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Secured operating regions of Slotted ALOHA in the presence of interfering signals from other networks and DoS attacking signals

Journal of Advanced Research (2011) 2, 207–218

Cairo University

Journal of Advanced Research

Secured operating regions of Slotted ALOHA
in the presence of interfering signals from other networks
and DoS attacking signals
Jahangir H. Sarker, Hussein T. Mouftah

*

School of Information Technology and Engineering (SITE), University of Ottawa, Ottawa, Ontario, Canada K1N 6N5
Received 14 October 2010; revised 8 April 2011; accepted 10 April 2011
Available online 14 May 2011

KEYWORDS
Ad Hoc networks;
Attacking noise packets;
Interfering signals;

Multiple channels;
New packet rejection;
Retransmission trials;
Other networks;
Sensor networks;
Slotted ALOHA

Abstract It is expected that many networks will be providing services at a time in near future and
those will also produce different interfering signals for the current Slotted ALOHA based systems.
A random packet destruction Denial of Service (DoS) attacking signal can shut down the Slotted
ALOHA based networks easily. Therefore, to keep up the services of Slotted ALOHA based systems by enhancing the secured operating regions in the presence of the interfering signals from other
wireless systems and DoS attacking signals is an important issue and is investigated in this paper.
We have presented four different techniques for secured operating regions enhancements of Slotted
ALOHA protocol. Results show that the interfering signals from other wireless systems and the
DoS attacking signals can produce similar detrimental effect on Slotted ALOHA. However, the
most detrimental effect can be produced, if an artificial DoS attack can be launched using extra false
packets arrival from the original network. All four proposed secured operating regions enhancement techniques are easy to implement and have the ability to prevent the shutdown of the Slotted
ALOHA based networks.
ª 2011 Cairo University. Production and hosting by Elsevier B.V. All rights reserved.

* Corresponding author. Tel.: +1 613 562 5800x2173; fax: +1 613
562 5664.
E-mail addresses: jsarker@site.uottawa (J.H. Sarker), mouftah@site.
uottawa.ca (H.T. Mouftah).
2090-1232 ª 2011 Cairo University. Production and hosting by
Elsevier B.V. All rights reserved.
Peer review under responsibility of Cairo University.
doi:10.1016/j.jare.2011.04.008

Production and hosting by Elsevier

Introduction
To improve the secured transmission over vulnerable wireless
networks, assessment of the wireless multiple access schemes
in the presence of jamming or attacking signals is an important
issue [1]. It is well known that the Code Division Multiple Access (CDMA) system has a special resistance against the interference signals from other networks and the attacking signals.
Thus the CDMA scheme may be the first choice as a multiple
access scheme in the presence of interference signals from other
networks or/and attacking signals. The attacker should spread
its energy evenly over all degrees of freedom in order to minimize the average capacity of the original signals [2,3]. In a sim-




208
plified CDMA transmission system, with the knowledge of
spreading code, the receiver is able to detect the users’ signals
from interfering signals from other networks and attacking signals. Using the attacker state information and the effects of
fading, the channel capacity can be enhanced further. For
enhancing uplink channel capacity, the attacker state information is more important than that of the effects of fading.
Preventing the attacking signals becomes very difficult, if the
attackers use the same code as the legal users and transmit. A
specific Frequency Hoping Speed Spectrum (FHSS) technique
can prevent this type of attack [4]. However, a specific FHSS
technique is inefficient for a large number of mobile nodes.
An innovative message-driven frequency hopping was introduced and analyzed to improve the system capacity [5]. The
mobile nodes can exploit channel diversity in order to create
wormholes in hostile jamming or attacking environment, [6].
In infrastructure-less wireless Ad Hoc and sensor a network,
mobile nodes not only behave as transmitters and receivers but
also as network elements, i.e., switches or routers, without any
established network infrastructure. As a result, low power
consumption systems are becoming important for infrastructure-less wireless Ad Hoc and sensor networks. The Slotted
ALOHA is the most spectral and power efficient multiple access scheme [7,8]. Although, the CDMA has especial resistance
against interference and attacking signals, Slotted ALOHA is a
widely used random access protocol not only for its simplicity
also for its higher spectral and power efficiency.
The Slotted ALOHA multiple access schemes is used exclusively in newly developed Radio Frequency Identification
(RFID) technology [9]. The Slotted ALOHA is also used as
a part of different multiple access protocols especially for the
control channels in many new wireless technologies. For instance, it is used in the random access channels of Global System for Mobile (GSM) communications [10] and its extension
General Packet Radio Services (GPRS) [11,12], Wideband
Code Division Multiple Access (WCDMA) system [13],
cdma2000 [14,15], IEEE 802.16 [16], IEEE 802.11 [17], etc. A
smart power saving jammer or attacker can attack only in
the signaling channels, instead of attacking whole channels
[18–20]. Therefore, defending the control channels from external and internal attacks [21] are very important issue. If the total network is based on Slotted ALOHA based protocol, then
defending the network against the DoS attack is one of the
most important factors [22,23] and has been discussed in this
paper.
A special type of Denial of Service (DoS) attack, called random packet destruction that works by transmitting short periods of noise signals is considered as attacking signals. This
random packet destruction DoS packets can effectively shut
down Slotted ALOHA based networks [9] and the networks
use the Slotted ALOHA based signaling channels [10–20].
One of the main drawbacks of Slotted ALOHA is its excessive
collisions at higher traffic load condition. The current antiattack measures such as encryption, authentication and authorization [24,25] cannot prevent these types of attacks. Since the
random packet destruction DoS packets increase the collision
further, the receiver cannot read the message packets.
The effect of attacking noise packet signals on the Slotted
ALOHA scheme without autonomic is investigated [26–28].
The stability of Slotted ALOHA in the presence of attacking
signals is presented in Sarker and Mouftah [29], where dynamic channel load and jamming information are needed to

J.H. Sarker and H.T. Mouftah
maximize the channel throughput, which makes the system
implementation difficult. Recently, there has been an increasing interest in the autonomic networks, i.e., networks should
be self-stabilized without the use of feedback information
[30]. Excellent work in self-stabilized Slotted ALOHA without
the use of feedback information is presented in Bing [31],
where the effect of attacking signals is not considered. A
self-stabilized random access protocol in the presence of random packet destruction DoS attack for infrastructure-less
wireless autonomic networks is presented in Sarker and
Mouftah [32]. In this paper we have investigated the combined
effect of the interfering signals from other networks and the
DoS attacking signals on Slotted ALOHA. Three different
types of noises are considered in this paper. First, noise related
to interfering packets from the same network. Second, noise
related to interfering packets from the other networks and
third, noise related to attacking packets from DoS attack.
The contributions of this paper are outlined as follows.
(1) The throughput of Slotted ALOHA in the presence of
the interfering signals from other networks and the random packet destruction DoS attack is presented.
(2) It is shown that for any positive value of message packet
arrival rate, the throughput decreases with the increase
of the interfering signals from other networks’ signal
rate. Similarly, the throughput decreases with the
increase of the random packet destruction DoS attacking packet rate.
(3) A sufficient number of channels can prevent the shutdown of Slotted ALOHA in the presence of interfering
signals from other networks or/and the random packet
destruction DoS attack by reducing the collisions.
(4) In the presence of other message packets, a message
packet is captured, if its power is higher than the message
capture ratio times of all other interfering message packets’ power for a certain section of time slot to lock the
receiver. Similarly, a message packet is captured, in the
presence of interfering packets from other networks, if
its power is higher than the interfering capture ratio times
of the power of the interfering packets from other networks. At the same way, a message packet is captured,
in the presence of attacking noise packets, if its power
is higher than the attacking capture ratio times of other
attacking noise packets’ power. Results show that a
lower value of the message capture ratio is the most
effective solution comparing with the interfering packet
capture ratio or the attacking packet capture ratio.
(5) The approximate value of the number of channels that
provides the maximum throughput is derived.
(6) The security improvement region using the number of
retransmission trials control is presented.
(7) The security improvement region using the new packet
rejection is also presented.
Rest of the paper is organized as follows. The system model
and assumptions are described in the next section. The third
section shows the security improvement using multiple channels and capture effects. The security improvement by limiting
the number of retransmission trials is evaluated in the fourth
section. The fifth section presents the security improvement
by new packet rejection. The conclusion is provided in the last
section.


Secured operating regions of Slotted ALOHA in the presence of interfering signals from other networks and attacking signals 209
System model and assumptions
Let us consider a system, where a base station is located in the
middle of a very large number of users having mobile units
(nodes). Assume that the average value of the new message
packet arrival rate from all active mobile nodes per time slot
is k packet per time slot. In Slotted ALOHA, the throughput
initially increases with the increase of the new packet generation rate, k. The throughput reaches its maximum value for
a certain value of the new packet generation rate from all active nodes. The throughput collapse and reaches to zero, if
the new packet generation rate increases further. The throughput collapse is known as the security or stability problem in
Slotted ALOHA. The reason for throughput collapse is excessive collision. The throughput collapse can be prevented by
reducing the new packet arrival rate per slot. The packet rejection can provide one of the solutions and is considered in this
paper.
Assuming that the new packet rejection probability is a.
The new packet transmission rate per time slot is kð1 À aÞ.
Let there be L parallel Slotted ALOHA based channels. The
mobile nodes can transmit their packets selecting any of the
L channels by random selection, without the knowledge of
other mobile units’ activeness. During the transmission of
packets, each mobile node adjusts their packet size to fit into
the time slots. Since the average new message packet transmission rate from all active mobile nodes per time slot is kð1 À aÞ
packet per time slot and the channel selection is random, the
new packet transmission rate from all mobile nodes is
k
ð1 À aÞ packets per time slot.
L
It is well known that the Slotted ALOHA’ performance is
degraded due to excessive collision. The interference from
other networks can produce packets to increase the collision
farther. Let the interference from other networks’ packet arrival to the base station be Poisson Point Process with an average rate of I packet per time slot. The probability that m
packets are transmitted to the same slot from other networks
as jamming is
INm ¼

Im ÀI
e
m!

ð1Þ

In the first collision reducing technique, we have used multiple parallel Slotted ALOHA Slotted channels instead of single
channel Slotted ALOHA channel. For doing that the message
packets can be transmitted in a multiple L-channel Slotted
ALOHA system. Then we have the possibility of reducing collisions. In multiple L-channel Slotted ALOHA system, interference from other networks’ jamming packets will transmit
to all L channels uniformly. Let the probability that i interference from other networks’ jamming packets out of m jamming
packets be transmitted at the same slot of an L-channel Slotted
ALOHA system
mÀi
  i 
m
1
1
ð2Þ
INmji ¼

L
L
i
Now form total probability theory, the probability that i
interference from other networks’ jamming packet are transmitted to the same slot is
  i 
mÀi
1
X
Jm ÀI m
1
1
ðI=LÞi ÀðI=LÞ
INi ¼
e
e

¼
ð3Þ
m!
i!
L
L
i
m¼i

The attacking noise packets can also collide with message
packets to reduce the performance of Slotted ALOHA. Therefore, attacking signals are made to produce dummy packets/
noise packets of the same size to increase the collision farther
[22,23]. In addition, assume that the attacking signals are not
producing noise packets in each slot for two reasons. First, it
will be detected immediately and will be removed. Second, it
will dissipate more energy and will die soon. Let the attacking
packet arrival to the base station be also Poisson Point Process
with an average rate of J packet per time slot. The probability
that n packets are transmitted to the same slot from the attacking node (or nodes) is
An ¼

Jn ÀJ
e
n!

ð4Þ

In multiple L-channel Slotted ALOHA system, the attacker
packets need to transmit all L channels separately. The attacker should spread its energy evenly over all degrees of freedom
in order to minimize the average capacity [2,3]. Let us assume
that the attacking packets also transmitted at L parallel Slotted
ALOHA channels to increase the collision. The effect of receiver noise has not been considered in this analysis, since it is
very small compared to the collision.
The probability that j attacking noise packets out of n
attacking noise packets will be transmitted at the same slot
of an L-channel Slotted ALOHA system is
Anjj ¼

nÀj
  j 
n
1
1

L
L
j

ð5Þ

From total probability theory, the probability that j attacking noise packets are transmitted to the same slot is
  j 
nÀj
1
X
Jn ÀJ n
1
1
ðJ=LÞj ÀðJ=LÞ
Aj ¼
e
e

¼
ð6Þ
n!
j!
L
L
j
n¼j
If the base station can receive only one message packet per
time slot in the presence of interfering packets from other networks and attacking noise packets, then the slot is considered
as successful. Let a maximum of r retransmission trials be allowed. Assume the retransmitted packets are also Poisson arrival [33]. Thus, the aggregate message packet arrival rate is
G packet per time slot. If any message packet also selects L
channels by random selection, the aggregate message packet
arrival rate per time slot is G/L. The system model and
assumptions is presented in Fig. 1.
Attacking signal
with rate J

Other networks’
jamminging signal I

Success

Active

λ

λ

L

L

Total rejection

(1 − α )

Retransmissions =

λα

λ

L

L

r

(1 − α )∑{1 − P( Su)}i
i =0

+
Yes
Retransmission rejection =

λ
L

Fig. 1

L- parallel
channels

G/L

+

Retransmission
trials<=r
No

(1 − α ){1 − P(Su)}

r +1

System model and assumptions.


210

J.H. Sarker and H.T. Mouftah

Probability of success
The radio channel is characterized by fading of the receiving
signal, resulting from vector addition of several reflected, scattered or diffracted multi-paths. The fading is assumed to be
slow, affecting all bits in a packet in the same way, and flat,
implying sufficiently low bit rates. With these assumptions
the received signal envelop r is constant over each packet
and approximately Rayleigh distributed [34]
 2
2r
r
fðrÞ ¼
r!0
ð7Þ
exp À
P0
P0

A message packet is successfully received in a time slot, if four
conditions are fulfilled. First, the receiver will select a message
packet in the presence of message packets from the same network, interfering packets from other networks and attacking
noise packets. Second, there exists the probability that the message packet is captured in the presence of interfering packets
from other networks. Third, there exists the probability that
the message packet is captured in the presence of other attacking
noise packets. Fourth, the probability that the message packet is
captured in the presence of other interfering message packets
from the same network exists. Therefore, the probability that a
message packet is successfully transmitted can be written as

PðSuÞ ¼ Pðthe selected packet will be a message packetÞ
à Pða message packet is captured in the presence of interfering packets from other networksÞ
à Pða message packet is captured in the presence of attacking noise packetsÞ
à Pða the message packet is captured in the presence of other interfering message packetsÞ
¼ PM Ã PCI Ã PCA Ã PCM

ð10Þ

where P0 is the average power of the received packets. The corresponding instantaneous power distribution (i.e., power distribution of the packets) can easily been shown to be [34]
fðpÞ ¼



1
p
p!0
exp À
P0
P0

ð8Þ

In the following analysis it is assumed that packet collision in a slot is the sole cause of packet loss. This, of
course, is not strictly true since deep fades also contribute
to packet loss, due to an increase error rate, even without
packet collision. In a well-designed system, the probability
of such events is generally order of magnitude smaller than
that of packet collision.
In a Rayleigh fading channel the probability that the
power of a message packet is higher than that of the power
of an attacking packet is ½ [32]. In the same way, it can be
shown that the probability that the power of an attacking
packet is higher than that of the power of a message packet
is also ½.
The probability that a test message packet will be selected from all three types of packets is the ratio of the total
number of message packets per time slot and the total number of message packets per time slot plus the total number
of interfering packets from other networks per time slot
and plus the total number of attacking noise packets per
time slot. Therefore, the probability that a selected test
packet is a message packet is

According to our assumption, the power distribution of a
message packet, the power distribution of an interfering packet
from other networks and the power distribution of an attacking packet are the same. The capture effect of a message packet
in the presence of interfering packets from other networks is
defined in the following way. In case of a message packet collision with interfering packets from other networks, a message
test packet is captured if its power is zf times higher than the
combined power of all interfering packets from other networks
transmitted on the same slot as message (selected by receiver)
packet is being transmitted, during a ‘certain section of time
slot’, to lock the receiver. Note that, capture ratio zf and
‘certain section of time slot’ both are affected by modulation
and coding technique [34]. Using the procedure presented in
Sarker and Mouftah [32], it can be shown that the probability
of a message packet is captured against all interfering packets
from other network transmitted to the same slot is

PCI ¼ exp À

I=L
1 þ 1=zf


ð11Þ

The capture effect of a message packet in the presence of
attacking packets is defined in the following way. In case of
a message packet collision with attacking packets, a message
test packet is captured if its power is za times higher than that
of all attacking interfering packets transmitted on the same
slot as message (selected by receiver) packet is being transmitted, defined as the attacking packet capture ratio, during a

PM ¼ Pðthe selected packet is a message packetÞ
¼

Total number of message packets
Total of message packets þ Total of interfering packets from other networks þ Total of attacking packets
1
P

1
P
a¼0

a

a ðG=LÞ
eÀðG=LÞ þ
a!

‘certain section of time slot’, to lock the receiver. The probability that a message packet is captured against all attacking
packets transmitted to the same slot is [32]

a

a ðG=LÞ
eÀðG=LÞ
a!

a¼0
1
P

b¼0

b

b ðI=LÞ
eÀðJ=LÞ þ
b!

G=L
G
¼
¼
G=L þ I=L þ J=L G þ I þ J

1
P
c¼0

c

c ðJ=LÞ
eÀðJ=LÞ
c!
ð9Þ


PCA ¼ exp À

J=L
1 þ 1=za


ð12Þ


Secured operating regions of Slotted ALOHA in the presence of interfering signals from other networks and attacking signals 211
The evaluation procedure of ‘‘za’’ is presented in Sarker and
Mouftah [32]. At the same way, a message test packet is captured, if its power is zm times higher than that of all other interfering packets transmitted from the same network defined as
the attacking packet capture ratio, during a ‘certain section
of time slot’, to lock the receiver. The probability that a message packet is captured against all other interfering packets
transmitted from the same network is [32,34]


G=L
PCM ¼ exp À
ð13Þ
1 þ 1=zm
Finally the probability of success of a message packet in the
presence of interfering packets transmitted from the other networks, attacking noise packets and interfering packets transmitted from the same network is


G
I=L
exp À
PðSuÞ ¼ PM Ã PCI Ã PCA Ã PCM ¼
GþIþJ
1 þ 1=zf




J=L
G=L
exp À
:
ð14Þ
 exp 1 þ 1=za
1 þ 1=zm
The probability of failure of any message packet is
1 À PðSuÞ. This unsuccessful part Lk ð1 À aÞf1 À PðSuÞg will be
transmitted during first retransmission time. The probability
of two successive failures is f1 À PðSuÞg2 . So the second time
retransmission part is Lk ð1 À aÞf1 À PðSuÞg2 , and so on. In general, kth time retransmission part is Lk ð1 À aÞf1 À PðSuÞgk . Let
the total number of retransmissions of a packet be r (one transmission followed r retransmission trials). The total mean offered
traffic from all active users is then given by
r
G X
k
¼
ð1 À aÞf1 À PðSuÞgk
L k¼0 L

ð15Þ

Simplifying Eq. (15) and combining with Eq. (14), we can
write
G
k
G
G
PðSuÞ ¼ ð1 À aÞ½1 À f1 À PðSuÞgrþ1 Š )
L
L
L GþIþJ






I=L
J=L
G=L
 exp exp À
exp À
1 þ 1=zf
1 þ 1=za
1 þ 1=zm
&


k
G
I=L
exp À
¼ ð1 À aÞ 1 À 1 À
L
GþIþJ
1 þ 1=zf



'rþ1 #
J=L
G=L
ð16Þ
exp À
 exp 1 þ 1=za
1 þ 1=zm
Eq. (16) is the basic equation of retransmission cut-off and
new packet rejection algorithm of multiple L-channels Slotted
ALOHA in the presence of interfering packets transmitted
from other networks and attacking noise packets.
Security improvement using multiple channels and capture
effects
The probability of success of L-channel Slotted ALOHA system in the presence of interfering packets transmitted from
other networks and attacking noise packets is derived in Eq.
(14). The throughput per slot of L-channel Slotted ALOHA
system is defined as the multiplication of average traffic arrival
rate per time slot and the probability of success in the presence
of interfering packets transmitted from other networks and
attacking noise packets. Thus, the throughput is







G
G2
I=L
J=L
PðSuÞ ¼
exp À
exp À
L
1 þ 1=zf
1 þ 1=za
LðG þ I þ JÞ
&


G=L
k
G
¼ ð1 À aÞ 1 À 1 À
 exp 1 þ 1=zm
L
GþIþJ





'rþ1 #
I=L
J=L
G=L
 exp ð17Þ
exp À
exp À
1 þ 1=zf
1 þ 1=za
1 þ 1=zm

Eq. (17) is the basic equation for the throughput of a message packet. Articulately, the new packet generation rate k,
number of channels L, new packet rejection probability a, capture ratios, zf , za , zm , interfering packets from other networks’
generation rate, I, attacking signal generation rate, J, and
number of retransmission trials, r, play important role in this
equation.
Fig. 2 shows the throughput of Slotted ALOHA in the presence of interfering packets from other networks and attacking
signals. But in this section, we will limit our discussion only to
the effect of L-channels and capture ratios. Therefore, we will
consider only the first two methods of secured transmission in
Slotted ALOHA. The first method is to use multiple channels
and the second method is to lower the capture ratios.
Fig. 2 shows the throughput per slot, S with the variation of
aggregate message packet arrival rate, G for different values of
attacking packets rates of J. From Fig. 2 we can make the following conclusions:
1. The throughput per slot S of 1-channel without capture
effects, za ¼ 1; zf ¼ 1; zm ¼ 1, Slotted ALOHA system
is very low in the presence of interfering signals from other
networks and attacking noise packet signal (Fig. 2b comparing with Fig. 2a). Because of that the current 1-channel
without capture effects Slotted ALOHA based networks [9–
17] can be shut down very easily. A lower message capture
ratio, zm ¼ 1, can increase the channel throughput significantly at all traffic load (Fig. 2c).
2. A lower interfering capture ratio, zf ¼ 1, can increase the
channel throughput slightly. A lower interfering capture
ratio is only effective, if the interfering signals rate from
other networks, I is high (Fig. 2 d comparing with Fig. 2c).
3. Similarly, a lower attacking capture ratio, za ¼ 1, can
increase the channel throughput slightly. If the attacking
signals rate, J is high only then a lower attacking capture
ratio is effective (Fig. 2e comparing with Fig. 2d).
4. If 5-channels are used instead of 1-channel then the
throughput per slot increases significantly, even under the
high interfering signals from other networks and attacking
signals (Fig. 2f comparing with Fig. 2b).
5. Since the throughput per slot, S, does not collapse even
with a high interfering signals rate from other networks, I
and attacking noise packet generation rate, J, with a lower
message capture ratio, zm ¼ 1, the security of Slotted
ALOHA system can be enhanced by lowering the capture
ratios Fig. 2c–e.
6. Since the throughput per slot, S, does not collapse with a
high message packet arrival rate, G, even with a high
interfering signal rate from other networks, I and a high
attacking noise packet generation rate, J, with a higher
number of channels, L, the security of Slotted ALOHA
system can be enhanced using multiple L-Slotted ALOHA
channels.


212

J.H. Sarker and H.T. Mouftah
0.6

0.6

z a = ∞ , z f = ∞, z m = ∞
0.5

I=0, L=1

0.4

Throughput per slot S

J=0
0.1
0.2
0.3
0.5
1

0.3
0.2
0.1
0

0

3

6

9

12

0.4
J=0
0.3

0.1
0.2

0.2

0.3
0.5

0.1

1

0

15

0

3

(a) 1-channel, without interference, without capture

12

0.2
0.4

0.3

0.3

0.5

0.2

1

3

6

9

12

15

Message packet generation rate G

(d) interfering packet capture
Fig. 2

0.5
0.2

1

0

zf =1

6

9

12

15

(c) message packet capture

zm = 1

0.6

z a = ∞, z f = ∞, z m = ∞

I=0.3, L=1

I=0.3, L=5

0.5

J=0
0.1
0.2

0.4

0.3

0.3

0.5

1

0.2

0

3

Message packet generation rate G

J=0
0.1
0.2
0.3
0.5
1

0.4
0.3
0.2
0.1

0.1

0.1

0

0.3

0.3

0

15

0.5

Throughput per slot S

Throughput per slot S

9

z a = 1, z f = ∞, z m = 1

z a = ∞, z f = 1, z m = 1
I=0.3, L=1

0.1

0.2

(b) I=0.3

J=0

0.5

0.4

0.1

0.6

0.6

I=0.3, L=1

0.1

Message packet generation rate G

Message packet generation rate G

0

6

z a = ∞, z f = ∞, z m = 1

J=0

0.5

I=0.3, L=1

Throughput per slot S

Throughput per slot S

0.5

z a = ∞, z f = ∞ , z m = ∞
Throughput per slot S

0.6

0

0

3

6

9

12

15

Message packet generation rate G

(e) attacking packet capture

za = 1

0

3

6

9

12

15

Message packet generation rate G

(f) 5-channel, without capture

Throughput per slot with the variation of message packet arrival rate.

7. There exists an optimum point where throughput per time
slot, S, is maximum for given values of message packet generation rate, G, interfering signals rate from other networks, I, and attacking packet generation rate, J.
Differentiating Eq. (17) with respect to attacking noise
packet generation rate, J, we obtain

the increase of attacking noise packet generation rate, J. Exactly in the same way, it can be shown that for any positive value of message packet generation rate, G, and the attacking
noise packet generation rate, J, the throughput, S, decreases
with the increase of interfering packets arrival rate from other
networks, I. However, the numerical results of these two results have already been depicted in Fig. 2.




" 




#
dS
I=L
G=L
1=L
G2
J=L
G4
J=L
exp À
exp À
À
À
exp
À
¼ exp À
dJ
1 þ 1=zf
1 þ 1=zm
1 þ 1=za LðG þ J þ IÞ
1 þ 1=za
1 þ 1=za
LðG þ J þ IÞ2
"
#

 
 
 

I=L
G=L
J=L
1=L
G2
G4
À
¼ exp À
exp À
exp À
À
1 þ 1=zf
1 þ 1=zm
1 þ 1=za
1 þ 1=za LðG þ J þ IÞ LðG þ J þ IÞ2
ð18Þ
It is clear from Eq. (18) that for any positive value of message packet generation rate, G, and interfering packets arrival
rate from other networks, I, the throughput, S, decreases with

Differentiating Eq. (17) with respect to message packet generation rate, G, we get


 
"




#
dS 1
J=L
I=L
G2
G=L
1=L
ðG þ J þ IÞð2GÞ À G2
G=L
¼ exp À
exp À
exp À
À
þ
exp À
dG L
1 þ 1=za
1 þ 1=zf G þ J þ I
1 þ 1=zm
1 þ 1=zm
1 þ 1=zm
ðG þ J þ IÞ2
!






1
J=L
G=L
G
G=L
ðG þ J þ IÞð2Þ À G
À
¼ exp À
exp À
þ
L
1 þ 1=za
1 þ 1=zm G þ J þ I
1 þ 1=zm
ðG þ J þ IÞ
ð19Þ


Secured operating regions of Slotted ALOHA in the presence of interfering signals from other networks and attacking signals 213
The optimum throughput per time slot is shown in Figs. 3
and 4 using Eq. (21). Figs. 3 and 4 show that the optimum
throughput can be increased significantly using lower capture

Now putting the differentiation result Eq. (19) equal to
zero, we obtain the optimum value of the message packet arrival rate from all active mobile nodes,

Gopt ¼

fLð1 þ 1=zm Þ À ðJ þ IÞg Æ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
fLð1 þ 1=zm Þ À ðJ þ IÞg2 þ 8LðJ þ IÞð1 þ 1=zm Þ

ð20Þ

2

ratios and multiple channels. The conclusions of Figs. 3 and
4 are almost same as the conclusions drawn from Fig. 2.

Using the value of optimum message packet arrival rate,
Gopt , in Eq. (17), we can obtain the optimum throughput per
time slot as







ðGopt Þ2
J=L
I=L
Gopt =L
exp À
exp À
exp À
1 þ 1=za
1 þ 1=zf
1 þ 1=zm
LðGopt þ J þ IÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2
fLð1 þ 1=zm Þ À ðJ þ IÞg þ fLð1 þ 1=zm Þ À ðJ þ IÞg2 þ 8LðJ þ IÞð1 þ 1=zm Þ
¼ &
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi'
2L Lð1 þ 1=zm Þ þ ðJ þ IÞ þ fLð1 þ 1=zm Þ À ðJ þ IÞg2 þ 8LðJ þ IÞð1 þ 1=zm Þ

Sopt ¼


 exp 
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3
2



ÀfLð1 þ 1=zm Þ À ðJ þ IÞg þ fLð1 þ 1=zm Þ À ðJ þ IÞg2 þ 8LðJ þ IÞð1 þ 1=zm Þ
J=L
I=L
5
exp À
exp4
1 þ 1=za
1 þ 1=zf
2Lð1 þ 1=zm Þ

0.8

0.8

0.7

z a = ∞, z f = ∞

0.7

z a = ∞, z f = ∞

0.7

0.6

z m = ∞, I = 0

0.6

zm = ∞, I

0.6

0.5
0.4

J=0

0.3

1

2

0.2

5
10
20

0.1
0

10

20

30

40

0.5

J=0

0.3

5

10

5
10
20

0.3
0.2

z a = ∞, z f = 1

0.1

z m = 1, I = 2
10

20

30

40

50

L

(d) interfering packet capture
Fig. 3

20

0.3

z m = 1, I

0

10

20

30

40

50

0

10

20

1

30

J=0

zf =1

5
10
20

0.4
0.3
0.2

z a = 1, z f = ∞

0.1

z m = 1, I = 2

0

0

10

20

50

zm = 1

0.8

1

0.7

0.5

40

(c) message packet capture

2

0.6

= 2

L

(b) with interference, I=2
J=0

0.6

0

0.4

0

0.7

0.4

10

Number of channels L

2

0.5

5
0.5

z a = ∞, z f = ∞

0

Throughput per slot Sopt

0.7

2

0.1

20

0.8

1

1

0.1
50

0.8

J=0

J=0

0.2

0.2

(a) without interference, without capture

Throughput per slot Sopt

2

0.4

Number of channels L

0

1

= 2

Throughput per slot Sopt

0

0.8

Throughput per slot Sopt

Throughput per slot Sopt

Throughput per slot Sopt

ð21Þ

30

Number of channels L

(e) attacking packet capture

40

50

2
5

0.6

10
0.5

20

0.4
0.3
0.2

z a = 1, z f = 1

0.1

z m = 1, I = 2

0

0

10

20

30

40

L

za = 1

(f) with capture effects

The maximum throughput, Sopt with the variation of number of channels L.

50


214

J.H. Sarker and H.T. Mouftah

Now the question that may arise is: what is the optimum
number of channels, L that provides maximum throughput.
To answer this question, the optimum L can be obtained by
setting Eq. (19) is equal to zero. Therefore, the optimum number of channels is
Lopt ¼

GðG þ J þ IÞ
ð1 þ 1=zm ÞðG þ 2J þ 2IÞ

ð22Þ

rate from all active users in a given time slot is less than or
equal to the optimum packet arrival per time slot, the system
is secured. This assumption is reasonable for Slotted ALOHA
system [33].
The optimum throughput per slot of L-channels Slotted
ALOHA system with and without limiting the number of
retransmission trials can be obtained from Eqs. (18 and 21) as







È
Érþ1 i
kopt h
ðGopt Þ2
J=L
I=L
Gopt =L
1 À 1 À Popt ðSuÞ
exp À
¼
exp À
exp À
L
LðGopt þ J þ IÞ
1 þ 1=za
1 þ 1=zf
1 þ 1=zm
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2
fLð1 þ 1=zm Þ À ðJ þ IÞg þ fLð1 þ 1=zm Þ À ðJ þ IÞ2 þ 8LðJ þ IÞgð1 þ 1=zm Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼
2LfLð1 þ 1=zm Þ þ ðJ þ IÞ þ fLð1 þ 1=zm Þ À ðJ þ IÞg2 þ 8LðJ þ IÞð1 þ 1=zm Þg
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3
2




fLð1 þ 1=zm Þ À ðJ þ IÞg þ fLð1 þ 1=zm Þ À ðJ þ IÞg2 þ 8LðJ þ IÞð1 þ 1=zm Þ
J=L
I=L
5
exp À
exp 4À
 exp 1 þ 1=za
1 þ 1=zf
2Lð1 þ 1=zm Þ

Sopt ¼

ð23Þ
Eq. (22) shows that the optimum number of channels, Lopt ,
increases linearly with the increase of aggregate message traffic
arrival rate, G. The Lopt , increases further with the increase of

The optimum probability of success can be obtained from
Eqs. (23) and (20) as







Sopt =L
Gopt
J=L
I=L
Gopt =L
exp À
exp À
exp À
Popt ðSuÞ ¼
¼
Gopt
1 þ 1=za
1 þ 1=zf
1 þ 1=zm
Gopt þ J þ I
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Lð1 þ 1=zm Þ À ðJ þ IÞ þ fLð1 þ 1=zm Þ À ðJ þ IÞg2 þ 8LðJ þ IÞð1 þ 1=zm Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼
Lð1 þ 1=zm Þ þ ðJ þ IÞ þ fLð1 þ 1=zm Þ À ðJ þ IÞg2 þ 8LðJ þ IÞð1 þ 1=zm Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3
2




fLð1 þ 1=zm Þ À ðJ þ IÞg þ fLð1 þ 1=zm Þ À ðJ þ IÞg2 þ 8LðJ þ IÞð1 þ 1=zm Þ
J=L
I=L
5
exp À
exp 4À
 exp 1 þ 1=za
1 þ 1=zf
2Lð1 þ 1=zm Þ
ð24Þ
message packet capture ratio, zm . However, the same decreases
with the increase of interfering packet arrival rate from other
networks, I, or/and attacking packet arrival rate, J.
Security improvement by limiting the number of retransmission
trials
In a normal data transmission system, every packet must be
transmitted successfully. On the other hand, in the case of contention based access protocol or for real-time data transmission, we can cut the retransmission number, which will avoid
the undesirable stability or security problem of Slotted ALOHA [35]. Over a long time period, the total offered traffic load
G will have an optimum value Gopt depending on the other
parameters like new packet generation rate per time slot and
the number of retransmission trials. In the case of access or
real-time traffic transmission packets are identical in nature
for each user and the access procedure is limited by time.
For a secured operation of L-channels Slotted ALOHA type
system, with a higher value of new packet generation rate
per time slot, the retransmission trials should be controlled.
The purpose of the retransmission trial control is to get the
optimum value of offered traffic load from all users Gopt , which
will make the system secured or stable. Here, in this paper a
simplified assumption is considered: if the traffic generation

Therefore, the optimum throughput per slot of L-channels
Slotted ALOHA system by limiting the number of retransmission trials can be obtained from Eq. (17) as
h
i
k
Sopt ¼ opt
1 À f1 À Popt ðSuÞgrþ1
L
ð25Þ
k
Sopt
¼
or opt
rþ1
L
1Àf1ÀPopt ðSuÞg
where the values of Sopt and Popt ðSuÞ are given in Eqs. (23) and
(24), respectively. The main purpose of our system model is to
maximize the throughput per slot, S by adjusting the transmission trials, r and the new packet generation rate per slot, k=L,
for a given interfering packet arrival rate from other networks,
I, and attacking packet arrival rate, J. We have already derived
the maximum throughput of L-channels Slotted ALOHA system Sopt in Eq. (23). And it occurs when the aggregate traffic
generation rate, Gopt , which is shown in Eq. (20).
Eq. (25) is the basic equation for the secured transmission
method. The secured transmission method can be stated as follows: For a call establishment system design or for a real-time
traffic transmission, the time out is the most important parameter. This time out is the time to transmit the access information from mobile to base station plus the switching time. From
the value of the time to transmit the access information or realtime transmission plus the propagation delay, we can find the
maximum allowable retransmission trials, r, i.e., how many


Secured operating regions of Slotted ALOHA in the presence of interfering signals from other networks and attacking signals 215
retransmission trials are possible for a given time. From this
value of r, L, I, J, za , zf and zm we can find the optimum
new packet generation rate per time slot, kopt =Lper time slot
using Eqs. (23)–(25). Fig. 5 shows the variation of optimum
new packet generation rate per time slot, kopt =L, with the variation of number of channels, L, using Eqs. (23)–(25).
Without any retransmission attempts (r ! 0 the optimum
new packet generation rate per time slot can be obtained using
Eq. (25) as
 
Lð1 þ 1=zm Þ À ðJ þ IÞ þ
kopt
Sopt
¼
¼
L r!0 Popt ðSuÞ

UR ¼


Lð1 þ 1=zm Þ À ðJ þ IÞ þ
1
Sopt
¼
1 À a Popt ðSuÞ

Ar ¼



kopt
À
L
r!0


¼

Lð1 þ 1=zm Þ À ðJ þ IÞ þ

ð30Þ

2Lð1 À aÞ
ð27Þ



Fig. 6 shows clearly that by increasing the value of a (new packet
rejection probability), the secured operating regions with and
without retransmission cut-off can be increased significantly.
From Fig. 6, it can be said that the maximum value of the new
packet generation rate per slot, kopt =L with new packet rejection
is

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
fLð1 þ 1=zm Þ À ðJ þ IÞg2 þ 8LðJ þ IÞð1 þ 1=zm Þ

where the value of Sopt is given in Eq. (23). Therefore, the security improvement area by limiting the number of retransmission trials, r is

kopt
L

ð26Þ

2L

 
kopt
¼ Sopt
L r!1



The secured operating region of L-channel Slotted ALOHA system with and without limiting the retransmission trials is depicted in Fig. 6. Please note that here the y-axis should be
Gopt
. The value of Gopt is given in Eq. (20).
multiplied by XÃ ¼ 1Àa

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
fLð1 þ 1=zm Þ À ðJ þ IÞg2 þ 8LðJ þ IÞð1 þ 1=zm Þ

Eq. (26) shows the limit of the proposed solution with
retransmission cut-off scenario.
In the other extreme, without any retransmission cut-off
(r fi 1) the optimum new packet generation rate per time slot
can be obtained using Eq. (25) as



slot G/L, by limiting the number of retransmission trials and
new packet rejection is shown in Eq. (16). Combining Eqs.
(16 and 17) and after simplification we can write


kopt
1
Sopt
¼
ð29Þ
È
É
L
1 À a 1 À 1 À Popt ðSuÞ rþ1

Comparing Eqs. (26) and (30), the upper limit of the new packet generation rate per slot with new packet rejection is 1=ð1 À aÞ
times higher than that of the without new packet rejection.
On the other hand the new packet generation rate per slot,
without limiting the number of retransmission trials with new
packet rejection is

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
fLð1 þ 1=zm Þ À ðJ þ IÞg2 þ 8LðJ þ IÞð1 þ 1=zm Þ

r!1

The shaded parts indicated in Fig. 5 show the secured region by limiting the number of retransmission trials, r. The
lower most parts of the figures show the secured transmission
region without limiting the number of retransmission trials.
Increasing the number of channel, L or/and reducing the capture ratios, za , zf and zm are not enough to obtain a higher secured transmission operating region. Limiting the number of
retransmission trials can increase the secured transmission
operating region significantly.
Security improvement by new packet rejection
The main purpose of this paper is to obtain the secured transmission of L-channel Slotted ALOHA system. It is already
shown that if L-channel Slotted ALOHA system provides
maximum throughput then the system is secured. If limiting
the retransmission trials is not sufficient for obtaining a secured stabilized L-channel Slotted ALOHA system, then it
can be achieved by the expense of newly generated packet
rejection.
The maximum throughput per slot of a L-channel Slotted
ALOHA is Sopt is derived in Eq. (23), and it occurs when the
aggregate traffic generation rate, Gopt , which is shown in Eq.
(20). The aggregate message packet generation rate per time

2L

LR ¼

Sopt
1Àa

À Sopt

ð28Þ

ð31Þ

It can be said that the new packet generation rate per slot
without limiting the number of retransmission trials with
new packet rejection is 1=ð1 À aÞ times higher than that of
the without new packet rejection, by comparing Eqs. (27)
and (31).
From Fig. 6, we can conclude that, the aggregate message
packet generation rate, G, never reaches its optimum point,
if the new packet generation rate per slot, kopt =L, is less than
LR packet per time slot. The reason is that, we started to get
the result of Eq. (31) with the aggregate message packet generation rate, Gopt . So, it is unnecessary to control the retransmission attempt for a secured operation of L-channels Slotted
ALOHA, if the new packet generation rate per slot, kopt =L,
is less than LR packet per time slot, where a is the newly generated packet rejection probability.
Conclusions
In this paper, an analytical approach for secured operating regions of Slotted ALOHA in the presence of interfering signals
from other networks and DoS attacking signals has been


216

J.H. Sarker and H.T. Mouftah

L =20
10

5
3
2
1

L =20

L =20

L =20

10

10

10
5

5
5

3
3

3

2
2

2

1

1

1

Fig. 4

The maximum throughput, Sopt .

investigated. The performance evaluations presented in this
paper are based on the numerical analysis.
The security improvement of L-channels Slotted ALOHA
in the presence of interfering signals from other networks
and random attacking noise packets signals is studied in this
paper. The current security protected measures such as encryption makes the packets unreadable by unauthorized users. The
authentication technique is used to protect the system from
illegal users and authorization separates the legal users. However, in a Slotted ALOHA based network, the interfering signals from other networks and the random packet destruction
DoS attacking noise packets may collide with message packets
and reduces the secured transmission. Therefore, the current
security measures such as encryption, authentication and
authorization cannot prevent those types of attack. One of
the main drawbacks of Slotted ALOHA protocol is its excessive collisions.
In this paper, we have used four different techniques for
security improvement of Slotted ALOHA by reducing the
collisions. Since the interfering signals from other networks
and the random packet destruction DoS attacking noise
packet increase the collision, we intend to use multiple channels in the Slotted ALOHA protocol to reduce the collisions
in the first technique. The use of multiple channels in the
Slotted ALOHA protocol reduces three types of packet colli-

sions. First type of collision is the collision between two or
more message packets. The second type of collision is the collision between a message packet and one or more interfering
packets from other networks. The third type of collision is
the collision between a message packet and one or more
other attacking noise packets.
In the second security improvement technique, we have
shown the effects of capture ratios in the presence of interfering signals from other networks and the random packet
destruction DoS attacking noise packet. A lower message capture ratio can increase the throughput and maximum throughput significantly. A lower interfering capture ratio can increase
the throughput and maximum throughput only if the rate of
interfering signals from other networks’ packets rate is high.
Exactly same conclusion is applied for a lower attacking capture ratio.
In the third technique, we have used retransmissions cut-off
by limiting the number of retransmission trials. The retransmissions cut-off technique can limit the aggregate packet flow
and form the optimum message packet flow in the presence of
interfering signals from other networks and the attacking noise
packet. It is possible that the third technique called retransmissions cut-off technique is not enough to control the flow of
message packets. Because of that the fourth technique called
new packet rejection probability is introduced. The secured


4

z a = ∞, z f = ∞ , z m = ∞
I = 0, J = 0

3.6
3.2
2.8
2.4
2

Unsecured operating region

1.6

r =0

1.2
Secured region with limiting r

0.8
0.4
0

r =∞

Secured region without limiting r
1

11

21

31

41

z a = ∞, z f = ∞ , z m = ∞

3.6
3.2

I = 2.5, J = 0

2.8
2.4
2
Unsecured operating region

1.6

r =0

1.2
Secured region with limiting r

0.8

r =∞

0.4
0

51

1

Number of channels L

Secured region without limiting r
11
21
31
41
51

Optimum new packet generation
rate per slot λopt / L

4

Optimum new packet generation
rate per slot λopt / L

Optimum new packet generation
rate per slot λopt / L

Secured operating regions of Slotted ALOHA in the presence of interfering signals from other networks and attacking signals 217
4

z a = ∞, z f = ∞ , z m = ∞

3.6
3.2

I = 2.5, J = 2.5

2.8
Unsecured operating region

2.4
2
1.6

r =0

1.2
0.8

Secured region with limiting r

r=∞

0.4
0

Secured region without limiting r
1

11

21

31

41

51

Number of channels L

Number of channels L

(a) z a = ∞ , z f = ∞ z m = ∞ , I=0, J=0 (b) z a = ∞ , z f = ∞ z m = ∞ , I=2.5, J=0 (c) z a = ∞ , z f = ∞ z m = ∞ , I=2.5, J=2.5
4
3.6

z a = ∞ , z f = ∞, z m = 1

3.2

I = 2.5, J = 2.5

2.8
Unsecured operating region

2.4

r =0

2
Secured region with limiting r

1.6
1.2

r =∞

0.8
0.4
0

Secured region without limiting r
1

11

21

31

41

z a = ∞, z f = 1, z m = 1

3.6
3.2

I = 2.5, J = 2.5

2.8
Unsecured operating region

2.4

r =0

2
1.6

Secured region with limiting r

1.2

r=∞

0.8
0.4

51

0

Secured region without limiting r
1

11

21

31

41

51

Optimum new packet generation
rate per slot λopt / L

4

Optimum new packet generation
rate per slot λopt / L

Optimum new packet generation
rate per slot λopt / L

4

z a = 1, z f = 1, z m = ∞

3.6
3.2

I = 2.5, J = 2.5

2.8
2.4
Unsecured operating region

2
1.6

r =0

1.2
0.8

Secured region with limiting r

0.4
0

1

Number of channels L

Number of channels L

r =∞

Secured region without limiting r
11

21

31

41

51

Number of channels L

(d) z a = ∞ , z f = ∞ , z m = 1 , I=2.5, J=2.5 (e) z a = ∞ , z f = 1 , z m = 1 , I=2.5, J=2.5 (f) z a = 1 , z f = 1 , z m = ∞ , I=2.5, J=2.5
Fig. 5

Security improvement by limiting the retransmission trials.

References

2.4

Optimum new packet generation
rate per slot λopt / L

2.2
2
1.8

Unsecured operating region

1.6

X*

1.4
1.2

Secured region
with limiting r

1
0.8

Gopt

0.6
0.4

0

Sopt

Secured region
without limiting r

0.2
1

10

100

Retransmission trials r+1

Fig. 6 Secured and unsecured operating regions of multichannel
L-Slotted ALOHA in the presence of interfering signals from
other networks and attacking noise packets. The y-axis should be
Gopt
.
multiplied by XÃ ¼ 1Àa

operating regions can be increased 1=ð1 À aÞ times by using
this fourth technique. Where a is the new packet rejection
probability.

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