derived in detail in Refer-
ence [245] and makes use of results from Reference [244] to provide a value for the MAI
variance for the case of random coding and equal power reception. Extending SGA
to unequal power reception is fairly straightforward. Here, the system model consid-
ered is briefly introduced and the SGA expression used for performance assessment

is reported. A detailed derivation of SGA in the unequal power case was provided in
Reference [31].
The transmitted signal of user k, using BPSK modulation, is written as
s
k
(t) =

2P
k
a
k
(t)b
k
(t) cos(ω
c
t + θ
k
), (5.1)
with spreading sequence (also called direct or signature sequence) a
k
(t), data sequence
b
k
(t), carrier frequency ω
c
, transmitted power level P
k
and carrier phase θ
k
. The data

sequence b
k
(t) is made up of positive or negative rectangular pulses of unit amplitude
and bit duration T
b
. The sequence a
k
(t), on the other hand, is also made up of rectan-
gular pulses of unit amplitude, but now with duration T
c
, the so-called chip duration.
Random direct sequences are used, i.e. Pr{a
(.)
j
=+1}=Pr{a
(.)
j
=−1}=0.5, where a
(.)
j
is an arbitrary chip j of the direct sequence. This is why the section title refers to random
codes. The length of the sequence is equal to the spreading factor or processing gain
4
X = T
b
/T
c
, with X ∈ℵ. In other words, the sequence is randomly chosen to spread the
first bit, but repeated for subsequent bits.
In a system in which K simultaneous mobile users transmit according to Equation (5.1),

the total signal received at the base station can be written as
r(t) = n(t) +
K

k=1

2
P
k
α
k
a
k
(t − τ
k
)b
k
(t − τ
k
) cos(ω
c
t + ϕ
k
)(5.2)
where τ
k
is the propagation delay, α
k
is the propagation attenuation experienced, such
that the received power level amounts to P

k

k
,andn(t) is the additive white Gaussian
noise (AWGN). Here, ϕ
k
= θ
k
− ω
c
τ
k
+ ψ
k
with 0 ≤ θ
k
< 2π and ψ
k
the phase-shift due
to fading. If user i is to be detected, we can assume ϕ
i
= τ
i
= 0, as only relative delays
and phase angles need to be considered. On the other hand, on the uplink of a mobile
communication system it is rather difficult to achieve ϕ
k
= τ
k
= 0fork = 1 ...K, k = i.

Instead, carrier phases ϕ
k
are assumed to be uniformly distributed in the interval [0, 2π),
and chip delays τ
k
in the interval [0,T
b
)fork = i.
4
If FEC coding is used, the redundancy introduced through coding may be considered as part of the processing
gain, in which case spreading factor X and processing gain are not equal.
228
5 MODELS FOR THE PHYSICAL LAYER AND FOR USER TRAFFIC GENERATION
In the following, it is assumed that the dominant interference contribution is MAI, and
AWGN is ignored. The average BER or probability of bit error P
e
can then be calculated
using
P
e
≈ Q(SNR), (5.3)
with Q(x) =
1




x
e
−u

2
2
du, (5.4)
and the short-term average signal-to-noise-ratio
SNR =





3X ·
P
i
α
i

K
k=1
k=i
P
k
α
k
(5.5)
For equal power reception, that is if P
k

k
= P for k = 1, 2,...,K, Equation (5.5)
reduces to the well known

SNR =

3X
K − 1
.(5.6)
Similar expressions reported in Reference [246] and summarised in Reference [248]
can be obtained, if either phases, or chips, or both together are aligned. These cases are
not relevant for the uplink considered here, but may be of interest for the downlink of
a mobile communications system and, with limitations, for the uplink of synchronous
CDMA systems such as the synchronous UTRA TDD mode envisaged to be introduced
as part of further UMTS developments.
5.2.3 Deriving Packet Success Probabilities
Transmitting a packet of length L bits over a memoryless binary symmetric communica-
tion channel with average probability of data bit success Q
e
= 1 − P
e
yields a probability
of packet success Q
pe
of
Q
pe
=
e

i=0

L
i


(
1 − Q
e
)
i
(
Q
e
)
L−i
,(5.7)
when a block code is employed which can correct up to e errors. At this point, a problem
ignored until now pertaining to both SGA and IGA needs to be addressed. Normally, phys-
ical layer design parameters will be chosen such that, at least for mobiles at moderate
speed, the channel is quasi static during the transmission of a burst or packet
5
. Because
of this, the assumption underlying these approximations, that delays and phases of the
interfering users are random, is violated. While they may be randomly selected at the
start of a packet, they will essentially remain constant over its duration. This in turn
will introduce dependencies between bits in errors or, put differently, the channel will
have memory. If one bit is in error, there is an increased likelihood that the next bit
5
In the physical layer context, a packet is equivalent to a burst, i.e. a data unit transmitted in a single
(code-)time-slot.
5.2 ACCOUNTING FOR MAI GENERATED BY RANDOM CODES
229
will also be in error, which has also implications on the calculation of the success
rate of packets that are protected by error coding. Use of SGA or IGA to establish P

e
and subsequent use of Equation (5.7) to establish Q
pe
could therefore cause inaccurate
results.
Such issues are addressed in detail in Reference [246], where a method for calcu-
lating packet success probabilities using IGA, which correctly accounts for bit-to-bit error
dependence, is introduced. It is shown that, in systems appling error correction coding,
the techniques that ignore error dependencies are optimistic for a lightly loaded channel
and pessimistic for a heavily loaded channel. In other words, if error dependencies were
correctly accounted for, the slope of Q
pe
[K] depicted in Figure 5.2 (see next subsec-
tion) would be flatter. For two reasons, these issues are ignored in the following and
Equation (5.7) is resorted to for the calculation of Q
pe
: application of interleaving and
coding over several bursts should at least partially eliminate error dependencies
6
.Further-
more, the impact of flatter slopes on system performance will be investigated anyway in
the context of power control errors.
5.2.4 Importance of FEC Coding in CDMA
According to Lee, coding is always beneficial and sometimes crucial in CDMA applica-
tions [6]. Results reported in References [137,247,250,251] confirm this statement
7
.
In digital cellular communication systems currently operational, a combination of convo-
lutional coding and block coding is often used. Typically, a Viterbi decoder carries the
main burden of error correction at the receiving end, thus convolutional coding is applied

for error protection. Block coding is then applied in the shape of cyclic redundancy checks,
i.e. some parity bits are added, which allow in GSM for instance detection of whether a
voice frame is bad or good.
For mathematical convenience, the focus here is on block FEC coding only. As in
References [137] and [247], the Gilbert–Varshamov-Bound is used to account for the
redundancy required to correct a certain number of errors and assess the code-rate r
c
which maximises the normalised throughput S.Oncer
c
is determined, a BCH code [252]
with appropriate parameters is selected.
The bandwidth-normalised throughput S is defined as
S =
r
c
· K
pe max
X
,(5.8)
with
K
pe max
= max
K=1,2,...

K


P
pe

[
K
]


P
pe

max

,(5.9)
where P
pe
[K] = 1 − Q
pe
[K]isthepacket error probability,andK
pe max
is the number of
users supported at a certain tolerated maximum packet error probability (P
pe
)
max
. Equal
power reception is assumed to determine P
pe
[K] using Equation (5.7).
6
For a detailed discussion on coding, channel memory, and interleaving, see e.g. Chapter 4 in Reference [3].
7
Prasad identified certain scenarios in which it makes more sense to increase the processing gain through

increase of the spreading factor X rather than to introduce FEC redundancy [26, p. 128]. However, in most
scenarios considered, his investigations also underlined the benefits of FEC coding.
230
5 MODELS FOR THE PHYSICAL LAYER AND FOR USER TRAFFIC GENERATION
The Gilbert–Varshamov bound is employed to account for redundancy. According to
Reference [253], for any integer d and L with 1 ≤ d ≤ L/2, there is a binary (L, B)
linear code with a minimum Hamming distance d
min
≥ d, such that
r
c
≥ 1 − h

d − 1
L

,(5.10)
where h(p) =−p log(p) − (1 − p) log(1 − p) is the binary entropy function, L is the
size of the packet and r
c
the code-rate.
Such a code will correct at least
e =
d
min
− 1
2
(5.11)
errors and have B = r
c

· L message bits.
Figure 5.1 shows S as a function of r
c
for X = 7andX = 63, and values of one per
cent and one per thousand for (P
pe
)
max
, respectively. For reasons outlined in Chapter 7,
the number of message bits B is kept at 224. The steplike behaviour of the effective
throughput can be explained by the fact that K
pe max
can only be increased by steps of
one user at a time, hence decreasing the code-rate will reduce S in spite of increasing e,
as long as no additional user can be supported. The reason why the throughput is higher
for X = 7 is because self-interference is ignored. Subtracting the desired user, e.g. in the
denominator of Equation (5.6), increases the SNR the more, the lower X.
Irrespective of X, the optimal code-rate is in the range of 0.4 to 0.6. BCH codes
are efficient block codes and therefore often used. A possible BCH code with r
c
=
0.45 which supports around 224 message bits is one with L = 511 bits, B = 229 bits
Normalised throughput
S
0.9 0.7 0.5 0.3 0.1
0.15
0.2
0.25
0.3
0.35

0.4
0.45
0.5
0.55
Code-rate
r
c
B
= 224 bits for Gilbert-varshamov bound
(
L
,
B
,
e
) = (511,229,38) for BCH code
X
= 7, (
P
pe
)
max
= 0.01
X
= 63, (
P
pe
)
max
= 0.01

X
= 7, (
P
pe
)
max
= 0.01, BCH
X
= 63, (
P
pe
)
max
= 0.01, BCH
X
= 7, (
P
pe
)
max
= 0.001
X
= 63, (
P
pe
)
max
= 0.001
X
= 7, (

P
pe
)
max
= 0.001, BCH
X
= 63, (
P
pe
)
max
= 0.001, BCH
Figure 5.1 Impact of the code-rate on the normalised throughput
5.2 ACCOUNTING FOR MAI GENERATED BY RANDOM CODES
231
and the capability of correcting up to e = 38 errors [252], henceforth referred to as
(511, 229, 38) BCH code. With this code and X = 7, the packet error rate is zero for
K ≤ 7, while P
pe
[8] = 2.6 × 10
−4
,andP
pe
[9] = 1.40%. With K ≥ 10, the performance
degrades rapidly, and success rates are smaller than 1% for K ≥ 15, as illustrated in
Figure 5.2.
These calculations are intended to establish the best set of design parameters for MD
PRMA as investigated in Chapters 7 and 8. With 
K
a random variable describing the

number of terminals per time-slot, S should be maximised taking the distribution of 
K
experienced under MD PRMA operation into account, rather than trying to maximise
r
c
· K
pe max
in Equation (5.8) as done here. However, given that access control in MD
PRMA and therefore the distribution of 
K
may depend on Q
pe
[K], while Q
pe
[K]in
turn depends on r
c
, this will prove rather difficult. On the other hand, efficient access
control should ensure that K is equal to or close to K
pe max
most of the time. Note
also that in References [137] and [251], throughput was maximised over the code-rate
for slotted ALOHA and packet CDMA, respectively, thus taking protocol operation into
account. The optimum range of values for r
c
was found to be between 0.4 to 0.6 and 0.4
to 0.7 respectively, which is in agreement with the above findings.
5.2.5 Accounting for Intercell Interference
Consider a cellular environment with R equally loaded cells and K simultaneously active
transmitters in each cell, all cells sharing the same spectrum, thus operating at a frequency

reuse factor of one. Assume that perfect power control is employed, such that all the
packets transmitted by terminals within any given cell, say for instance cell i, can be
received by their base station at equal power level P
0
. On the other hand, terminals
served by cells other than cell i are power-controlled by their respective base station. The
0
1
6 7 8 9 10111213141516
Users
K
per time-slot
Packet success probability
Q
pe
[
K
]
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Figure 5.2 Q
pe
[K] for equal power reception, a (511, 229, 38) BCH code, and X = 7

232
5 MODELS FOR THE PHYSICAL LAYER AND FOR USER TRAFFIC GENERATION
power level P
(k,j)
i
received at base station i from a mobile k served by base station j
will depend on the propagation attenuation from that mobile to both base stations i and
j. Assuming perfect power control, the average SNR according to Equation (5.5) can be
rewritten for test cell i as
SNR =











3P
0
X
(K − 1)P
0
  
Intracell
+
R


j=1
j=i
K

k=1
P
(k,j)
i
  
Intercell
.(5.12)
If a propagation model with distance-independent pathloss coefficient γ
pl
and log-
normal shadowing is considered, the attenuation is [254]
α(r, ζ) = r
γ
pl
10
ζ/10
,(5.13)
where r is the distance between MS and BS, and ζ is the attenuation in dB due to
shadowing. The spatial distribution of ζ is commonly modelled as a Gaussian random
variable with zero mean and standard deviation σ
s
. Typically, for mobile communications,
γ
pl
= 4 (lower for small cells) and σ

s
is around 8 dB. With r
(k,j)
i
the distance from the
mobile (k, j ) being considered to the base station in cell i,andr
(k,j)
j
that to the base
station in the serving cell j , as illustrated in Figure 5.3, P
(k,j)
i
amounts to
P
(k,j)
i
= P
0
r
γ
pl
(k,j)
j
· 10
ζ
(k,j )j
/10
r
γ
pl

(k,j)
i
· 10
ζ
(k,j )i
/10
.(5.14)
As mobile (k, j ) moves, P
(k,j)
i
varies not only because of the changing distances, but
also due to the values of the shadowing attenuation, which can change if the mobile
moves far enough.
Consider a snapshot value of the level of intercell interference at base station i,
normalised to the total received power of the K mobiles served by this base station,
i
intercell
=
1
P
0
K
R

j=1
j=i
K

k=1
P

(k,j)
i
.(5.15)
If each mobile is served by the base station with minimum attenuation, such that
the intercell interference level is minimised, the normalised level of intercell interference
I
intercell
averaged over shadowing and the mobile locations will depend only on the spatial
distribution of mobiles in the cell, the pathloss coefficient γ
pl
, and the shadowing standard
deviation σ
s
. In particular, since γ
pl
is assumed to be distance-independent, I
intercell
does
not depend on the cell radius r
0
. It is therefore possible to account for the average
intercell interference without having to consider more than one cell by evaluating
I
intercell
for the distribution of mobiles being considered and the values chosen for γ
pl
and σ
s
.
Equation (5.12) then simplifies to

SNR =

3X
(K − 1) + K · I
intercell
.(5.16)
5.2 ACCOUNTING FOR MAI GENERATED BY RANDOM CODES
233
Mobile (
k
,
j
)
Cell
j
r
(
k
,
j
)
j
r
(
k
,
j
)
i
Cell

i
r
0
First tier of cells
interfering to cell
i
Second tier of cells
interfering to cell
i
Figure 5.3 Centre test cell i with radius r
0
and two tiers of interfering cells. Mobile (k, j ) is
served by cell j and creates interference to cell i. Base stations are at cell centres
Due to perfect power control, the common reference power level P
0
disappears. I
intercell
must be denormalised with K, which is the average channel load or the average number
of users per time-slot in each cell. The attentive reader will have observed a discontinuity:
so far, K stood for the number of simultaneously active transmitters in each cell, whether
test cell or interfering cells. Now, for Equation (5.16) to be useful for MAC performance
investigations, a distinction is being made between the average channel load
K (here to
be assumed the same in the test cell and interfering cells) and the instantaneous number
of users K in the test cell. Equation (5.16) can therefore be used to account for slot-to-
slot load fluctuations caused by the MAC operation in the test cell, while ignoring such
fluctuations in interfering cells.
In Reference [251],
I
intercell

was evaluated based on an approach outlined in Refer-
ence [9] for a cellular system with hexagonal cells, uniform distribution of mobiles and
various values of the pathloss coefficient γ
pl
. Shadowing was not considered, which means
that the nearest base station is also the serving base station. Ganesh et al. report
I
intercell
=
0.37 for γ
pl
= 4, which is typical for large cells, and I
intercell
= 0.75 for γ
pl
= 3, which
is more representative for smaller cells, e.g. microcells. Similar values (slightly higher
for γ
pl
= 4) appear also in Reference [255], where Newson and Heath approximated the
hexagonal cells by circular cells of equal area and integrated the interference numerically
over two tiers of interfering cells. In Reference [254], on the other hand,
I
intercell
= 0.44
and 0.77, respectively (i.e. considerably higher for γ
pl
= 4), although the scenario consid-
ered by Viterbi et al. appears to be the same as that by Ganesh et al. at first glance.
234

5 MODELS FOR THE PHYSICAL LAYER AND FOR USER TRAFFIC GENERATION
A possible reason for these discrepancies could be the exact spatial distribution of the
interference considered. Although in all references, uniform distribution is considered,
it is not clear from References [9] and [251] whether this is on a per-cell-basis or over
all cells. If Monte Carlo snapshot simulations are performed to assess the interference,
during which mobiles are repeatedly redistributed over the test area, and the observed
interference is averaged in the end, this matters. In Reference [254] for instance, the
authors clearly specified that they considered a uniform distribution over the whole area
of interfering cells, such that only the expected number of mobiles per cell is the same,
while the actual value may fluctuate.
It is well known that CDMA capacity suffers under unequal cell load (see for instance
Reference [112], where possible approaches to mitigate the problem through adaptive
adjustment of the reference power level at each cell are discussed). Therefore, if inter-
ference is averaged over snapshots with unequally loaded cells, one will have to expect
higher interference levels than when averaging over snapshots with an equal number of
mobiles per cell, even if in both cases the cells are on average equally loaded. Indeed,
we managed to reproduce the lower values reported in Reference [251] through Monte
Carlo simulations considering the first two tiers of interfering cells
8
(see Figure 5.3)
when for every snapshot the same number of mobiles were uniformly distributed in
each cell.
Both in References [254] and [255] intercell interference levels are also evaluated when
shadowing is considered. This case is more intricate, since the base station with the lowest
attenuation is not necessarily the nearest base station, which complicates the evaluation of
I
intercell
. In fact, while Newson and Heath carried out a numerical integration for the case
without shadowing, they had to resort to Monte Carlo simulations when accounting for
shadowing. In Reference [254], the selection of the serving base station is constrained to

one of a limited set of N
c
nearest base stations (N
c
= 1, 2, 3 or 4), which may not include
the base station with lowest attenuation. Shadowing from a given mobile to different
base stations is assumed to be partially correlated, while no correlation is considered
in Reference [255]. In both cases,
I
intercell
≈ 0.55 for γ
pl
= 4, σ
s
= 8dB,andN
c
= 4
(the latter only relevant for Reference [254]). It appears that the constraint to N
c
= 4in
Reference [254], which should increase
I
intercell
compared to the scenario considered by
Newson and Heath, is offset by the partial correlation of the shadowing, which reduces
I
intercell
.
For the results presented in Chapter 7,
I

intercell
values of 0.37 and 0.75 were used for
γ
pl
= 4 and 3 respectively, having the no-shadowing case in mind, where all cells are
always equally loaded. In Reference [254], for γ
pl
= 4, shadowing with σ
s
= 10 dB, and
the case where the serving base station must be among the three base stations closest to
the mobile being considered (i.e. N
c
= 3), I
intercell
is also reported to be 0.75. Therefore,
the results reported in Chapter 7 for γ
pl
= 3 without shadowing could also be taken to
stand for γ
pl
= 4 with shadowing under the conditions just outlined.
Recall that
I
intercell
denotes the normalised averaged intercell interference level assuming
the same constant number of simultaneously active mobiles
K in every cell. Given
the large number of interfering mobiles, we would expect I
intercell

, the random variable
describing instantaneous intercell interference levels, to be log-normally distributed. On
top of movement of mobiles and fluctuating shadowing attenuation, the fact that the
8
With γ
pl
≥ 3, interference from outside the first two tiers of cells is negligible.

×