Communications and Network, 2013, 5, 578-583
http://dx.doi.org/10.4236/cn.2013.53B2104 Published Online September 2013 (http://www.scirp.org/journal/cn)
Protection Model of Security Systems Based on
Neyman-Person Criterion
Haitao Lv, Ruimin Hu, Jun Chen, Zheng He
National Engineering Research Center for Multimedia Software, Wuhan University,
Wuhan, China
Received May 2013
ABSTRACT
In this paper security systems deployed over an ar e a are rega r ded abstractly as a diagram of security network. We pro-
pose the Neyman-Pearson protection model for security systems, which can be used to determine the protection proba-
bility of a security system and find the weakest breach path of a security network. We present the weakest breach path
problem formulation, which is defined by the breach protection probability of an unauthorized target passing through a
guard field, and provide a solution for this problem by using the Dijkstra’s shortest path algorithm. Finally we study the
variation of the breach pro tection prob ability w ith the change of th e parameters of the model.
Keywords: Security System; Protection Probability; Security Network; Breach Protection Probability; Breach Path
1. Introduction
The society security problem has been attached impor-
tance by national governments. In order to maintain so-
cial public safety, many security systems have been con-
structed in cities in the world. With the rapid develop-
ment of information technology, especially the Internet
of Things and cloud computing, the security system is
getting more and more complex, which consists of the
intrusion alarm system, the video surveillance system,
the access control system, the explosion-proof security
check system, etc. Security systems are deployed at dif-
ferent positions in an area, which can communicate and
share data each other through the internet, and complete
protection tasks cooperatively. In this paper, a large and
complex security system is regarded as abstractly as a
diagram of a security network. As shown in Figure 1,
there is a security netwo rk consisted of some security
systems in the guard zone, where each of yellow filled
circles represents a security system, and every triangle
represents a protection target.
For a security network, depending on the protection
ranges and the protection coverage sch emes of security
systems, as well as the deployment-density of the guard
field, the protection coverage area may contain vulnera-
ble paths. The probability that an unauthorized target
traverses the region through a vulnerable path gives in-
sight about the level of security provided by the security
network. In this paper, the protection mo del of security
systems based on Neyman-Pearso n Criterion is proposed.
The protection probability at any position in a guard field,
which is provided by a security system, can be calculated
by the model. The weakest breach pa th of a security net-
work, which is defined by the breach protection proba-
bility of an unauthorized target passing through a guard
field, can be found.
Figure 1. The abstract diagram of a security network.

H. T. LV ET AL.
579
The r eminder of this paper is organized as follows: In
the next section, the related work about seurity systems is
introduced. In Section 3, the protection model is pro-
posed, the weakest breach path of a security network is
described and the method based on Dijkstra’s shortest
path algorithm to find the weakest breach path is put
forward. After presenting the details of the problems for-
mally, th e results ar e simulated and analyzed in Section 4.
Finally, conclusions are drawn in Section 5.
2. Related Work
In 1970’s, U.S. Department of Energy’s Sandia National
Laboratories [1] first introduced the basic concepts of the
Physical Protection System, from which th e security system
evolved. Subsequently, the U.S. Department of Energy
put forward a model named adversary sequence diagram
(ASD) [2], which was applied to the field of nuclear
facilities protection. ASD can recognize vulnerability of
physical protection systems by analyzing how hypothet-
ical enemies might achieve their objects through various
barriers. The path th at is most easily broken through is
considered weakest.
In 1981, Doyon [3] presen ted a probabilistic network
model for a system consisting of guards, sensors, and
barriers. He determined analytic representations for de-
termining probabilities of intruder apprehension in dif-
ferent zones between site entry and a target object. In
1997 Kobza and Jacobson [4] have presented probability
models for access security systems with particular appli-
cations to aviation security. In 1998, Hicks et al. [5] Pre-
sented a cost and performance analysis for ph ysical pro-
tection systems. He considered the systems-level perfor-
mance metric is risk, which is defined as follows.
( )()
1RiskpAp EC= ×−×


(1)
where
( )
pA
is Probability of Attack,
( )
pE
is Prob-
ability of System Effectiven ess, and C is Consequence.
After the events of September 11, 2001, public safety
becomes the issue co ncerned by countries in the world.
The concep t of Physical Protection Syste m has been
changed. Some researchers from USA and Australia con-
side red that a ph ysical protection system is made up of
people, architectures and electronic devices. So the con-
cept of Security Sys te m was born. Many researcher s were
interested in assess the protection effectiveness of secu-
rity systems through risk analysis. In 2004, Fische r [6]
presented a very subj ectiv e risk analysis approach to rank
threats using a probability matrix, criticality matrix, and
vulnerability matrix. In 2006, Zhihua Chen [7] in Chi-
nese Peop l e ’s P ub lic Security Univ ersity h as proposed
performance evaluation index and evaluation methods of
crime prevention system for the assessment of the effec-
tiveness and vulnerability of crime prevention system. This
method is a qualitative assessment to the crime preven-
tion system based on management science. In 2007, Gar-
cia [8] gave an integrated approach f or designing physi-
cal security systems. The protection effectiveness of a
physical protection system was defined as the cumulative
probability of detection fro m th e star t of an adversary
path to th e point determined by the time available for
response. In 2009, Jonathan Pollet and Joe [9] Cummins
put forward a performance assessment framework of the
Security Systems, which considered not only the charac-
teristics of the system, also the ris k outsid e the system.
In recent years, some researcher s considered that the re
were enormous uncertainty in the vulnerability evalua-
tion of security systems, and they put forward some me-
thods to reduce unc er tain ty. In 2011, Xu peida [10]
thought that each individual component of the security
system w as modeled, and he used the Dempster-Shafer
(D-S) evidence theory to analyze potential threats. Some
literatures also proposed methods such as bounded inter-
vals [11], exoge n ous dynamics [12], games of imperfect
information [13-15], to characterize uncertainty in vul-
nerability analysis.
3. Protection Model and the Weakest Breach
Path Problem Formulation
3.1. Neyman-Pearson Protectio n Model
In our research, th er e is a basic assumption that is a secu-
rity system can eliminate any threat as long as a threat is
detected. If a security system finds a threat, it will sound
alarm. So each security system has its own false alarm
rate, and it is regarded abstractly as the process of deci-
sion. The optimal decision rule that maximizes the detec-
tion probability subject to a maximum allowable false
alarm rate α that is given by the Neyman-Pearson lemma
[16]. Two hypotheses that represent th e ab sen ce and pr e s-
ence of an unauthorized object are set up. The model
computes the likelihood ratio of the respective probabili-
ty density functions, and compares it against a threshold
which is configured, and the false alarm constraint is sa-
tisfied. The process of a security system finding threats
can be considered as the process signal reception. Sup-
pose that an unauthorized object is a passive signal re-
ception th a t happens in the presence of additive white
Gaussian noise (AWGN) with zero mean and variance
2
n
σ
, as well as path-loss with path loss exponent
η
. Every
breach protection decision is based on the processing of
L data samples. If samples are collected fast enough, the
distance between a security system and a object can be
considered constant throughout the observation period.
Let
vi
d
be the Euclidean distance between the grid point
v
and the security system
i
. Based on Neyman-Pearson
Criterion with false alarm rate
α
, the pro tection probabil-
ity of an unauthorized object at grid point
v
by the se-

H. T. LV ET AL.
Copyright © 2013 SciRe s. CN
580
curity system
i
is defined as follows.
( )
()
1
11
vi vi
p Ld
η
αγ
−−
=−Φ Φ−−
. (2)
Where
( )
xΦ
is the cumulative distribution function
of the zero-mean, unit-variance Gauss i an random varia-
ble at point
x
.
2
n
A ψ
γσ =. (3)
γ
controls the per-datum signal-to-noise power r atio
where the security system transmits information with
power
ψ
, and A is a cons tant, which is regarded as sig -
nal propagation losses, emergency res o urces, infor mation
communication, etc.
3.2. Grid-Based Guard Field
In ord er to simplify the formulations, we consider the
guard field as a cross-connected grid. A sa mple field
model is pre s ented in Figure 2.
The guard field model consists of the grid points and
two auxiliary nodes which are the starting and the desti-
nation points. The aim of the target is to go through the
guard field from the starting point that represents the
insecure side to the destination point that represents the
secure side. The horizontal axis is divided into N − 1 and
the vertical axis is divided into M − 1 equal parts . Thus,
there are
NM×
grid points plus the starting and desti-
nation points. For the sake of simplifying the notation,
instead of us ing two dimensional grid point indices
( )
,
vv
xy
where
0,1, ,1
v
xN= −
and
0,1, ,1
v
yM= −
,
we utilize a kind of one dimensional grid point index
v
which is calculated as
1
vv
v yN x= ++
. The index of the
starting point is defin ed as
0v=
, a nd th e index of the
destination point is
1v NM= +
. We use the connection
matrix
( )( )
,
22
vw NM NM
cC
+× +
∈
to represent the connections
Figure 2. A sample field model constructed to find the vul-
nerable pat h for the guar d field whe re the le ngth is 8 m, the
width is 4 m, and the grid size is 1m (N = 8, M = 4).
of the grid points. The matrix
,vw
c
is defi n ed as defin ed
in Equation (4).
( )
,
10, 1,
10 0
111
0
v wvw
w
vw
v
ifv wNMandxxyyD
if vand y
cifwNMandyM
otherwise
 <<+ −−∈
 = =

= =+ =−



.
(4)
{{ 1,0,1}{ 1,0,1}}{(0,0)}D= −×−− which is the set of
possible difference-tuples of the two-dimensional grid
point indices excluding
vw=
.
Using the grid-based field model above, the protection
probabilities can be computed for every grid point through
the Neyman-Pearson Protection Model of security sys-
tems.
3.3. The Weakest Breach Path Problem
The weakest breach path problem can be defined as fin ding
the permutation of a subset of grid points
12
{,,, }
k
V vvv= with which an object traverses from
the starting point to the destination point with the least
probability of be in g detected. The nodes
1i
v−
and
i
v
are connected to each other where 1,1
ii
vv
c−=. The miss
probability p of the most vulnerable pathVis defined as
follows.
( )
1i
i
v
vV
p pn
∈

= −


∑. (5)
Where vi p is the breach protection probability asso-
ciated with the grid point
i
vV∈
, n is the number of
i
v
.
The weakest breach path can be find by solving the
following optimization problem as defined in Equation
(6).
ij
x
denotes the edg e which originates from the ith
node and sinks in the jth node, s is the starting node and
d is the destination node and C is as defined in Equation
(4). In this formulation , the aim is to maximize the miss
probability P defined in Equation (6). By using the
logarithm function the optimization problem defined in
Equation (7) can be changed to a linear program, where
the aim is to find the minimum value.
( )
()( )
( )( )
,,
,,
max 1
1;1;1, 2,,
01, 2,,,
11
0
i
i
v ij
vV
sj id
s jCidC
ij ki
ij Cki C
ij
ij
pxsubject to
xx iNM
xx iNM
ifith andjth nodes areonthepath andc
xotherwise
∈
∈∈
∈∈
−
= = ∀ =
−= ∀ =
=

=

∑
∑∑
∑∑


(6)
( )
,
minlog(1 )
i ij
ij C
px
∈
−−
∑
(7 )
0
1
9
17
25
2
10
18
26
3
11
19
27
4
12
20
28
5
13
21
29
6
14
22
30
7
15
23
31
8
16
24
32
33 Guard Fi el dS ECURE SIDE
INSECURE S IDES TARTING P O INT
DE STINA TION P O INT

H. T. LV ET AL.
581
4. Simulation and Analysis
The grid-based field can be regard abstractly as a graph,
so Dijkstr a’s shortest path algorithm can be employed to
solve the weakest breach path problem too. The protec-
tion probability asso ciated with the grid points cannot be
used as we ights of the grid points. Consequently, the
weights of the grid points need be converte d to a new
measure dv, which is defined as log(1 )
vv
dp=−−
. This
algorithm finds the path with the smallest negative loga-
rithm value that is equal to be the weakest breach path. A
sample security systems co verag e graph and the weakest
breach path is shown in Figure 3. Using the two-dimen-
sional field model and adding the protection prob ab ility
as the third axis.
Valleys and hills of protection probabilities are shown
in Figure 3. The weakest breach path follows the valleys
because the valleys denote the low protection probabili-
ties.
4.1. Effects of Parameters on the Protection
Probability
In this subsection, the effects of the Neyman-Pearson
Protection Model parameters on the breach protection
probability are analyzed. The deployment of security sys-
tems is random with uniform distribution. The parame-
ters are shown in Table 1. The figures, which are pre-
sented in the following are the averages of 100 runs, de-
pict how the environmental properties and th e toleran c e
to the false alarms affect the vulnerability of a security
network.
Ten security systems are deplo yed in a field where the
parameters are same as in Ta ble 1. The effect of the false
Figure 3. A sample of a guard field and vulnerable path
where the length i s 101 m, the widt h is 60 m, and grid si ze is
1 m. Twenty security systems are deployed in this field
randomly. The Neyman-Pearson Protection Model is confi-
gured with L = 100, R = 9, α = 0.01, η = 2, γ = 20 db. The
breach probability is 0.0639.
alarm rate, α, on the breach protection probability P is
shown in Figure 4, which essen tially r epr esents the net-
work operating characteristics. With greater tolerance to
false alarms, the P performance improves, and hence the
protection range becomes larger. Sufficiently high signal
noise ratio is necessary for an acceptable level of breach
protection probability, whi ch is relatively insensitive to
the false alarm rate.
As shown in Figure 4, the false alarm rate α has a
great effect on the breach protection probability P, and as
α increases, the breach probability decreases, which re-
flects the protection probability v
p of an unauthorized
object at grid
v
increases.
Although α is very in fluentia l on br each protection
probability, η does not have an appreciable impact when
the signal noise ratio is small. When the values of γ
become large, η significantly increases the breach protec-
tion probability as shown in Fig ure 5.
This is due to the fact that the protection probability is
inversely proportional to the distance on the order of η.
The effect of η is very significant when
4 η ≤
.
As shown in Figu re 6, wh en the signal noise ratio γ
incre a ses, the breach protection probability decreases,
Table 1. Parameter values used in the simulations for Ney-
man-Pearson Protection Model.
Parameters Value
Length of the field 51 m
Width of the fileld 41 m
Grid Size 1 m
Numbers of Security Systems 10
α 0.1
η 2
γ 20 db
L 100
Figure 4. The effect of α on th e b rea ch p rot ecti on probability.
020 40 60 80 100
0
10
20
30
40
50
60
0.92
0.94
0.96
0.98
1
x
v
y
v
Protec tion P robabl i ty
Most V ul nerabl e P ath
Secure S i de
Insecure Si de
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
B reac h Prot ec tion Probabil i t y
α
ϒ =20db
ϒ =5db
ϒ =60db

H. T. LV ET AL.
Copyright © 2013 SciRe s. CN
582
Figure 5. The effect of η on the breach protection probability,
Figure 6. The effect of γ on the breach protection probability .
which indicates that the protection probability of the se-
curity netwo rk improves and the protection performance
increases. If targets are closer to security systems, signal
noise ratio has more influence on the protection probabil-
ity. When th e parameter
3 η ≥
, the effect of the signal
noise ratio γ becomes very small.
4.2. Effects of Number of Security Systems on
the Protection Probability
While analyzing the required number of security systems
for a given breach probability, one case of random dep-
loyment is considered. The case is assumed that security
systems are uniformly distributed alon g bo th the vertical
and horizontal axes . The effect of numbers of security
system in a field on the breach protection prob ability is
shown in Figure 7.
As the density of security systems increases in a field,
the breach protection probability tends to stabilize, which
approximates the zero. The results suggest that there is a
Figure 7. The effect of numbers of security systems on the
breach protection probability.
saturation point after which randomly placing more secu-
rity systems does not significantly impact the breach pro-
tection probability of a secu rity network in a field.
When the signal noise ratio is same, α affects the
breach protection probability more than
η
(see Figures 4
and 5), so the false alarm rate α is more influential here
too. The rapid d ecreas e in the breach protection probabil-
ity can be explained by the fact as the density of security
systems is saturated in a field, grid points are cover ed
with high protection probabilities. Consequently, at the
beginning, an additional security system decreases the
breach protection probability considerably, however, once
the saturation is reached, the affection of numbers of
security systems is not so large anymore.
5. Conclusion
In this paper, we put forward the Neyman-Pearson pro-
tection model of security systems that can be employed
to find the weakest breach path of a security network. In
order to f in d the weakest breach path, we apply Dijkstra’s
shortest path algorithm by using the negative log of the
breach protection probability as the grid point weights.
Finally, the effect of para met ers of N eyman-Pearson pro-
tection model on breach protection probability is studied
by MATALAB simulation. The simulation experiments
show that the false alarm rate is the most influential pa-
rameter on the breach protection prob ability.
6. Acknowledgements
Thanks for the assistance from National Science Founda-
tion of China (No. 61170023), the Major Nat ional Science
and Technology Special Projects of China (2010ZX
03004-003-03, 2010ZX03004-001-03), National Nature
Science Foundation of China (No. 60832002). The au-
thors would like to thank Ruimin Hu. IEEE Member, pro-
12345678910
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
B reac h Protect i on Probabil i t y
η
ϒ =20db
ϒ =5db
ϒ =60db
051015 2025 30 35 40 4550
0.2
0.22
0.24
0.26
0.28
0.3
0.32
0.34
0.36
B reac h Protect i on Probabil i t y
γ
η=2
η=3
η=4
010 20 30 4050 60 70 80 90 100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Numbers of S ecurity S yst em s
B reac h Prot ec tion Probabil i t y
α =0.9 η =2 γ =20
α =0.9 η =3 γ =20
α =0.9 η =2 γ =60

H. T. LV ET AL.
583
fessor of School of Computer, Wuhan University, Ren
Pin teaching assistant Department of Electrical Engi-
neering and Computer Science, Northwestern University
USA, for their thoughtful comments.
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