 Research
 Open Access
Crosslayer design for decentralized detection in WSNs
 Ashraf Tantawy^{1}Email author,
 Xenofon Koutsoukos^{2} and
 Gautam Biswas^{2}
https://doi.org/10.1186/16876180201443
© Tantawy et al.; licensee Springer. 2014
 Received: 8 October 2013
 Accepted: 10 March 2014
 Published: 31 March 2014
Abstract
A wireless sensor network (WSN) deployed for detection applications has the distinguishing feature that the sensors cooperate to perform the detection task. Therefore, the decoupled and maximum throughput design approaches typically used to design communication networks do not lead to the desired optimal detection performance. Recent work on decentralized detection has addressed the design of media access control (MAC) and routing protocols for detection applications by considering independently the quality of information (QoI), channel state information (CSI), and residual energy information (REI) for each sensor. However, little attention has been given to integrate the three quality measures (QoI, CSI, and REI) in the system design. In this work, we present a crosslayer approach to design a QoI, CSI, and REIaware transmission control policy (XCP) that coordinates communication between local sensors and the fusion center, in order to maximize the detection performance. We formulate and solve a constrained nonlinear optimization problem to find the optimal XCP design variables, for both ALOHA and timedivision multiple access (TDMA) sensor networks. We show the detection performance gain compared to the typical decoupled and maximum throughput design approaches, without utilizing additional network resources. We compare ALOHA and TDMA MAC schemes and show the conditions under which each transmission scheme outperforms.
Keywords
 Decentralized detection
 Wireless sensor networks
 Networked information fusion
 Multiaccess communication
1 Introduction
The deployment of wireless sensor networks (WSNs) in decentralized detection applications is motivated by the availability of lowcost sensors with computational capabilities, combined with advances in communication network technologies. In decentralized detection (DD), multiple sensors collaborate to distinguish between two or more hypotheses, and the classical problem is to find the local sensor detection strategies (quantization rules) to minimize a systemwide cost function using different network topologies and channel models [1, 2]. Despite the fact that this classical problem is insightful, current research on detection using modern WSNs has shifted the focus away from this classical quantization problem for two main reasons: (1) performance loss due to quantization decays rapidly with the number of information bits in the packet payload [3, 4], and (2) the payload of a packet could be considered large enough to represent local sensor information with adequate accuracy, as additional bits in the payload are unlikely to affect power or delay, given the relatively large packet overhead [5, 6] (e.g., IEEE 802.15.4 standard has a minimum of 9 bytes for the medium access control (MAC) overhead [7]). On the other hand, the deployment of WSNs in detection applications brings new challenges to the field. In addition to the design of signal processing algorithms at the application layer that has been previously addressed [8], protocols for other communication layers have to be optimized to maximize the detection performance.
The layered approach commonly adopted to design wireless networks may not be appropriate for detection applications. Although the layered approach provides simplicity in the design due to the decoupling of system layers, it neither provides the optimal resource allocation nor exploits the application domain knowledge. As an example, throughput is a common performance metric used to design media access control protocols. In DD applications, maximizing the throughput is not the prime objective, rather maximizing the quality of the information received that yields the best detection performance is the prime objective. Accordingly, a crosslayer design approach is desired for efficient implementation of WSNs in decentralized detection applications.
 1.
Digital transmission. Although uncoded analog transmission is optimal in a sensor network under certain conditions (see, e.g., [9]), digital transmission is still the choice for costeffective, commercial offtheshelf deployments of sensor network applications.
 2.
Slotted ALOHA/timedivision multiple access (TDMA) MAC. The traditional assumption of a dedicated orthogonal channel between each sensor node and the fusion center may not be feasible in practice. On the other hand, random access techniques and TDMA are frequently used MAC protocols. Therefore, tuning of the protocol parameters to optimize the detection performance can be done in practice without a need to redesign the system.
 3.
Single hop networks. We focus on the case where sensor nodes cannot communicate with each other to form a multihop network to the fusion center, e.g., radiofrequency identification (RFID) sensors communicating to an RFID reader. Preliminary results for multihop tree networks are presented in [10, 11].
The rest of the paper is organized as follows: Section 2 summarizes the related work. Section 3 presents the detection problem formulation for WSNs. Section 4 explains the derivation of the system model. Section 5 presents the solution of the optimization problem. Section 6 presents the performance comparison between the proposed design approach and the classical design approaches. Section 7 presents a performance comparison between the TDMA and ALOHA networks. The work is concluded in Section 8.
2 Related work
Early work on crosslayer design has focused on the design of channelaware decentralized detection schemes [12]. More recent work on channelaware design considers sequential detection schemes [13]. The crosslayer design approach has been recently explored for the design of MAC and routing protocols for detection applications. Cooperative MAC, where individual sensor transmissions are superimposed in a way that allows the fusion center to extract the relevant detection information, is considered in [14]. This approach leads to significant gains in performance when compared to conventional architectures allocating different orthogonal channels for each sensor. However, technical issues such as symbol and phase synchronization have to be taken into account for practical implementations [5, 15]. Datacentric MAC, where existing protocols are tuned/modified for optimal performance, represents a viable alternative to cooperative MAC and, therefore, has gained considerable attention recently. Decision fusion over slotted ALOHA MAC employing a collision resolution algorithm is studied in [16], where the objective is to analyze the performance, rather than to design the MAC layer, in order to optimize the detection performance. A more thorough investigation of the design of MAC transmission policies to minimize the error probability has been considered in [17], where the system model includes the MAC and detection application layers, excluding the physical channel model, and assuming a stochastic MAC policy. Although stochastic transmission policy results in performance gains compared to deterministic policies, the extension of this framework to include the physical channel model is challenging mathematically.
The integration of the channel model and the MAC layer in the context of distributed estimation has been considered in [18], where analog transmission of sensor data is assumed. The crosslayer approach is also considered in [6], where an integrated model for the physical channel and the queuing behavior for sensors is developed. The design problem is to choose the code rate and the number of sensors to minimize the error probability for an frequencydivision multiple access (FDMA) system, where orthogonal channels are used between sensors and the fusion center.
Routing for decentralized detection has been considered separately from the MAC design problem. Energyefficient routing for signal detection in WSNs is considered in [19], where the objective is to find the optimal route for local data from a target location to the fusion center. Cooperative routing for distributed detection in large sensor networks is studied in [20] using a link metric that characterizes the detection error exponent. For a survey on the interplay between signal processing and networking in sensor networks, see [21] and the references therein.
 1.
Integrated model for the detection system. We integrate the physical layer, MAC layer, and the detection application layer in one unified system model.
 2.
Integration of different sensor quality measures. The model captures three sensor quality measures, namely, quality of information (QoI), channel state information (CSI), and residual energy information (REI). In addition, delay for detection and network lifetime are considered as additional design constraints.
 3.
Design of a complete transmission control policy. We design an optimal transmission control policy that includes not only the transmission probabilities but also the communication rate and the energy allocation for each sensor. The authors are not aware of a literature work on crosslayer design for detection applications, where an integrated model is developed that captures different communication layers and several sensor quality measures, while simultaneously considering different transmission design parameters as well as the delay for detection.
 4.
Nonasymptotic analysis. We assume a finite number of sensor nodes and do not resort to asymptotic analysis as commonly adopted in detection studies. Therefore, the analysis results are applicable on smallscale and largescale sensor networks.
 5.
Enhanced detection performance. Without additional resources, the proposed design approach outperforms the classical decoupled and maximum throughput design approaches.
 6.
Slotted ALOHATDMA comparative analysis. We show the conditions under which each transmission scheme outperforms. These conditions represent a guideline for the designer to choose between the two protocols based on the available system resources and design constraints.
The work presented in this paper is an extension of our previous work in [22], where only ALOHA networks were considered, and the energy allocation scheme was fixed. In addition, a more detailed simulation experiment and a full comparison between ALOHA and TDMA sensor networks are presented.
3 Problem formulation
Initially, the fusion center broadcasts a message containing the location of the resolution cell to be surveyed, soliciting information from different sensors. Each sensor responds with the following information: (1) channel state between the sensor and the fusion center, which could be estimated using channel measurement techniques [23, 24]; (2) signaltonoise ratio for the reflected probing signal used by the sensor to illuminate the target, which could be estimated from sensor location (estimated using different localization methods [25]), resolution cell location, and channel measurement techniques; and (3) the energy reserve, representing the REI, which could be estimated by the sensor from the battery charging state. We call this communication process the handshaking cycle, which starts with the broadcast message from the fusion center and ends when the fusion center receives the information from all sensors. This handshaking cycle is repeated periodically to cope with changes in the environment that impact the sensor quality measures. Therefore, the overhead of the handshaking cycle is proportional to the environment dynamics, i.e., how fast the environment changes to affect the quality measures for sensors. We ignore the handshaking overhead in the development of the system model in Section 4. In Section 4.6, we provide an upper bound on the environment dynamics such that the handshaking overhead could be safely ignored without affecting the model accuracy.
After the handshaking cycle, the fusion center calculates the optimal transmission control policy for each sensor based on the quality measures received. The values of the XCP variables are then sent back to the sensors that have reliable quality measures to contribute to the detection task. The resulting values of the XCP variables are stored in a lookup table in the sensor memory (for each resolution cell), which remain valid for the given location as long as the quality measures for each sensor have not changed from the last run of the optimization algorithm. The detection process then proceeds as follows: the fusion center broadcasts a message to initiate a detection cycle for a specific resolution cell. The local sensors selected by the optimization algorithm will sample the environment by collecting a number of observations x_{ i }, form a data packet, and communicate their messages directly to the fusion center over the MAC channel. Finally, the fusion center makes a final decision after a fixed amount of time representing the maximum allowed delay for detection.
4 System model
4.1 Wireless channel model
where d_{0} is a reference distance corresponding to a point located in the far field of the transmit antenna, λ_{ p } is the wavelength of the propagating signal, ρ_{c} is the path loss exponent, and ${X}_{{\sigma}_{\mathrm{c}}}\sim \mathcal{N}(0,{\sigma}_{\mathrm{c}}^{2})$.
where P_{t} is the average transmitted signal power, and Φ(.) is the cumulative distribution function for the standard normal PDF. We note that the CSI relevant to our model is represented by the statistics σ_{c}, μ_{c}, and N_{0}. These statistics are required to be estimated by each sensor, and no instantaneous channel state information is required for the XCP design. Since we assume fixed nodes and a slowly varying channel, the estimation process could be executed less frequently to save sensor node resources. This is particularly important in wireless sensor networks since the estimation of the channel state is both time and power consuming.
It should be highlighted that the largescale fading model presented here allows us to obtain the closed form solution in (2). More complex fading models, e.g., smallscale fading, can be integrated similarly, but they may allow only numerical solutions.
4.2 Media access control protocol model
 1.
Slotted ALOHA. Sensor attempts transmission in every slot during the detection cycle, with probability q _{ i }, despite the state of their last transmission.
 2.
TDMA. Sensor transmits to the fusion center only in its dedicated time slot, assigned using the fixed assignment TDMA scheme.
Unequal priority TDMA, where a single sensor may be assigned more than one time slot, could also be used and may lead to a better detection performance. However, the resulting optimization problem is an integer programming problem that is generally hard to solve in real time. Therefore, only equalpriority TDMA is considered in this work. Without loss of generality, we assume that L = m N, where m is a positive integer, i.e., at each detection cycle, all sensors transmit the same number of times. This assumption facilitates the comparison with the slotted ALOHA scheme. The decision takes place at the end of the detection cycle by the FC. The process repeats for every detection request initiated by the fusion center. To simplify the analysis, the MAC protocol does not consider the acknowledgement slots and any protocol specifics required for synchronization or rate negotiation. Also, we ignored the packet overhead, which is a reasonable approximation for practical WSN protocols with large packet payload.
 1.
Slotted ALOHA. At any given time slot, the probability of a successful packet transmission by sensor i is given by ${q}_{i}\prod _{j\ne i}(1{q}_{j})$. Further, this packet will be successfully received by the fusion center if the state of the physical channel between the sensor and the fusion center is ON during this time slot.
 2.
TDMA. Since collisions are eliminated, the probability of successful packet transmission depends solely on the physical channel condition, as given by (2).
4.3 Energy model
where ${\mathcal{E}}^{0}=\sum _{i=1}^{N}{e}_{i}^{0}$ is the total initial energy in all sensors at the time of deployment, ${\mathcal{E}}^{\mathrm{w}}=\sum _{i=1}^{N}{e}_{i}^{\mathrm{w}}$ is the total wasted energy remaining in sensor nodes when the network dies, f_{ r } is the average sensor reporting rate defined here as the number of detection cycles per unit time, and ${\mathcal{E}}^{\mathrm{r}}=\sum _{i=1}^{N}{e}_{i}^{\mathrm{r}}$ is the expected reporting energy consumed by all sensors in one detection cycle.
Our objective is to allocate the reporting energy ${e}_{i}^{\mathrm{r}}$ for each sensor in such a way that maximizes the detection performance. In what follows, we derive the energy constraints for the sensor network.
4.3.1 Total energy constraint
4.3.2 Individual energy constraints
 1.
ALOHA: We note that the expected number of transmissions by sensor i during a detection cycle is Lq _{ i }. Therefore, T = (τ / L)L q _{ i } = τ q _{ i }, and we get ${P}_{\mathrm{t}}^{i}={e}_{i}^{\mathrm{r}}/\tau {q}_{i}$.
 2.
TDMA: Since we assume L = m N, each sensor transmits m times, and we get ${P}_{\mathrm{t}}^{i}={\mathit{\text{Ne}}}_{i}^{\mathrm{r}}/\tau $.
We note that in the above discussion, we neglected the energy consumed by each sensor to report its quality measures to the fusion center. This energy component could be included in the analysis by subtracting it from the initial sensor energy. However, for slowly varying environments, where the sensor characteristics need to be updated less frequently, this energy component could be neglected compared to the periodic sensor reporting energy.
4.4 Sensing model
where ε is the amplitude of the emitted signal at the object, η is a known attenuation coefficient, typically between 2 and 4, and w_{ i } is an additive white Gaussian noise with zero mean and variance ${\sigma}_{s}^{{i}^{2}}$. We note that the above model considers passive sensing [25]. In the active sensing case, the observation model is given by x_{ i } = ζ ε_{ tr } / (2d_{ i })^{η/2} + w_{ i }, where ζ is a known reflection coefficient at the object, ε_{ tr } is the amplitude of the signal transmitted by the active sensor (illuminating signal), and 2d_{ i } is the round trip distance traveled by the signal [19]. We note that the two observation models differ only in the scaling factor ζ / 2^{η/2}. Therefore, without loss of generality, we assume the active sensing model in the following discussion. If passive sensing is assumed, then the detection problem will be slightly different than the problem presented here, since the amplitude of the source signal, ε, is unknown and has to be estimated from sensor observations.
where μ^{ i } = ζ ε_{ tr } / (2d_{ i })^{η/2}, and n_{ i } is the number of independent and identically distributed (IID) observations obtained by sensor i at each time slot. We note that noise samples are independent across sensors, i.e., the observations at local sensors are independent across time and space, but not necessarily identically distributed since some sensors may be closer to the resolution cell, and noise variances are assumed unequal. In the following, we designate the vector of sensor observations at time slot k by x_{ i }[k] = [x_{ i }[1, k] x_{ i }[2, k] … x_{ i }[n_{ i }, k]]. We note that x_{ i }[k] has the multivariate Gaussian distribution $\mathcal{N}(0,\mathbf{C})$ under hypothesis ${\mathcal{H}}_{0}$ and $\mathcal{N}(\mu ,\mathbf{C})$ under hypothesis ${\mathcal{H}}_{1}$, where μ = [μ^{1}μ^{2} … μ^{ N }] and $\mathbf{C}={\sigma}_{s}^{{i}^{2}}I$.
Proposition 1.
where n_{ s } = L for ALOHA and m for TDMA, and r_{ i }[k] is a Bernoulli random process representing the success (r_{ i } = 1) or failure (r_{ i } = 0) of receiving a packet from sensor i in communication slot k. The sample space and probability measure of r_{ i }are defined as${\Omega}_{{r}_{i}}=\{0,1\}$and P[r_{ i } = 1] = λ_{ i }, respectively, where λ_{ i }is given by (4).
Proof.
4.5 Measurement of detection performance
Under nonGaussian assumptions, there is no general result that enhancement of the deflection coefficient will lead to a better performance in terms of the ROC curve. However, it is likely that more separation between the two density functions will lead to a better detection performance.
Proposition 2.
where λ_{ i }is given by (4).
Proof.
From (26), (27), and (29), we get (25).
We note that the quantity ${D}_{i}={n}_{i}{\left({\mu}^{i}/{\sigma}_{s}^{i}\right)}^{2}$ represents the signaltonoise ratio at sensor i, and we adopt it as a measure of the sensor QoI. From (25), we note that the overall deflection coefficient at the fusion center is simply a weighted sum of the individual deflection coefficients for each sensor, where the weights are the probabilities of successful packet transmission for each sensor, and the deflection coefficient in case of a collision is set to 0.
We note that ${c}_{i}={\epsilon}^{2}/{\sigma}_{s}^{{i}^{2}}{d}_{i}^{\eta}$; therefore, the signal amplitude at the object to be detected appears as a scaling factor only in the objective function. This means that the signal amplitude does not affect the optimal operating point for the system. However, the amplitude does affect the detection performance, as intuitively expected. We further note that the objective function does not depend directly on L and n_{ i }. Rather, from the optimal communication rates and (3), L and n_{ i } could be arbitrarily chosen such that L n_{ i } = τ R_{ i } / b for any nonzero communication rate, i.e., R_{ i } > 0, n_{ i } ≥ 1, and consequently L ≤ τ R_{ i } / b.
Model parameters for the wireless sensor network
Parameter  Description  Calc  Simulation value (Section 6)  Notes  Layer 

W  Channel bandwidth  G  10^{3} Hz  Physical layer  
N _{0}  Noise power spectral density  E  10^{9} W/Hz  CSI  
μ _{c}  Mean path loss  C (1)  [40, 45], [45, 50], [50, 55], [55, 60], [60, 65] dB  CSI  
[65, 70], [70, 75] dB  
σ _{c}  Path loss standard deviation  E  [5, 8] dB  CSI  
e ^{r}  Reporting energy  D  Design variable  
R  Communication bit rate  D  Design variable  
L  Number of communication slots  C (3)  Ln_{ i } = τ R_{ i } / b  MAC layer  
b  Number of encoding bits/observations  G  16 bits  
q  Retransmission probability  D  Design variable  
τ  Delay for detection  G  1:150 s  BJA01196xml0x.png Application layer\c:IMG  
l _{t}  Network lifetime  G  100:500 days  
l  Sensor lifetime  G  0.7* l_{t}  
n  Number of observations  C (3)  Ln_{ i } = τ R_{ i } / b  
c = (μ / σ_{ s })^{2}  Signaltonoise ratio  G  [2,3.2], [2.5,3.5], [0.06,0.08], [1,1.4], ×10^{3}  QoI  
[0.5,0.7], [0.12,0.16], [0.03,0.04] ×10^{3}  
e^{0}  e^{w}  Net sensor useful energy  G  10^{4} J  REI  
f _{r}  Sensor reporting rate  G  200 cycles/day 
4.6 Handshaking overhead
Example 1.
5 Transmission control policy design for optimal detection
I is the identity matrix, 0(1) is the vector/matrix of all zeros (ones), with appropriate dimensions, and $\epsilon =\left[\begin{array}{llll}{\epsilon}_{1}& {\epsilon}_{2}& \dots & {\epsilon}_{N}\end{array}\right]$. We note that our objective function is not convex. Instead of following the classical approach to simplify the system model to obtain a convex function, which may ignore important system dependencies and may lead to a less accurate model, our approach is to analyze the optimization problem to obtain a possible set of candidate points that may speed up the convergence process for existing numerical optimization algorithms, then resort to simulation experiments for performance evaluation.
The given KKT conditions cannot be solved analytically. However, the optimization problem could be solved efficiently using a variety of constrained optimization algorithms, e.g., the interior point method. The following two theorems provide a possible set of candidate points for a local maximizer, hence speeding up the convergence process and providing a set of initial points for the optimization algorithm.
Theorem 1.
Proof.
When q_{ i } = 1, the objective function reduces to ${D}^{2}=(\tau /b){c}_{i}{R}_{i}\Phi \left({\rho}_{i}\right)\prod _{j\ne i}(1{q}_{j})$. Any value of q_{ j } ≠ 0 will cause the objective function value to decrease. Physically, q_{ j } ≠ 0 corresponds to a guaranteed collision, i.e., loss of information. Therefore, q_{ j } = 0. Since R_{ j } and ${e}_{j}^{\mathrm{r}}$ do not affect the objective function, we arbitrarily set R_{ j } = 0. ${e}_{j}^{\mathrm{r}}$ should be set to 0 to save the energy budget for the noncontributing sensor. The solution q_{ j } = R_{ j } = e j r = 0 for j ≠ i could be shown to satisfy the KKT conditions by direct substitution. The objective function monotonically increases with ${e}_{i}^{\mathrm{r}}$. Therefore, ${e}_{i}^{\mathrm{r}}$ should be set to its maximum value, i.e., ${e}_{i}^{\mathrm{r}}=min(\epsilon ,{\epsilon}_{t})$. Finally, optimal R_{ i } is set to maximize the objective function and, hence, given by (63).
We conclude that we have a set of N candidate points, (q_{ i } = 1, q_{ j } = 0, j ≠ i), for a local maximum, which could be checked easily in N time steps, in addition to the computations required to find the optimal communication rate, which could be implemented efficiently for a singlevariable function.
Theorem 2.
A candidate point for a local maximizer of the objective function in (43) is when a subset of the sensors, defined by the index set${\mathcal{S}}_{\epsilon}$, transmit with their maximum energy, while all other sensors remain silent. Optimal design variables for the active sensors are at x^{∗}, where ∇J(x^{∗}) = 0. The unallocated energy is equal to${\epsilon}_{\mathrm{t}}\sum _{i\in {\mathcal{S}}_{\epsilon}}{\epsilon}_{i}$.
Proof.
For the active sensors, 0 < q_{ i } < 1. The total energy constraint is inactive, i.e., ${\nu}_{{e}_{\mathrm{T}}}=0$. If ${e}_{i}^{\mathrm{r}}<{\epsilon}_{i}$, then from the complementary slackness condition ${\nu}_{{e}_{i}^{0}}={\nu}_{{e}_{i}^{1}}=0$. From the stationarity condition, we get ${\nabla}_{{e}_{i}^{\mathrm{r}}}J=0$, but ${\nabla}_{{e}_{i}^{\mathrm{r}}}J=0$ if and only if ${e}_{i}^{\mathrm{r}}=0$, a contradiction. Therefore, the only option left is ${e}_{i}^{\mathrm{r}}={\epsilon}_{i}$. In this case, the constraint ${e}_{i}^{\mathrm{r}}\le {\epsilon}_{i}$ is active; hence, ${\nu}_{{e}_{i}^{0}}=0$. From the stationarity condition, we get ${\nabla}_{{e}_{i}^{\mathrm{r}}}J{}_{{e}_{i}^{\mathrm{r}}={\epsilon}_{i}}={\nu}_{{e}_{i}^{1}}$, which satisfies the dual feasibility condition, since the lefthand side is ≥0. Therefore, this point is a candidate for a local maximizer.
We conclude that all active sensors in this case should transmit with maximum energy. Since all other constraints are inactive, all Lagrange multipliers are equal to 0, and therefore, from the stationarity condition, the optimal values for q and R are equal to the stationary point x^{∗}, where ∇J(x^{∗}) = 0.
Theorem 2 results in few candidate points if the individual energy constraint for each sensor is a large fraction of the total energy constraint, such that only few sensors consume the total energy budget. Otherwise, the number of candidate points will be prohibitively large. The solution of the optimization problem is summarized in Algorithm 1, where Theorem 1 is used, and the called optimization algorithm is any optimization method of choice, e.g., interior point method.
Algorithm 1 Optimization Problem Solution
6 Performance comparison
where ε_{ i } is the same for all sensors. Since l < l_{t}, we have ε_{t} / N < ε_{ i }, and therefore, the equal energy allocation is feasible in this case. This case is the one considered in the numerical example in Section 6.4. Finally, we present an upper bound on the system performance to better assess how well the CLD performs compared to the best possible performance, achievable only in theory.
6.1 Maximum throughput design
In maximum throughput designs, the design variables R_{ i } and q_{ i } are chosen to maximize the throughput in (65). The maximum throughput design thus does not consider the QoI for each sensor. This is clearly shown by comparing (65) to (30), where we note that the maximum throughput design is equivalent to the CLD if all sensors have the same quality of information.
6.2 Decoupled design
In practice, λ is predetermined from the application. However, to make a fair comparison, we use the value of λ that maximizes the deflection coefficient in (68), i.e., λ = arg max_{ λ }D^{2}, 0 ≤ λ ≤ 1.
6.3 Performance upper bound
For the given problem setup and for a given energy and delay constraints, the upper bound on the performance is when there are no channel drops or contentions between sensors, i.e., all observations generated locally at each sensor are received successfully at the fusion center. Mathematically, this case is equivalent to a realization of the physical channel were the channel state is ON for all transmissions. In this case, the detection problem reduces to the classical centralized shiftinmean Gaussian detection problem, where the ROC curve is given by (24).
6.4 Simulation results
In this section, we evaluate the proposed crosslayer design approach for the system in Figure 1, as compared to the classical approaches summarized in Section 6, via a numerical example. We consider a network with 70 sensors (N = 70) deployed for detection, with parameter values as shown in Table 1. To avoid manual entry of parameter values for the 70 sensors, the mean path loss, path loss variance, and the signaltonoise ratio for each sensor are generated using uniform random number generators. The evaluation is performed both numerically and through Monte Carlo Simulation (MCS) experiments.
6.4.1 Numerical evaluation
We use Algorithm 1 to calculate the optimal solution for the CLD in (30) and the maximum throughput design in (65), where the interior point method is used as the core optimization algorithm. The interior point method is also used to find the optimal probability of successful packet transmission for the decoupled design in (68).
6.4.2 Simulation study
We use the optimal solution for the design variables obtained from the numerical evaluation to set up an MCS for the wireless network, as follows:

Hypothesis. MCS is performed for both ${\mathcal{H}}_{0}$ and ${\mathcal{H}}_{1}.$ to evaluate the deflection coefficient, ROC curve, and probability of error.

Sensors. Observations are generated locally at each sensor for each communication slot. For ALOHA channels, each sensor attempts transmission randomly according to its retransmission probability. For TDMA channels, each sensor transmits in its allocated slots only.

Communication channel. The channel state for each sensor is simulated for each detection cycle.

Fusion center. The fusion center performs the likelihood ratio test on the observations received. Equivalently, the fusion center calculates the test statistic in (15) and compares it to a threshold value.

Performance evaluation. The deflection coefficient is evaluated statistically according to (23). The ROC curve is evaluated by running MCS for different threshold γ values.
We run the MCS experiment 5,000 times for each delay for detection/network lifetime values to obtain accurate results.
6.4.3 Deflection coefficient
Figure 7, top right graph, shows the deflection coefficient as it varies with the network lifetime, where delay for detection is set to 50 s. The results are similar to the delay for detection study, where the proposed CLD approach outperforms the maximum throughput and decoupled design approaches. Equivalently, for the same deflection coefficient, the network lifetime with the CLD is longer. The MCS results are superimposed on the numerically obtained curves, verifying the correctness of the analysis. The TDMA sensor network exhibits a similar behavior as illustrated in Figure 7, bottom left and right graphs. The MCS results are shown for the CLD design only to avoid cluttering the figures.
6.4.4 ROC curves
In practice, a family of these ROC curves are provided for different values of the delay for detection and network lifetimes. The operating point is located on a specific ROC curve, and the relevant values of the detector threshold and the WSN design variables are set accordingly.
7 Slotted ALOHATDMA comparison
Figure 10, top right graph, shows the deflection coefficient for different lifetime values. Similarly, the TDMA outperforms the ALOHA for lifetime values greater than the threshold lifetime ${\mathcal{L}}_{\text{th}}$. The threshold lifetime gets higher as the delay for detection decreases. For the given numerical example, the threshold lifetime ${\mathcal{L}}_{\text{th}}\approx 285$ days. Since the performance degrades with increasing network lifetime, the deflection coefficient at the threshold lifetime may be below the minimum design value, and therefore, TDMA may not be a feasible design option. For example, in Figure 10, top right graph, the minimum detection performance is specified by D^{2}=6, and therefore, the ALOHA is the design option. At the threshold lifetime, D^{2}≈5.2, which is below the minimum design requirement, and therefore, TDMA cannot be used with such design requirements. However, for scarce energy applications, the threshold lifetime gets smaller, so that TDMA maybe the only viable design option to extend the network lifetime, on the expense of degraded detection performance.
Figure 10, bottom left graph, summarizes the performance comparison in the delay lifetime twodimensional space. The curve represents the boundary between the ALOHA and TDMA regions. For any pair of (delay, lifetime) in the ALOHA region, the ALOHA sensor network has a superior performance and similarly for the TDMA region. The figure could be augmented by the contour lines for the deflection coefficient for both ALOHA and TDMA to show the performance measure value. Using the deflection coefficient values, the designer can check whether the selected operating point satisfies the minimum performance requirement. Figure 10, bottom right graph, shows the performance regions with the contour lines for the ALOHA region.
8 Conclusion
In this paper, we pursued a crosslayer, modelbased approach to design a singlehop ALOHA and TDMA WSNs deployed for detection applications. We developed an integrated model for the detection system that includes the communication network, sensing, and energy models. We considered the QoI, CSI, and REI quality measures in the design process. We designed a complete transmission control policy that includes the transmission probabilities, communication rate, and energy allocation for each sensor. We showed a significant performance increase over the decoupled and maximum throughput design approaches with equal energy allocation scheme, for both ALOHA and TDMA networks.
The TDMA sensor network is easier to design than the ALOHA network, since one of the design variables is omitted (retransmission probability). However, we showed in this paper that the ALOHA network outperforms TDMA for small to moderate delays. For large delays, TDMA outperforms the ALOHA network unless the network lifetime is reduced. The designer chooses the best option based on the delay and lifetime constraints, in addition to the minimum allowed performance measure.
The crosslayer design approach results in a nocost performance increase, since the designer obtains a performance increase for the same delay and lifetime constraints. However, the crosslayer design has its own pitfalls. First, a mathematical model that captures the interrelationships between different layers has to be developed. This model is, in general, complex, and it maybe required to go through the design process several times to refine the assumptions in order to obtain a tractable model. Second, the optimization problem obtained has to be solvable in real time with existing optimization algorithms. This is not always possible, as the optimization problem complexity is closely coupled to the model complexity. Finally, the optimality of the design depends on the availability of the global information in real time. This assumption may not always be true in practice. Despite these pitfalls, the crosslayer design complexity is justified when it is desired to optimize the performance with limited system resources that cannot be replenished (e.g., remote WSN in a battlefield). The decoupled approach, on the other hand, maybe justified for systems with enough resources such that the performance loss could be compensated by additional resource allocation.
Several extensions could be made to the work presented in this paper. Multihop sensor networks could be addressed instead of singlehop networks. Smallscale fading could be incorporated in the system model, providing a more general model that is applicable in a variety of sensor network applications. Finally, other channel access schemes could be considered, e.g., FDMA, CDMA, and SDMA.
Declarations
Acknowledgements
This work is supported in part by the National Science Foundation (CNS1238959, CNS1035655).
Authors’ Affiliations
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