Energy aware optimization model in chain based routing

Mobile Netw Appl (2014) 19:249–257 DOI 10.1007/s11036-014-0497-8 Energy-Aware Optimization Model in Chain-Based Routing Nguyen Thanh Tung Published online: 26 March 2014 # Springer Science+Business Media New York 2014 Abstract Sensor networks are deployed in numerous military and civil applications, such as remote target detection, weather monitoring, weather forecast, natural resource exploration and disaster management Despite having many potential applications, wireless sensor networks still face a number of challenges due to their particular characteristics that other wireless networks, like cellular networks or mobile ad hoc networks not have The most difficult challenge of the design of wireless sensor networks is the limited energy resource of the battery of the sensors This limited resource restricts the operational time that wireless sensor networks can function in their applications Routing protocols play a major part in the energy efficiency of wireless sensor networks because data communication dissipates most of the energy resource of the networks The above discussions imply a new family of protocols called chain-based protocols In the protocols, all sensor nodes sense and gather data in an energy efficient manner by cooperating with their closest neighbors The gathering process can be done until an elected node calculates the final data and sends the data to the base station In our works, we have proposed two methods to optimize the lifetime of chain-based protocols using Integer Linear Programming (ILP) formulations Also, a method to determine the bounds of the lifetime for any energy-efficient routing protocol is presented Finally, simulation results verify the work in this chapter Furthermore, previous researches assume that the base station position is randomly placed without optimization In our works, a non convex optimization model has been developed for solving the base station location optimization problem N T Tung (*) International School, Vietnam National University, 144 Xuan Thuy Street, Cau Giay District, Ha Noi, Vietnam e-mail: tungnt@isvnu.vn Keywords Sensor Routing Chain based routing Linear programming Non convex optimization Introduction Lindsey et al [5] proposed one type of chain-based protocol called PEGASIS (Power-Efficient Gathering in Sensor Information Systems), which is near optimal for gathering data in sensor networks PEGASIS forms a chain among sensor nodes so that each node will receive data from a close neighboring node and transmit data to another close neighbor Gathered data moves from a sensor node to the nearest neighbor, is aggregated with the neighbor’s data, and eventually reaches a determined Cluster-Head (CH) before finally being transmitted to the Base Station (BS) Figure illustrates the ideas of the PEGASIS protocol In this round of data transmission, Node is elected as the CH Node transmits data to Node 4, and Node fuses the data with its own data and transmits the fused data to Node Similarly, Node transmits data to Node 2, and Node transmits the fused data to Node Finally, Node fuses the data of the other nodes with its own data and transmits the final fused data to the base station The data fusion function can be any function e.g minima, maxima and average, depending on the specific applications as discussed in [1–3] Nodes take turns equally to be the CH so that the energy spent by each node is balanced In other words, each node becomes a CH once for every n rounds of data transmission, where n is the number of sensor nodes The authors in [5] showed that building a chain to minimize the energy consumption is similar to the traveling salesman problem [6], which is known to be NP-complete They proposed a greedy algorithm starting from the furthest node from the base station until a near optimal chain is built as follows: 1) Add the node furthest from the base station to the chain 250 Mobile Netw Appl (2014) 19:249–257 BS Fig A reconstructed chain from PEGASIS method N5 N4 N3 In each round, a sensor node must be selected as the CH Each sensor node receives data from its downstream neighbor, fuses with its own data to generate a single packet of the same length, and transmits the fused data to its upstream neighbor on the chain This process is illustrated in Fig below When Node is selected as the CH, Node fuses data with Node Node fuses its data with Node Node fuses its data with Node and Node and transmits the data to the base station N2 Problem formulation N1 : Cluster-head 2) This node finds a closest node from it that is not already in the chain (Closest Euclidean distance) 3) Repeat until all nodes are added to the chain Figure shows the formation of a chain with five sensor nodes Node connects to Node 2, Node connects to Node 3, Node connects to Node and Node connects to Node In many applications, the data reporting of all sensor nodes is critical as in medical applications or in security applications The above PEGASIS protocol tries to ensure that every node can become a CH equally This is not appropriate for optimum system lifetime Sensor nodes that are far away from the base station will consume more energy than closer nodes to send data to the base station Also, nodes that have too little energy should not become CHs As an equal selection of CHs will result in a reduced lifetime, a formulation to determine the CH pattern among all sensor nodes is presented below In the next section, we have proposed two methods to optimize the lifetime using Integer Linear Programming (ILP) formulations [7, 8] The first method is applied for chain-based routing, the second method can be applied for any routing including chain-based routing 2.1 Method Let us define n to be the number of sensor nodes, and xj to be the number of rounds node j becomes a CH In chain-based Fig Greedy algorithm to build a chain by PEGASIS method Fig Data moving from all sensor nodes to the CH node Mobile Netw Appl (2014) 19:249–257 251 routing, only one CH is selected each round Therefore, there are n possible choices of CHs The problem for the selection of the CHs is formulated as follows: Maximize : n X xj j¼1 n X Subject to : cij x j E i : iẵ1n 1ị jẳ1 x j Z ỵ : jẵ1n where cij is the energy usage of Node i to send a unit of data in a round, when Node j becomes CH and Ei to be the initial energy storage of Node i The above Linear Programming problem tries to maximize the total number of rounds of transmitting data by all sensor nodes under the battery-constraint of all sensor nodes The energy coefficients cij of each non CH node include the energy dissipation for the node to receive data from its downstream neighbor and to send the fused data to its upstream neighbor in the chain The energy coefficients of each CH node in the formula include the energy dissipation for the node to receive data from its downstream neighbors and to send the fused data to the base station The diagram in Fig shows that when Node becomes a CH, c24 includes the energy dissipation to receive data from Node and to send the fused data to Node c44 includes the energy dissipation to receive data from Node and Node and to send the fused data to the base station 2.2 Method The problem of finding the optimal routing to achieve the maximum network lifetime in a sensor network was studied as a constrained linear program optimization in [12–15] and [16] In this work, the authors find the maximum lifetime that could be achieved by any routing cost or balancing scheme for optimizing average flows between nodes Similar to work in [12–15] and [16], we model the routing problem as below: A set of Ns sensors is deployed in a region in order to monitor some physical phenomenon The complete set of sensors that has been deployed can be referred as S={s1……sN} Sensor i generates traffic at a rate of Q bps All of the data that is generated must eventually reach a single data sink, labeled s0 Let qi,j be traffic on the link (i,j) during the time T Each node i has the initial battery energy of Ei, and the amount of energy consumed in transmitting a packet across link L(i,j) is eli,j Maximize : Subject to : T N X j¼1 n X iẳ1 n X q j;i ỵ QT ẳ N X qi; j : iẵ1N jẳ0 qi; j el i; j < ẳ E i : iẵ1n 2ị qi;0 ẳ Qn iẳ1 qi; j >ẳ : i; jẵ1n A new heuristic solution Problem formulation (1) and (2) can be solved by Linear Programming solvers These solvers are not always available and it is not easy to build these solvers inside sensors Therefore, a heuristic RE_chain algorithm is proposed In the RE_chain algorithm, the CH positions are reallocated among the sensor nodes so that the minimum residual energy of all sensor nodes is maximized The heuristic algorithm (RE_chain) is given as below: RE_chain: In every round of data transmission to the base station, select a sensor node as a leader for the chain in order to maximize the minimum residual energy of all sensor nodes after sending data for the round Given: N s f(s) Fig Energy consumption coefficients of every sensor depends on the position of the CH S0 the number of sensor nodes indexed from to N A current CH solution The minimum residual energy of all nodes with solution s Best solution so far 252 Mobile Netw Appl (2014) 19:249–257 Ratio between the number of rounds of RE_chain and RE_with ILP RE_chain algorithm: Initialization: s0 ← 1.002 For (s from to N) 0.998 If δ>0 then s0 =s Ratio δ ¼ f ðsÞ− f ðs0 Þ 0.996 Ratio 0.994 0.992 Result S0 is the CH solution obtained from the RE_chain algorithm 0.99 0.988 20 40 60 80 100 Network topology Simulation results To evaluate the performance of RE_chain and compare the performance with that of PEGASIS and LEACH protocol [1], a number of simulators in Visual C++ were developed The comparison between the system lifetime from Problem formulation (1) and that of RE_chain is also performed In the first set of simulations, the performance of RE_chain is compared to the solution given by Formulation (1) In the simulations, 100 random 100-node sensor networks are generated Each node begins with J of energy The network settings for the simulations in this section are given below The energy model was used in [1, 3, 5, 9–11, 16] Network size (100m×100m) Base station (50m,300m) Number of sensor nodes 100 nodes Data message size: 4000 bits Broadcast message: 200 bits Energy message: 20 bits Position of sensor nodes: Uniform placed in the area Energy model: Eelec =50∗10−9J, εfs =10∗10−12J/bit/m2 and εmp =0.0013∗10−12J/bit/m4 Figure shows the ratio of the number of rounds of RE_chain and the Linear Programming solution of Formulation (1) From the simulation result, it can be said that RE_chain performs within % of the Linear Programming solution It is also of interest to compare the performance of RE_chain, PEGASIS, and LEACH on the network topologies On average, LEACH, PEGASIS, and RE_chain perform 602, 890, and 1,305 rounds respectively (Fig 6; Table 1) Fig Ratio of the number of rounds between RE_chain and ILP model round, every node must transmit its packet and some node must receive it As the total transmission energy of a message is calculated by: E t ẳ kEelec ỵ amp kd n And the reception energy is calculated by: E r ¼ kEelec where Eelec is the energy dissipation of the electronic circuitry to encode or decode a bit, k is message size, εamp is the amplifier constant and d is the distance between the transmitter and the receiver On average, each non-CH node spends two times the energy for electronics and some additional energy sending data to its neighbor depending on how far the node transmits As a result, the total energy consumption of any chain built will be at least two times the energy of the electronics multiplied by the number of sensor nodes Therefore, in the bounded solution, the authors set the energy usage of non CH nodes to two times the energy usage for electronics Determination of bounds for the lifetime from any routing algorithm The authors in [5] proposed a method to determine the upper bounds of the chain-based routing system lifetime In each Fig Number of rounds over 100 random 100-node networks Mobile Netw Appl (2014) 19:249–257 253 Example 2: Table Results for Fig Protocol PEGASIS RE_chain LEACH Mean Variance 90 % confidence interval of the sample means 890.3 84.9 (876, 904) 1305.4 174.5 (1276, 1335) 602.3 62.5 (592, 613) Maximize : xj jẳ1 n X Subject to : x1 ỵ x2 Subject to : 1x1 ỵ 2:8x2 6:1 0:5x1 ỵ 0:5x2 ≤ 6:5 The solutions of the bound Table method are the solutions of the ILP formulation (1) with the following coefficients: n X Maximize cij x j ≤ E i : iẵ1n 3ị jẳ1 x1 ; x2 Z ỵ The sum of coefficients of constraint functions is: 1+2.8+ 0.5+0.5=4.8 The optimum objective for the example is The two simple examples show that smaller sum of the coefficients of the ILP problem does not necessarily mean that the better objective solution can be obtained In other words, for the chain-based routing problem, minimizing the total energy dissipation for gathering data does not always guarantee an optimum solution x j Z ỵ : jẵ1n If (Node i is a non CH) then cij ẳ 2kE elec : i; jẵ1n; i j else cij ẳ 2kE elec ỵ mp kd 4to BS ; i¼ j where dto_BS is the distance from Node i to the base station The discussion above is true for the minimum energy consumption of any chain but is not always true for the maximum lifetime problem In other words, a smaller total of energy coefficients in Formulation (3) not always provide a better optimum We show that by an example below: Consider two ILP problems Determination of absolute upper bounds for the lifetime of any routing problem The bounds calculated by the method in (3) are called the minimum energy bounds As discussed in the two ILP examples above, the bounds cannot be proven to be the upper bounds of the system lifetime given by any chain-based routing In the section, a new method to determine the absolute upper bounds of the system lifetime given by any routing protocol is presented On any routing protocol, any sensor must send data once in each round, and at least a sensor node must deliver data to the base station Let us consider Formulation (1), in which the energy coefficients are determined by: Maximize : n X xj jẳ1 Example 1: Maximize : x1 ỵ x2 n X Subject to : cij x j ≤ Ei : iẵ1n jẳ1 x j Z ỵ : jẵ1n Subject to : 1x1 ỵ 1:1x2 6:1 1:2x1 ỵ 1:3x2 6:5 x1 ; x2 Z ỵ If (Node i is a non CH) then cij ¼ kE elec : ∀i; j∈½1…nŠ; i≠ j else The sum of coefficients of constraint functions is: 1+1.1+ 1.2+1.3=4.6 The optimum objective for the example is cij ẳ 2kE elec ỵ mp kd 4to BS ; iẳ j 4ị 254 Mobile Netw Appl (2014) 19:249–257 where dto_BS is the distance from Node i to the base station Let O be the optimum solution of the ILP problem (4) Then O is the upper bound for the lifetime that can be achieved by any routing method Therefore, any feasible solution O' obtained by any routing algorithm will satisfy: Proof Theorem is stated and proved below to simplify the process: As a result, O from Formulation (4) is the upper bound of any possible feasible solution (End of proof) Theorem Consider two ILP problems with the same objective function and the same variables, if the set of coefficients of ILP problem is smaller than the set of coefficients of ILP problem respectively for all of these coefficients, then the optimal solution of Problem is higher than that of Problem 6.1 Consider two ILP problems Problem Maximize : n X xj j¼1 n X Subject to : Optimization of the Base station location Previous researches in [1, 3, 4, 10–16] assume that S the base station position is randomly placed without optimization Actually, the location needs to be optimized In order to minimize the complexities of the problem, the wireless radio energy dissipation model is not used yet A very simple energy usage model is given below Assume that the energy to transmit a unit of data is proportional to the square of the distance to a destination, and there is no energy spent at the destination E S ị ẳ d ; E Dị ẳ 0; for α > cij x j ≤ E i : iẵ1m 5ị jẳ1 x j Z ỵ : j∈½1…nŠ , where S denotes a source node, D denotes a destination node, E(S) is the energy usage of node and d is the distance from S to D The Linear Programming model (2) becomes: ð7aÞ Maximize : T Problem Maximize : O0 ≤ O n X xj j¼1 Subject to : N X q j;i ỵ QT ẳ j¼1 n X Subject to : c0 j x j E i : iẵ1m i 6ị jẳ1 x j Z ỵ : jẵ1n N X jẳ1 If c ij ≤cij,∀i∈[1…m],∀j∈[1…n], then O2 ≥O1 Proof Since c′ ij ≤cij ∀i∈[1…m],∀j∈[1…n] and O1 is the optimal solution of Problem 1, then O1 is a feasible solution of Problem because O1 satisfy all constraints of (5) Since is the optimal solution of Problem 2, O2 ≥ O1 (End of proof) Using the result from Theorem 1, the energy coefficient c′ ij from any constructing algorithm: c′ ij ≥cij,∀i,j∈[1…n] of Formulation (4) qi; j : iẵ1N 7bị jẳ0 h i qi; j d i; j ỵqi;0 xi X ị2 ỵ yi Y ị2 < ẳ E i : iẵ1N Š n X i¼1 Definition O1 is the optimal solution of Problem (5) O2 is the optimal solution of Problem (6) N X qi;0 ¼ Qn q; T ; X ; Y : Variable ð7cÞ This problem is difficult due to Constraints (7c), that are non convex Finding efficiently a solution of Problem (7) is a challenge We propose a solution approach based on DC (Difference of Convex) Programming and DCA (DC Algorithm) They are introduced by Pham Dinh in 1985 and have been extensively developed by Le Thi and Pham Dinh since 1994 [17–28] In the literature, several work in non convex optimization have been developed for solving the optimization problems Several approximations have been proposed including Mobile Netw Appl (2014) 19:249–257 255 Concave exponential approximation and logarithmic approximation of Weston, piecewise concave approximation [25–27] A common point of these approximations is that the resulting optimization problems are all DC programs and one can investigate DCA, an efficient method in nonconvex programming framework for solving them Thanks to a new result concerning with exact penalty techniques in DC programming [23], we first reformulate Problem (10) as a DC program then apply DCA to the resulting problem Despite its local character, DCA with a good initial point quite often converges to global solutions in practice Now, consider the left hand side of the difficult constraints (7c) Let f i ðzÞ ¼ N X h i qi; j d i; j ỵqi;0 xi X ị2 ỵ yi Y ị2 jẳ1 Where ta ≤min i z∈K g(z) and tb ≤max i max z∈K h(z) Problem (9) is written as: Min : −T ð10aÞ Â ÃN Subject to : ðz; t; sịK t a t b ẵ0; 10bị gi zịt i : iẵ1N 10cị t i ỵ si hi zị ẳ : iẵ1N 10dị < ẳ Ei : iẵ1N Where =max i max{h(z)−ti} Using penalty techniques (see [23]), Problem (10) is equivalent to: It can be decomposed as follows:   ! ρ  2 ρ  2 f i ðzÞ ¼ z − z − f ðzÞ 2 2 f i zị ẳ gi zịhi zị 2 jzj Where gzị ẳ is convex and hzị ¼ jzj − f ðzÞ is also convex with a sufficiently large number ρ Constraints (7c) are thus rewritten as(a DC constraints N By denoting K ¼ Min : T ỵ N X ri t i ỵ si hi zịị 11aị iẳ1 N Subject to : z; t; sịK t a ; t b ẵ0; 11bị gi zịt i : iẵ1N 11cị t i ỵ si hi zị : iẵ1N 11dị z ẳ q; T ; X ; Y ị : q j;i ỵ QT ẳ jẳ1 N qi; j : iẵ1N jẳ0 Problem (7) becomes: Max : T Subject to : z∈K gi ðzÞ−hi ðzÞ≤ : iẵ1N 8aị 8bị 8cị Or Where ri is sufficiently large Problem (11) has the convex feasible set while whose objective function is concave, which is easily transformed to a DC function There are many ways to decompose this function We can choose and test DCA with different DC decomposition ways (possibly use a similar way as did for f(x) in (10c)) Thus, we already transformed the origin problem to a DC program All we have to now is to use DCA Max : T ð9aÞ Â ÃN Subject to : ðz; t Þ∈K  t a ; t b 9bị gi zịt i : iẵ1N Š ð9cÞ Conclusion ð9dÞ This paper has focused on a new family of routing protocols for sensor networks: chain-based routing protocols In chainbased routing, nodes form a chain connecting all nodes in the t i −hi ðzÞ ≤ : ∀i∈½1…N Š 256 network Data are gathered from all sensor nodes and move along the chain toward an elected sensor The role of the elected node is rotated between all sensor nodes to increase the network lifetime Chain-based routing exploits the data aggregation capability of sensor networks at maximum When data are gathered from all sensor nodes, the data are aggregated with the data from their neighbors into a single message The process is repeated until a single message is collected at the elected sensor node The previous chain-based routing (PEGASIS) selects the CH nodes uniformly among all sensor nodes It is demonstrated in this chapter that the selection is a bad practice to ensure a good lifetime Depending on the energy usage of each sensor to send data to its neighbors and to the base station, the sensor nodes should be elected as a leader differently The paper has then proposed a method to optimize the selection of the CH among all sensor nodes using Linear Programming formulations As it is not always practical to the Linear Programming formulation, a simple heuristic method called RE_chain is proposed to calculate the selection Simulations showed that RE_chain performs very closely to the Linear Programming formulation The performance of RE_chain was then compared to that of LEACH, PEGASIS This was shown that RE_chain improves the system lifetime significantly than that of PEGASIS Also, it was observed that RE_chain performs about times better than LEACH Although the actual optimal lifetime for chain-based methods is unknown, two methods were proposed to compute the upper bounds for the lifetime of any routing method (including chain-based routing and cluster-based routing) The first method is called “minimum energy bound” and the second one is called “absolute bound” It was proved that the absolute bounds are the upper bound for any routing method Furthermore, previous researches assume that the base station position is randomly placed without optimization In our works, a non convex optimization model has been developed for solving the base station location optimization 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