Fault Tolerant Coverage Models for Sensor Networks Wolfgang W. Bein∗ School of Computer Science University of Nevada Las Vegas, NV
[email protected]
Doina Bein School of Computer Science University of Nevada Las Vegas, NV
[email protected]
Srilaxmi Malladi Department of Computer Science Georgia State University, GA
[email protected]
Abstract We study the coverage problem from the fault tolerance point of view for sensor networks. Fault tolerance is a critical issue for sensors deployed in places where they are not easily replaceable, repairable and rechargeable. The failure of one node should not incapacitate the entire network. We propose three 1-fault tolerant models, and we compare them to each other, as well as with the minimal coverage model [11]. To study the reliability of proposed models, we develop the Markov model for each of them and calculate the reliability assuming a constant failure rate. We show that the most unreliable model among these three models is the hexagonal model, and the improved model is the most reliable on long term. For short time from the start, the square model is more reliable, but after a short while, the improved model becomes and remains the better one. Keywords: Coverage, efficiency, fault tolerance, Markov model, reliability, smart sensors, sensor network.
1
Introduction
If all sensors deployed within a small area are active simultaneously, an excessive amount of energy is used, redundant data is generated, and packet collision can occur on transmitting data. At the same time, if areas are not covered, events can occur without being observed. A density control function is required to ensure that a subset of nodes is active in such a way that coverage and connectivity are maintained. Coverage refers to the total area currently monitored by active sensors; this needs to include the area required to be covered by the sensor networks. Connectivity refers to the connectivity of the sensor network modeled as a graph. The nodes in a wireless environment are greatly dependent on the battery life and power. Therefore, minimizing energy consumption for the network while keeping its functionality is a major objective in designing a robust, reliable network. But sensors are prone to failures and disconnection. Only minimal coverage of a given region without redundancy would make such a network unattractive from a practical point of view. Therefore it is necessary to not only design for minimal coverage, on the other hand fault tolerance features must be viewed in light of the additional sensors and energy used. Sensors coupled with integrated circuits, known as smart sensors, provide high sensing from their relationship with each other and with higher level processing layers. A smart sensor is specifically designed for the targeted application [5]. Smart sensors find their applications in a wide variety of fields such as military, civilian, biomedical as well as control systems, etc. In military applications, sensors can track troop movements and help decide deployment of troops. In civilian applications, sensors can typically be applied to detect pollution, burglary, fire hazards and the like. Wireless body sensors implanted in the body must be energy efficient, utilize bandwidth, 1
robust, lightweight and fault tolerant as they are not easily replaceable, repairable and rechargeable. Bio sensors need a dynamic, self-stabilizing network. Related work Given a sensor network deployed in a target area, [3] focused on finding whether each point of the area is covered by at least K sensors. [12] extends the problem further and focuses on selecting a minimum size set of sensor nodes which are connected in such a way that each point inside the area covered by the entire sensor network is covered by at least K sensors. Starting from the uniform sensing range model [11], two models are proposed using sensors with different sensing ranges [10]. Variable sensing range is novel, unfortunately both models are worse in terms of achieving a better coverage. Also, the second model in [10] requires (for some sensors) that the communication range to be almost six times larger than the sensing range, otherwise connectivity is not achieved. A relay node, also called in the literature, gateway [1] or application node [6], acts as clusterhead in the corresponding cluster. In [2] a fault-tolerant relay node placement scheme is proposed for wireless sensor networks, and a polynomial time approximation algorithm is presented to select a set of active nodes, given the set of all the nodes. In [7] the project of building a theoretical artificial retina made up of smart sensors is described. The sensors should form a tapered array that should rests on retina and produce electrical signals which are converted by the underlying tissue into chemical signals to be sent to the brain. The sensor array is therefore used for both reception and transmission in a feedback system. The challenges with these sensors are the wireless networking, distribution, placement and continuing operation of these sensors. Motivation and Contributions We are interested in the bio-medical domain where applications of sensors are relatively new. Sensors are already applied to monitor temperature level, glucose level, organs and its implants, and to detect external agents in the body in connection with cancer and other health abnormalities. We study the coverage problem from the fault tolerance point of view. The goal of our paper is to propose several sensor array placement schemes, which are fault tolerant. Fault tolerance is a critical issue depending on where the sensors are employed. The failure of one node should not incapacitate the entire network. Thus despite the presence of limited number of failed sensors, the system continues to function. We propose three 1 fault tolerant models, and we compare with each other and with the minimal coverage model from [11]. To study the reliability of proposed models, we develop the Markov model for each of them and calculate the reliability assuming a constant failure rate. We show that the most unreliable model among these three models is the hexagonal model, and the improved model is the most reliable on long term. For short time from the start, the square model is more reliable, but after a short while, the improved model becomes and remains the better one. Outline of the paper Section 2 presents the various parameters for the sensor nodes, followed by the proposed three models. In Section 3 we compare them among each other, and with the minimal coverage model from [11]. We finish with concluding remarks in Section 4.
2
Fault Tolerant Models
Two parameters are important for a sensor node: the wireless communication range of a sensor rC , and the sensing range rS . They generally differ in values, and a common assumption is that rC ≥ rS . Obviously, two nodes u and v, whose wireless communication ranges are rCu , respectively rCv , can communicate directly if dist(u, v) ≤ min(rCu , rCv ). In [11], it is proved that if all the active sensor nodes have the same parameters (radio range rC and sensing range rS ) and the radio range is at least twice of the sensing range rC ≥ 2 × rS , complete coverage of an area implies connectivity among the nodes. Therefore, under this assumption, the connectivity problem reduces to the coverage problem. There is a trade-off between minimal coverage and fault tolerance. For the same set of sensors, a fault tolerant model will have a smaller area to cover. Or, given an area to be covered, more sensors will be required, or the same number of sensors but with a higher values for the parameters. 2
A model is k fault tolerant if by removal of any k nodes, the network preserves its functionality. A k fault tolerant model for the coverage problem will be able to withstand k removals: by removing any k nodes, the covered region remains the same. A 0 tolerant model will not work in case of any removal of a node. A straightforward approach is to either double the number of sensors in each point, or to double the sensor parameters for some sensors of the minimal coverage model to make it 1 tolerant. Similar actions can be taken for a 1 tolerant model to be 2 tolerant and so on. In order for a k fault-tolerant model to be worthwhile, it has to be better than the straightforward approach. We propose three 1 fault tolerant models: square, hexagonal and improved 8-node model (obtained by adding one more bigger range sensor in the middle of a 7-node minimal coverage model). We assume the sensing range of the sensors in square, hexagonal, and partially in √ the improved 8-node model to be r. In the improved 8-node model, there is one sensor whose sensing range is r 3. In the first model, the basic structure is composed of four sensors arranged in a square-like structure of side r. In the second model, the basic structure is composed of six sensors arranged in a regular hexagon-like structure of side r. The third model comprises sensors of different sensing √ range. The basic structure is composed of seven sensors arranged in a regular hexagon-like structure of side r 3 and one more sensor placed in the center of the hexagon. Based on the topology of these models, the assumption that the communication range is greater than twice the sensing range is enough guarantees the connectivity of the network.
2.1 Square Fault Tolerant Model The basic structure for the first model is drawn in Figure 1(a). A
B
(a) Four sensors in a square arrangement
C
(b) Selected areas A, B, and C
Figure 1. Square fault tolerant model square The square surface S4 = r2 is partitioned into an area covered by exactly two sensors S2s , an area covered square square by exactly three sensors S3s , and an area covered by exactly four sensors S4s . In order to calculate the values for those areas, let A, B, and C to be some disjoint areas as drawn in Figure square square square 1(b). We observe that S2s = 4SA , S3s = 8SB , S4s = 4SC . We can derive the following system of equations:
8 > SA + 2SB + SC > > > > < SB + SC + 14 > > > > > : 4SB + 4SC + SA
=
r2 4
=
Πr 2 8
=
Πr 2 3
−
√ r2 3 4
8 > SA > > > > < ⇒ SB > > > > > : SC
√ r2 3 4
=
r2 −
=
− r2 +
=
r2 4
2
−
−
√ r2 3 4
√ r2 3 4
+
Πr 2 6
+
Πr 2 24
Πr 2 12
8 square √ 2 > S2s = 4r 2 − r 2 3 − 2Πr > 3 > > > < √ 2 square ⇒ S3s = −4r 2 + 2r 2 3 + Πr 3 > > > > > √ 2 : square S4s = r 2 − r 2 3 + Πr 3
Therefore, given a 2D-region of dimension (rN ) × (rM ), with N and M strictly positive integers, we can derive the following results. The number of sensors required is (N + 1) × (M + 1). The ratio between the 2 sensor area used and the area covered is (N +1)(M+1)Πr . The area covered by two sensors is = (N +1)(M+1)Π N Mr 2 NM √ 2 square N M S2s = N M (4r2 − r2 3 − 2Πr ). 3 √ 2 square The area covered by three sensors is N M S3s = N M (−4r2 + 2r2 3 + Πr3 ). √ 2 square The area covered by four sensors is N M S4s = N M (r2 − r2 3 + Πr3 ).
3
2.2 Hexagonal Fault Tolerant Model The basic structure for the second model is drawn in Figure 2(a). A
B
(a) Six sensors in a regular hexagonal arrangement
(b) Selected areas A and B
Figure 2. Hexagonal fault tolerant model √
hexagon , and an The hexagonal surface S6 = 3 2 3 is partitioned into an area covered by exactly two sensors S2s hexagon area covered by exactly three sensors S3s . In order to calculate the values for those areas, let A and B be some disjoint areas as drawn in Figure 2(b). We hexagon hexagon observe that S2s = 6SA and S3s = 6SB . We can derive the following system of equations:
8 > < SA + SB > :
1 S 2 B
+
√ r2 3 4
√ r2 3 4
=
Πr 2 6
=
⇒
8 > < SA = > :
SB =
√ 3r 2 3 4 Πr 2 3
−
−
Πr 2 3
√ r2 3 2
⇒
8 hexagon > = < S2s
√ 9r 2 3 2
− 2Πr 2
> : S hexagon = −3r 2 √3 + 2Πr 2 3s
Therefore, given a 2D region of dimension (rN ) × (rM ), with N and M strictly positive integers, we can derive √ ⌋. the following results. The number of sensors required is 2 + 4⌊ N2 ⌋⌊ 4M 3 The ratio between the sensor area used and the area covered is
4M √ ⌋)Πr 2 (2+4⌊ N 2 ⌋⌊ 3
N Mr 2
=
4M √ Π(2+4⌊ N 2 ⌋⌊ 3 ⌋)
NM
.
2.3 Improved 8-Node Model We now consider instead of the minimal coverage model of three nodes, the seven-node model obtained by overlapping three three-node models (see Figure 3(a)). The minimal coverage √ model can be improved to be 1 fault tolerant by adding a sensor whose sensing range is increased from r to r 3 and we obtain the improved 8-node model (see Figure 3(b)).
(a) 7-node minimal coverage model
(b) Fault tolerant improved 8-node model
Figure 3. Improved 8-node model versus 7-node minimal coverage model √
eight , an area The hexagonal surface S8 = 3 2 3 is partitioned into an area covered by exactly two sensors S2s eight eight covered by exactly three sensors S3s , and an area covered by exactly four sensors S4s (see Figure 4).
4
B
A
C
D E
F
Figure 4. Partitioning the Improved 8-node model In order to calculate the values for those areas, let A, B, C, D, E and F be some disjoint areas as drawn in eight eight eight Figure 4. We observe that S2s = SF + 2SD , S3s = 2SE + SC + 2SB , and S4s = SA We can derive the following system of equations: 8 > SA + 2SB > > > > > > > > > SA + SC > > > > > > > > > SA + 2SB + SC + 2SD + 2SE + SF > > < > SA + SB + SC + SD + SE > > > > > > > > > 3SE + 3SF > > > > > > > > > 2SE + SF > > :
=
3Πr 2 4
=
Πr 2 6
= = = =
3r
−
−
2√
√ 3r 2 3 4
√ r2 3 4
3
4 Πr 2 6 (r
⇒
√ 2√ 3) 3 4
Πr 2 6
8 > SA = > > > > > > > > > SB = > > > > > > > > > < SC =
> > > SD = > > > > > > > > > SE = > > > > > > > : SF =
Πr 2 2
−
√ 3r 2 3 4
Πr 2 8 √ r2 3 2 √ r2 3 2 Πr 2 6
− −
−
√ r2 3 2
Πr 2 3 7Πr 2 24
⇒
8 improved > < S2s =
√ 9r 2 3 2
− 2Πr 2
> : S improved = −3r 2 √3 + 2Πr 2 3s
√ r2 3 4
−
Πr 2 6
Given a 2D region of dimension (rN ) × (rM ), with N and M strictly positive integers, we can derive the √ ⌋. following results. The number of sensors required is 2 + 6⌊ N2 ⌋⌊ 4M 3 The ratio between the sensor area used and the area covered is
3
4M √ ⌋)Πr 2 (2+6⌊ N 2 ⌋⌊ 3
N Mr 2
=
4M √ Π(2+6⌊ N 2 ⌋⌊ 3 ⌋)
NM
.
Comparative Results
We compare the the minimum coverage model [11], the 8-node improved model, the square model, and the hexagonal model in terms of: the model efficiency (the area covered vs. the portion of the sensors area used for coverage), the probability for the model to function, and the reliability. Table 1 contains some comparative results, and the reliability for each model is calculated in Subsection 3.1, using Markov model of reliability. Let f. t. be a short form of fault tolerant. Covered area denotes the area covered by the polygonal line formed by the sensors. Portion used denotes the portion of the sensor areas used for covering that area; this value helps in calculating the energy used for covering the region. Efficiency is defined as the ratio between the previous two values (the covered area and the portion of the sensor area used), and denotes the efficiency of using a particular model. Max. nodes to fail denotes the maximum number of nodes that can fail and still the coverage is available over the number of nodes in the model. Prob. to function denotes the probability for the model to be functional. Assume that all the sensors, independent of their sensing range, have the probability p to fail, 0 ≤ p ≤ 1, thus the probability to function for sensor is 1 − p. Also we assume that any two failures are independent one another. The values for the probability functions in case of the square and hexagonal model from Table 1 follow. The probability to function in case of square model is Psquare = (1−p)4 +4p(1−p)3 +2p2 (1−p)2 = (1−p)2 (1+2p−p2). In case of the hexagonal model, the probability to function is Phexa = (1−p)6 +6p(1−p)5 +15p2 (1−p)4 +2p3 (1−p)3 = (1 − p)2 (1 + 3p + 6p2 − 8p3 ). 5
Min. cov. 0 f. t.
8-node Improved 1 f. t.
Square 1 f. t.
Hexagonal 1 f. t.
No. sensors
7
8
4
6
model covered area
√ 9 3r 2 2
√ 9 3r 2 2
r2
√ 3r 2 3 2
fraction used
3Πr2
6Πr2
Πr2
2Πr2
efficiency
√ 3 3 2Π
√ 9 3 12Π
≃ 0.827
1 Π
≃ 0.413
≃ 0.318
√ 3 3 4Π
≃ 0.413
max. nodes to fail
0/7
7/8
2/4
3/6
prob. to function
(1 − p)7
1 − p + p(1 − p)7
(1 − p)2 (1 + 2p − p2 )
(1 − p)3 (1 + 3p − 2p3 )
Table 1. Comparisons among the four models From Table 1 we note that the minimal coverage model has the best efficiency, followed by the 8-node improved and hexagonal (the same value), and the square model. As far as the probability to function, the minimal model has the lowest value. For 0 ≤ p ≤ 0.082, the square model is better than the hexagonal, that is better than the improved model. For 0.083 ≤ p ≤ 0.348, the square model is better than the improved model, that is better than the hexagonal model. For p ≥ 0.349, the improved model is better than the square model, that is better than the hexagonal model.
3.1 Reliability of the Proposed Models Reliability is one of the most important attributes of a system. Markov modeling is the most commonly used analytical technique for complex systems [8, 9, 4], and uses system state and state transitions. The state of the system comprises all it needs to be known to fully describe it at any given instant of time [4]. Each state of the Markov model is a unique combination of faulty and non-faulty modules. There is one state called “F” which is the failed state (the system does not function anymore). A state transition occurs when one or more modules had failed or had recovered (if possible), and it is characterized by probabilities (to fail or to recover). The exponential failure law states that the reliability of a system varies exponentially as a function of time, for a constant failure rate function. The reliability of the system at time t, R(t), is an exponential function of the failure rate, λ, that is a constant: R(t) = e−λt where λ is the constant failure rate. If we assume that each module (in our case, each sensor) in the models we have proposed obeys the exponential failure law and has a constant failure rate of λ, the probability of each module (sensor) being failed at some time t + ∆t, given that the module (sensor) was operational at time t, is given by 1 − e−λ∆t . For small values of ∆t, the expression reduces to 1 − e−λ∆t ≈ λ∆t. In other words, the probability that a sensor will fail within the time period ∆t is approximately λ∆t. In our models, we make three assumptions. We assume that the system starts in a state with no failures (perfect state). Second, we assume that a failure is permanent: once a module has failed, it does not recover. Third, we assume that there is one failure at a time. 3.1.1
Square Fault Tolerant Model
The Markov model of the square fault tolerant model is shown in Figure 5(a). It can be reduced further as follows. The single state in which all four sensors are operational can be called “4”. The four states in which a sensor has failed can be reduced to one state called “3”. The two states in which two opposite sensors have failed can be
6
reduced to one state called “2”. The reduced Markov model is shown in Figure 5(b). 1−3λ∆ t
2 λ∆ t
λ∆ t 1−3λ∆ t λ∆ t 1−4λ∆ t
1−2λ∆ t
λ∆ t
2 λ∆ t 2 λ∆ t
λ∆ t
2 λ∆ t λ∆ t 1−3λ∆ t
1−2λ∆ t
1−4λ∆t
F
2 λ∆ t
1−3λ∆t 1
1
4
2 λ∆ t
4λ∆t
2λ∆t
3
F
λ∆t
λ∆ t
2λ∆t 2 1−2λ∆t
1−3λ∆ t
(a) Markov model of the square model
(b) Reduced Markov model of the square model
Figure 5. Markov models of the square model The equations of Markov model of the square fault tolerant model can be easily written from the reduced state diagram shown in F igure 5(b). For each state S, the probability of the system being in that state at some time t + ∆t depends on the probability that the system was in that state S at time t and on any probability that the ′ ′ system was in another state S and it has transition from S to S. The reliability of the system is the probability of the system to be in any of the non-failed states Rsquare (t) = 1 − PF (t) = P4 (t) + P3 (t) + P2 (t). We obtain the system of equations below, with the initial values for P4 (0) = 1, P3 (0) = P2 (0) = PF (0) = 0. Taking the limit as δt approaches 0 results in a set of differential equations to which Laplace transformation can be applied. Then we applied the reverse Laplace transformation. 8 P4 (t + ∆t) > > > > > > > < P3 (t + ∆t)
> > P2 (t + ∆t) > > > > > : PF (t + ∆t)
=
(1 − 4λ∆t)P4 (t)
=
(1 − 3λ∆t)P3 (t) + 4λ∆tP4 (t)
⇒
=
(1 − 2λ∆t)P2 (t) + λ∆tP3 (t)
=
PF (t) + 2λ∆t)P3 (t) + 2λ∆tP2 (t)
8 P4 (t) > > > > > > > > < P3 (t)
> > P2 (t) > > > > > > : PF (t)
=
e−4λt
=
4e−3λt − 4e−4λt
=
2e−2λt − 4e−3λt + 2e−4λt
=
1 − 2e−2λt + e−4λt
We obtain the reliability of the system to be Rsquare (t) = 2e−2λt − e−4λt .
3.2 Hexagonal Fault Tolerant Model The Markov model of the hexagonal fault tolerant model has more than 30 states. Some of the states (as graphs) are isomorphic with each other, so we can reduced them as follows. The single state in which all six sensors are operational can be called “6”. The six states in which a sensor has failed can be reduced to one state called “5”. The states in which two sensors adjacent to a working sensor have failed can be reduced to one state called “4a”. The states in which two diametrical opposite sensors have failed can be reduced to one state called “4b”. The
7
1−4λ∆ t
4b
1−5λ∆ t 1−6λ∆t
λ∆ t
6
6 λ∆ t
4 λ∆ t
5
2 λ∆ t 2 λ∆ t
1−4λ∆ t
3 λ∆ t
4a
F
1
λ∆ t 3 λ∆ t
3 1−3λ∆ t
Figure 6. Reduced Markov model of the hexagonal model states in which three non-consecutive sensors have failed can be reduced to one state called “3”. The reduced Markov model is shown in Figure 3.2. The equations of Markov model of the hexagonal fault tolerant model can be easily written from the reduced state diagram shown in Figure 3.2. For each state S, the probability of the system being in that state at some time t + ∆t depends on the probability that the system was in that state S at time t and on any probability that ′ ′ the system was in another state S and it has a transition from S to S. The reliability of the system is the probability of the system to be in any of the non-failed states Rhexagon (t) = 1 − PF (t) = P6 (t) + P5 (t) + P4a (t) + P4b (t) + P3 (t). We obtain the system of equations below, with the initial values for P6 (0) = 1, P5 (0) = P4a (0) = P4b (0) = P3 (0) = PF (0) = 0. Taking the limit as δt approaches 0 results in a set of differential equations to which Laplace transformation can be applied. We apply then the reverse Laplace transformation. 8 P6 (t + ∆t) = (1 − 6λ∆t)P6 (t) > > > > > > > P5 (t + ∆t) = (1 − 5λ∆t)P5 (t) + 6λ∆tP6 (t) > > > > > > > > < P4a (t + ∆t) = (1 − 4λ∆t)P4a (t) + 2λ∆tP5 (t)
> > P4b (t + ∆t) = (1 − 4λ∆t)P4b (t) + λ∆tP5 (t) > > > > > > > > P3 (t + ∆t) = (1 − 3λ∆t)P3 (t) + λ∆tP4a (t) > > > > > : PF (t + ∆t) = PF (t) + λ∆t(2P5 (t) + 3P4a (t) + 4P4b (t) + 3P3 (t))
⇒
8 P6 (t) > > > > > > > > P5 (t) > > > > > > > > < P4a (t) > > P4b (t) > > > > > > > > P3 (t) > > > > > > : PF (t)
=
e−6λt
=
6e−5λt − 6e−6λt
=
6e−4λt − 12e−5λt + 6e−6λt
=
3e−4λt − 6e−5λt + 3e−6λt
=
2e−3λt − 6e−4λt + 6e−5λt − 2e−6λt
=
1 − 2e−3λt + e−4λt + 6e−5λt − 2e−6λt
We obtain the reliability of the system to be Rhexagon (t) = 2e−3λt + 3e−4λt − 6e−5λt + 2e−6λt .
3.3 Improved 8-node Model The Markov model of the improved 8-node model has more than 40 states. Some of the states (as graphs) are isomorphic with each other, so we can reduced them as follows. We label the states by the number of sensors of √ each type, first the sensor whose sensing range is r, second the sensor whose sensing range is√r 3. The single state in which all eight sensors are operational can be called “(7,1)”. The state in which the r 3 sensor has failed is
8
called “(7,0)”. The seven states in which a r sensor has failed can be reduced to one state called “(6,1)”. The states in which two r sensors have failed can be reduced to one state called “(5,1)”. The states in which three r sensors have failed can be reduced to one state called “(4,1)”. The states in which four r sensors have failed can be reduced to one state called “(3,1)”. The states in which five r sensors have failed can be reduced to one state called “(2,1)”. The states in which six r sensors have failed can be reduced to one state called “(1,1)”. The states in which all r sensors have failed can be reduced to one state called “(0,1)”. The reduced Markov model is shown in Figure 3.3. 1 −(λ 1+ 5 λ ) ∆ t 1 −(λ 1+ 6 λ )∆ t
7 λ∆ t
(7,1)
1 −(λ 1+ 7λ ) ∆ t
(5,1)
6λ∆ t
(6,1) λ1 ∆ t
λ1 ∆ t
1 −(λ + 4 λ ) ∆ t 1
(4,1) 4 λ∆ t
λ1 ∆ t
λ1 ∆ t
1 −(λ + 3 λ ) ∆ t 1
λ1 ∆ t
F 7λ∆ t
(3,1)
λ1 ∆ t
3λ∆ t
λ1 ∆ t
λ1 ∆ t
(7,0)
5 λ∆ t
(0,1) (1,1)
λ∆ t
(2,1)
1 −(λ + 2 λ ) ∆ t 1
2 λ∆ t
1 −(λ 1+ λ ) ∆ t
Figure 7. Reduced Markov model of the improved 8-node model √ We assume that the r 3 sensor has the constant failure rate λ1 , while a r sensor has the constant failure rate λ. The equations of Markov model of the improved 8-node fault tolerant model can be written from the reduced state diagram shown in Figure 3.3. For each state S, the probability of the system being in that state at some time t + ∆t depends on the probability that the system was in that state S at time t and on any probability that ′ ′ the system was in another state S and it has transition from S to S. The reliability of the system is the probability of the system to be in any of the non-failed states Rimproved (t) = 1 − PF (t) = P(7,1) (t) + P(7,0) (t) + P(6,1) (t) + P(5,1) (t) + P(4,1) (t) + P(3,1) (t) + P(2,1) (t) + P(1,1) (t) + P(0,1) (t). We obtain the system of equations below, with the initial values for P(7,1) (0) = 1, P(7,0) (0) = P(6,1) (0) = P(5,1) (0) = P(4,1) (0) = P(3,1) (0) = P(2,1) (0)P(1,1) (0) = P(0,1) (0) = PF (0) = 0.
9
8 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > :
P(7,1) (t + ∆t)
=
(1 − (λ1 + 7λ)∆t)P(7,1) (t)
P(7,0) (t + ∆t)
=
(1 − 7λ∆t)P(7,0) (t) + λ1 ∆t)P(7,1) (t)
P(6,1) (t + ∆t)
=
(1 − (λ1 + 6λ)∆t)P(6,1) (t) + 7λ∆tP(7,1) (t)
P(5,1) (t + ∆t)
=
(1 − (λ1 + 5λ)∆t)P(5,1) (t) + 6λ∆tP(6,1) (t)
P(4,1) (t + ∆t)
=
(1 − (λ1 + 4λ)∆t)P(4,1) (t) + 5λ∆tP(5,1) (t)
P(3,1) (t + ∆t)
=
(1 − (λ1 + 3λ)∆t)P(3,1) (t) + 4λ∆tP(4,1) (t)
P(2,1) (t + ∆t)
=
(1 − (λ1 + 2λ)∆t)P(2,1) (t) + 3λ∆tP(3,1) (t)
P(1,1) (t + ∆t)
=
(1 − (λ1 + λ)∆t)P(1,1) (t) + 2λ∆tP(2,1) (t)
P(0,1) (t + ∆t)
=
(1 − λ1 ∆t)P(0,1) (t) + λ∆tP(1,1) (t)
PF (t + ∆t)
=
PF (t) + 7λ∆tP(7,0) (t) + λ1 ∆t(P(6,1) (t) + P(5,1) (t) + P(4,1) (t) + P(3,1) (t) + P(2,1) (t) + P(1,1) (t) + P(0,1) (t))
Assuming λ1 = λ, and taking the limit as δt approaches 0 results in a set of differential equations to which Laplace transformation can be applied. Then we apply the reverse Laplace transformation. 8 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > :
P(7,1) (t)
=
e−8λt
P(7,0) (t)
=
e−7λt − e−8λt
P(6,1) (t)
=
7e−7λt − 7e−8λt
P(5,1) (t)
=
21e−6λt − 42e−7λt + 21e−8λt
P(4,1) (t)
=
35e−5λt − 105e−6λt + 105e−7λt − 35e−8λt
P(3,1) (t)
=
35e−4λt − 140e−5λt + 210e−6λt − 140e−7λt + 35e−8λt
P(2,1) (t)
=
21e−3λt − 105e−4λt + 210e−5λt − 210e−6λt + 105e−7λt − 21e−8λt
P(1,1) (t)
=
7e−2λt − 42e−3λt + 105e−4λt − 140e−5λt + 105e−6λt − 42e−7λt + 7e−8λt
P(0,1) (t)
=
e−λt − 7e−2λt + 21e−3λt − 35e−4λt + 35e−5λt − 21e−6λt + 7e−7λt − e−8λt
PF (t)
=
1 − e−λt − e−7λt + e−8λt
We obtain the reliability of the system to be Rimproved (t) = e−λt + e−7λt − e−8λt .
3.4 Simulation Results We drew the reliability functions of the three models for values of λ ∈ {0.005, 0.01, 0.02, 0.05} (Figure 8). As λ draws closer to 1, the difference between reliability values for the three models become smaller. In Figure 8(a), we can see better that the reliability values for the hexagonal model are always smaller than the other two models. From the starting point in time (time = 0), the square model has a better reliability than the improved model. But once the two reliability values drew closer to the value 0.7, the models switch and remains so thereafter: the better reliability is achieved by the improved model, while the square model has a worse reliability. This conclusion is consistent among all the four figures.
4
Conclusion
We study the coverage problem from the fault tolerance point of view for sensor networks. Fault tolerance is a critical issue for sensors depending on where the sensors are employed. The failure of one node should not
10
Reliability (units)
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
R square R hexagon R improved
0
100
50
150
200
Time (units)
0
R square R hexagon R improved
50
100
150
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
R square R hexagon R improved
0
50
Time (units)
Time (units)
(b)λ = 0.01
(c)λ = 0.02
Reliability (units)
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
Reliability (units)
Reliability (units)
(a)λ = 0.005
100
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
R square R hexagon R improved
50 Time (units)
(d)λ = 0.05
Figure 8. Reliability chart (constant lambda) incapacitate the entire network. Wireless body sensors have to be energy efficient, utilize bandwidth, robust, lightweight and fault tolerant as they are not easily replaceable, repairable and rechargeable. We propose three 1 fault tolerant models, and we compare them among themselves, and with the minimal coverage model. We note that the minimal coverage model has the best efficiency, followed by the 8-node improved and hexagonal (the same value), and the square model. As far as the probability to function, the minimal model has the lowest value. For 0 ≤ p ≤ 0.082, the square model is better than the hexagonal, that is better than the improved model. For 0.083 ≤ p ≤ 0.348, the square model is better than the improved model, that is better than the hexagonal model. For p ≥ 0.349, the improved model is better than the square model, that is better than the hexagonal model. To study the reliability of proposed models, we develop the Markov model for each of them and calculate the reliability assuming a constant failure rate. We show that the most unreliable model among these three models is the hexagonal model, and the improved model is the most reliable on long term. For short time from the start, the square model is more reliable, but after a short while, the improved model becomes and remains the better one. We are currently working on algorithms to move sensors in order to preserve the network functionality when more than a fault occurs. If the network layout is composed by hundreds of such proposed models, in some cases sensors need to be moved to cover areas left uncovered by faulty or moving sensors.
References [1] G. Gupta and M. Younis. Fault-tolerant clustering of wireless sensor networks. Proceedings of IEEE Wireless Communications and Networking Conference (WCNC), pages 1579–1584, 2003. [2] B. Hao, J. Tang, and G. Xue. Fault-tolerant relay node placement in wireless sensor networks: formulation and approximation. IEEE Workshop on High Performance Switching and Routing (HPSR), pages 246–250, 2004.
11
[3] C. Huang and Y. Tseng. The coverage problem in a wireless sensor network. ACM International Workshop on Wireless Sensor Networks and Applications (WSNA), pages 115–121, 2003. [4] B. Johnson. Design and Analysis of Fault-Tolerant Digital Systems. Addison-Wesley Publishing Company, Inc., 1989. [5] A. Moini. Vision chips or seeing silicon. Department of Electrical and Electronics Engineering, University of Adelaide, Australia, http://www.iee.et.tu-dresden.de/iee/eb /analog/papers/mirror/visionchips/vision chips/smart sensors.html, 1997. [6] J. Pan, Y. Hou, L. Cai, Y. Shi, and S. Shen. Topology control for wireless sensor networks. Proceedings of ACM MOBICOM, pages 286–299, 2003. [7] L. Schwiebert, S. Gupta, and J. Weinmann. Research challenges in wireless networks of biomedical sensors. ACM Sigmobile Conference, pages 151–165, 2001. [8] M. Shooman. Probabilistic Reliability: An Engineering Approach. McGraw-Hill, New York, 1968. [9] K. Trivedi. Probability and Statistics with Reliability, Queuing, and Computer Science Applications. Prentice-Hall, Englewood Cliffs, N.J., 1982. [10] J. Wu and S. Yang. Coverage issue in sensor networks with ajustable ranges. International Conference on Parallel Processing (ICPP), pages 61–68, 2004. [11] H. Zhang and J. Hou. Maintaining sensing coverage and connectivity in large sensor networks. Proceedings of NSF International Workshop on Theoretical and Algorithmic Aspects of Sensor, Ad Hoc Wireless, and Peer-to-Peer Networks, 2004. [12] Z. Zhou, S. Das, and H. Gupta. Connected k-coverage problem in sensor networks. International Conference on Computer Communications and Networks (ICCCN), pages 373–378, 2004.
12
