IEEE
TRANSACTIONS
ON
VEHICULAR TECHNOLOGY,
VOL.
41,
NO.
1,
FEBRUARY
1992
I1
Near-Far Effects in Land Mobile Random
Access Networks with Narrow-Band
Rayleigh Fading Channels
Jean-Paul
M.
G.
Linnartz,
Member,
ZEEE,
Ramin Hekmat, and Robert-Jan Venema
Abstract-The near-far effect of random access protocols in
mobile radio channels with receiver capture is investigated.
To
this end, the probability of successful reception of a packet from
a terminal at a known distance from the central receiver is
obtained taking into account Rayleigh fading,
UHF
propagation
attenuation, and the statistics of contending packet traffic in
radio nets employing slotted
ALOHA,
carrier sense multiple
access
(CSMA)
or inhibit sense multiple access
(ISMA)
protocols.
Various models of receiver capture are compared, namely packet
error rates for synchronous detection in slow- and fast-fading
channels, and the probability that the signal-to-interference ratio
is
above a required threshold.
I.
INTRODUCTION
YNAMIC multiple access
to
a common receiver via fad-
D
ing radio channels is an important issue to the efficiency
of
mobile data networks. In this paper the case of mobile trans-
mitters sending data packets to a single (fixed) base station
over a common radio channel is studied. Examples of popular
protocols to provide random access for a large number of users
are slotted ALOHA [1]-[16], carrier sense multiple access
(CSMA) 1171, [18] and related techniques such as inhibit sense
multiple access (ISMA)
[
191-[21], idle-signal casting multiple
access (ICMA) 1221 and busy channel multiple access [23].
According
to
the ALOHA protocol, any terminal is allowed to
transmit a packet of data without considering whether another
terminal is already transmitting on the common inbound
(mobile-to-base) channel. Overlapping transmissions called
“collisions” severely reduce the throughput of a channel if
the offered traffic load is high. A refinement of the ALOHA
protocol is slotted ALOHA. In the latter protocol all packets
are transmitted within time slots defined by the base station.
This reduces the probability of interference between terminals.
Manuscript received November
9,
1990;
revised February
15,
1991
and
July
23,
1991.
J.-P.
M.
G. Linnartz was with the Telecommunications and Traffic-Control
Systems Group, Delft University
of
Technology, 2600
GA
Delft, The Nether-
lands.
He is now with the Department
of
Electrical Engineering and Computer
Science, University of California, Berkeley, CA 94720.
R. Hekmat was with the Telecommunications and Traffic-Control Systems
Group, Delft University of Technology, 2600 GA Delft, The Netherlands.
He is now with the
PTT
Research Neher Laboratories, Leidschendam, The
Netherlands.
R.-J.
Venema was with the Telecommunications and Traffic-Control Sys-
tems
Group,
Delft University
of
Technology, 2600 GA Delft, The Netherlands.
He is now with the Royal Netherlands Navy, Air Force Base Valkenburg,
The
Net herlands.
IEEE
Log Number
9105387.
In CSMA, each mobile terminal first senses the radio
channel, and no packet transmission is initiated when the
channel is already busy.
As
this dramatically reduces the
number of collisions, the throughput is significantly higher
than in slotted ALOHA networks [17]. However, in mobile
radio nets with fading channels, a mobile terminal might
not be able to sense a transmission by another (remote)
terminal. This effect, known as the hidden terminal prob-
lem,
is
avoided in ISMA, where the base station transmits
a “busy” signal to inhibit all other mobile terminals from
transmitting as soon as an inbound packet is being received
[21].
Murase and Imamura [22] addressed a possible extension
of the feedback signaling:
in idle-signal casting multiple
access with collision detection (ICMA-CD) the base station
broadcasts “idle,” “busy,” or “stop” (i.e., collision) messages.
Andrisano
et
al.
[23] studied the case that idlelbusy feedback
information is broadcast by on-off keying of a carrier, and
they assessed probabilities of erroneously detecting feedback
messages. However, even if signaling messages on this feed-
back channel are always received correctly by all mobile
terminals, collisions can nonetheless occur in ISMA for two
reasons:
i)
(re)transmissions of new or rescheduled packets
can start during the delay in reception of the inhibit signal,
and ii) packets from two or more persistent terminals, awaiting
the channel
to
become idle, can collide immediately after the
termination
of
the previous packet transmission. Considering
these collisions, papers such as
[
171 determine the successful
packet traffic based on the two assumptions that, firstly, a data
packet is always received correctly in the absence of collisions
and, secondly, all packets involved in a collision are lost. In
mobile radio nets both assumptions should be reconsidered.
Channel imperfections may cause loss of a data packet even
if no interference from other terminals occurs; further, each
received signal experiences fading, such that the received
power can substantially vary for each terminal position. In the
event of a collision, the strongest contending signal may then
capture the receiver. Compared to remote terminals, nearby
terminals thus experience a higher probability of success in
transmitting a packet. This near-far discrimination has been
studied previously for slotted ALOHA, e.g.,
[l],
[2], 141, [5],
but the performance of CSMA and ISMA protocols, which are
also commonly used (see e.g., [25]), received relatively little
attention. This has motivated the authors to present results
for unslotted nonpersistent and p-persistent ISMA over fading
radio channels and to compare these with the performance
0018-9548/92$03.00
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1992
IEEE
78
IEEE
TRA
NSACTIONS
ON
VEHICULAR
TECHNOLOGY, VOL.
41,
NO.
1,
FEBRUARY
1992
of slotted ALOHA. The results presented here for ISMA also
apply for those CSMA nets where the hidden terminal problem
is negligible.
Section I1 analyzes packet traffic over the radio channel.
The probability of overlapping transmissions is studied for
one particular “test” packet with
a
priori
known properties,
such as the area mean received power. Our method, presented
in brief in [24], differs from other papers on ISMA and
CSMA, where the Poisson distributed arrivals of packets at
the receiver are studied without any specific knowledge about
each packet [17], [20], [21]. Probabilities of a successful
transmission, conditional on the position of the transmitter of
the test packet, are developed to investigate the near-far effect.
Section I11 summarizes the mechanisms of the mobile fading
considered. Various models for receiver capture in a narrow-
band channel are reviewed and extended in Sections IV and
V: in Section IV, block error probabilities are considered to
evaluate the probability of capture, and in Section V a packet is
considered to be received correctly if the signal-to-interference
ratio is above a certain capture threshold. Results are compared
and discussed in Section VI, and conclusions are given in
Section VII.
11.
PACKET
TRAFFIC
All data packets are assumed to be uniform duration,
equal to the (normalized) unit of time and, in the event of
slotted
ALOHA,
equal to the duration of a time slot. If a
packet is received correctly, the base station transmits an
acknowledgment over the outbound channel. Assuming that
these acknowledgments are never lost, the mobile terminal
deems its packet to be lost if no acknowledgment is received,
and retransmits the packet after waiting a random time.
An
infinite population of (independent) terminals is assumed,
and the probability that one particular terminal attempts a
(re)transmission during a certain interval of time is assumed
to offer an infinitely small contribution to the total traffic. The
total attempted packet traffic is denoted as
Gt,
expressed in the
average number of attempted packets transmissions per unit of
time (ppt). This (composite) traffic
Gt
includes not only the
initial attempts to transmit newly arrived packets, but also in-
cludes attempts to retransmit previously unsuccessful packets.
We assume that the number of (re)transmission attempts during
a time interval of duration
T
is Poisson distributed, with mean
TGt.
The probability of
n
attempts during
T
is
P,(n)
=
___
(GtT)n
exp(-GtT)
(1)
n!
with
(n
=
0,1,2,.
.
.).
We address steady-state operation of
the channel, i.e.,
Gt
is assumed to be constant with time.
Such stable behavior of the net [13] requires random waiting
times to be sufficiently long to ensure uncorrelated interference
during the initial and any successive transmission attempts.
The average number
of
attempted packet transmissions
originating from a (normalized) distance
r
(0
<
r
<
GQ)
from
the central receiver, per (normalized) unit of time and per
range
0
<
r
<
1
(see Section 111). The total attempted packet
traffic
Gt
is found from (polar) integration of
G(r)
over the
service area, viz.
(2)
Gt
=
2mG(r)dr.
i
0
The following probabilities of successful reception are de-
Q(r)
is the unconditional probability of the successful
reception of a test packet generated in a terminal
at a distance
r
from the central base station, taking
account of the probability of permission to transmit
and averaged over the number of interfering packets
n
and over the unknown positions of the interfering
terminals.
qn(r)
is
the probability of correct reception of a test
packet transmitted from a distance
r,
given the
number of interfering packets
n,
but averaged over
the unknown positions from which these interfering
signals originate.
is the probability of successful reception given that
the test packet is transmitted in the presence of
n
interfering signals, averaged over the unknown
positions of all mobile terminals, including the one
transmitting the test packet. Hence
fined.
qn
Qn
-
-
Gt 727rrqn(r)G(r) dr.
(3)
03
ir
(3)
qn
=
Gt 2mqn(r)G(r)dr.
Cn+l
is the expectation value of the number of correctly
received packets in the event that
n
+
1
packets
collide. If receiver capture
is
mutually exclusive for
each of the
n
+
1
packets, as assumed in [1]-[7],
[11], [w, [15], [20], and [21],
cn+i
=
(n
+
1)qn.
In Section IV, this is discussed in more detail.
The total throughput
St
of the network is defined as the
expected number of transmitted packets per unit of time that
are detected correctly at the base station, i.e.:
A.
Slotted
ALOHA
-
According to the
ALOHA
protocol, a mobile terminal
transmits its packets regardless of other transmissions in the
same time slot. The probability
Pn(n)
that the test packet
experiences interference from
n
other (contending) signals in
the same time slot is assumed to be Poisson distributed, with
mean
Gt,
thus with
T
=
1
inserted in
(1).
The probability
Q(r)
of a successful transmission is
(normalized) unit of -area, is denoted as
G(r)
[l], [2]. The
03
unit of distance (thus also the unit
of
area) is normalized to
Q(r)
=
Pn(n)qn(r).
(5)
ensure that the major part of the traffic is transmitted in the
n=O
LINNARTZ
et
al.:
EFFECTS
IN
LAND
MOBILE
RANDOM
ACCESS
NETWORKS
79
The total packet throughput results from
00
03
n=O
i=l
where
i
represents the total number of packets in a time slot
(i
=
0,1,...
and
CO
=
0).
B.
Inhibit Sense Multiple Access (ISMA)
In ISMA, the radio system is supplemented by an outbound
signaling of the status of the channel: either “busy” or “idle.”
When the base station receives an inbound packet, a “busy”
signal is broadcast to all mobiles after a processing delay
d.
This delay is normalized to the unity duration of each data
packet
(0
5
d
<
1).
After termination of all
n
+
1
contending
transmissions, the base station starts transmitting an “idle”
signal after a delay also of duration
d.
In CSMA, the delay
is
mainly caused by the time a mobile terminal takes to switch
from reception to transmission mode (powerup), after sensing
the radio channel for carriers from other active terminals [25].
The busy period is the period during which the base station
broadcasts a busy signal plus the preceding signaling delay
d.
For memoryless Poisson arrivals, the expected duration
I
of
the idle period between two busy periods equals the average
lapse of time until a new packet arrival occurs, thus
I
=
Gtl
[17]. The average duration
B
of the busy period depends on
the signaling delay d and on the persistency p in scheduling
inhibited packets [17]. We address an unslotted version of
p-
persistent CSMA and ISMA. If the channel
is
idle, the packet
is transmitted immediately. If the channel is busy, the terminal
performs a binary random experiment: with probability p the
terminal transmits the packet as soon as the channel becomes
idle. Such an attempt is considered successful unless the packet
is destroyed in a collision. Alternatively, with probability
1
-
p,
the packet arriving at an instant when the receiver is busy
is rescheduled with a random delay. Such an inhibited packet
is considered to be unsuccessful and a next attempt
is
to
be performed after waiting a random time. For nonpersistent
CSMA and ISMA, rescheduling always occurs if the channel
is busy at the instant of sensing.
I)
Unslotted Nonpersistent ISMA:
If a packet arrives at a
nonpersistent terminal when the base station transmits a “busy”
signal (denoted by event
HB),
the attempt is considered to
have failed. The packet is rescheduled for later transmission.
With probability
I/(I
+
B),
a test packet transmitted from a
distance
T
starts at an instant when the channel
is
idle. This
event is denoted as
HI.
A collision can occur if one or more
other terminals start transmitting during the time delay d of the
inhibit signal. The conditional probability of
n
transmissions
overlapping with the test packet that initiated the busy period
is
(7)
Altematively, the test packet itself starts during a period
of
duration
d
when the channel is busy because of a transmission
by another terminal, but seems idle since the inhibit signal
is
not yet being broadcast. This event, denoted as
Hd,
occurs
with probability d/(B
+
I).
This packet thus experiences
interference from at least one contender. The additional,
n
-
1
contending signals arrive independently of this event,
so
they are Poisson distributed. The conditional probability of
n
interferers
is
found from
(dGt)n-l
(n
-
l)!
(8)
Rn(nJHci)
=
~
exp
(
-
dG
t
where
(n
=
1,2,
.
. .).
Taking account of the above three
possible events
HB, HI,
and
Hd,
the unconditional probability
of successful transmission
Q(T)
is
The probability of capture
qn(r)
depends, among other things,
on the probability to acquire receiver synchronization, which,
in general, depends on the channel status
(Hd
or
HI)
at the
arrival of the packet. This is elaborated in Section IV for a
simplified synchronization model. The busy period was shown
in [17] to be
of
average duration
(10)
1
B
=
1
+
2d
-
-
[l
-
exp(-dGt)].
Gt
The total channel throughput
St
is found from the integration
00
After interchanging the order of integration and the sum-
mation, we recover the expression used in [20], [21] by
considering Poisson arrivals of packets at the receiver in the
base station, viz.:
1
O0
dnG:
St
=
-
exp(-dGt)C
--Cn+1
(12)
n!
n=O
B+I
where we assumed
Cn+l
=
qn~~I
+
nqnlHd.
For instantaneous
inhibit signaling (d
+
0),
collisions can never occur in non-
persistent ISMA, and (12) reduces to
St
+
Gt(l
+
Gt)-’,
in
agreement with
[17].
2)
Unslotted p-Persistent ISMA:
We now consider unslotted
p-persistent ISMA
(0
5
p
5
1)
without signaling delay
(d
=
0),
although in practice only nonzero values of
d
can
be achieved. The case d
=
0
thus represents an upper-
bound on throughput performance. A busy period can consist
of a number of packet transmissions in succession because
a terminal may start transmitting as soon as the previous
transmission by another mobile station is terminated. If
a packet arrives during an idle period (event
HI),
the
probability of correct reception of this initial packet is
qo(r).
During the transmission of this packet, a random number
of
IC
(IC
=
0,1,2,.
.
.)
terminals sense the channel busy with
80
IEEE
TRANSACTIONS
ON
VEHICULAR TECHNOLOGY, VOL.
41,
NO.
1,
FEBRUARY
1992
probability
Pn(k),
with
T
=
1
in (1). When the channel goes
idle, each of the
k
terminals starts transmitting with probability
p.
For a test packet arriving during a busy period (event
HB),
the probability of
n
interfering packets is thus
In
particular, the probability that the busy period is terminated,
i.e.,
none of the
k
terminals starts transmitting,
is
The probability
P,
(m)
of continuing transmissions during
m
(m
=
0;
1,
.
.
.)
units of time, concatenated to the initial packet
is
On the average, a busy period thus has the total duration
B
4
E,,,
[
1
+
m]
03
=
1
+
exp(-pGt)
=
exI.'(+pGt) (16)
m[l
-
exp(-pGt)],
m=O
where
E,
denotes the expectation over
m.
The probability of
a successful transmission
Q(T)
is
Using
I
=
Gtl,
(13),
and (16),
(17)
becomes
After integration, the total channel throughput
St
is obtained
from
In the special case
p
=
0,
we recover the result for unslotted
nonpersistent ISMA without signaling delay
(d
=
0).
The
classical case of 1-persistent CSMA on wired channels is
found by inserting
qo
=
1
and
qn
=
0
The throughput then becomes
Gt
+
G:
st
=
1
+
Gt exp(Gt)
This agrees with expressions derived for
receiver without capture [17].
for
n
2
1
in (19).
Poison arrivals at a
III.
CHANNEL
MODEL
The normalized local-mean power
pj
received from the jth
mobile terminal
(j
=
0,1,.
.
.)
at a (normalized) distance
rj
from the central receiver is taken to have the form [26]:
If the position of the terminal is unknown, the probability
density function (PDF) of the mean power
pj
is
found from
121
27rrj
G(
rj
)
Gt
In the following, we assume the quasi-uniform spatial distri-
bution of the offered channel traffic suggested in [2], and also
in [SI,
[91,
[111,
[151,
[161, namely:
This is an approximation of the exactly uniform distribution
G(T)={$>
o<r<l
0,
elsewhere
by a smooth analytical function. The main part of the traffic
thus arrives from the normalized distance
0
<
T
<
1,
whereas
beyond
T
=
1
the intensity
of
the attempted traffic rapidly
decreases. The main reason to consider
a
quasi-uniform,
rather than an exactly spatial distribution is the convenient
analytical expression found for the PDF of the joint power of
n
uncorrelated signals, viz. [2]:
-_
where
Pt
(Pt
=
CP,)
is
the local-mean power of the joint
interference signal.
Further, Rayleigh fading in a narrow-band channel is as-
sumed [26]. The instantaneous amplitude
pj
of the jth carrier
is Rayleigh distributed, with PDF:
for
0
<
pj
<
30.
This Rayleigh distribution is due to Gaussian
in-phase and quadrature carrier components
<j
and
tj,
respectively, with identical with zero mean and a vari-
ance equal to the local-mean power. The corresponding
total instantaneous (in-phase plus quadrature) power
pj
(pj
=
1/2p;
=
1/2cj2
+
1/2(;)
received from the jth mobile
terminal
is
exponentially distributed about the mean power,
viz.:
with
p,
2
0.
Combining the statistical fluctuations caused by
the spatial distribution of the mobile terminals (22) and by
Rayleigh fading, the unconditional PDF's of the amplitude
pj
and the in-phase component
(j
are
LINNARTZ
et
al.:
EFFECTS
IN
LAND MOBILE RANDOM ACCESS NETWORKS
81
Ec~r
Id>?#,
~2-3P
6:
._
-
\>?-
2~
-724
L
-PATH
LOSS
-
Y
FADING
3ATH
LOS5
P
-
RAYLEIGH
';lc
7w
%
?-I,HTk
*'
FADING
Fig.
1.
Correlation receiver
for
coherent detection
of
BPSK
in a mobile radio channel with multiple interferers and Gaussian
noise.
-
2
3
-
.
-
-
p.-
"-1
-
PATH
LOSS
-
RAYLEIGH
p.
Fr
I'
->.
iFj
FACING
for
0
<
pj
<
m,
and
COSw,t INTEGRATOR
v
COHERENT
BPSK
DETECTOR
for
-m
<
(j.
<
m,
respectively
[8].
Any retransmitted or
rescheduled packet is assumed to experience uncorrelated
fading and path loss.
Iv.
PROBABILITY
OF
PACKET ERROR
In this section, a test packet of
L
bits is assumed to
have captured the receiver if and only if the bit sequence
detected by the receiver entirely matches the bit sequence in
the test packet. The assumption to require
all
bits in the test
packet to be received correctly is pessimistic. Most practical
implementations of packet radio employ some form of forward
error correction coding, which allows a test packet to be
received correctly as long as the number of errors is within the
error correcting capability of the code
[7],
[lo],
[14],
[15].
The
case of fast fading, i.e., with signal amplitudes independent
from bit to bit, is studied considering two different models:
model
LA
describes a receiver that perfectly locks to the test
packet, whereas in model
1.B
the receiver selects one favorite
packet out of the
n
+
1
contending signals. In the latter case,
the test packet is always assumed to be lost if the receiver
happens to lock to another signal.
With slow fading, which will be studied for five cases
(model
1I.A-ILE),
the amplitude and phase of each signal are
assumed constant for the entire duration of the packet. Models
IIA
and
1I.B
correspond to the receiver synchronization
behavior assume in
1.A
and
I.B,
respectively. The models
1I.C-1I.E
are introduced to approximate the effect that the
carrier recovery in the receiver is impaired by interfering
signals.
A.
Receiver
Model
The classical correlation receiver
[27]
is considered, which
is known to be optimal for time invariant channels with addi-
tive white Gaussian noise (AWGN), but free of interference.
We study the performance
of
this receiver with
n
cochannel
signals interfering with the test signal (Fig.
1).
At least during
one bit, the phase
Oj
and the amplitude
pj
of each signal
is assumed to remain constant. For each of the interfering
signals, exactly overlapping bit periods are assumed, i.e., no
bit synchronization offset
is
taken into account. The received
signal
w(t)
is on the form
w(t)
=
POKO
cos(w
.t
+
eo)
n
+
pj~j
cos(w
.t
+
Oj)
+
n(t)
(281
j=1
where
~j
(~j
=
fl)
represents phase reversals due to BPSK
modulation of the jth carrier and
n(t)
is the AWGN, bandpass
filtered for the transmission bandwidth
BT
in any practical
receiver. The test signal is denoted by index
0.
Fig.
2
illustrates
the phase-quadrature constellation for a test packet in the
presence of three interferers and (bandlimited) AWGN. Three
idealized cases for the phase of the local oscillator in the
coherent detector are compared, namely, a receiver locked
to the test signal (Fig. 2(a)), a receiver locked to one of
the interferers (Fig. 2(b)), and a receiver with arbitrary (but
constant) phase (Fig. 2(c)). The latter two events correspond
to extreme cases of carrier phase errors in the receiver, caused
by signals competing with the test packet.
In the detector, the received composite signal
w(t)
is multi-
plied by a locally generated cosine
(2
cos
w,t)
and integrated
over the entire (normalized) bit duration
Tb (TbL
=
1).
The re-
ceived energy per bit
is
E,,
=
poTb
cos280
=
1/2pi~b
cOs28,,.
The decision variable for synchronous bit extraction from a
test packet in the presence of
n
interferers
is
w
=
72w(t)
cos(w
.t)
dt
Tb
0
n
=
po~o
cos(00)
+
pj~j
cos(8j)
+
ni.
(29)
j=1
In a Rayleigh fading channel, both the in-phase components
cj
(Cj
=
pj
cos
Oj)
of the
n
interferers with random phase
relative to the local oscillator in the receiver, and the noise
sample
ni,
are independent Gaussian variables. Phase reversals
of bit synchronous interference do not change the Gaussian
PDF. The rate of fading determines the correlation of ampli-
tude and phase in successive bits. Fast and slow fading are
distinguished.