Optimal Packet Size of Underwater Sensor Networks

SHU Yuanzhong（舒遠仲），WANG Yagang（王亞鋼）

School of Information Engineering，Nanchang Hangkong University，Nanchang 330063，China

Introduction

With the continuing heaing up of “ocean heat”and“heat sensors”，many achievements of the underwater sensor networks have been made，including the study of underwater sensor network protocols，routing，and other aspects of targeting［1］.However，the research of parameters communication optimization about underwater sensor network node is little currently.This paper helps to improve the communication performance［2］.Therefore，it has important theoretical research and practical applications.

Data transmission between nodes is generally required to subcontract processing，and data packet size directly affects the size of communication performance.The mainly research of optimal data packet size was that，in 2008，a cross-layer packet size optimization method was made by Vuran et al.［3］By optimizing the three different object functions，the optimal data packet size under the different conditions was solved.This numerical analysis method is applies to solving the optimal packet size，because the numerical analysis method is an ergodic way，which is difficult to meet real-time solving requirements.In 2012，some researchers［4］used the simultaneous perturbation stochastic approximation（SPSA）algorithm to solve the optimal real-time data packet size in order to optimize the energy efficiency target.However，this study is for terrestrial wireless sensor networks，and underwater sensor networks have their own unique characteristics，such as high latency and high bit error rate［5］.

Therefore，this article is to optimize the characteristics of underwater sensor networks，and introduce SPSA algorithm to optimize the model in real-time to solve the optimal packet size.

1 A Single Hop Communication Model of Energy Efficiency and Throughput（EET）

The model is based on the establishment of a single-hop communication between two nodes in the underwater sensor networks for subcontracting data processing before data transmission between nodes.Generally，apacket is com-posed of header，payload and trailer［6］，as shown in Fig.1，αbit，l bit andτbit.

Fig.1 Packet components

Therefore node transmits 1bit data needing energy consumption as follows：

where，Etand Errepresent the energy consumptions for sending and receiving nodes，and Edecdenotes the energy of decoding apacket.

The attenuation loss of underwater acoustic communication［7］is

where xis the distance between two nodes；f means the frequency；k means the coefficient，which is the geometric description of energy extending，k ＝2is a spherical expansion，k ＝1is a cylindrical extension，and k ＝1.5 means the actual propagation extension［8］；and a（f）is the absorption coefficient.

where Prerepresents the received power of the receiving node；P0represents the input power of the receiving node to receive the information required；R represents the data transmission rate；Ptst／Prstindicates the power of sending or receiving components；Ttst／Trstindicates the time of sending or receiving components；nis the size of the packet；n＝α＋l ＋τ.Take forward error correction（FEC）as an example.Energy consumption of decoding Edecand packet size n are related to the error correction t bits；Eaddand Emultrespectively mean the energy consumption in the decoding process for the addition and multiplication.

Combining formulas（3）and（4），we obtain the energy for 1bit data transmission between two nodes：

由此可見，在全面深化改革的今天，憲法解釋作為憲法適應機制的重要方式，已經(jīng)受到人們普遍關注，而且在一定程度上被“激活”了。

From the parameters k1and k2，the following formulas are obtained.

where k1represents the energy of transmitting 1bit information，k2represents the energy of sending and receiving module，and k1and k2are constants for a given sensor node.

From the above，when nodes need to transfer N bit data，the number of data packets is「N／l?.Before transferring data，the sender needs to send a request to send（RTS）data packet to the receiver，and then begins to send data until it receives the CTS data packet from the receiver.After each data packet transmission，the receiver replies an acknowledgement（ACK）packet，and the sender sends the next packet after receiving an ACK packet［9］.Assume RTS，CTS and ACK packet sizes are q bit.When the N bit data are transferred，energy consumption for the transmission of these control packets is

Energy for the transmission of data is

Therefore，the total energy consumption is

In the underwater environment，as the speed of acoustic signal propagation is much lower than the electromagnetic signal，the collision problem is a two-dimensional function of time and space.The collisions of the data packet are related not only with the time of transmitting packet between the sending and the receiving nodes，but also with the Euclidean distance from both sides.

Motion increases the uncertainty of the space，so that the collision avoidance mechanism complicates the design.Therefore，this paper uses the reservation media access control（MAC）protocol.Control package needs to reach within the listening window of the receiving node.When a node is affected by the movement，the available receiving windows become fewer，which increase probability of the control packet collision and the reservation failure.In addition，the motion makes the control packet unable to reach the destination node within specified time slot，so the reservation fails.

Therefore the probability of the node，Pre，which can receive packet，is closely related with underwater signal attenuation，time and distance.

Take BCH （Bose，ray-Chaudhuri，Hocquenghem，a kind of coding method）as an example.Within the scope of its error correction capability，the receiver can recover the data correctly.If the data packet exceeds the error correction capability，it will be lost.When the capability of error correction is t，the probability of correctly receiving data packet［10］is

Underwater communication delay is

Therefore，EET model is

This paper aims to solve the optimal payload size l＊for the maximum Targetat the same time.Optimal packet size can be calculated after getting l＊.

2 A Data Packet Length Optimization Algorithm Based on SPSA

The SPSA algorithm is a kind of stochastic optimization algorithm，introduced by Spall in 2005［11］.The objective function value may be obtained only by calculating or measuring each point without the gradient of the objective function［12］.

This algorithm has the characteristics as follows.

（1）Since each dimension of the argument is synchronous disturbance，the algorithm is particularly suitable for the argument which is the case of high-dimensional.（2）The value of the objective function with noise can also be applied in the algorithm.（3）The algorithm is applicable for global optimization and local optimization.（4）Compared with other random optimization methods，this algorithm is easier to be achieved.（5）Compared with other evolutionary algorithms，this algorithm has a faster convergence speed.

Since SPSA algorithm is designed for solving the minimum objective function，we get optimization objective function based on formula（17）：

When the algorithm is in the iterative process，the payload lkis not necessarily apositive integer multiple of 8.To make the SPSA algorithm run smoothly，we relax integer constraint of formula（17），and only at the end of the algorithm，the optimum integer solution which satisfies the formula（17）in the vicinity of the optimal real solutions.

Firstly，an initial point is chosen within［0，N］，which is the initial length of the payload l0，and then the iteration starts.Through symmetric Bernoulli distribution，we can determine the perturbation direction of the current iteration point lkand the perturbation amplitude is ck.

where c（c ＞0）is a constant，γ ＝0.101，and k is the number of iteration.Thus，after the disturbance objective function values Y（lk＋ckΔk）and Y（lk-ckΔk）can be obtained through two points corresponding to the disturbance，these two points will be determined by the disturbance gradient gkas the estimate of the gradient at lk.

Then factor akcan be determined by formula（21），so the iterative point can be updated.

where a（a ＞0）is a constant，α ＝0.602，A ≤10%Imax，Imaxrepresents the expected iterative sum，and kis the iterative number.

The next iteration payload length lk＋1was obtained from formula（22）：

If the value of iteration reaches Imaxor the evaluative magnitude of iterative point is small enough（≤0.1bit），then the algorithm stops iteration，otherwise it continues.Ibis the iterative sum in actual iterative process，then Ib≤Imax.In addition，the iterative point of the search scope is constrained，if lk≤0，thenlk＝1；if lk＞N，thenlk＝N.If the disturbance point falls beyond（0，N］，treatment is the same as the iterative point.

3 Experiments

Definition（EET accuracy）Suppose that the data to be transmitted is N bit long and the initial parameters of r group are selected randomly.By using the SPSA algorithm，we can educe the corresponding optimal payload length ls1，ls2，…，lsrrespectively，their EET values Ys1，Ys2，…，Ysr，and its average EET value.If using numerical analysis technique，we can find out the optimal payload length lvand its corresponding ECE Yv，then the average accuracy of EET based on SPSA algorithm is defined as

The closerφis to 1，the more accurate the SPSA algorithm solution is.

Suppose the data Nthat will be transmitted is 640，1000 or 1500bits long.The positions N／4，N／2and 3 N／4points can be selected as three different initial iteration point l0respectively.For each group of Nbit length data and each initial iteration point l0，10different initial iterative steps U are choosen respectively.As shown in Table 1，the performance of the algorithm is tested from a statistical point of view.

Table 1 Initial iteration points and the initial steps of a combination of circumstances

Table 1 contiuned

4 Conclusions

Assuming using forward error correction coding，and t＝2，the experimental results are analyzed.

The experimental results in Table 2shows that，for each different initial iterative point，the most optimal length of the payload is solved and the average accuracy of EET is more than 0.999，observed from the row direction of the table.This illustrates the SPSA algorithm in this case still has very high accuracy of EET.

Table 2 Optimization results

SPSA algorithm that is better reflected in the objective function value than the numerical analysis method has less calculation，high efficiency and good real-time performance.

In addition，the average EFT accuracy and the efficiency of computing are comparatively affected by choice of initial iterative point for each group of N bit length data in the column direction of Table 2.But overall，the average accuracy of EET is still very close to 1，and calculation performance is relatively high.Thus，choosing a different initial iterative point does not have a negative impact on the performance of SPSA algorithm.

As can be seen from the above three graphics（Figs.2-4），for different initial iterative points，the algorithm can quickly and efficiently get the optimal solution finally.

Fig.2 Evolution curve of the average EET with the initial point of N／2

Fig.3 Evolution curve of the average EET with the initial point of 3 N／4

Fig.4 Evolution curve of the average EET with the initial point of N／4

［1］Heidemann J，Stojanovic M，Zorzi M.Underwater Sensor Networks：Applications，Advances and Challenges［J］.Philosophical Transactions of the Royal Society A：Mathematical，Physical and Engineering Sciences，2012，370（1958）：158-175.

［2］Zhao Y G，Govindan R，Estrin D.Residual Energy Scans for Monitoring Wireless Sensor Networks［C］.IEEE Wireless Communications and Networking Conference WCNC'02，Orange County Convention Center，Orlando，USA，2003：17-21.

［3］Vuran M C，Akyildiz I F.Cross-Layer Packet Size Optimization for Wireless Terrestrial，Underwater，and Underground Sensor Networks［C］.The 27th IEEE Conference on Computer Communications，Phoenix，AZ，2008：226-230.

［4］Xia N，F(xiàn)eng R J，Xu L N.SPSA Based Packet Size Optimization Algorithm in Wireless Sensor Networks［M］／／Proceedings of Wireless Algorithms，Systems，and Applications.Berlin Heidelberg：Springer-Verlag，2012：112-119.

［5］Akyildiz I F，Pompili D，Melodia T.Underwater Acoustic Sensor Networks：Research Challenges ［J］.Ad Hoc Networks，2005，3（3）：257-279.

［6］Sankarasubramaniam Y，Akyildiz I F，McLaughlin S W.Energy Efficiency Based Packet Size Optimization in Wireless Sensor Networks ［C］.Proceedings of IEEE International Workshop on Sensor Network Protocols and Applications，Anchorage，USA，2003：1-8.

［7］Brekhovskikh L M，Lysanov Y P.Fundamentals of Ocean Acoustics［M］.New York：Springer，2003.

［8］Kilfoyle D B，Baggeroer A B.The State of the Art in Underwater Acoustic Telemetry［J］.IEEE Journal of Oceanic Engineering，2000，25（1）：4-27.

［9］Xu J F，Li K Q，Min G Y.Reliable and Energy-Efficient Multipath Communications in Underwater Sensor Networks［J］.IEEE Transactions on Parallel and Distributed Systems，2012，23（7）：1326-1335.

［10］Guo Z，Wang B，Xie P，et al.Efficient Error Recovery with Network Coding in Underwater Sensor Networks［J］.Ad Hoc Networks，2009，7（4）：791-802.

［11］Spall J C.Introduction to Stochastic Search and Optimization［M］.Hoboken，NJ：John Wiley ＆Sons，2005.

［12］Zhang H J，Zhao J，Geng T.Convergence Accelerated by the Improvements of Step Size and Gradient in SPSA ［C］.2011 Chinese Control and Decision Conference （CDCC），Mianyang，China，2011：1-6.