Distributed optimization of lifetime and throughput with power consumption balance opportunistic routing in dynamic wireless sensor networks

International Journal of Distributed Sensor Networks

2016, Vol. 12(10) Ó The Author(s) 2016 DOI: 10.1177/1550147716671144 ijdsn.sagepub.com

Distributed optimization of lifetime and throughput with power

consumption balance opportunistic routing in dynamic wireless sensor networks

Jian Sun, Junni Zou and Liwan Huang

Abstract

This article studies a joint performance optimization on the network lifetime and network throughput for an energy- constrained dynamic wireless sensor network. We propose a fully distributed power consumption balance opportunistic routing scheme to cope with the dynamic network that is not considered in the classic low-energy adaptive clustering hierarchy routing and evenly allocate power consumption among the sensor nodes for obtaining longer network lifetime.

Moreover, a fully distributed optimization solution, whose distinctive feature to the Lagrange dual approach is capable of handling the changing network, is developed to achieve joint performance optimization of objectives. We mathematically prove the convergence of the proposed solution and analyze its computational complexity. Extensive simulation results illustrate the effective measures to deal with the varying network of power consumption balance opportunistic routing and the best tradeoff performance achieved by the proposed solution and evaluate the more positive impact on the net- work lifetime of power consumption balance opportunistic routing than the existing routings and better ability to adapt the dynamic network of the proposed solution than the Lagrange dual approach.

Keywords

Network lifetime, wireless sensor networks, opportunistic routings, distributed optimization, dynamic networks

Date received: 21 May 2016; accepted: 1 September 2016 Academic Editor: Federico Barrero

Introduction

In a wireless sensor network (WSN), each sensor node is tasked to capture data and deliver them over wireless channels to the sink nodes for further data analysis and decision-making. In general, maximizing the network lifetime and network throughput are critical issues in WSNs. Nevertheless, the prolongation of the network lifetime always leads to the decrease in network throughput. How to make a tradeoff between the net- work lifetime and network throughput for the overall system performance remains a challenging problem.

Moreover, WSNs usually comprise a large number of sensor nodes deployed randomly in a highly dynamic

and hostile environment, resulting in the frequent change in network topology. In this article, for a joint network lifetime and throughput optimization under dynamic network settings, we focus on an opportunis- tic routing and a distributed optimization solution.

Department of Communication Engineering, Shanghai University, Shanghai, China

Corresponding author:

Junni Zou, Department of Communication Engineering, Shanghai University, Shanghai 200072, China.

Email: [email protected]

Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (http://www.uk.sagepub.com/aboutus/

openaccess.htm).

Low-energy adaptive clustering hierarchy (LEACH) routing is the classic clustering-based protocol that uti- lizes randomized rotation of local cluster heads to extend the network lifetime. However, LEACH routing does not take account of the dynamic networks.

Moreover, neglecting the influence of transmission dis- tance and the periodical clustering of LEACH routing limit maximization of the network lifetime. To over- come these problems, we propose a fully distributed power consumption balance (PCB) opportunistic rout- ing scheme. It enables network topology update to cope with the dynamic network. For longer network lifetime, it abolishes clustering scheme and weakens the impact of long-distance transmission on node power consump- tion to save node energy. Furthermore, it adopts a new next-hop selection strategy to equally utilize power among the sensor nodes.

The Lagrange dual approach is frequently used to achieve the optimal solutions. But it could not handle the changing network. To solve this problem, we develop a fully distributed optimization solution, which is capable of tackling the changing networks, to achieve the joint performance optimization of the network life- time and network throughput. The proposed solution decomposes the original problem into two stages. In the first stage, the optimization is executed at each indi- vidual node for maximizing the node lifetime and throughput. The second stage utilizes the results of the first stage to bring global optimization on the network lifetime and throughput.

The remainder of this article is organized as follows:

section ‘‘System modeling’’ describes a generalized sys- tem model. Section ‘‘PCB opportunistic routing proto- col’’ introduces the PCB opportunistic routing scheme.

Section ‘‘Problem statement’’ formulates the proposed distributed optimization solution. In section ‘‘Theoretic analysis,’’ the convergence performance and computa- tional complexity of the proposed solution are theoreti- cally analyzed, and we prove Lemma 1 for the next section. Extensive simulation results are presented and discussed in section ‘‘Simulation results.’’ Finally, sec- tion ‘‘Concluding remarks’’ concludes this article.

Related work

In the past, a variety of routing and rate control schemes have been proposed to maximize the lifetime of WSNs.^1–10 Madan and Lall,¹ He et al.,² and Cetin et al.³studied distributed routing algorithms for maxi- mizing the lifetime. Wang et al.⁴proposed a cross-layer design approach for minimizing the energy consump- tion of a multiple-source and single-sink WSN. Zhu et al.⁵ exploited the interaction between the network lifetime maximization and fair rate allocation.

Furthermore, many researches concentrated on LEACH routing protocol to prolong the network

lifetime. Based on LEACH routing, Li et al.⁶proposed an improved cluster head reappointment algorithm (LEACH-R), to overcome the shortcoming of LEACH that each node would be frequently elected as the clus- ter head for several times and consumed some energy for this. Haneef et al.⁷ utilized redundancy of the deployed nodes to increase energy efficiency in multi- group-based LEACH, outperforming LEACH at the network lifetime. A protocol named Bayesian network model LEACH realized uniform distribution of cluster heads that is not guaranteed in LEACH to extend the network lifetime in Ghasemzadeh et al.⁸Quynh et al.⁹ presented energy and load balance LEACH routing (EL-LEACH), which showed a cluster head selection strategy for covering defects brought by the original method of LEACH, to achieve better energy consump- tion, load balance, and network lifetime. Ahmad et al.¹⁰controlled election and selection of cluster heads to uniform load on cluster heads and took the free association mechanism to remove back transmissions in adaptive clustering habit routing scheme for mini- mizing the overall energy consumption of the network.

Whereas these studies are all on the basis of the static network and do not consider the dynamic settings.

Thus, we propose a fully distributed PCB opportunistic routing to handle the varying network and obtain lon- ger network lifetime than the existing routings.

How to maximize network throughput has been studied by extensive works. And most of them take the Lagrange dual approach to achieve the optimal solu- tions. Fida et al.¹¹used a route selection metric based on the reception probability of Rician fading channel for avoiding the low-throughput routes of the conven- tional route selection metrics to optimize network throughput. Sappidi et al.¹²utilized in-network compu- tation for statistical functions to reduce the volume of traffic transmitted, achieving maximized throughput in a WSN. The research results of Verdone et al.¹³showed that the density of the sensor nodes and the query inter- val decided optimized area throughput for different scales in a multi-sink clustered WSN. Based on the pro- posed mobile-relay-assisted data collection model, the achievable throughput capacity of large-scale WSNs is analyzed by choosing appropriate mobility parameters in Liu et al.¹⁴Xie et al.¹⁵solved the throughput perfor- mance deterioration problem by referring to the notion of fairness and demonstrated the availability of throughput maximum under an equal proportion of channel occupancy time for each contending node.

Moreover, the tradeoff between throughput optimiza- tion and energy consumption optimization is consid- ered. Ren and Liang¹⁶ proposed a throughput maximized Medium Access Control (MAC) protocol, redividing each pico-net into several subsets in which communication pairs can make communication simul- taneously, to maximize throughput and achieve short

latency. However, these works are also founded on the static network and do not consider the condition that the change in a network happens. Hence, in this article, a fully distributed optimization solution, which has bet- ter ability to adapt the dynamic network than the Lagrange dual approach, is proposed to optimize the network lifetime and throughput simultaneously.

System modeling

A WSN could be modeled as a directed graph G(V , Z), where V is the set of network nodes and Z is the set of directed links between the nodes. The set V includes two disjoint subsets S and R representing the sensor nodes and sink nodes, respectively. Each sensor node has a maximum transmission range d_max. A directed link ( j, i)2 Z exists between the jth node and the ith node if their distance d_jisatisfies d_ji dmax.

Suppose that there are N nodes in a WSN. The ith node has a N-dimensional vector H_i to reflect its com- munication relationship with the other nodes. Let H_i^j be the jth element of H_i and H_i^j= 1 if the jth node could communicate with the ith node, or else H_i^j= 0.

Assume that the ith node has a N-dimensional vector fi

that represents data transmission rates from the other nodes to the ith node. The jth element f_i^jof firepresents the data transmission rate from the jth node to the ith node. Let f_i^j= 0 if the jth node does not send data to the ith node. Thus, the node throughput at the ith node is f_i= H_i f_iand the traffic of the neighbors around the ith node is B_i=P

j2bif_j, where b_iis the set of neighbor nodes around the ith node. Accordingly, the network throughput is formulated as Gnet=P

i2Rf_i.

To denote the latency resulting from channel conten- tion, we introduce a concave increasing latency function l( )¹⁷over one hop, whose input is the data load of the medium. Accordingly, we define the cost function C_iof the ith node as the latency experienced by all communi- cations that run through the ith node during channel contention, that is, C_i= l(f_i+ B_i)f_i. Then, the average cost C of all nodes is formulated as C = (1=N )P

i2VCi. Moreover, define V_P= (1=n_S)P

i2S(P_i P)² as the variance of power consumption among the sensor nodes, where nSis the number of sensor nodes, Piis the power consumption of the ith sensor node, and

P = (1=nS)P

i2SPi is the average power consumption of the sensor nodes. The smaller the VP, the more equal the power usage among the sensor nodes. Assume that the ith sensor node has an initial energy E_i and a life- time T_i= E_i=P_i, then the network lifetime is defined as Tnet= mini2ST_i= mini2SE_i=P_i.

PCB opportunistic routing protocol

Compared with LEACH routing protocol, the pro- posed PCB opportunistic routing scheme makes two

improvements. The first is to achieve longer network lifetime, and the second is adapt to the dynamic net- work. To these ends, first, it cuts down the periodical clustering of LEACH routing to save the power con- sumption. Second, it takes into account the impact of transmission distance on power consumption. As is well known, the shorter the transmission distance, the less the transmit power required. In LEACH routing, a sensor node directly sends its data to the cluster head regardless of their distance. When the sensor node is far away from its cluster head, it uses a large amount of transmit power, which may lead to its early death and decrease Tnet. PCB opportunistic routing with smaller d_max than LEACH routing allows the commu- nication between the sensor nodes and decomposes one-hop long-distance transmission into multi-hop short-distance transmission so as to avoid the early death in LEACH routing. Third, when forwarding the received data, it uses a new next-hop selection strategy to achieve equal power consumption among the sensor nodes. Based on the above three, the proposed PCB opportunistic routing outperforms LEACH routing at the network lifetime. Fourth, it supports the changing network, which is not considered in LEACH routing.

We now introduce some definitions and notations of PCB opportunistic routing. For a minimal number of hops to the sink node, the sensor nodes are classified into several levels. As shown in Figure 1, the 1st, 3rd, and 4th sensor nodes are considered as the third-level nodes because they need a minimum of three hops to reach the 10th sink node. Thus, the second, fifth, sixth, and seventh sensor nodes that need two hops are the second-level nodes. Following the same principle, the eighth and ninth sensor nodes are the first-level nodes.

Figure 1. Topology of the WSN.

Furthermore, we define the sensor node as an exclu- sive node when the number of the next low-level nodes it can connect is one. In Figure 1, the fourth node in the third level is an exclusive node as it only connects to one second-level node, that is, the seventh node.

Similarly, the second, fifth, and seventh nodes in the second level are all exclusive nodes.

PCB opportunistic routing consists of three pro- cesses, that is, initialization, next-hop selection, and network topology update. The way of initialization uses LEACH protocol. It means that all nodes in a ran- domly distributed WSN (i.e. Figure 1) make sure their communication relationships in the ad hoc way. Within the maximum transmission range d_max, the ith sensor node tries to communicate with the other nodes by one-hop or multi-hop for H_i, determine its own level as well as levels of its neighbors, and decide whether it is an exclusive node.

When selecting the next hop, a sensor node follows the five principles that are listed below:

1. Next-hop selection starts from the highest level node, in a decreasing order to its next low-level node over the next hop, then one by one, and finally ends at the sink node. Next-hop selection among the sensor nodes in the same level is not allowed.

2. The exclusive nodes have the priority on next- hop selection. Afterward, other sensor nodes in the same level select their own next-hop nodes, one by one, from near to far around the exclu- sive node.

3. The sensor node selects only one next-hop node.

4. The sensor node selects the next-hop node whose node throughput is the least among the next low-level nodes of its neighbors.

5. The sensor node selects the nearest next low- level node under the circumstance that the next low-level nodes of its neighbors have the same node throughput.

Note that when next-hop selection process is fin- ished, it will maintain stable until the network variation happens.

Take Figure 1 as an example. Assume that the amount of captured data at each sensor node is equal.

Based on the principles above, the fourth exclusive node in the highest level first selects the seventh node in the next low level. Then, obeying principle (4), the third node in the highest level selects the sixth node in the next low level. According to principle (5), the first node in the highest level selects the second node in the next low level; moreover, the second and fifth exclu- sive nodes in the second level select the eighth node in the first level, and the seventh exclusive node in the second level selects the ninth node in the first level.

Thereafter, following principle (4), the sixth node in the second level selects the ninth node with the least node throughput among the first-level nodes of its neighbors. Finally, the 8th and 9th nodes in the first level send data to the 10th sink node. To make a com- parison, assume that in LEACH routing, the first, second, fifth, and eighth sensor nodes are in the same cluster and the eighth sensor node is elected as the cluster head. The first node would use a lot of power to transmit data to the cluster head for the long dis- tance, which would lead to its early death. However, in PCB opportunistic routing, the first node could send its data to the much nearer second node, which could forward data of the first node to the eighth node, so as to save much power of the first node avoiding its early death.

Network topology update, including re-initialization and re-selection of the next hop, is triggered when the change in the network occurs. The sensor nodes that detect the change would notify the sink node, and then the sink node sends re-initialization messages to all the sensor nodes, telling them to do initialization and next- hop selection processes again.

Problem statement Optimization problem

The total power of a sensor node is mainly used for two functions: (1) data transmission and (2) data reception.

Based on the power consumption model widely used in WSNs,^2,5 the transmission power consumption at the ith sensor node can be denoted as P^t_i= e_ijP

j:(i, j)2Zf_jⁱ, where e_ij2 (0, 1) is the transmission energy consump- tion coefficient of link (i, j). The reception power con- sumption at the ith sensor node can be formulated as P^r_i= j fi, where j2 (0, 1) is the energy consumption coefficient of the radio receiver. Therefore, the total power consumption of the ith sensor node is given by Pi(fi) = P^t_i+ P^r_i= eijP

j:(i, j)2Zf_jⁱ+ j fi.

Maximizing the network lifetime and network throughput are two contradictory objectives. To make a tradeoff, we use a simple and efficient weighting method,¹⁸to combine these two objectives into a single objective function. That is, maxfaTnet+ (1 a)Gnetg, where a2 ½0, 1 is a weighted system parameter.

Mathematically, the joint optimization problem can be formulated as follows

P1 : max aTf _net+ (1 a)Gnetg ð1Þ s.t.

1. Tnet= mini2STi= mini2S(E_i=Pi);

2. Gnet=P

i2Rfi;

3. l(fi+ B_i) Ti, 8i 2 S;

4. fi+P

p2S

P

(p, q)2C(j, i)f_q^p A, 8(j, i) 2 Z.

Constraint (3) shows that the latency of the ith sen- sor node should be no more than node lifetime T_i. Constraint (4) is the wireless network channel interfer- ence constraint. It specifies that the sum of needed transmission data and interference data must be no more than maximized rate A of the wireless shared medium, and C(j, i) is the set of links that would inter- fere with the communication between the jth node and the ith node when it is alive.¹⁹

In Problem P1, Gnet of the objective function is not concave, thus this problem is not a convex problem that is usually difficult to solve in practice. Therefore, we introduce a new variable G_net^ln = ln Gnet,¹⁵which is directly proportional to Gnet. Gnet achieves its maxi- mum when G^ln_netarrives at the maximum. However, the order of magnitude of G^ln_net is much less than Tnet. We define a coefficient l_f to adjust the order of magnitude of G_net^ln to make l_f  G^ln_net and Tnet comparable. Thus, the original objective function is rewritten as maxfaTnet+ (1 a)  l_f  G^ln_netg. Moreover, we use G_net^ln = ln Gnet= ln P

i2Rf_i to replace the original con- straint (2). Then, to match up with the expression form of G^ln_net, we take the logarithm of constraint (4). Thus, constraint (4) is equally changed to ln (f_i+P P p2S

(p, q)2C(j, i)f_q^p) ln A.

After the above transformation, Problem P1 becomes

P2 : max aT _net+ (1 a)  lf  G^ln_net

ð2Þ s.t.

1. Tnet= mini2STi= mini2S(E_i=Pi);

2. G^ln_net= ln G_net= lnP

i2Rf_i; 3. l(fi+ B_i) Ti, 8i 2 S;

4. ln (f_i+P

p2S

P

(p, q)2C(j, i)f_q^p) ln A, 8(j, i) 2 Z.

Proposed distributed optimization solution

According to the definitions of T_net and G_net^ln, we know that the optimal T_net and G^ln_netcome out from the opti- mal node lifetime T_i and node throughput f_i. In order to obtain the optimal Tnetand G^ln_net, we first need to find out the optimal Ti and fi. Hence, the solution of Problem P2 can be divided into two stages. The first stage, as shown in subproblem P2a, fulfills at each indi- vidual node to find out the optimal Tiand fi. In the sec- ond stage, as shown in subproblem P2b, the sink node collects the results of the first stage and determines the optimal Tnetand G_net^ln

P2a : max aT _i+ (1 a)  l_f  ln f_i

ð3Þ s.t.

1. T_i= (E_i=P_i), 8i 2 S;

2. l(fi+ B_i) Ti, 8i 2 S;

3. ln (f_i+P

p2S

P

(p, q)2C(j, i)f_q^p) ln A, 8(j, i) 2 Z.

P2b : max aT _net+ (1 a)  lf  G^ln_net ð4Þ s.t.

1. Tnet= mini2ST_i= mini2S(E_i=P_i);

2. G^ln_net= ln Gnet= lnP

i2Rf_i.

Note that ln f_i is directly proportional to f_i and f_i acquires its maximum when ln f_ireaches the maximum.

Based on the results of subproblem P2a sent from each sensor node, the sink node determines the final solution by max Tnet= mini2ST_i^ and max G^ln_net= lnP

i2Rf_i^, where T_i^ and f_i^ are the optimal lifetime and throughput of the ith node obtained at the first stage, respectively. Namely, subproblem P2b can be solved by

max aT _net+ (1 a)  lf  G^ln_net

= a mini2ST_i^+ (1 a)  lf  lnX

i2R

f_i^ ð5Þ

It is noted that the solution to P2b can only be obtained at the sink node during the second stage. To solve subproblem P2a in a fully distributed way, we employ the adaptive penalty-based distributed stochas- tic optimization algorithm.²⁰In particular, P2a can be solved as an unconstrained optimization problem by penalty functions as follows

Wi(fi) = maxfiaTi+ (1 a)  lf  ln fi + h d^EP TiEi

Pi

 

+ h d^IP½l(fi+ Bi) Ti

+ h d^IP ln fi+X

p2S

X

(p, q)2C(j, i)

f_q^p

!

 ln A

" # ð6Þ

where h.0 is a scalar parameter that controls the rela- tive importance of adhering to constraints. The corre- sponding penalty functions are

d^EP(x) = 0, x = 0

\0, x6¼ 0, d^IP(x) = 0, x 0

\0, x\0





where EP and IP denote the equality penalty and inequality penalty, respectively. In practice, we use d^SEP( ) and d^SIP( ) to approximate d^EP( ) and d^IP( ), respectively. Here, d^SEP( ) and d^SIP( ) are formulated as

d^SEP(x) =  x², d^SIP(x) = min (0, x³)

where SEP and SIP denote the simulated equality pen- alty and simulated inequality penalty, respectively. For simplifying expression, we define

Ji(f_i) = max_fi aTi+ (1 a)  lf  ln fi

 

= aEi eij X

j:(i, j)2Z

f_jⁱ+ j fi

" #1

+ (1 a)  lf  ln fi

ð7Þ mi(fi) = d^SEP TiEi

P_i

 

+ d^SIP½l(fi+ Bi) Ti

+ d^SIP ln fi+X

p2S

X

(p, q)2C(j, i)

f_q^p

!

 ln A

" #

ð8Þ Hence, equation (6) is converted into Wi(fi) = Ji(fi) + h mi(fi).

To find f_i^, the following iterations run with a con- stant step-size m²⁰

1_{i, n}= fi, n1+ m crfiJi(fi, n1) ð9aÞ c_{i, n}= 1_{i, n}+ mhrfimi(1_{i, n}) ð9bÞ

fi, n=X

j2b_i

a^j_ic_{j, n} ð9cÞ

where n is the iteration number, 1_{i, n} and c_{i, n} are the intermediate variables, m is the constant step-size, and fa^j_ig is the set of non-negative combination coefficients of the neighbors around the ith node, which satisfy

a_i^j= 0, when j62 bi

X^N

j = 1

a^j_i= 1, i = 1, . . . , N

In addition, crfiJi(f_{i, n1}) is rcfiJi(fi, n1) = (1 a) l_f

fi, n1

 aEij Pi(fi, n1)^2+ vi, n1

ð10Þ where vi, n1 is the random perturbation term (or gradi- ent noise) that makes 1, c, and f in the diffusion strate- gies (9a)–(9c) become random variables, and Pi(fi, n1) = e_ijP

j:(i, j)2Zf_jⁱ+ j fi, n1.

Because rfid^SEP(T_i (Ei=Pi)) = 0, rfimi(1_{i, n}) is denoted as

rfimi(1_{i, n}) = 3X³

k = 2

Dk(1_{i, n})² _Dk(1_{i, n}) ð11Þ

where D_k(1_{i, n}) = (∂Dk=∂1_{i, n}), k = 2, 3 and D2(1_{i, n}) = l(1_{i, n}+ B_i) Ei Pi(1_{i, n})^1, D_2(1_{i, n}) = _l(1_{i, n}+ B_i) + E_i j P_i(1_{i, n})^2, D₃(1_{i, n}) = ln (1_{i, n}+P

p2S

P

(p, q)2C(j, i)f_q^p)

 ln A, and _D3(1_{i, n}) = (1_{i, n}+P

p2S

P

(p, q)2C(j, i)f_q^p)^1.

It is worth mentioning that the first stage should also be conducted at the sink node. Since the sink node is supposed to be power infinite, lifetime maximization is no longer a problem, and throughput maximization becomes the sole objective of the sink node. Therefore, constraints (1) and (2) of subproblem P2a become inva- lid. For the sink node, subproblem P2a is rewritten as follows

P2a⁰: maxfln fig ð3#Þ s.t.

1. ln (f_i+P

p2S

P

(p, q)2C(j, i)f_q^p) ln A, 8(j, i) 2 Z Besides, equations (6)–(8) are re-depicted as follows

W⁰_i(f_i) = max_filn f_i + h d^IP ln f_i+X

p2S

X

(p, q)2C(j, i)

f_q^p

!

 ln A

" #

ð6#Þ

J_i⁰(f_i) = max_filn f_i ð7#Þ

m⁰_i(f_i) = d^SIP ln f_i+X

p2S

X

(p, q)2C(j, i)

f_q^p

!

 ln A

" #

ð8#Þ

Meanwhile, equation (9c) remains the same, equa- tions (9a) and (9b) are omitted, and this results in the worthlessness of equations (10) and (11). The reason is that equations (9a) and (9b) acquire c_{i, n}transmitted to the next hop; however, for the sink node, the next-hop nodes do not exist and equation (9c) leads to f_{i, n} for achieving the optimum f_i^. That is to say, equations (9a)–(9c) are converted into

fi, n=X

j2bi

a^j_ic_{j, n} ð9#Þ

It is noted that when the network changes, the opti- mization of the lifetime and throughput at each node could run with the new parameters offered by the net- work topology update process of PCB opportunistic routing. After that, the second stage finds out the opti- mal T_netand G^ln_netunder the new network.

Practical problem

The computational complexity of the proposed distrib- uted solution is determined by the computational com- plexity in two stages. At the first stage, the other nodes correlated to the ith node would obtain the optimums when the variable f_{i, n} converges to the optimum f_i^. Note that achieving the optimums of the correlated nodes is not conditional on converging to f_i^ at the ith node and vice versa. Achieving the optimums of the correlated nodes and converging to f_i^ at the ith node

are paratactic and simultaneous. It is worth mentioning that the optimization of each individual node is simul- taneous at the first stage. Thus, the computational com- plexity of the first stage is actually the maximum of the computational complexity of each node, which is O(n).

At the second stage, the sink node obtains the optimal solution of Problem P2 according to the received T_i^ and f_i^from all the nodes. The computational complex- ity of a mini2ST_i^ is O(1) and the computational com- plexity of (1 a)  lf  lnP

i2Rf_i^ is O(1) as well; thus, the computational complexity of the second stage is O(1) + O(1) = O(1). Hence, the computational com- plexity of the proposed distributed optimization solu- tion is O(n) + O(1) = O(n).

To realize the proposed distributed optimization solution, each node is considered as a processor of a distributed computation system. Assume that the pro- cessor of the ith node keeps track of variable f_i. At initi- alization phase, each node contacts with the other nodes to gain H_i, then the ith node obtains b_i and its path to the sink node. After the distributed optimiza- tion of each individual node at the first stage, the sec- ond stage brings the optimal solution to Problem P2 as shown in equation (5).

There are three points need to emphasize: (1) for the static network, H_i and b_i are fixed; (2) from equation (9c), we know the distributed optimization of each indi- vidual node is correlated, and thus, the other nodes correlated to the ith node would reach the optimums when the variable f_{i, n}converges to the optimum f_i^; and (3) at the first stage, the correlated nodes could com- municate with each other through established links in PCB routing to transfer needed c_{j, n}(j2 bi) in a distrib- uted manner, rather than send c_{i, n} to the sink node and then broadcast.

When the communication overhead issue²¹ is taken into account, all the update operations at the first stage can utilize those variables stored in the local node or link, except the updated rates f_jand c_{j, n}(j2 bi) that are needed to be transmitted by extra packets. For exam- ple, according to equations (9b) and (11), the definition of B_i, and equation (9c), to update c_{i, n} and f_{i, n}, the packet carrying f_j and c_{j, n}is only required to transmit along the link between the jth node and the ith node. If we adopt the float type in implementation, each f_j or c_{j, n} takes up only 4 bytes, thus it is negligible com- pared to the main data traffic. Roughly estimated, the time spent by the whole network to reach the stability is equal to the number of iterations required for con- vergence multiplying the update time interval of each iteration. It is found in Deb and Srikant²² that an update interval is about two to three times the one-way propagation delay of the particular receiver. An update interval is sufficient for the updated rates’ interaction between the nodes. Therefore, the entire overhead of the proposed distributed solution is quite small.

Theoretic analysis

As is stated above, subproblem P2b can be resolved eas- ily by choosing Tnetand calculating G_net^ln, which do not need iterations; thus, the convergence performance of the proposed distributed optimization solution is deter- mined by that of subproblem P2a.

Theorem 1. The iterations at each individual node in the first stage converge to an optimum f_i^.

Proof. According to the proposed distributed optimiza- tion solution, the optimization at each individual node is to seek the optimum of equation (6) obtained by the optimum f_i^. We introduce equations (12) and (13) to denote W_i(f_i) = Y_Ti+ Y_fi, where Y_Ti represents the set of the terms related to node lifetime, and Y_fi represents the set of the terms related to node throughput

Y_Ti= max_fiaT_i+ h d^SEP T_iE_i Pi

 

+ h d^SIP½l(fi+ Bi) Ti

ð12Þ

Y_fi= max_fi(1 a)  l_f  ln f_i + h d^SIP ln f_i+X

p2S

X

(p, q)2C(j, i)

f_q^p

!

 ln A

" #

ð13Þ

Assume that all constraints are satisfied, thus, Dk(f_i) 0, k = 2, 3. Note that 1 l(fi+ B_i) Ti, _l(f_i+ B_i) 0, €l(fi+ B_i) 0, and h, a are close.

Algorithm 1 Proposed distributed optimization solution 1. The first stage: distributed optimization of each individual node

2. Initialization

3. Set n = 1 for all i, j;

4. Fetch Hiand bistored in the local processor;

5. Receive fi, 0andfa^j_ig from sensor nodes;

6. Calculate fi, 0according to fi, 0= Hi fi, 0. 7. Repeat

8. Updating at theith node:

9. For the ith sensor node,

10. Fetch the random vi, n1of the ith sensor node;

11. Update the fi, nby equations (9a)–(9c).

12. For the ith sink node,

13. Update the fi, nby equation (9#).

14. end for 15. n = n + 1.

16. Until the variable fi, nconverges to the optimum f_i^. 17. Calculate T^_i of all sensor nodes and f_i^of all nodes according

18. To the definitions.

19. The second stage at the sink node 20. Receive T_i^and f_i^from all nodes;

21. Achieve the optimal solution to Problem P2 by equation (5).

Because _d^SEP(T_i (Ei=Pi)) = 0, the first-order deri- vative of equation (6) is

W__i(f_i) =∂Wi

∂f_i = _Y_Ti(f_i) + _Y_fi(f_i)

= 3hD₂(f_i)² _l(fi+ B_i)

+ EijPi(fi)^2 3hD 2(fi)² a + (1 a)l_f

fi

+ 3hD₃(f_i)² _D3(f_i) Because D₂(f_i)² 1 and all parameters are positive, 3hD2(f_i)² a.0 and _Dk(f_i) 0, k = 2, 3. Thus

Y_Ti(fi) = 3hD2(fi)² _l(fi+ Bi)

+ E_ijPi(f_i)^2 3hD 2(f_i)² a

 0 Y_fi(fi) = (1 a)l_f

f_i + 3hD3(fi)² _D3(fi) 0 Hence, Y__Ti(f_i) 0, _Y_fi(f_i) 0 and W__i(f_i) = _Y_Ti(f_i) + _Y_fi(f_i) 0.

The second-order derivative of equation (6) is W€i(fi) = ∂²Wi

∂f_i² = €YTi(fi) + €Yfi(fi)

= 3hD₂(f_i) 2 _ D₂(f_i)²+ D₂(f_i) €l(f_i+ B_i) + E_ij²Pi(f_i)^3 2a  6hD 2(f_i)²

 (1  a)l_f f_i² + 3hD3(fi) 2 _ D3(fi)²+ D3(fi) €D3(fi)

where D€₂(f_i) =€l(f_i+ B_i) 2E_i j² P_i(f_i)^3 and D€₃(f_i) = (fi+P

p2S

P

(p, q)2C(j, i)f_q^p)^2.

Since D₂(f_i)² 1 and all parameters are positive, 2a 6hD2(f_i)²\0 and €D_k(f_i) 0, k = 2, 3. Thus

Y€Ti(f_i) = 3hD₂(f_i) 2 _ D2(f_i)²+ D₂(f_i) €l(fi+ B_i) + E_ij²P_i(f_i)^3 2a  6hD 2(f_i)²

 0 Y€fi(fi) =  (1  a)l_f

f_i² + 3hD3(fi)

 2 _D 3(fi)²+ D3(fi) €D3(fi)

 0

Hence, €YTi(fi) 0, €Yfi(fi) 0 and €Wi(fi) = €YTi(fi) + Y€fi(fi) 0. _Wi(fi) will converge to 0. W_i(f_i^) = 0 and W_i(fi).0 for all fi6¼ f_i^. In equation (6), Wi(fi) increases with iterations and Wi(f_i^) gets the optimum of equa- tion (6). Therefore, the optimum of equation (6) and the optimum f_i^exist.

Considering another condition that the ith node is the sink node with infinite power, equation (6) is replaced by equation (6#). The first-order derivative of equation (6#) is

W__i⁰(f_i) =∂W⁰i

∂fi

= f_i^1+ 3hD₃(f_i)² _D₃(f_i)

Because all the parameters are positive and Dk(f_i) 0, _Dk(f_i) 0, k = 2, 3

f_i^1 0

3hD3(fi)² _D3(fi) 0 Hence, _W_i⁰(f_i) 0.

The second-order derivative of equation (6#) is W€⁰i(fi) =∂²W⁰i

∂fi2 = fi2

+ 3h D3(fi)

 2 _D ₃(f_i)²+ D₃(f_i) €D₃(f_i)

Because all the parameters are positive and D€k(f_i) 0, k = 2, 3

 fi2 0

3h D₃(f_i) 2 _ D₃(f_i)²+ D₃(f_i) €D₃(f_i)

 0 Hence, €W_i⁰(f_i) 0.

It is clear that equation (6#) has the same derivative properties as equation (6); thus, the optimum of equa- tion (6#) and the optimum fi^at the sink node also exist.

Above all, this theorem is established.

Theorem 2. Starting from any sufficiently small fi, 0 at the ith node, the proposed distributed optimization solution converges to the optimum f_i^ in finite iterations.

Proof. In fact, equation (6) could be written as W_i(f_i) = Y_Ti+ Y_fi at each sensor node. We take equa- tions (9a)–(9c) to run iterations for the optimum f_i^. The iterative interactions between the nodes would finally end with a set of optimums, at which Wi( ) is maximized and equation (6) converges.

For the terms related to node lifetime, Y_Ti(fi) 0, €YTi(fi) 0. It has _YTi(f_i^) = 0 and _YTi(fi).0 for all f_i6¼ f_i^. f_i continually increases as long as f_i6¼ f_i^ and terminates at f_i^, at which _Y_Ti(f_i^) = 0 and Y_Ti(f_i^) is maximized.

For the terms related to node throughput, Y_fi(f_i) 0, €Yfi(f_i) 0 and Yfi is a concave increasing function, whose maximum corresponds to the optimum f_i^. Whenever fi6¼ f_i^, fi, n increases to next fi, n + 1 such that Yfi(fi, n + 1).Yfi(fi, n).

When both YTiand Yfiare maximized, the distributed optimization at the ith sensor node converges to f_i^.

When the ith node is the sink node, equations (6#)–

(9#) replace equations (6)–(8) and (9a)–(9c), and the results still remain the same. Detailed proof for the sink node is omitted.

In addition, for easy comparison of the performance of PCB opportunistic routing with LEACH routing, to

be detailed in section ‘‘Simulation results,’’ we propose the following lemma.

Lemma 1. The average cost C of all the nodes is directly proportional to network throughput G_net.

Proof. According to definitions of C and G_net, we know G_net could be considered as the sum of sink node throughput f_i(i2 R), so the proportional relation between C and Gnet is the same as that between C and fi.

Differentiating C on fi, we get C(f_ i) = 1

N X

i2V

_l(f_i+ B_i) fi+ l(f_i+ B_i)

 

where all the variables and parameters are positive.

As we know l(f_i) is a concave increasing function of f_i, thus, _l(f_i+ B_i).0 and l(f_i+ B_i) 0. _C(fi).0. The average cost C of all the nodes is directly proportional to node throughput fi at the ith sink node. Hence, the average cost C of all the sensor nodes is directly pro- portional to network throughput Gnet.

Simulation results

In this section, we will evaluate the overall performance of PCB opportunistic routing scheme and the proposed distributed optimization solution. We consider static WSNs with 10 nodes randomly distributed in a square region of 50 m 3 50 m, as illustrated in Figure 1, where the 10th node is the sink, and the others are the sensor nodes. The sensor nodes follow the PCB opportunistic routing scheme to transmit sensor data to the sink node. Numerically, the values of all the related model parameters are tentatively listed in Table 1.

Convergence behavior of the proposed solution In accordance with subproblem P2a, Figure 3 shows the convergence behavior of node lifetime and node throughput in Figure 1. Specifically, Figure 3(a) illus- trates the iterations of each sensor node lifetime, and the convergence of each node throughput is shown in Figure 3(b). It can be seen that both the optimization goals can achieve optimal values after about 500 Table 1. Configuration of model parameters in the WSN.

Para. Description Value

A Capacity of the wireless shared medium 5 Mbps

N Number of nodes 10

Ei Initial power of the ith sensor node 1 MJ e_ij Transmission energy cost of link (i, j) 0.8 J/Mb j Consumption cost of the radio receiver 0.2 J/Mb l_f Adjustment coefficient of G^ln_net 1310⁶

a Weighted system parameter 0.5

m Iteration step-size 1.15

h Scalar weight of constraints’ importance 0.4

Figure 2. Topology of the changed WSN: (a) the topology of the WSN in Figure 1 changed by the locomotion of the second node and (b) the topology of the WSN in Figure 1 changed by the death of the ninth sensor node.

iterations. Based on the performance of node lifetime and node throughput optimization, it is obvious that the proposed distributed optimization solution can converge to a steady state after a relatively short period of time.

Impact of the dynamic network

In this section, we take Figure 1 and 2 as an example to study the network topology update of PCB opportunis- tic routing. Figure 2(a) and (b) shows network topol- ogy updates of PCB opportunistic routing under the condition that the node moves and the condition that the sensor node dies, respectively. Precisely, in contrast to Figure 1, the variation in Figure 2(a) is the locomo- tion of the second node, which results in its larger dis- tance to the fifth node and losing communication with the eighth node. The sensor nodes, which detect the change, inform the sink node by multi-hop transmis- sion, and then the sink node sends re-initialization mes- sages to all the sensor nodes. Re-initialization starts, where the second node builds communication with the third node. After that, the next-hop selection process brings us some changes that the first node sends data to the fifth node instead of the second node, and the second node transmits data to the sixth node rather than the eighth node. Moreover, when compared to Figure 1, the change in Figure 2(b) is the death of the ninth sensor node. It leads to that the 6th node, the 7th node, the 8th node, and the 10th node could not com- municate with the 9th node. Therefore, the fourth node becomes the fourth-level node and the seventh node is considered as the third-level node in re-initialization.

Also, it is given by the next-hop selection strategy that

the seventh node sends data to the sixth node, and the sixth node transmits data to the eighth node. To sum- marize, the performance of the network topology update in PCB opportunistic routing in Figure 2 demonstrates that PCB opportunistic routing is capable of handling the dynamic WSN.

The comparisons of re-convergence behavior of the proposed solution and the Lagrange dual approach under the dynamic network are illustrated in Figures 4 and 5. Figure 4 is the comparison under the locomotion of the node, that is, Figure 2(a), and Figure 5 is the comparison under the death of the sensor node, that is, Figure 2(b). The final values of Figure 3(a) are the ini- tial values of any sub-figure about the network lifetime in Figures 4 and 5, and the final values of Figure 3(b) are the initial values of any sub-figure about network throughput in Figures 4 and 5. Specifically, Figure 4(a) and (b) expresses iterations of each sensor node lifetime and each node throughput in the proposed solution under the locomotion of the node, respectively.

Apparently, node lifetime re-converges to a steady state after about 6000 iterations and node throughput re- converges after about 8000 iterations. However, Figure 4(c) and (d) indicates that optimization goals do not converge after 10,000 iterations in the Lagrange dual approach. Moreover, Figure 5(a) and (b) shows itera- tions of each sensor node lifetime and each node throughput in the proposed solution under the death of the sensor node, respectively. Unlike Figure 4(a) and (b), node lifetime re-converges to a steady state after about 6500 iterations in Figure 5(a) and node through- put re-converges after about 7600 iterations in Figure 5(b). The death of the ninth sensor node results in smaller node lifetime optimums and node throughput Figure 3. Convergence behavior of the proposed distributed optimization solution: (a) iterations of node lifetime and (b) node throughput.

optimums than the locomotion of the second node.

The reason is that the death of the ninth sensor node increases the burden of other sensor nodes to forward data and more power would be consumed to shorten node lifetime. The death of the ninth sensor node also decreases the paths to the sink node and this would increase the delay lowering the transmission efficiency to decrease node throughput optimums. Nevertheless, the locomotion of the second node only adjusts the amount of forwarding data of other sensor nodes and adjusts transmission paths rather than increases the burden to forward data and decreases the paths to the sink node. Thus, it is lighter than the death of the ninth sensor node that the extent of decreasing node lifetime optimums and node throughput optimums brought by the locomotion of the second node. Meanwhile, Figure 5(c) and (d) indicates that optimization goals still do

not converge after 10,000 iterations in the Lagrange dual approach. In conclusion, the results in Figures 4 and 5 demonstrate that the proposed distributed opti- mization solution outperforms the Lagrange dual approach at re-convergence under the dynamic network.

Tradeoff between network lifetime and throughput The impact of the weighted system parameter a on the tradeoff between the network lifetime maximum and network throughput maximum is illustrated in Figure 6, where Figure 6(a) and (b) shows the impact of a on the network lifetime and network throughput, respectively. We can observe that as a increases, the corresponding optimal network lifetime increases with the decrement of the optimal network throughput. On Figure 4. Re-convergence behavior of the proposed distributed optimization solution and the Lagrange dual approach for the locomotion of the node in the WSN: (a) and (b) the re-convergence process of the proposed distributed optimization solution and (c) and (d) the re-convergence process of the Lagrange dual approach.

the contrary, the network lifetime decreases and net- work throughput increases with the decrement of a.

Figure 6(c) indicates the tradeoff between the network lifetime and throughput when a varies. We can observe that when a increases to an extremely large value, for example, a = 0:99, the network lifetime approximately achieves its maximal value, since at this moment the network lifetime maximization problem is the domi- nant problem. For the same reason, when a decreases to a relatively small value, for example, a = 0:01, the original optimization problem transforms to the overall network throughput maximization problem. Finally, it should be noted that a = 0:5 in Figures 3, 4, and 7.

Impact of PCB opportunistic routing

To evaluate the impact of PCB opportunistic routing on the network lifetime, in Figure 7(a) and (c) we

investigate the performance of variance V_P of power consumption among the sensor nodes and network life- time T_net for different routing schemes, that is, PCB opportunistic routing, LEACH-R routing,⁶ EL- LEACH routing,⁹and LEACH routing. The compari- sons of average cost C of the nodes and network throughput G_netin these four routings are illustrated in Figure 7(b) and (d) for analyzing the influence of PCB opportunistic routing on network throughput. Note that the proposed distributed optimization solution is implemented in Figure 7.

Smaller V_P means more equal power consumption among the sensor nodes that brings larger T_net. It is seen that PCB opportunistic routing obviously outper- forms EL-LEACH routing, LEACH-R routing, and LEACH routing at V_P and Tnet. Precisely, V_P in PCB opportunistic routing is 33% smaller than EL-LEACH routing, 49% smaller than LEACH-R routing, and Figure 5. Re-convergence behavior of the proposed distributed optimization solution and the Lagrange dual approach for the death of the sensor node in the WSN: (a) and (b) the re-convergence process of the proposed distributed optimization solution and (c) and (d) the re-convergence process of the Lagrange dual approach.

67% smaller than LEACH routing. T_netof PCB oppor- tunistic routing is also 25% larger than EL-LEACH routing, 43% larger than LEACH-R routing, and 78%

larger than LEACH routing regardless of the network scale. Moreover, as the network scale increases, T_net becomes smaller with the increment of VP. However, no matter what the network scale is, larger C leads to higher G_net, which conforms to Lemma 1. G_net of PCB opportunistic routing is only 3% lower than LEACH routing since C in PCB opportunistic routing stays 5%

smaller than LEACH routing. On the other hand, Gnet

of PCB opportunistic routing is 18% higher than EL- LEACH routing and 11% higher than LEACH-R rout- ing. C in PCB opportunistic routing is 20% larger than EL-LEACH routing and 14% larger than LEACH-R routing. In addition, as the network scale increases, Gnetbecomes higher with the increment of C.

Impact of network scale

To evaluate the impact of the network scale, we vary the size of networks and accordingly realize PCB opportunistic routing and the proposed distributed optimization solution over larger networks, where 20 and 30 nodes are randomly located in square regions of 80 m 3 80 m and 100 m 3 100 m, respectively. In addition, the comparison of the achieved network life- time and throughput on different network scales is shown in Figure 7(c) and (d). We observe that as the network scale increases, the duty of data transmission at each sensor node becomes heavier, and thus sensor nodes will consume higher power in data transmission and reception, which results in the performance with higher network throughput and shorter network lifetime.

Figure 6. Impact of weighted system parameter a on (a) network lifetime, (b) network throughput, and (c) the tradeoff between network lifetime and network throughput.

Concluding remarks

In this article, we have investigated the tradeoff optimi- zation between the network lifetime and throughput for dynamic WSNs. By the PCB opportunistic routing scheme, which could cope with the dynamic network and aims to prolong the network lifetime, and the pro- posed distributed optimization solution, which can tackle the changing network, we solve the tradeoff topic in a fully distributed manner. The convergence of the proposed solution was mathematically proved and its computational complexity was analyzed as well.

Extensive numerical and simulation experiments evalu- ated the effectiveness of the proposed PCB opportunis- tic routing scheme on dealing with the varying network

and achieving longer network lifetime by comparing with the existing routing schemes, and we demonstrate that the proposed distributed optimization solution can provide the best tradeoff performance in dynamic net- work settings and has better ability to adapt the chang- ing network than the Lagrange dual approach. Besides, the PCB opportunistic routing scheme and the pro- posed solution both well support different network scales.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Figure 7. Comparison of PCB opportunistic routing, EL-LEACH routing, LEACH-R routing, and LEACH routing on different network scales: (a) variance of power consumption among sensor nodes, (b) average cost of nodes, (c) network lifetime, and (d) network throughput.