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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 2011Available online 14 May 2011
*
56
E-
uo
20
El
Pe
do
KEYWORDS
Ad Hoc networks;
Attacking noise packets;
Interfering signals;
Multiple channels;
New packet rejection;
Retransmission trials;
Other networks;
Sensor networks;
Slotted ALOHA
Corresponding author. Tel.:
2 5664.
mail addresses: jsarker@site
ttawa.ca (H.T. Mouftah).
90-1232 ª 2011 Cairo Un
sevier B.V. All rights reserve
er review under responsibilit
i:10.1016/j.jare.2011.04.008
Production and h
+1 613
.uottawa
iversity.
d.
y of Cair
osting by E
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 sys-
tems 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 enhance-
ment 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.
562 5800x2173; fax: +1 613
(J.H. Sarker), mouftah@site.
Production and hosting by
o University.
lsevier
Introduction
To improve the secured transmission over vulnerable wirelessnetworks, assessment of the wireless multiple access schemesin the presence of jamming or attacking signals is an important
issue [1]. It is well known that the Code Division Multiple Ac-cess (CDMA) system has a special resistance against the inter-ference signals from other networks and the attacking signals.
Thus the CDMA scheme may be the first choice as a multipleaccess scheme in the presence of interference signals from othernetworks or/and attacking signals. The attacker should spread
its energy evenly over all degrees of freedom in order to mini-mize the average capacity of the original signals [2,3]. In a sim-
208 J.H. Sarker and H.T. Mouftah
plified CDMA transmission system, with the knowledge of
spreading code, the receiver is able to detect the users’ signalsfrom interfering signals from other networks and attacking sig-nals. Using the attacker state information and the effects offading, the channel capacity can be enhanced further. For
enhancing uplink channel capacity, the attacker state informa-tion 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. Aspecific Frequency Hoping Speed Spectrum (FHSS) techniquecan 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 intro-duced and analyzed to improve the system capacity [5]. The
mobile nodes can exploit channel diversity in order to createwormholes 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 anyestablished network infrastructure. As a result, low powerconsumption systems are becoming important for infrastruc-
ture-less wireless Ad Hoc and sensor networks. The SlottedALOHA is the most spectral and power efficient multiple ac-cess scheme [7,8]. Although, the CDMA has especial resistance
against interference and attacking signals, Slotted ALOHA is awidely used random access protocol not only for its simplicityalso for its higher spectral and power efficiency.
The Slotted ALOHA multiple access schemes is used exclu-
sively in newly developed Radio Frequency Identification(RFID) technology [9]. The Slotted ALOHA is also used asa part of different multiple access protocols especially for the
control channels in many new wireless technologies. For in-stance, it is used in the random access channels of Global Sys-tem for Mobile (GSM) communications [10] and its extension
General Packet Radio Services (GPRS) [11,12], WidebandCode 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 inthe signaling channels, instead of attacking whole channels[18–20]. Therefore, defending the control channels from exter-nal and internal attacks [21] are very important issue. If the to-
tal network is based on Slotted ALOHA based protocol, thendefending the network against the DoS attack is one of themost important factors [22,23] and has been discussed in this
paper.A special type of Denial of Service (DoS) attack, called ran-
dom packet destruction that works by transmitting short peri-
ods of noise signals is considered as attacking signals. Thisrandom packet destruction DoS packets can effectively shutdown 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 excessivecollisions at higher traffic load condition. The current anti-attack measures such as encryption, authentication and autho-
rization [24,25] cannot prevent these types of attacks. Since therandom packet destruction DoS packets increase the collisionfurther, the receiver cannot read the message packets.
The effect of attacking noise packet signals on the SlottedALOHA 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 dy-namic channel load and jamming information are needed to
maximize the channel throughput, which makes the system
implementation difficult. Recently, there has been an increas-ing interest in the autonomic networks, i.e., networks shouldbe 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. Aself-stabilized random access protocol in the presence of ran-
dom packet destruction DoS attack for infrastructure-lesswireless autonomic networks is presented in Sarker andMouftah [32]. In this paper we have investigated the combined
effect of the interfering signals from other networks and theDoS attacking signals on Slotted ALOHA. Three differenttypes of noises are considered in this paper. First, noise related
to interfering packets from the same network. Second, noiserelated to interfering packets from the other networks andthird, 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 ofthe interfering signals from other networks and the ran-
dom 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’ signalrate. Similarly, the throughput decreases with theincrease of the random packet destruction DoS attack-ing packet rate.
(3) A sufficient number of channels can prevent the shut-down of Slotted ALOHA in the presence of interferingsignals 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 pack-ets’ power for a certain section of time slot to lock thereceiver. Similarly, a message packet is captured, in the
presence of interfering packets from other networks, ifits power is higher than the interfering capture ratio timesof the power of the interfering packets from other net-works. At the same way, a message packet is captured,
in the presence of attacking noise packets, if its poweris higher than the attacking capture ratio times of otherattacking noise packets’ power. Results show that a
lower value of the message capture ratio is the mosteffective solution comparing with the interfering packetcapture ratio or the attacking packet capture ratio.
(5) The approximate value of the number of channels thatprovides 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 modeland assumptions are described in the next section. The thirdsection shows the security improvement using multiple chan-
nels and capture effects. The security improvement by limitingthe number of retransmission trials is evaluated in the fourthsection. The fifth section presents the security improvement
by new packet rejection. The conclusion is provided in the lastsection.
ActiveL- parallel channels
Retransmission trials <=r
+
+
Total rejection
Retransmission rejection =
Retransmissions =
Yes
No
Success
G/L( )αλ −1L
L
λα
L
λ
( ) { }∑=
−−r
i
iPL 0
)Su(11 αλ
( ){ } 1)Su(11 +−− rPL
αλ
Attacking signal with rate J
Other networks’ jamminging signal I
Fig. 1 System model and assumptions.
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 messagepacket arrival rate from all active mobile nodes per time slotis k packet per time slot. In Slotted ALOHA, the throughput
initially increases with the increase of the new packet genera-tion rate, k. The throughput reaches its maximum value fora certain value of the new packet generation rate from all ac-
tive nodes. The throughput collapse and reaches to zero, ifthe new packet generation rate increases further. The through-put collapse is known as the security or stability problem in
Slotted ALOHA. The reason for throughput collapse is exces-sive collision. The throughput collapse can be prevented byreducing the new packet arrival rate per slot. The packet rejec-
tion can provide one of the solutions and is considered in thispaper.
Assuming that the new packet rejection probability is a.The new packet transmission rate per time slot is kð1� aÞ.Let there be L parallel Slotted ALOHA based channels. Themobile nodes can transmit their packets selecting any of theL channels by random selection, without the knowledge of
other mobile units’ activeness. During the transmission ofpackets, each mobile node adjusts their packet size to fit intothe time slots. Since the average new message packet transmis-
sion rate from all active mobile nodes per time slot is kð1� aÞpacket per time slot and the channel selection is random, thenew packet transmission rate from all mobile nodes iskLð1� aÞ packets per time slot.
It is well known that the Slotted ALOHA’ performance isdegraded due to excessive collision. The interference fromother networks can produce packets to increase the collision
farther. Let the interference from other networks’ packet arri-val to the base station be Poisson Point Process with an aver-age rate of I packet per time slot. The probability that m
packets are transmitted to the same slot from other networksas jamming is
INm ¼Im
m!e�I ð1Þ
In the first collision reducing technique, we have used multi-ple parallel Slotted ALOHA Slotted channels instead of singlechannel Slotted ALOHA channel. For doing that the message
packets can be transmitted in a multiple L-channel SlottedALOHA system. Then we have the possibility of reducing col-lisions. In multiple L-channel Slotted ALOHA system, inter-
ference from other networks’ jamming packets will transmitto all L channels uniformly. Let the probability that i interfer-ence from other networks’ jamming packets out of m jammingpackets be transmitted at the same slot of an L-channel Slotted
ALOHA system
INmji ¼m
i
� �1
L
� �i
1� 1
L
� �m�ið2Þ
Now form total probability theory, the probability that iinterference from other networks’ jamming packet are trans-mitted to the same slot is
INi ¼X1m¼i
Jm
m!e�I
m
i
� �1
L
� �i
1� 1
L
� �m�i
¼ ðI=LÞi
i!e�ðI=LÞ ð3Þ
The attacking noise packets can also collide with message
packets to reduce the performance of Slotted ALOHA. There-fore, 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, itwill be detected immediately and will be removed. Second, itwill dissipate more energy and will die soon. Let the attacking
packet arrival to the base station be also Poisson Point Processwith an average rate of J packet per time slot. The probabilitythat n packets are transmitted to the same slot from the attack-
ing node (or nodes) is
An ¼Jn
n!e�J ð4Þ
In multiple L-channel Slotted ALOHA system, the attacker
packets need to transmit all L channels separately. The attack-er should spread its energy evenly over all degrees of freedomin order to minimize the average capacity [2,3]. Let us assume
that the attacking packets also transmitted at L parallel SlottedALOHA channels to increase the collision. The effect of recei-ver 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 slotof an L-channel Slotted ALOHA system is
Anjj ¼n
j
� �1
L
� �j
1� 1
L
� �n�jð5Þ
From total probability theory, the probability that j attack-ing noise packets are transmitted to the same slot is
Aj ¼X1n¼j
Jn
n!e�J
n
j
� �1
L
� �j
1� 1
L
� �n�j
¼ ðJ=LÞj
j!e�ðJ=LÞ ð6Þ
If the base station can receive only one message packet per
time slot in the presence of interfering packets from other net-works and attacking noise packets, then the slot is consideredas successful. Let a maximum of r retransmission trials be al-
lowed. Assume the retransmitted packets are also Poisson ar-rival [33]. Thus, the aggregate message packet arrival rate isG packet per time slot. If any message packet also selects Lchannels by random selection, the aggregate message packet
arrival rate per time slot is G/L. The system model andassumptions is presented in Fig. 1.
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, scat-tered or diffracted multi-paths. The fading is assumed to beslow, 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 packetand approximately Rayleigh distributed [34]
fðrÞ ¼ 2r
P0
exp � r2
P0
� �r � 0 ð7Þ
PðSuÞ ¼ Pðthe selected packet will be a message packetÞ� Pða message packet is captured in the presence of interfering packets from other networksÞ� Pða message packet is captured in the presence of attacking noise packetsÞ� Pða the message packet is captured in the presence of other interfering message packetsÞ
¼ PM � PCI � PCA � PCM ð10Þ
where P0 is the average power of the received packets. The cor-responding instantaneous power distribution (i.e., power dis-tribution of the packets) can easily been shown to be [34]
fðpÞ ¼ 1
P0
exp � p
P0
� �p � 0 ð8Þ
In the following analysis it is assumed that packet colli-sion in a slot is the sole cause of packet loss. This, of
course, is not strictly true since deep fades also contributeto packet loss, due to an increase error rate, even withoutpacket collision. In a well-designed system, the probability
of such events is generally order of magnitude smaller thanthat of packet collision.
In a Rayleigh fading channel the probability that the
power of a message packet is higher than that of the powerof an attacking packet is ½ [32]. In the same way, it can beshown that the probability that the power of an attackingpacket is higher than that of the power of a message packet
is also ½.The probability that a test message packet will be se-
lected from all three types of packets is the ratio of the total
number of message packets per time slot and the total num-ber of message packets per time slot plus the total numberof interfering packets from other networks per time slot
and plus the total number of attacking noise packets pertime slot. Therefore, the probability that a selected testpacket is a message packet is
PM ¼ Pðthe selected packet is a message packetÞ
¼ Total number of message packets
Total of message packetsþ Total of interfering packets from other networksþ Total of attacking packets
P1a¼0
a ðG=LÞa
a!e�ðG=LÞ
P1a¼0
a ðG=LÞa
a!e�ðG=LÞ þ
P1b¼0
b ðI=LÞb
b!e�ðJ=LÞ þ
P1c¼0
c ðJ=LÞc
c!e�ðJ=LÞ
¼ G=L
G=Lþ I=Lþ J=L¼ G
Gþ Iþ Jð9Þ
Amessage packet is successfully received in a time slot, if four
conditions are fulfilled. First, the receiver will select a messagepacket in the presence of message packets from the same net-work, interfering packets from other networks and attackingnoise packets. Second, there exists the probability that the mes-
sage packet is captured in the presence of interfering packetsfrom other networks. Third, there exists the probability thatthemessage packet is captured in the presence of other attacking
noise packets. Fourth, the probability that themessage packet iscaptured in the presence of other interfering message packetsfrom the same network exists. Therefore, the probability that a
message packet is successfully transmitted can be written as
According to our assumption, the power distribution of amessage packet, the power distribution of an interfering packetfrom other networks and the power distribution of an attack-ing packet are the same. The capture effect of a message packet
in the presence of interfering packets from other networks isdefined in the following way. In case of a message packet col-lision with interfering packets from other networks, a message
test packet is captured if its power is zf times higher than thecombined power of all interfering packets from other networkstransmitted on the same slot as message (selected by receiver)
packet is being transmitted, during a ‘certain section of timeslot’, to lock the receiver. Note that, capture ratio zf and‘certain section of time slot’ both are affected by modulationand coding technique [34]. Using the procedure presented in
Sarker and Mouftah [32], it can be shown that the probabilityof a message packet is captured against all interfering packetsfrom other network transmitted to the same slot is
PCI ¼ exp � I=L
1þ 1=zf
� �ð11Þ
The capture effect of a message packet in the presence ofattacking packets is defined in the following way. In case of
a message packet collision with attacking packets, a messagetest packet is captured if its power is za times higher than thatof all attacking interfering packets transmitted on the same
slot as message (selected by receiver) packet is being transmit-ted, defined as the attacking packet capture ratio, during a
‘certain section of time slot’, to lock the receiver. The probabil-ity that a message packet is captured against all attackingpackets transmitted to the same slot is [32]
PCA ¼ exp � J=L
1þ 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 cap-tured, if its power is zm times higher than that of all other inter-fering packets transmitted from the same network defined asthe attacking packet capture ratio, during a ‘certain section
of time slot’, to lock the receiver. The probability that a mes-sage packet is captured against all other interfering packetstransmitted from the same network is [32,34]
PCM ¼ exp � G=L
1þ 1=zm
� �ð13Þ
Finally the probability of success of a message packet in the
presence of interfering packets transmitted from the other net-works, attacking noise packets and interfering packets trans-mitted from the same network is
PðSuÞ ¼ PM � PCI � PCA � PCM ¼G
Gþ Iþ Jexp � I=L
1þ 1=zf
� �
� exp � J=L
1þ 1=za
� �exp � G=L
1þ 1=zm
� �: ð14Þ
The probability of failure of any message packet is1� PðSuÞ. This unsuccessful part k
Lð1� aÞf1� PðSuÞg will be
transmitted during first retransmission time. The probability
of two successive failures is f1� PðSuÞg2. So the second timeretransmission part is k
Lð1� aÞf1� PðSuÞg2, and so on. In gen-
eral, kth time retransmission part is kLð1� aÞf1� PðSuÞgk. Let
the total number of retransmissions of a packet be r (one trans-mission followed r retransmission trials). The totalmean offeredtraffic from all active users is then given by
G
L¼Xrk¼0
kLð1� aÞf1� PðSuÞgk ð15Þ
Simplifying Eq. (15) and combining with Eq. (14), we can
write
G
LPðSuÞ¼ k
Lð1�aÞ½1�f1�PðSuÞgrþ1�)G
L
G
GþIþJ
�exp � I=L
1þ1=zf
� �exp � J=L
1þ1=za
� �exp � G=L
1þ1=zm
� �
¼ kLð1�aÞ 1� 1� G
GþIþJexp � I=L
1þ1=zf
� ���
�exp � J=L
1þ1=za
� �exp � G=L
1þ1=zm
� ��rþ1#
ð16Þ
Eq. (16) is the basic equation of retransmission cut-off andnew packet rejection algorithm of multiple L-channels SlottedALOHA 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 sys-
tem in the presence of interfering packets transmitted fromother 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 arrivalrate per time slot and the probability of success in the presenceof interfering packets transmitted from other networks and
attacking noise packets. Thus, the throughput is
S¼ G
LPðSuÞ¼ G2
LðGþIþJÞ exp �I=L
1þ1=zf
� �exp � J=L
1þ1=za
� �
�exp � G=L
1þ1=zm
� �¼ kLð1�aÞ 1� 1� G
GþIþJ
��
�exp � I=L
1þ1=zf
� �exp � J=L
1þ1=za
� �exp � G=L
1þ1=zm
� ��rþ1#ð17Þ
Eq. (17) is the basic equation for the throughput of a mes-sage packet. Articulately, the new packet generation rate k,number of channels L, new packet rejection probability a, cap-ture ratios, zf, za, zm, interfering packets from other networks’generation rate, I, attacking signal generation rate, J, andnumber of retransmission trials, r, play important role in thisequation.
Fig. 2 shows the throughput of Slotted ALOHA in the pres-ence of interfering packets from other networks and attackingsignals. But in this section, we will limit our discussion only to
the effect of L-channels and capture ratios. Therefore, we willconsider only the first two methods of secured transmission inSlotted 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 fol-lowing conclusions:
1. The throughput per slot S of 1-channel without capture
effects, za ¼ 1; zf ¼ 1; zm ¼ 1, Slotted ALOHA systemis very low in the presence of interfering signals from othernetworks and attacking noise packet signal (Fig. 2b com-
paring with Fig. 2a). Because of that the current 1-channelwithout 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 signifi-cantly at all traffic load (Fig. 2c).
2. A lower interfering capture ratio, zf ¼ 1, can increase thechannel throughput slightly. A lower interfering capture
ratio is only effective, if the interfering signals rate fromother 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 attackingsignals rate, J is high only then a lower attacking captureratio is effective (Fig. 2e comparing with Fig. 2d).
4. If 5-channels are used instead of 1-channel then thethroughput per slot increases significantly, even under thehigh 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, Iand attacking noise packet generation rate, J, with a lower
message capture ratio, zm ¼ 1, the security of SlottedALOHA system can be enhanced by lowering the captureratios Fig. 2c–e.
6. Since the throughput per slot, S, does not collapse with ahigh message packet arrival rate, G, even with a highinterfering signal rate from other networks, I and a high
attacking noise packet generation rate, J, with a highernumber of channels, L, the security of Slotted ALOHAsystem can be enhanced using multiple L-Slotted ALOHA
channels.
Message packet generation rate G
Thr
ough
put p
er s
lot S
1
J=00.10.20.30.5
0 3 6 9 12 150
0.1
0.2
0.3
0.4
0.5
0.6
∞=∞=∞= mfa zzz ,,
I=0, L=1
0 3 6 9 12 150
0.1
0.2
0.3
0.4
0.5
0.6
Message packet generation rate G
Thr
ough
put p
er s
lot S
∞=∞=∞= mfa zzz ,,
I=0.3, L=1
1
J=0
0.10.2
0.30.5
0 3 6 9 12 150
0.1
0.2
0.3
0.4
0.5
0.6
Message packet generation rate G
Thr
ough
put p
er s
lot S
I=0.3, L=1J=0
0.1
0.2
0.3
0.5
1,, =∞=∞= mfa zzz
1
(a) 1-channel, without interference, without capture (b) I=0.3 (c) message packet capture 1=mz
1,1, ==∞= mfa zzz
0 3 6 9 12 150
0.1
0.2
0.3
0.4
0.5
0.6
I=0.3, L=1
J=0
0.1
0.2
0.3
0.5
Thr
ough
put p
er s
lot S
Message packet generation rate G
1
0 3 6 9 12 150
0.1
0.2
0.3
0.4
0.5
0.6
Message packet generation rate G
Thr
ough
put p
er s
lot S
J=0
0.1
0.2
0.3
0.5
1,,1 =∞== mfa zzz I=0.3, L=1
1
0 3 6 9 12 150
0.1
0.2
0.3
0.4
0.5
0.6
Message packet generation rate G
Thr
ough
put p
er s
lot S
∞=∞=∞= mfa zzz ,,I=0.3, L=5
1
J=00.10.20.30.5
(d) interfering packet capture 1=fz 1=az (f) 5-channel, without capture (e) attacking packet capture
Fig. 2 Throughput per slot with the variation of message packet arrival rate.
212 J.H. Sarker and H.T. Mouftah
7. There exists an optimum point where throughput per timeslot, S, is maximum for given values of message packet gen-eration rate, G, interfering signals rate from other net-
works, I, and attacking packet generation rate, J.
Differentiating Eq. (17) with respect to attacking noise
packet generation rate, J, we obtain
dS
dJ¼ exp � I=L
1þ 1=zf
� �exp � G=L
1þ 1=zm
� �� 1=L
1þ 1=za
� �G2
LðGþ Jþ IÞ exp � J=L
1þ 1=za
� �� G4
LðGþ Jþ IÞ2exp � J=L
1þ 1=za
� �" #
¼ exp � I=L
1þ 1=zf
� �exp � G=L
1þ 1=zm
� �exp � J=L
1þ 1=za
� �� 1=L
1þ 1=za
� �G2
LðGþ Jþ IÞ �G4
LðGþ Jþ IÞ2
" #
ð18Þ
It is clear from Eq. (18) that for any positive value of mes-sage packet generation rate, G, and interfering packets arrivalrate from other networks, I, the throughput, S, decreases with
dS
dG¼ 1
Lexp � J=L
1þ 1=za
� �exp � I=L
1þ 1=zf
� �G2
Gþ Jþ Iexp � G=
1þ 1
�"
¼ 1
Lexp � J=L
1þ 1=za
� �exp � G=L
1þ 1=zm
� �G
Gþ Jþ I� G=L
1þ 1=z
��
the increase of attacking noise packet generation rate, J. Ex-actly in the same way, it can be shown that for any positive va-lue of message packet generation rate, G, and the attacking
noise packet generation rate, J, the throughput, S, decreaseswith the increase of interfering packets arrival rate from othernetworks, I. However, the numerical results of these two re-sults have already been depicted in Fig. 2.
Differentiating Eq. (17) with respect to message packet gen-eration rate, G, we get
L
=zm
�� 1=L
1þ 1=zm
� �þ ðGþ Jþ IÞð2GÞ � G2
ðGþ Jþ IÞ2exp � G=L
1þ 1=zm
� �#
m
�þ ðGþ Jþ IÞð2Þ � G
ðGþ Jþ IÞ
�ð19Þ
Secured operating regions of Slotted ALOHA in the presence of interfering signals from other networks and attacking signals 213
Now putting the differentiation result Eq. (19) equal to
zero, we obtain the optimum value of the message packet arri-val rate from all active mobile nodes,
Gopt ¼fLð1þ 1=zmÞ � ðJþ IÞg �
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifLð1þ 1=zmÞ � ðJþ IÞg2 þ 8LðJþ IÞð1þ 1=zmÞ
q2
ð20Þ
Using the value of optimum message packet arrival rate,Gopt, in Eq. (17), we can obtain the optimum throughput pertime slot as
Number of channels L
Thr
ough
put p
er s
lot S
opt
Thr
ough
put p
er s
lot S
opt
J=0
1
52
1020
0,
,
=∞=
∞=∞=
Iz
zz
m
fa
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Number o
Thr
ough
put p
er s
lot S
opt
0 10 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
J=0 1 2
z
z
(a) without interference, without capture (b) with inte
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
2,1
1,
==
=∞=
Iz
zz
m
fa
L
5
10
20
J=0 1 2
0 10 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Number o
Thr
ough
put p
er s
lot S
opt
J=0 1 2
(d) interfering packet capture 1=fz (e) attacking pack
Fig. 3 The maximum throughput, Sopt with
Sopt ¼ðGoptÞ2
LðGopt þ Jþ IÞ exp � J=L
1þ 1=za
� �exp � I=L
1þ 1=zf
� �exp �
�
¼Lð1þ 1=zmÞ � ðJþ IÞf g þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifLð1þ 1=zmÞ � ðJþ IÞg2 þ 8L
q�
2L Lð1þ 1=zmÞ þ ðJþ IÞ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifLð1þ 1=zmÞ � ðJþ IÞg2 þ 8
q�
� exp � J=L
1þ 1=za
� �exp � I=L
1þ 1=zf
� �exp
�fLð1þ 1=zmÞ �24
The optimum throughput per time slot is shown in Figs. 3
and 4 using Eq. (21). Figs. 3 and 4 show that the optimumthroughput can be increased significantly using lower capture
ratios and multiple channels. The conclusions of Figs. 3 and4 are almost same as the conclusions drawn from Fig. 2.
Thr
ough
put p
er s
lot S
opt
Thr
ough
put p
er s
lot S
opt
f channels L L
510
20
30 40 50
2,
,
=∞=
∞=∞=
Im
fa z
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
2,1
,
==
∞=∞=
Im
fa
z
zz
rference, I=2 (c) message packet capture 1=mz
J=0
J=0 1 2
510
20
1 2
510
20
2,1
,1
==
∞==
Iz
zz
m
fa
30 40 50
510
20
f channels L L0 10 20 30 40 50
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
2,1
1,1
==
==
Iz
zz
m
fa
et capture 1=az (f) with capture effects
the variation of number of channels L.
Gopt=L
1þ 1=zm
�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðJþ IÞð1þ 1=zmÞ
�2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLðJþ IÞð1þ 1=zmÞ
�
ðJþ IÞg þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifLð1þ 1=zmÞ � ðJþ IÞg2 þ 8LðJþ IÞð1þ 1=zmÞ
q2Lð1þ 1=zmÞ
35
ð21Þ
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 bysetting Eq. (19) is equal to zero. Therefore, the optimum num-ber of channels is
Lopt ¼GðGþ Jþ IÞ
ð1þ 1=zmÞðGþ 2Jþ 2IÞ ð22Þ
Sopt ¼kopt
L1� 1�PoptðSuÞ
rþ1h i¼ ðGoptÞ2
LðGopt þ Jþ IÞ exp � J=L
1þ 1=za
� �exp � I=L
1þ 1=zf
� �exp � Gopt=L
1þ 1=zm
� �
¼fLð1þ 1=zmÞ � ðJþ IÞg þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifLð1þ 1=zmÞ � ðJþ IÞ2 þ 8LðJþ IÞgð1þ 1=zmÞ
q� �22LfLð1þ 1=zmÞ þ ðJþ IÞ þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifLð1þ 1=zmÞ � ðJþ IÞg2 þ 8LðJþ IÞð1þ 1=zmÞ
qg
� exp � J=L
1þ 1=za
� �exp � I=L
1þ 1=zf
� �exp �
fLð1þ 1=zmÞ � ðJþ IÞgþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifLð1þ 1=zmÞ � ðJþ IÞg2 þ 8LðJþ IÞð1þ 1=zmÞ
q2Lð1þ 1=zmÞ
24
35ð23Þ
Eq. (22) shows that the optimum number of channels, Lopt,increases linearly with the increase of aggregate message trafficarrival rate, G. The Lopt, increases further with the increase of
PoptðSuÞ¼Sopt=L
Gopt
¼ Gopt
GoptþJþ Iexp � J=L
1þ1=za
� �exp � I=L
1þ1=zf
� �exp � Gopt=L
1þ1=zm
� �
¼Lð1þ1=zmÞ�ðJþ IÞþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifLð1þ1=zmÞ�ðJþ IÞg2þ8LðJþ IÞð1þ1=zmÞ
qLð1þ1=zmÞþðJþ IÞþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifLð1þ1=zmÞ�ðJþ IÞg2þ8LðJþ IÞð1þ1=zmÞ
q
� exp � J=L
1þ1=za
� �exp � I=L
1þ1=zf
� �exp �
fLð1þ1=zmÞ�ðJþ IÞgþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifLð1þ1=zmÞ�ðJþ IÞg2þ8LðJþ IÞð1þ1=zmÞ
q2Lð1þ1=zmÞ
24
35ð24Þ
message packet capture ratio, zm. However, the same decreaseswith the increase of interfering packet arrival rate from othernetworks, 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 betransmitted successfully. On the other hand, in the case of con-tention based access protocol or for real-time data transmis-
sion, we can cut the retransmission number, which will avoidthe undesirable stability or security problem of Slotted ALO-HA [35]. Over a long time period, the total offered traffic load
G will have an optimum value Gopt depending on the otherparameters like new packet generation rate per time slot andthe number of retransmission trials. In the case of access orreal-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 typesystem, 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 theoptimum value of offered traffic load from all users Gopt, which
will make the system secured or stable. Here, in this paper asimplified assumption is considered: if the traffic generation
rate from all active users in a given time slot is less than or
equal to the optimum packet arrival per time slot, the systemis secured. This assumption is reasonable for Slotted ALOHAsystem [33].
The optimum throughput per slot of L-channels Slotted
ALOHA system with and without limiting the number ofretransmission trials can be obtained from Eqs. (18 and 21) as
The optimum probability of success can be obtained fromEqs. (23) and (20) as
Therefore, the optimum throughput per slot of L-channelsSlotted ALOHA system by limiting the number of retransmis-sion trials can be obtained from Eq. (17) as
Sopt ¼ koptL
1� f1� PoptðSuÞgrþ1h i
orkoptL¼ Sopt
1� 1�PoptðSuÞf grþ1ð25Þ
where the values of Sopt and PoptðSuÞ are given in Eqs. (23) and(24), respectively. The main purpose of our system model is tomaximize the throughput per slot, S by adjusting the transmis-
sion 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 derivedthe maximum throughput of L-channels Slotted ALOHA sys-
tem Sopt in Eq. (23). And it occurs when the aggregate trafficgeneration 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 fol-lows: For a call establishment system design or for a real-timetraffic transmission, the time out is the most important param-
eter. This time out is the time to transmit the access informa-tion from mobile to base station plus the switching time. Fromthe value of the time to transmit the access information or real-
time transmission plus the propagation delay, we can find themaximum 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 optimumnew packet generation rate per time slot, kopt=Lper time slotusing Eqs. (23)–(25). Fig. 5 shows the variation of optimumnew packet generation rate per time slot, kopt=L, with the var-
iation 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
kopt
L
� �r!0
¼ Sopt
PoptðSuÞ¼
Lð1þ 1=zmÞ � ðJþ IÞ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifLð1þ 1=zmÞ � ðJþ IÞg2 þ 8LðJþ IÞð1þ 1=zmÞ
q2L
ð26Þ
Eq. (26) shows the limit of the proposed solution withretransmission cut-off scenario.
In the other extreme, without any retransmission cut-off(r fi1) the optimum new packet generation rate per time slot
can be obtained using Eq. (25) as
UR ¼ 1
1� a
� �Sopt
PoptðSuÞ¼
Lð1þ 1=zmÞ � ðJþ IÞ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifLð1þ 1=zmÞ � ðJþ IÞg2 þ 8LðJþ IÞð1þ 1=zmÞ
q2Lð1� aÞ ð30Þ
kopt
L
� �r!1¼ Sopt ð27Þ
where the value of Sopt is given in Eq. (23). Therefore, the secu-rity improvement area by limiting the number of retransmis-
sion trials, r is
Ar ¼ kopt
L
� �r!0
� kopt
L
� �r!1¼
Lð1þ 1=zmÞ � ðJþ IÞ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifLð1þ 1=zmÞ � ðJþ IÞg2 þ 8LðJþ IÞð1þ 1=zmÞ
q2L
� Sopt ð28Þ
The shaded parts indicated in Fig. 5 show the secured re-gion by limiting the number of retransmission trials, r. Thelower most parts of the figures show the secured transmissionregion without limiting the number of retransmission trials.
Increasing the number of channel, L or/and reducing the cap-ture ratios, za, zf and zm are not enough to obtain a higher se-cured transmission operating region. Limiting the number of
retransmission trials can increase the secured transmissionoperating region significantly.
Security improvement by new packet rejection
The main purpose of this paper is to obtain the secured trans-
mission of L-channel Slotted ALOHA system. It is alreadyshown that if L-channel Slotted ALOHA system providesmaximum throughput then the system is secured. If limitingthe retransmission trials is not sufficient for obtaining a se-
cured stabilized L-channel Slotted ALOHA system, then itcan be achieved by the expense of newly generated packetrejection.
The maximum throughput per slot of a L-channel SlottedALOHA is Sopt is derived in Eq. (23), and it occurs when theaggregate traffic generation rate, Gopt, which is shown in Eq.
(20). The aggregate message packet generation rate per time
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
L¼ 1
1� a
� �Sopt
1� 1� PoptðSuÞ rþ1 ð29Þ
The secured operating region ofL-channel SlottedALOHA sys-tem with and without limiting the retransmission trials is de-
picted in Fig. 6. Please note that here the y-axis should bemultiplied by X� ¼ Gopt
1�a. The value of Gopt is given in Eq. (20).
Fig. 6 shows clearly that by increasing the value of a (new packetrejection probability), the secured operating regions with andwithout retransmission cut-off can be increased significantly.
FromFig. 6, it can be said that themaximumvalue of the newpacket generation rate per slot, kopt=Lwith new packet rejectionis
ComparingEqs. (26) and (30), the upper limit of the newpack-et generation rate per slot with new packet rejection is 1=ð1� aÞtimes 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 newpacket rejection is
LR ¼ Sopt
1� að31Þ
It can be said that the new packet generation rate per slotwithout limiting the number of retransmission trials with
new packet rejection is 1=ð1� aÞ times higher than that ofthe without new packet rejection, by comparing Eqs. (27)and (31).
From Fig. 6, we can conclude that, the aggregate messagepacket generation rate, G, never reaches its optimum point,if the new packet generation rate per slot, kopt=L, is less thanLR packet per time slot. The reason is that, we started to get
the result of Eq. (31) with the aggregate message packet gener-ation rate, Gopt. So, it is unnecessary to control the retransmis-sion 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 gen-erated packet rejection probability.
Conclusions
In this paper, an analytical approach for secured operating re-gions of Slotted ALOHA in the presence of interfering signalsfrom other networks and DoS attacking signals has been
L =20
10
5
3
2
1
L =20
10
5
3
2
1
L =20
10
5
3
2
1
L =20
10
5
3
2
1
Fig. 4 The maximum throughput, Sopt .
216 J.H. Sarker and H.T. Mouftah
investigated. The performance evaluations presented in thispaper are based on the numerical analysis.
The security improvement of L-channels Slotted ALOHA
in the presence of interfering signals from other networksand random attacking noise packets signals is studied in thispaper. The current security protected measures such as encryp-
tion makes the packets unreadable by unauthorized users. Theauthentication technique is used to protect the system fromillegal users and authorization separates the legal users. How-
ever, in a Slotted ALOHA based network, the interfering sig-nals from other networks and the random packet destructionDoS attacking noise packets may collide with message packetsand reduces the secured transmission. Therefore, the current
security measures such as encryption, authentication andauthorization cannot prevent those types of attack. One ofthe main drawbacks of Slotted ALOHA protocol is its exces-
sive 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 networksand the random packet destruction DoS attacking noisepacket increase the collision, we intend to use multiple chan-
nels in the Slotted ALOHA protocol to reduce the collisionsin the first technique. The use of multiple channels in theSlotted ALOHA protocol reduces three types of packet colli-
sions. First type of collision is the collision between two ormore message packets. The second type of collision is the col-lision between a message packet and one or more interfering
packets from other networks. The third type of collision isthe collision between a message packet and one or moreother attacking noise packets.
In the second security improvement technique, we haveshown the effects of capture ratios in the presence of interfer-ing signals from other networks and the random packet
destruction DoS attacking noise packet. A lower message cap-ture ratio can increase the throughput and maximum through-put significantly. A lower interfering capture ratio can increasethe 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 cap-ture ratio.
In the third technique, we have used retransmissions cut-offby limiting the number of retransmission trials. The retrans-missions cut-off technique can limit the aggregate packet flow
and form the optimum message packet flow in the presence ofinterfering signals from other networks and the attacking noisepacket. It is possible that the third technique called retransmis-
sions cut-off technique is not enough to control the flow ofmessage packets. Because of that the fourth technique callednew packet rejection probability is introduced. The secured
1 10 1000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
Opt
imum
new
pac
ket g
ener
atio
n
rat
e pe
r sl
ot
Retransmission trials r+1
Secured region without limiting r
X*
Secured region with limiting r
Unsecured operating region
Sopt
Gopt
λ opt /
L
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
multiplied by X� ¼ Gopt
1�a.
1 11 21 31 41 510
0.4
0.8
1.2
1.6
2
2.4
2.8
3.2
3.6
4O
ptim
um n
ew p
acke
t gen
erat
ion
r
ate
per
slot
Number of channels L
∞=∞=∞= mfa zzz ,,
Secured region without limiting r
I = 0, J = 0
Secured region with limiting r
Unsecured operating region
∞=r
0=r
Opt
imum
new
pac
ket g
ener
atio
n
rat
e pe
r sl
ot
Number of channels L1 11 21 31 41 51
0
0.4
0.8
1.2
1.6
2
2.4
2.8
3.2
3.6
4
∞=∞=∞= mfa zzz ,,
I = 2.5, J = 0
∞=r
0=r
Secured region without limiting r
Secured region with limiting r
Unsecured operating region
1 11 21 31 41 510
0.4
0.8
1.2
1.6
2
2.4
2.8
3.2
3.6
4
Number of channels L
Opt
imum
new
pac
ket g
ener
atio
n
rat
e pe
r sl
ot
∞=∞=∞= mfa zzz ,,
Secured region without limiting r
I = 2.5, J = 2.5
Secured region with limiting r
Unsecured operating region
∞=r
0=r
(a) ∞=az , ∞=fz ∞=mz , I=0, J=0 (b) ∞=az , ∞=fz ∞=mz , I=2.5, J=0 (c) ∞=az , ∞=fz ∞=mz , I=2.5, J=2.5
1 11 21 31 41 510
0.4
0.8
1.2
1.6
2
2.4
2.8
3.2
3.6
4
Number of channels L
Secured region without limiting r
I = 2.5, J = 2.5
1,, =∞=∞= mfa zzz
Opt
imum
new
pac
ket g
ener
atio
n
rat
e pe
r sl
ot
0=r
∞=r
Secured region with limiting r
Unsecured operating region
1 11 21 31 41 510
0.4
0.8
1.2
1.6
2
2.4
2.8
3.2
3.6
4
Number of channels L
Opt
imum
new
pac
ket g
ener
atio
n
rat
e pe
r sl
ot
1,1, ==∞= mfa zzz
I = 2.5, J = 2.5
Unsecured operating region
Secured region without limiting r
∞=r
0=r
Secured region with limiting r
1 11 21 31 41 510
0.4
0.8
1.2
1.6
2
2.4
2.8
3.2
3.6
4
∞=== mfa zzz ,1,1
I = 2.5, J = 2.5
∞=r
0=r
Secured region without limiting r
Unsecured operating region
Number of channels L
Opt
imum
new
pac
ket g
ener
atio
n
rat
e pe
r sl
ot
Secured region with limiting r
(d) ∞=az , ∞=fz , 1=mz , I=2.5, J=2.5 (e) ∞=az , 1=fz , 1=mz , I=2.5, J=2.5 (f) 1=az , 1=fz , ∞=mz , I=2.5, J=2.5
λ opt /
Lλ o
pt /
L
λ opt /
L
λ opt /
L
λ opt /
L
λ opt /
LFig. 5 Security improvement by limiting the retransmission trials.
Secured operating regions of Slotted ALOHA in the presence of interfering signals from other networks and attacking signals 217
operating regions can be increased 1=ð1� aÞ times by usingthis fourth technique. Where a is the new packet rejectionprobability.
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