**4. Routing**

Data transmission in WSNs, also referred to as the routing problem, is one of the most widely studied problems in WSN. Different to the previous section, we focus here on the main proposed models and give some analysis on their use. The models and methods used for solving routing problems in WSN can be roughly divided in two main groups. The first group includes related shortest and spanning tree models, while the second group is centered around flow models and comprises a range of different minimum cost/maximum multicommodity flow models. While abundant work relating to such problems exists for wired networks, some new challenges have appeared for wireless networks, and especially for WSNs. The nature of some of these problems can change quite radically when they are placed in a WSN context and new requirements are introduced. These requirements include sensors' energy constraints, the interference caused by the broadcast nature of transmissions over wireless links, as well as data compression, aggregation and processing constraints. For instance, in traditional formulations of the network flow problem, link capacity is a strong constraint, while in WSN this constraint is frequently supplanted by the node energy constraint. Another important difference between these two paradigms is the inclusion of the dynamic topology models and the need for distributed solutions for wireless sensor networks.

#### **4.1 Shortest Path and Spanning Tree based models**

Shortest Path Tree (SPT) and Minimum Spanning Tree (MST) remain widely used models for routing design, even in WSNs. The goal of a SPT is to find a path of minimum cost from a specified *source node* to another specified *sink node*, assuming that each edge has an associated cost. In the WSN context the edge cost usually represents the power that would be consumed by the transmitting node when sending a packet to the node at the opposite end of the edge. Distributed routing algorithms based on Dijkstra, Bellman-Ford or Chandy-Misra's distributed algorithms can thus be employed (Rodoplu & H., 1999; Yilmaz & Erciyes, 2010). One of the disadvantages of SPT is the unbalanced load between the sensors and the disparity in the energy used by them that such methods can lead to. To overcome this problem, different strategies are proposed. In (Yilmaz & Erciyes, 2010) every node can regenerate a path when a fault occurs or available energy is depleted. Other works consider edge cost to be a combination of several metrics such as residual energy, buffer size, or the number of neighboring nodes.

16 Will-be-set-by-IN-TECH

based on its local information and to identify the cover with the smallest impact in order to be part of it. Finally, there is also a communication cost corresponding to the negotiation phase where nodes attempt to obtain a stable solution. In (Cardei & Cardei, 2008; Zou & Chakrabarty, 2005) the same problem is discussed and an additional constraint imposed: each set is required to be connected with the base station. In (Cardei & Cardei, 2008) the problem is formulated as Integer Linear Programming. It is first centrally solved using ILOG CPLEX, and then via a distributed approach. In the distributed case each node needs to know not only its own coordinates but also those of the given targets and base station. The initialization phase has a considerable communication cost resulting from exchanging the list of targets that the two-hop neighbors cover, the status of every node, and the synchronization message. This initialization phase includes the creation of the cover sets, while the subsequent phase finds the relaying nodes for connecting the cover with the base station (one node in the cover

Data transmission in WSNs, also referred to as the routing problem, is one of the most widely studied problems in WSN. Different to the previous section, we focus here on the main proposed models and give some analysis on their use. The models and methods used for solving routing problems in WSN can be roughly divided in two main groups. The first group includes related shortest and spanning tree models, while the second group is centered around flow models and comprises a range of different minimum cost/maximum multicommodity flow models. While abundant work relating to such problems exists for wired networks, some new challenges have appeared for wireless networks, and especially for WSNs. The nature of some of these problems can change quite radically when they are placed in a WSN context and new requirements are introduced. These requirements include sensors' energy constraints, the interference caused by the broadcast nature of transmissions over wireless links, as well as data compression, aggregation and processing constraints. For instance, in traditional formulations of the network flow problem, link capacity is a strong constraint, while in WSN this constraint is frequently supplanted by the node energy constraint. Another important difference between these two paradigms is the inclusion of the dynamic topology

Shortest Path Tree (SPT) and Minimum Spanning Tree (MST) remain widely used models for routing design, even in WSNs. The goal of a SPT is to find a path of minimum cost from a specified *source node* to another specified *sink node*, assuming that each edge has an associated cost. In the WSN context the edge cost usually represents the power that would be consumed by the transmitting node when sending a packet to the node at the opposite end of the edge. Distributed routing algorithms based on Dijkstra, Bellman-Ford or Chandy-Misra's distributed algorithms can thus be employed (Rodoplu & H., 1999; Yilmaz & Erciyes, 2010). One of the disadvantages of SPT is the unbalanced load between the sensors and the disparity in the energy used by them that such methods can lead to. To overcome this problem, different strategies are proposed. In (Yilmaz & Erciyes, 2010) every node can regenerate a path when a fault occurs or available energy is depleted. Other works consider edge cost to be a combination of several metrics such as residual energy, buffer size, or the number of

constructs a spanning tree that includes the target set and the BS).

models and the need for distributed solutions for wireless sensor networks.

**4.1 Shortest Path and Spanning Tree based models**

**4. Routing**

neighboring nodes.

Fig. 9. Shortest path (b) and minimum spanning tree (c) for the graph shown in (a)

Going further, WSN brings new constraints which may modify the nature of the problem. For instance, many applications of WSN require that the intermediate or relay nodes aggregate the data, while the criterion used is minimizing energy consumption. For (Cristescu et al., 2006) the joint problem of data aggregation and routing is NP-hard, and their heuristic combines an MST with an SPT. Normally, in cases where there is a high aggregation coefficient, the amount of traffic increases slightly from the source to the sinks, and an MST is a good compromise. However, where the aggregation coefficient is low, routes need to be found that minimize the number of hops, and therefore an SPT should be constructed. MST is the tree structure which minimizes the sum of edge costs, and the problem is polynomial. The difference between a shortest path tree and a minimum spanning tree is shown in Fig. 9.

Minimizing the total energy consumption is, however, not enough, since some nodes deplete their energy faster than others and may cause network partition. To balance the energy consumption, one strategy is to minimize the maximum energy consumption of the nodes. This problem has been modeled by (Gagarin et al., 2009) as the minimum degree spanning tree (MDST), which is an NP-hard optimization problem. Variations of this problem are encountered in the literature, in (Erciyes et al., 2008; Huang et al., 2006). A joint routing and data aggregation problem is also discussed in (Karaki et al., 2009) for a two-tier network, and some heuristic algorithms such as GA and greedy are proposed. From a distributed perspective, adapted versions of Prim's and Kruskal's algorithms have been proposed in (Attarde et al., 2010). In the distributed versions of SPT a node need only communicate to its neighbors information concerning the cost of links. Each node decides to communicate with the node that provides the minimal cost to the base station. An ACK mechanism is needed to dictate the end of the process. It may be remarked here that almost all the above cited models lead to single path routing schemes. They have the great advantage of being simple from an implementation point of view, while their main drawback is their difficulty in embracing additional requirements, energy consumption in particular. We now present some flow-based models that can model such requirements in a suitable way.

#### **4.2 Flow-based models**

The need to include energy/capacity constraints leads naturally to the use of flow models. Particularly for the WSN, routing problems are formulated as MultiCommodity Flow Problems (MCFPs). The commodity is a source-destination pair, and we are faced with an MFCP whenever several commodities share the network resources. In an MCFP the commodities will have different sources and/or destinations, but they are bound together

∑ *j*∈*Ni*

cost of energy for every node and repeat the process.

2010), while Mehrjoo et al. (2011) proposes genetic algorithms.

versions of routing problems. All this will be in the focus of this paragraph.

**4.2.2 Enhanced flow based models**

*xij* = *maxj*∈*Ni*

 *xji*, *yi* 

Review of Optimization Problems in Wireless Sensor Networks 171

The routing problem with data aggregation for lifetime maximization in a network has been formulated by Xue et al. (2005) as a concurrent multicommodity flow problem. Here the flow constraint implies that the amount of the flow commodities transmitted from a sensor node cannot be less than the sensor's data. They propose a polynomial time approximation scheme, strongly inspired by the Garg-Konemann algorithm. In outline, their algorithm is as follows: construct the shortest path between every source and the sink, initialize a cost unit flow for every node, push the maximum possible flow along the path for every commodity, update the

As regards routing paths, the routing schemes can use several paths (in other words perform multipath routing), or a single path (single-path routing.) Although requiring routing via a single path would appear preferable for WSN, adding such a constraint to the mathematical model gives rise to NP-hard problems. Worth citing here are two approaches proposed for WSN that attempt to circumvent the computational burden of such models while providing simplicity in implementation. The first approach computes a solution involving multiple paths, but uses only one single path at a time. Hou et al. (2004) propose an algorithm to solve the problem in two phases. In the first phase a solution is found for the multipath routing problem. Consequently every node knows the set of the relaying nodes and the respective amount of information to send to them. In the second phase one node, according to some local rule, will select one of its relaying nodes and will transmit to it the whole amount of information to be sent in this round. The second approach, in stark contrast to the first approach just described where routing takes place from the sensors to the BS (i.e. flat routing), may be seen as hierarchical routing, in that it decomposes the data transmission into two levels and thus converges to a cluster-based scheme. Each cluster head (CH) receives the data from the nodes of its cluster and from the other CHs, and transmits this data to another CH in the direction of the BS. Bari et al. (2008) consider a two-tier heterogeneous network containing powerful relay nodes which form a connected network that can relay data to the BS. They formulate the optimization problem as follows: knowing the positions of sensors and relay (CH) nodes, how should the network be clustered in order to maximize its lifetime? A sensor is not obliged to transmit directly to the CH, and sensors may have different amounts of flow to transmit. The problem is formulated as a max-min LP. Because the decision variables can take only binary values (1 if the sensor belongs to a given cluster and 0 otherwise) and the flow rate variable corresponds to a number of bits, we are dealing with an ILP problem. The heuristics presented for this problem are centralized. Other centralized techniques for solving the clustering problem in WSN are based on Fuzzy Logic (FL) (Anno et al., 2007; Ran et al.,

Advances in technology and the broad range of applications for WSN have given rise to new QoS requirements and made routing a more complex matter. Interference, delay and questions of reliability may all place additional constraints and lead to more elaborate and challenging

Radio interference has a significant impact on the performance of WSN as it affects the functioning of both MAC and routing protocols, and directly affects the transmission capacity

∀*i* ∈ *N* (7)

insofar as they share the same link capacities. Regarding commodities, a WSN gives rise to either single-sink or multi-sink models, and in the case of single-sink models all commodities will have the same extremity, namely the base station. In the following subsection 4.2.1 we discuss some basic versions of flow models used for routing path calculation in WSN. Then, in subsection 4.2.2 some further extended routing problems are presented.

#### **4.2.1 Conventional flow models in WSN**

A standard flow problem in WSN (regardless of whether it is a multicommodity flow problem) includes two type of constraints, namely the flow conservation constraint and the energy constraint.

$$\sum\_{j \in \mathcal{N}\_l} x\_{\vec{\imath}j}(t) = \sum\_{j \in \mathcal{N}\_l} x\_{\vec{\imath}i}(t) + y\_{\vec{\imath}}(t) \qquad \forall i \in \mathcal{N}\_\prime \,\forall j \in \mathcal{N}\_{\vec{\imath}\prime} \,\forall t \in T\_\prime \tag{5}$$

$$\sum\_{\mathbf{i}\in T} \sum\_{\mathbf{j}\in N\_{\mathbf{i}}} \mathbf{x}\_{\mathbf{i}\mathbf{j}}(T) \* \mathbf{e}\_{\mathbf{i}\mathbf{j}} \le E\_{\mathbf{i}} \tag{6}$$

where *t* (respectively *T*) is a time instance (respectively the network lifetime), *N* the set of sensors, *Ni* the set of neighboring nodes of *i*, *xij* the flow over the edge *ij* (that is to say the data transmitted over this link), *yi* the data generated by node *i*, *eij* the energy consumed in transmitting a unit flow and *Ei* the initial energy of the sensor. The flow conservation constraint, Equation (5), shows that the total amount of flow that a sensor receives plus the amount of data that it generates is equal to the amount of information that it transmits. The second constraint given in Equation (6) is the capacity constraint, which is related to energy. This constraint implies that the energy consumed by a sensor for transmitting the flow throughout the lifetime of the network must be less than its initial energy. In standard network flow problems this constraint is usually related to link capacity.

One of the first works to formulate this problem in terms of Integer Linear Programming is to be found in (Chang & Tassiulas, 2004). The flow is represented here by the number of packets and the transmission energy is calculated based on the distance between the nodes (and hence assuming a power control mechanism). The optimal solution of this problem gives an upper bound for network lifetime. While the problem of lifetime or flow maximization under these constraints can be solved in polynomial time for continuous values of flow *x*, the integer version is shown to be strongly NP-hard in Bodlaender et al. (2010). The distributed version of this problem is discussed in (Madan & Lall, 2006), where the subgradient algorithm is used to solve the problem. At each iteration the algorithm estimates the gradient value at a given point of the objective function and determines the next point to be considered, until the optimum is reached. The distributed implementation of this algorithm requires that every node keeps track of two variables, namely the *flow rate* of every outgoing link and the *network lifetime*. These variables are updated during each iteration of the algorithm based on their previous values and the subgradient function values (also a function of flow rates and network lifetime) are calculated according the information received from neighbor nodes. Subgradient methods are also used by Rabbat & Nowak (2004) as convenient tools for designing a distributed approach in sensor networks. Another characteristic of WSNs is the data aggregation applied by nodes. This phenomenon can easily be taken into account by slightly modifying the conservation flow constraint. For instance, in Cheng et al. (2009) each node sends the maximum amount of information between the received and the generated data set as in Equation (7).

18 Will-be-set-by-IN-TECH

insofar as they share the same link capacities. Regarding commodities, a WSN gives rise to either single-sink or multi-sink models, and in the case of single-sink models all commodities will have the same extremity, namely the base station. In the following subsection 4.2.1 we discuss some basic versions of flow models used for routing path calculation in WSN. Then,

A standard flow problem in WSN (regardless of whether it is a multicommodity flow problem) includes two type of constraints, namely the flow conservation constraint and the energy

where *t* (respectively *T*) is a time instance (respectively the network lifetime), *N* the set of sensors, *Ni* the set of neighboring nodes of *i*, *xij* the flow over the edge *ij* (that is to say the data transmitted over this link), *yi* the data generated by node *i*, *eij* the energy consumed in transmitting a unit flow and *Ei* the initial energy of the sensor. The flow conservation constraint, Equation (5), shows that the total amount of flow that a sensor receives plus the amount of data that it generates is equal to the amount of information that it transmits. The second constraint given in Equation (6) is the capacity constraint, which is related to energy. This constraint implies that the energy consumed by a sensor for transmitting the flow throughout the lifetime of the network must be less than its initial energy. In standard

One of the first works to formulate this problem in terms of Integer Linear Programming is to be found in (Chang & Tassiulas, 2004). The flow is represented here by the number of packets and the transmission energy is calculated based on the distance between the nodes (and hence assuming a power control mechanism). The optimal solution of this problem gives an upper bound for network lifetime. While the problem of lifetime or flow maximization under these constraints can be solved in polynomial time for continuous values of flow *x*, the integer version is shown to be strongly NP-hard in Bodlaender et al. (2010). The distributed version of this problem is discussed in (Madan & Lall, 2006), where the subgradient algorithm is used to solve the problem. At each iteration the algorithm estimates the gradient value at a given point of the objective function and determines the next point to be considered, until the optimum is reached. The distributed implementation of this algorithm requires that every node keeps track of two variables, namely the *flow rate* of every outgoing link and the *network lifetime*. These variables are updated during each iteration of the algorithm based on their previous values and the subgradient function values (also a function of flow rates and network lifetime) are calculated according the information received from neighbor nodes. Subgradient methods are also used by Rabbat & Nowak (2004) as convenient tools for designing a distributed approach in sensor networks. Another characteristic of WSNs is the data aggregation applied by nodes. This phenomenon can easily be taken into account by slightly modifying the conservation flow constraint. For instance, in Cheng et al. (2009) each node sends the maximum amount of information between the received and the generated

*xji*(*t*) + *yi*(*t*) ∀*i* ∈ *N*, ∀*j* ∈ *Ni*, ∀*t* ∈ *T*, (5)

*xij*(*T*) ∗ *eij* ≤ *Ei* ∀*i* ∈ *N*, ∀*j* ∈ *Ni*, (6)

in subsection 4.2.2 some further extended routing problems are presented.

**4.2.1 Conventional flow models in WSN**

∑ *j*∈*Ni*

∑ *t*∈*T* ∑ *j*∈*Ni*

data set as in Equation (7).

*xij*(*t*) = ∑

*j*∈*Ni*

network flow problems this constraint is usually related to link capacity.

constraint.

$$\sum\_{j \in N\_l} x\_{ij} = \max\_{j \in N\_l} \left\{ x\_{j i \prime} y\_i \right\} \qquad \forall i \in N \tag{7}$$

The routing problem with data aggregation for lifetime maximization in a network has been formulated by Xue et al. (2005) as a concurrent multicommodity flow problem. Here the flow constraint implies that the amount of the flow commodities transmitted from a sensor node cannot be less than the sensor's data. They propose a polynomial time approximation scheme, strongly inspired by the Garg-Konemann algorithm. In outline, their algorithm is as follows: construct the shortest path between every source and the sink, initialize a cost unit flow for every node, push the maximum possible flow along the path for every commodity, update the cost of energy for every node and repeat the process.

As regards routing paths, the routing schemes can use several paths (in other words perform multipath routing), or a single path (single-path routing.) Although requiring routing via a single path would appear preferable for WSN, adding such a constraint to the mathematical model gives rise to NP-hard problems. Worth citing here are two approaches proposed for WSN that attempt to circumvent the computational burden of such models while providing simplicity in implementation. The first approach computes a solution involving multiple paths, but uses only one single path at a time. Hou et al. (2004) propose an algorithm to solve the problem in two phases. In the first phase a solution is found for the multipath routing problem. Consequently every node knows the set of the relaying nodes and the respective amount of information to send to them. In the second phase one node, according to some local rule, will select one of its relaying nodes and will transmit to it the whole amount of information to be sent in this round. The second approach, in stark contrast to the first approach just described where routing takes place from the sensors to the BS (i.e. flat routing), may be seen as hierarchical routing, in that it decomposes the data transmission into two levels and thus converges to a cluster-based scheme. Each cluster head (CH) receives the data from the nodes of its cluster and from the other CHs, and transmits this data to another CH in the direction of the BS. Bari et al. (2008) consider a two-tier heterogeneous network containing powerful relay nodes which form a connected network that can relay data to the BS. They formulate the optimization problem as follows: knowing the positions of sensors and relay (CH) nodes, how should the network be clustered in order to maximize its lifetime? A sensor is not obliged to transmit directly to the CH, and sensors may have different amounts of flow to transmit. The problem is formulated as a max-min LP. Because the decision variables can take only binary values (1 if the sensor belongs to a given cluster and 0 otherwise) and the flow rate variable corresponds to a number of bits, we are dealing with an ILP problem. The heuristics presented for this problem are centralized. Other centralized techniques for solving the clustering problem in WSN are based on Fuzzy Logic (FL) (Anno et al., 2007; Ran et al., 2010), while Mehrjoo et al. (2011) proposes genetic algorithms.

#### **4.2.2 Enhanced flow based models**

Advances in technology and the broad range of applications for WSN have given rise to new QoS requirements and made routing a more complex matter. Interference, delay and questions of reliability may all place additional constraints and lead to more elaborate and challenging versions of routing problems. All this will be in the focus of this paragraph.

Radio interference has a significant impact on the performance of WSN as it affects the functioning of both MAC and routing protocols, and directly affects the transmission capacity

Routing under the physical interference model is more complex. Wang et al. (2011) discuss a link scheduling problem where flow capacities are satisfied and the time taken for scheduling is minimized. In this case the channel capacity is variable over time due to SINR, and its

Review of Optimization Problems in Wireless Sensor Networks 173

where *Cij*(*t*) is the channel service of link (*i*, *j*) during time *t*, and *B* is the channel bandwidth.

where *SINRij* is the *SINR* parameter for the link (*i*, *j*), *ωij* the gain of the fading channel for the link *ij*, *Pi* the power transmission of node *i*, *ωkjPk* measures the interference of the other links over the link (*ij*) and *Na* is the floor noise which is a constant. The channel service calculated in each time slot is used as parameter to bound the link data rate. The problem is

Interference can be more easily modeled in a protocol context. Wang et al. (2008) study the routing problem in the presence of interference by scheduling the nodes in accordance with the TDMA approach. The constraint added for the interference implies that the sum of the number of times a link is scheduled plus the sum of the number of times that all the links in its interference zone are scheduled in the time frame has to be smaller than the frame size, as

<sup>∑</sup>*k*∈*V*+/{*i*}(*ωkj*(*t*)*Pk*) + *Na*

*B* · *log*(1 + *SINRij*(*τ*))*dτ* (9)

) ≤ *S* (11)

(10)

integral gives the service provided by the channel as expressed in Equation (9).

0

*SINRij* <sup>=</sup> *<sup>ω</sup>ij*(*t*)*Pi*

*N*(*e*) + ∑

*e*�∈*I*(*e*)

where *N*(*e*) is the number of times that the edge *e* is scheduled in the time frame, *I*(*e*) is the subset of links of the original graph that can be influenced from *e* transmissions and *S* is the

We shall now focus on how WSN takes some QoS requirements and their associated metrics into consideration. We begin with a discussion of QoS metrics and the computational complexity that they introduce. Different metrics have different composition rules. Metrics such as delay, delay jitter and cost are additive (an additive metric is a metric which obeys the additive rule, meaning that the path metric is equal to the sum of the metric links that compose the relevant path). A multiplicative metric is a metric which obeys the multiplicative rule, meaning that the path metric is equal to the product of the link metric for all the links that compose the relevant path. Metrics like reliability (the probability that the transmission was successful) can thus be seen to be multiplicative. Finally, concave metrics obey the concave rule, meaning that the path metric is equal to the minimum (or maximum) link metric for all the links that compose the relevant path. Bandwidth is an example of a concave metric. Fig. 10 illustrates the concept of multicommodity flows in a graph and QoS multipath routing with

In (Wang & Crowcroft, 1996) it is shown that the problem of finding a path which satisfies *N* additive metrics, and/or *K* multiplicative metrics (where *N* and *K* are positive integers) is NP-hard, while it becomes polynomial when one is concave and the other additive or

*N*(*e* �

*Cij*(*t*) = *<sup>t</sup>*

solved off-line using the column generation method.

in Equation (11).

two metrics.

multiplicative.

number of time slots in the frame.

of links. In contrast to traditional networks where the capacity of links is determined by physical parameters only, in wireless communications radio interference strongly affects the transmission capacity of links that are located close to one another. The models we have cited above assume that the quantity of information generated is sufficiently low, or the channel capacity sufficiently high, for transmission capacity not to be an issue. But this assumption clearly does not always hold, and capacity constraints over links are sometimes unavoidable. It should be noted that IEEE 802.15.4 defines data rates of 20, 40, or 250 Kb/s for the physical layers. Channel capacity may therefore represent a strong constraint where huge amounts of data need to be transmitted, or when many sources have to transmit simultaneously. Interference needs to be taken into account because of the high bit error rates that it may cause. The capacity of wireless channels is calculated from the Shannon-Hartley formula given in Equation (8).

$$\mathbf{C} = \mathbf{B} \cdot \log\_2 \left( 1 + \frac{\mathbf{S}}{N} \right) \tag{8}$$

where *C* is the channel capacity (in bits per second), *B* the channel bandwidth (Hz) and *S*/*N* the signal-to-noise ratio.

From the point of view of computational complexity, including this constraint in the model makes the problem NP-hard, as shown in (Jain et al., 2003). More precisely, they show that the problem of finding a maximal flow for a source-destination pair under the interference constraint is equivalent to the Minimum Independent Set problem in a graph, and therefore NP-hard.

Krishnamachari & Ordonez (2003) add the link capacity constraint to the basic version of the flow problem with the goal of maximizing the throughput or minimizing the overall energy consumption. To ensure that the solution will not generate scenarios in which the traffic load is unfair for the nodes in the network, the flow transmitted by a node has to be less than a given fraction of the total flow generated by the network. Patel et al. (2006) add the following two constraints to the basic version of the routing problem: (i) the link capacity constraint where the rate (the number of packets per unit time) at each link has to be smaller than its capacity, and (ii) the node capacity constraint where the number of packets that a node can process in a unit time has to be smaller than its given capacity. The proposed algorithm is centralized and aims to find a maximum flow with the smallest possible energy cost. It is a kind of combination of maximum flow (getting as much flow as possible from the source to the sink) and shortest path (traveling from the source to the sink with minimum cost). The problem addressed in (Xu et al., 2008) has the same structure as that found in Patel et al. (2006), but the objective is utility maximization, which is a nonlinear convex function of the transmission rate. The problem is solved using the Lagrangian method. This method attempts to decompose the problem into a number of sub-problems via a Lagrange multiplier and to solve each of them separately. In these problems it is assumed that the bandwidth *B* is shared between different node channels, or that the nodes use the whole bandwidth but are already scheduled in order to avoid interference.

There are two possible ways of modeling a successful transmission in the presence of interference: i) the physical context, which requires that the Signal-to-Interference and Noise Ratio (SINR) given in Equation (10) exceeds a certain threshold; ii) the protocol context, where no two neighboring nodes may transmit at the same time.

20 Will-be-set-by-IN-TECH

of links. In contrast to traditional networks where the capacity of links is determined by physical parameters only, in wireless communications radio interference strongly affects the transmission capacity of links that are located close to one another. The models we have cited above assume that the quantity of information generated is sufficiently low, or the channel capacity sufficiently high, for transmission capacity not to be an issue. But this assumption clearly does not always hold, and capacity constraints over links are sometimes unavoidable. It should be noted that IEEE 802.15.4 defines data rates of 20, 40, or 250 Kb/s for the physical layers. Channel capacity may therefore represent a strong constraint where huge amounts of data need to be transmitted, or when many sources have to transmit simultaneously. Interference needs to be taken into account because of the high bit error rates that it may cause. The capacity of wireless channels is calculated from the Shannon-Hartley formula given in

*C* = *B* · log2

 1 + *S N* 

where *C* is the channel capacity (in bits per second), *B* the channel bandwidth (Hz) and *S*/*N*

From the point of view of computational complexity, including this constraint in the model makes the problem NP-hard, as shown in (Jain et al., 2003). More precisely, they show that the problem of finding a maximal flow for a source-destination pair under the interference constraint is equivalent to the Minimum Independent Set problem in a graph, and therefore

Krishnamachari & Ordonez (2003) add the link capacity constraint to the basic version of the flow problem with the goal of maximizing the throughput or minimizing the overall energy consumption. To ensure that the solution will not generate scenarios in which the traffic load is unfair for the nodes in the network, the flow transmitted by a node has to be less than a given fraction of the total flow generated by the network. Patel et al. (2006) add the following two constraints to the basic version of the routing problem: (i) the link capacity constraint where the rate (the number of packets per unit time) at each link has to be smaller than its capacity, and (ii) the node capacity constraint where the number of packets that a node can process in a unit time has to be smaller than its given capacity. The proposed algorithm is centralized and aims to find a maximum flow with the smallest possible energy cost. It is a kind of combination of maximum flow (getting as much flow as possible from the source to the sink) and shortest path (traveling from the source to the sink with minimum cost). The problem addressed in (Xu et al., 2008) has the same structure as that found in Patel et al. (2006), but the objective is utility maximization, which is a nonlinear convex function of the transmission rate. The problem is solved using the Lagrangian method. This method attempts to decompose the problem into a number of sub-problems via a Lagrange multiplier and to solve each of them separately. In these problems it is assumed that the bandwidth *B* is shared between different node channels, or that the nodes use the whole bandwidth but are already

There are two possible ways of modeling a successful transmission in the presence of interference: i) the physical context, which requires that the Signal-to-Interference and Noise Ratio (SINR) given in Equation (10) exceeds a certain threshold; ii) the protocol context, where

(8)

Equation (8).

NP-hard.

the signal-to-noise ratio.

scheduled in order to avoid interference.

no two neighboring nodes may transmit at the same time.

Routing under the physical interference model is more complex. Wang et al. (2011) discuss a link scheduling problem where flow capacities are satisfied and the time taken for scheduling is minimized. In this case the channel capacity is variable over time due to SINR, and its integral gives the service provided by the channel as expressed in Equation (9).

$$\mathbb{C}\_{ij}(t) = \int\_0^t \mathbb{B} \cdot \log(1 + SINR\_{ij}(\tau)) d\tau \tag{9}$$

where *Cij*(*t*) is the channel service of link (*i*, *j*) during time *t*, and *B* is the channel bandwidth.

$$SNR\_{ij} = \frac{\omega\_{ij}(t)P\_l}{\sum\_{k \in V^+ / \{i\}} (\omega\_{kj}(t)P\_k) + N\_a} \tag{10}$$

where *SINRij* is the *SINR* parameter for the link (*i*, *j*), *ωij* the gain of the fading channel for the link *ij*, *Pi* the power transmission of node *i*, *ωkjPk* measures the interference of the other links over the link (*ij*) and *Na* is the floor noise which is a constant. The channel service calculated in each time slot is used as parameter to bound the link data rate. The problem is solved off-line using the column generation method.

Interference can be more easily modeled in a protocol context. Wang et al. (2008) study the routing problem in the presence of interference by scheduling the nodes in accordance with the TDMA approach. The constraint added for the interference implies that the sum of the number of times a link is scheduled plus the sum of the number of times that all the links in its interference zone are scheduled in the time frame has to be smaller than the frame size, as in Equation (11).

$$N(e) + \sum\_{e' \in I(e)} N(e') \le S \tag{11}$$

where *N*(*e*) is the number of times that the edge *e* is scheduled in the time frame, *I*(*e*) is the subset of links of the original graph that can be influenced from *e* transmissions and *S* is the number of time slots in the frame.

We shall now focus on how WSN takes some QoS requirements and their associated metrics into consideration. We begin with a discussion of QoS metrics and the computational complexity that they introduce. Different metrics have different composition rules. Metrics such as delay, delay jitter and cost are additive (an additive metric is a metric which obeys the additive rule, meaning that the path metric is equal to the sum of the metric links that compose the relevant path). A multiplicative metric is a metric which obeys the multiplicative rule, meaning that the path metric is equal to the product of the link metric for all the links that compose the relevant path. Metrics like reliability (the probability that the transmission was successful) can thus be seen to be multiplicative. Finally, concave metrics obey the concave rule, meaning that the path metric is equal to the minimum (or maximum) link metric for all the links that compose the relevant path. Bandwidth is an example of a concave metric. Fig. 10 illustrates the concept of multicommodity flows in a graph and QoS multipath routing with two metrics.

In (Wang & Crowcroft, 1996) it is shown that the problem of finding a path which satisfies *N* additive metrics, and/or *K* multiplicative metrics (where *N* and *K* are positive integers) is NP-hard, while it becomes polynomial when one is concave and the other additive or multiplicative.

**5. Open issues and concluding remarks**

design and network cross layer design.

optimization problem models remains a challenge.

whether these strategies might be appropriate for WSN.

addressed.

There are several issues in WSN which are still open or which have not been sufficiently

Review of Optimization Problems in Wireless Sensor Networks 175

• Dynamicity is one of the most noticeable characteristics of WSN and also one of the biggest challenges. The term covers such phenomena as node failure, link fluctuations, node attacks and mobile nodes. Many studies in routing, coverage, scheduling or topology control have attempted to find solutions where these events occur, but including them in

• We consider that scalability is an important issue which is frequently neglected when solution methods are proposed. The eventually changes in network dimensioning may sometimes require to resolve the problem or to sufficiently increase the computation time. We observe this particularly in relation to issues related to multi-sink/multicommodity

• With respect to coverage problems, there are several potential directions that have not been fully explored. These include solving the deployment problem in the presence of obstacles, taking into account the restrictions for node placement and 3*D* deployments. In routing and topology control, cooperative decision-making strategies and opportunistic approaches also need to be modeled and examined in optimization problems, since in both areas some of the problems discussed here have been successfully addressed through opportunistic approach. But not many theoretical works have been undertaken in relation to this paradigm. Many questions remain open. For instance, in what scenarios should an opportunistic approach be favored over other approaches? How close is an opportunistic approach solution likely to be to the optimal solution? Routing in opportunistic networks <sup>1</sup> adopts a people-centric approach to model the network semantics (Verdone & Fabri, 2010). This routing group is classified as sociability-based routing and has been modeled in (Yoneki et al., 2007) based on human behavior characteristics. They propose a Socio-Aware Overlay (multi-point event dissemination using an overlay constructed by closeness centrality nodes in communities) for publish/subscribe communication. It is not clear

• Another crucial issue is the difference that still exists between theoretical studies and practical implementations in WSN. Some theoretical studies have already presented models for cross-layer design, together with corresponding solutions. But many of them remain centralized and require off-line computation. We remark that in some mathematical formulations the variables are considered continuous, despite the discontinuous nature of the corresponding events such as power transmission and flow. On the other hand, algorithms or protocols implemented in real hardware or tested in simulations do not address cross-layer design. They aim at distributed and on-line computations and handle mostly simplified problems. Moreover, in these works the analyses that might yield an optimal solution are neglected, and it is difficult to grasp the problem complexity and to know whether there is room for further improvement. Combining these two approaches is far from straightforward and calls for substantial work. We see as a primary concern in

<sup>1</sup> Examples of opportunistic networks are Delay Tolerant Networks (DTN) (Pelusi et al., 2006.) or Pocket Switched Networks, VANETs, networks composed of devices such as MP3 players, mobile telephones and PDAs which can communicate with each other by Bluetooth or Wi-Fi to share data, or even wireless

sensor networks which can send data using technologies such as GSM/UMTS, WiFi, etc.

(a) multicommodity flows in a network (b) QoS multipath routing with two metrics

Fig. 10. Multicommodity and multipath Routing

Most works dealing with QoS routing in WSN are concerned either with finding (disjoint) paths for guaranteeing network resilience (fault-tolerant network), or with finding a minimal number of paths such that QoS requirements are met. We recall that the problem of finding *k* disjoint paths (edge or vertex disjoint) such that the total cost of the paths is minimized has been shown in (Li et al., 1992) to be NP-hard, even for *k* = 2 in directed graphs. Heuristics therefore provide practical approaches for solving these kinds of problems. (Okdem & Karaboga, 2009) report an approach combining ACO with a tabu search. Each source node wishing to transmit data toward the BS has to launch *n* ants (*n* corresponds to the number of data packages that the source transmits). The ant's movement is based on the probabilistic decision where the heuristic value represents the estimation of the residual energy. After all the ants have completed their journey (from source to destination), each ant *k* deposits a quantity of pheromone equal to the inverse of the total number of nodes included in the path. This task is performed by sending ant *k* back to its source node following the arrival path. In this type of ACO each receiver node has to maintain a tabu list with the identities of the ants that it has encountered, enabling it to decide whether to accept the upcoming packet of ant *k*. Routing the information efficiently to guarantee the delay and reliability constraint is discussed in Saleem et al. (2010), who proposes a multi-agent approach for ant colony optimization (ACO). The movement of the ant is guided by the probabilistic decision rule, equation (4). The pheromone value corresponds to the end-to-end delay. The two heuristic evaluation parameters of every edge are determined by the residual energy at the extremity of the edge and its packet receive rate (PRR).

In (Bagula & Mazandu, 2008) the QoS routing problem is concerned with delay and reliability criteria. The goal is to find the smallest set of disjoint paths between a source and a destination such that both criteria are satisfied and energy consumption is minimized. Delay is a stringent metric, meaning that if the delay is not respected in any of the set paths, the packet is dropped. In contrast, the reliability of every source-destination connection obeys the multiplicative composition rule. Hence the more paths in the set, the more reliable the set will be. The problems of finding the path which minimizes the energy or the delay, or maximizes the reliability, taken separately, are solvable in polynomial time, but the problem considered in its entirety is not.

### **5. Open issues and concluding remarks**

22 Will-be-set-by-IN-TECH

(a) multicommodity flows in a network (b) QoS multipath routing with two metrics

Most works dealing with QoS routing in WSN are concerned either with finding (disjoint) paths for guaranteeing network resilience (fault-tolerant network), or with finding a minimal number of paths such that QoS requirements are met. We recall that the problem of finding *k* disjoint paths (edge or vertex disjoint) such that the total cost of the paths is minimized has been shown in (Li et al., 1992) to be NP-hard, even for *k* = 2 in directed graphs. Heuristics therefore provide practical approaches for solving these kinds of problems. (Okdem & Karaboga, 2009) report an approach combining ACO with a tabu search. Each source node wishing to transmit data toward the BS has to launch *n* ants (*n* corresponds to the number of data packages that the source transmits). The ant's movement is based on the probabilistic decision where the heuristic value represents the estimation of the residual energy. After all the ants have completed their journey (from source to destination), each ant *k* deposits a quantity of pheromone equal to the inverse of the total number of nodes included in the path. This task is performed by sending ant *k* back to its source node following the arrival path. In this type of ACO each receiver node has to maintain a tabu list with the identities of the ants that it has encountered, enabling it to decide whether to accept the upcoming packet of ant *k*. Routing the information efficiently to guarantee the delay and reliability constraint is discussed in Saleem et al. (2010), who proposes a multi-agent approach for ant colony optimization (ACO). The movement of the ant is guided by the probabilistic decision rule, equation (4). The pheromone value corresponds to the end-to-end delay. The two heuristic evaluation parameters of every edge are determined by the residual energy at the extremity

In (Bagula & Mazandu, 2008) the QoS routing problem is concerned with delay and reliability criteria. The goal is to find the smallest set of disjoint paths between a source and a destination such that both criteria are satisfied and energy consumption is minimized. Delay is a stringent metric, meaning that if the delay is not respected in any of the set paths, the packet is dropped. In contrast, the reliability of every source-destination connection obeys the multiplicative composition rule. Hence the more paths in the set, the more reliable the set will be. The problems of finding the path which minimizes the energy or the delay, or maximizes the reliability, taken separately, are solvable in polynomial time, but the problem considered in

Fig. 10. Multicommodity and multipath Routing

of the edge and its packet receive rate (PRR).

its entirety is not.

There are several issues in WSN which are still open or which have not been sufficiently addressed.


<sup>1</sup> Examples of opportunistic networks are Delay Tolerant Networks (DTN) (Pelusi et al., 2006.) or Pocket Switched Networks, VANETs, networks composed of devices such as MP3 players, mobile telephones and PDAs which can communicate with each other by Bluetooth or Wi-Fi to share data, or even wireless sensor networks which can send data using technologies such as GSM/UMTS, WiFi, etc.

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To conclude, wireless sensor networks represent an attractive research area due to several factors as the resource-constrained nature of sensor nodes, interference, data aggregation, power consumption model and the wide range of both commercial and military applications that this technology offers. Successful network design and deployment include understanding and modeling several problems related to these factors, which ultimately determine the available range and data rate of a WSN, as well as cost and battery lifetime. Therefore this study, intended to researchers and graduate students in computer science and fields related to operations research, information technology and applied mathematics, gives some highlights on a number of representative network problems in WSN and focuses on their respective optimization problems.

#### **6. References**


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24 Will-be-set-by-IN-TECH

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**Part 4** 

**Telecommunications** 

