Mathematical Problems in Engineering
() =(
1
,
2
,...,
𝑛
)is the set of target in the WSN on
the multicast model, where is the number of the
target and 2≤.
2.2. e Problem of WSN Reliability. In this paper, we studied
theinfrastructurecommunicationreliabilityofaWSN.Below
aretheassumptionsmadetoevaluatereliabilityinour
analysis:
() edges between two nodes are perfect;
() nodes failures are statistically independent;
() the fault-tolerant clustering protocol is executed to
keep the topology.
With the assumption that the links are imperfect, this type
of structure would rapidly increase the complexity in order to
compute the reliability of WSNs. In this study, we assume that
the links are perfect for the convenience of studying them;
however,ourfutureworkwillinvestigatetheevaluationof
WSN reliability with imperfect nodes as well as imperfect
links. Based on the above assumptions some denitions are
presented as follows.
Denition 1 (Reliability). e reliability is the probability that
the special elements will accomplish a required task within
a time threshold []. In this study, the reliability of a WSN
is an important measure in evaluating the performance of
aWSN.
Denition 2. e reliability of the WSN is the probability that
there exists an operational path between the sink node and
every node in the set of target nodes.
Denition 3. Given the WSN network = (,,,),
using random vector =(
0
,
1
,...,
𝑛−1
) and =
(
0
,
1
,...,
𝑚−1
) to, respectively, express the working state
of the network’s nodes and edges, and using (,),to
express the state of network , then the reliability of the
network ’s Boolean function is
𝑟
,=
1, if is at the state ,,
exists an operational path between
and every node in
0, else.
()
e reliability of a WSN under the multicast model is
𝑁
=
(𝑥,𝑦)∈Ω
Pr
𝑟
,=1,
()
where is the Cartesian product (×), which is the state
space of the network.
2.3. Ordered Binary Decision Diagram. An ordered binary
decision diagram (OBDD) [, ] provides a compact, canon-
ical, and eciently manipulative representation for Boolean
functions.
An OBDD for a nonconstant Boolean function (
1
,
2
,
...,
𝑛
)is a directed acyclic graph. e characteristics of the
OBDDcanbesummarizedasfollows:
() anOBDDhasthreetypesofnodes:root,terminal,and
internal. e root node has no parent or is without
input arcs, the terminal nodes have no child or out
arcs, and the internal nodes have two outgoing arcs;
() the root node of an OBDD represents the function ;
() there are two terminal nodes “” and “,” which
represent constant Boolean functions and ;
() each nonterminal node has four attributes:
𝑢
,var,
low, and high, where
𝑢
is the Boolean function of
the vertex and
𝑢
=(
1
,...,,...,
𝑛
);varis
thevariablelabelingthevertex;lowisthe-node
while var =0, the arc connecting the vertex and
low is -edge, and low =
𝑢
|
𝑢=0
as well as
𝑢
|
𝑢=0
=
(
1
,...,0,...,
𝑛
);highisthe-nodewhilevar=1,
the arc connecting the vertex and high is -edge, and
high =
𝑢
|
𝑢=1
as well as
𝑢
|
𝑢=1
=(
1
,...,1,...,
𝑛
);
then,
𝑢
=
⋅
𝑢
|
𝑢=0
+⋅
𝑢
|
𝑢=1
;
() given the variable ordering :
1
<
2
<⋅⋅⋅<
𝑛
,
variablesinanOBDDareorderedandallthepathsin
the OBDD keep the same variable ordering.
Given a Boolean function and any assignments to its
variables, the function value is determined by tracing a path
from the root to a terminal node following the appropriate
branch from each node. e branch depends on the variable
value of the assignments, and the function value under the
assignments is determined by its path’s terminal node.
For example, Figure shows the binary and the OBDD
for the Boolean function =
1
⋅
2
+
2
⋅
3
,inwhichthe
variable order is :
1
<
2
<
3
. It is obvious that the OBDD
is a directed acyclic graph and stores the same information
more compactly. We trace the path A → B → D → F
andreachtheterminalnode.us,thevalueofthefunction
=
1
⋅
2
+
2
⋅
3
of variable assignment (, , ) is .
3. A Solution Algorithm for the Problem of
WSN Reliability
3.1. Symbolic Formulation of the WSN on the Multicast Model.
WeconvertaWSNonthemulticastmodel,=(,,,),
toasymbolicOBDDformbyencodingtheverticesof
with a length-binary number, where =log
2
num
and
num
was the number of the vertices. Each encoded vertex
of corresponded to a vector of binary variables =
(
0
,...,
𝑛−1
).eedge(V
𝑖
,V
𝑗
)∈of can be represented by
abinaryvector(,)=(
0
,...,
𝑛−1
,
0
,...,
𝑛−1
),where=
(
0
,...,
𝑛−1
)and =(
0
,...,
𝑛−1
)are the binary encodings
of vertex and vertex , respectively. e connecting relation
between two nodes can be represented using the following
characteristic function:
,=
1, ,∈,
0, otherwise.
()
is function implies the relationship between the nodes and
edges of the WSN.
In this study, we assume that the operational probability
of all the nodes was .. en, in do not need to be