This section addresses the problem of routing in wireless sensor networks.
We are concerned in particular with the scenarios where several source nodes generate a certain quantity of information that can be transmitted, in the form of packets, from sensor to sensor over the wireless communication channels, and must eventually be collected by one or several sensors serving as sink nodes. The individual task of each sensor consists of selecting, among a list of neighbouring sensors, the best targets for the information packets it receives.
Ideally, these decisions should be made by the sensors without the supervision of a central unit, i.e. based on local observations of their own environments and interactions with neighbouring sensors.
The aim of the present section is to briey explain the close link that can be established between network routing and the constrained optimisation framework. The discussion is illustrated by a series of examples in which some typical routing problems are formulated as constrained optimisation problems.
The question of solving these problems is reported to Chapters 3, 4, and 5.
2.2.1 Routing as a constrained optimisation problem
Throughout the study we focus on the routing problems that can be stated as constrained optimisation problems and solved by distributed algorithms.
Many of these problems fall into the network utility maximisation (NUM) framework, discussed in detail in Chapter 3. In the NUM framework, a con-nected7 network withn sensors is regarded as a directed graphG = (N;E) of the type depicted in Figure 2.6, where N = {1, ..., n} is the set of the sensors (also called nodes or vertices), and E is a set containing l (directed) edges.
Every edge symbolises a wireless communication channel between two sensors.
2.2. Information routing 21 1♥
2♥
3♥
4♥
❅
❅
❅
❅❅❘ x12
x13✲
❄ x14
✛
x23
✻ x24
Figure 2.6: Directed graph of a network with4 nodes and5 edges
Direct transmission of information between two sensors is thus allowed only when the sensors are connected by an edge. If i, j ∈ N are two connected sensors, the edge binding i and j is unique and can be represented by an arbitrarily ordered pair (i, j) ∈ E, where i and j are called the origin and destination of the edge, respectively.
It is assumed that every sensor of the network is able to forward the mes-sages travelling along the network. In addition, each sensor can either generate its own ow of information (in that case it is called a source node), or collect messages generated by others (sink node). Information ow variables are as-signed to the edges of the network and denoted in this text by the vector x. They symbolise the number of network packets transmitted on each edge per unit of time. The aim of the routing process is to determine the optimal values for these variables: typically the average ow of information on each edge in slow-fading networks, or situation-dependent ow policies in fast-fading envi-ronments. For a start we rst consider anycast networks supporting a single information ow of undierentiated messages, in which the origin and nal destination of the packets are unimportant provided that they travel from a source node to a sink node. As seen later in this section, the extension to multicommodity networks, where each message is expected to reach a specic sink node, is straightforward.
The optimal ows are those which optimise a given objective function f of the variable xthe objective function is alternately referred to as the utility function or the cost function, subject to a collection of constraints dictated by the properties of the considered network and the problem specications.
The routing problem usually features a ow conservation constraint stated as a system of linear equations of the typeAx = s, where Ais the incidence ma-trix of the directed graph, and s is a vector containing the rate of information generated or collected by each sensor. Other types of constraints can be stated either as equality or inequality constraints, and typically include edge capac-ity constraints, which x the maximum information ows supported by each edge, or limitations on the power consumption at each node. An elementary
7A graph is connected if, for every pair of nodes(i, j), one can nd a directed path fromi toj or a directed path fromj toi.
routing problem for a slow-fading network is formulated in Example 2.1 for il-lustrative purposes. Multicommodity networks are addressed in Example 2.2.
Example 2.3 discusses a variant of the formulation of the network optimisa-tion problem where only positive ow variables are considered. Constraints are also introduced for the channel capacities and the power consumption of the sensors.
Example 2.1 (Network optimisation) Consider the network depicted in Figure 2.6 with N = {1,2,3,4} and E = {(1,2),(1,3),(1,4),(2,3),(2,4)}. Hence we have n = 4 and l = 5 for this example. The ow variable x = (x12, x13, x14, x23, x24) is a vector of Rl, where xij denotes the information ow on the edge (i, j) ∈ E. By convention, xij is positive if the sensor i transmits information toj, and negative ifj transmits to i. If, fori∈N, the variablesidenotes the quantity of information generated by nodeiper unit of time with∑
i∈Nsi= 0, we can writes= (s1, s2, s3, s4), wheres∈Rn. The ow conservation constraint takes the formAx=s, where the incidence matrixA is the n×l matrix such that, for anyi ∈N and j ∈E, Aij = 1 if i is the origin node of edge j, Aij=−1 ifi is the destination node ofj, andAij = 0otherwise , i.e.
A=
⎛
⎜
⎜
⎝
1 1 1 0 0
−1 0 0 1 1 0 −1 0 −1 0 0 0 −1 0 −1
⎞
⎟
⎟
⎠
. (2.7)
The routing problem in its most simple form can then be stated as minimise
x f(x)
subject to Ax=s (2.8)
where f is the cost function of the problem. If the cost function is set to f(x) = ∑
i∈Npi(x), wherepi(x)models the average power consumption of the sensori under the ow policyx(see also Example 2.3), then any solution of (2.8) minimises the total power consumption of the network.
Example 2.2 (Multicommodity ows) The problem (2.8) easily extends to multicommodity ows. Suppose that every message is given one specic destination (symbolised by a `commod-ity') among the sensors of the network, and denote by{1, ...,¯k}the set of possible destinations, and by sk = (sk1, ..., skn) the vector containing the rates of information generated by each node for the commodity k∈ {1, ...,¯k}. The basic routing problem is to nd thek¯ ows x1, ..., x¯k which optimise a given cost function of these ows under the ow conservation constraints for all the commodities,
i.e. minimise
x1,...,x¯k
f(x1, ..., x¯k)
subject to Axk =sk, k= 1, ...,¯k. (2.9) Example 2.3 (Network optimisation with nonnegative ows) In this example a version of (2.8) is formulated where the ow variable on any edge (i, j) ∈ E is duplicated into two non-negative terms ⃗xij and ji⃗x corresponding to opposite directions of transmission on(i, j). Since the ow vectorx= (⃗x, ⃗x)is now composed of⃗x= (⃗x12, ⃗x13, ⃗x14, ⃗x23, ⃗x24)and ⃗x= ( ⃗x21, ⃗x31, ⃗x41, ⃗x32, ⃗x42), the basic routing problem rewrites as
minimise
x≥0 f(x)
subject to A(⃗x− ⃗x) =s (2.10)
where the incidence matrix A was already dened in (2.7). The form of (2.10) is conducive to introducing additional constraints. A limitation on the capacity of each communication channel, for instance, is obtained by adding the constraint
x≤x,¯ (2.11)
2.2. Information routing 23
where x¯ is a positive constant vector with 2l components. In this example, the channel capacity vector x¯ is assumed to account for possible retransmissions due to packet losses, and thus models the maximum eective goodputs of the communication channels. Ifpi(x)denotes the expected power consumption of the sensor i ∈ N and p¯i a positive constant, constraints on the expected power consumption of the sensors can be stated as
pi(x)≤p¯i, ∀i∈N. (2.12)
One specicity of dealing with nonnegative ow variables is the possibility to use ane models for the total power consumption of the sensors. Assuming that the bit rate is xed for each packet transmission, a linear power consumption model is given by
pi(x) =∑
j:(i,j)∈E(¯eiij⃗xij+ ¯eiji ji⃗x ) +∑
j:(j,i)∈E(¯eiji⃗xji+ ¯eiij ij⃗x ), (2.13) where¯eiij is an estimate of the quantity of energy assumed in this model constant and independent of the ow variables consumed by the sensoriduring the transmission of one packet of information from i to j, and ¯eiji the expected quantity of energy consumed by i by transmission of one packet fromj toi.
The objective function is specied by the user and reects the purposes of the routing process. In wireless routing problems, it is usual to minimise the average power consumption of the sensors by settingf(x)to the expected total power consumption of the sensors under the ow policy x. Since in wireless network the sensors have limited battery supplies, routing policies where some sensors are overexploited with respect to their initial resources will lead these sensors to quickly run out of energy causing the routing policy to break down prematurely. An optimisation criterion more relevant to wireless networks is to maximise the lifetime of the network. It is convenient to dene the network lifetime as the time when a rst sensor has consumed all its energy supply8. For any sensor i ∈ N, let bi denote the initial energy supply of i, and pi(x) the expected power consumption of the sensor i ∈ N under the routing policy x. Under the assumption that the initial energy levels of the nodes are large compared to the energy required for the transmission of one packet, the network lifetime under x can be approximated by
tnet(x) ≈ inf
i∈N
bi
pi(x). (2.14)
It is explained in Example 2.4 how the problem of the maximisation oftnet can be stated as a constrained optimisation problem.
Example 2.4 (Network lifetime maximisation) The maximisation of the network lifetime ac-cording to (2.14) is facilitated by considering the `inverse lifetime' of the network, equal to1/tnet(x) underxand denoted by the nonnegative variable ˘x. The objective function then reduces tof(x) = ˘x
8More elaborate dentitions of the network lifetime have been proposed. In [ML06], for instance, the lifetime of the network is generalised to a function of the individual lifetimes of the sensors. The same study shows how the present framework can be extended to multicommodity ows or take packet losses into account.
and (2.14) translates into the system of inequalities 1/˘x ≤ bi/pi(x) for i ∈ N. The lifetime-maximisation routing problem is thus
minimise
x,˘x≥0 x˘
subject to Ax=s
pi(x)−bix˘≤0, ∀i∈N
(2.15) which can be solved using the techniques discussed in Chapter 3.
Ideally, the objective function and the constraints are formulated and ar-ranged so as to satisfy useful mathematical properties that make the opti-misation problem easier to solve. These properties, which include convexity and additive separability for the objective function and convexity and sparse dependency structure for the constraint functions, will be discussed in detail in Chapter 3.
2.2.2 Routing as a stochastic optimisation problem
Since the optimisation framework discussed in Section 2.2.1 considers envi-ronments with properties remaining constant over relatively long periods of time (slow fading), it proves inadequate for the wireless networks with quickly changing properties. These changes are typically due to variations in the quality of the transmission paths, as expected in the presence of fast fading communication channels, and uctuations in the availability and performance of the sensors, which remain subject to failures, or might be destined to cycli-cally enter sleeping phases so as to spare their energy resources. The networks with time-varying parameters are sometimes referred to as stochastic networks due to the random and often unpredictable nature of their characteristics.
In this section, we provide a few illustrative examples of basic optimisation problems inherent to stochastic networks, all concerned with optimal routing and ow allocation. The methods for solving stochastic optimisation problems will be discussed in Chapter 5.
The stochastic optimisation problem formulated in Example 2.5 is an ex-tension to stochastic environments of the basic network optimisation problem (Example 2.1). The varying properties of the network are assumed to be functions of an abstract random parameter ω, which embodies all the possi-ble changes in the network properties. The (nite or innite) set of possipossi-ble values for ω is denoted by Ω. No further assumption is made at this point on the probability distribution of ω, which can be known to the user or not.
Example 2.5 (Stochastic network optimisation) Consider a stochastic network with n sen-sors, l edges and directed graph G = (N;E). Assume that the properties of the network vary in function of some random parameterω taking its values in a setΩ. A basic routing problem is given
by minimise
x∈X E[f(x(ω), ω)]
subject to E[Ax(ω)−s(ω)] = 0 (2.16)
2.2. Information routing 25
whereA is the incidence matrix,f(·, ω)is a cost function, and s(ω)the vector containing the rate of generated information of each sensor. Notice that these last two quantities are dependent on the random parameterω∈Ω. ByX we denote a point-to-set function of the random parameterω such thatX(ω)⊂R is the set of the possible values forx(ω).
The unknownx in Example 2.5 is a situation-dependent ow allocation policy, also modelled as a function on the random parameter set Ω. The objective of the routing problem given in (2.16) is thus to derive the policy x(ω) which minimises the expectation, with respect to the uncertainty parameter ω, of a cost function f subject to the ow conservation constraint. The side con-straint x ∈ X, or equivalently x(ω) ∈ X(ω) for all ω ∈ Ω, is an additional restriction on the range of x(ω) for every value of the parameter ω. The eect of the ow conservation constraint is to stabilise the packet queues at every sensor by guaranteeing that the expected dierence between the ow of packets transmitted by a sensor and the ow of incoming packets is equal to the mean rate of information generated by the sensor. We will see in Chap-ter 5 that convexity of the cost function and the constraints remains a desired property when designing stochastic optimisation problems as it facilitates their analysis.
In the next examples, the possible network changes are specied by the set X. Example 2.6 considers the possibility of deeply fading communica-tion channels by temporarily setting to 0 the ow variables corresponding to channels that cannot be used for transmissions.
Example 2.6 (Deep fading) The network model with constant properties given in Example 2.1 can be modied so as to take into account temporary communication failures between sensors due to deep fading. We assume that each communication channel may, at any time, either allow for transmissions or not. A switching variable ωij is associated with each edge (i, j) ∈ E, and we consider that a transmission on (i, j) is possible if ωij = 1, and forbidden if ωij = 0. The set of all possible combinations channel states is modelled by the discrete set Ω = {0,1}l. In the present network, the switching variable for the whole network can be written as the vector ω = (ω12, ω13, ω14, ω23, ω24). The routing problem is then given by (2.16), where we dene
X :ω→X(ω) ={y∈Rl|yij = 0if ωij= 0, ∀(i, j)∈E}. (2.17) The role of the side constraint x ∈ X is thus to set to zero the ow variables corresponding to channels in a state of deep fading.
The setting where every sensor, together with its communication channels, may become unavailable due to a failure or a temporary sleeping state is discussed in Example 2.7.
Example 2.7 (Sleeping node cycles) Sleeping cycles are now considered for the sensors of the network given in Example 2.1, so that each sensor is alternately starting up and shutting down at will. It is assumed that the sleeping sensors are temporarily unable to either communicate with their neighbours, or generate or collect information. To every sensor i∈N, we associate the switching variableωi which indicates ifi is awake (ωi= 1) or asleep (ωi= 0). We setΩ ={0,1}n and write
ω = (ω1, ω2, ω3, ω4) for the four-sensor network of Figure 2.6. The routing problem with sleeping node cycles is given by (2.16), whereX is redened as
X :ω→X(ω) ={y∈Rl|yij = 0if ωi= 0or ωj= 0, ∀(i, j)∈E}. (2.18) Since sleeping nodes generate no information, the function s should also satisfy, for every i∈ N, si(ω) = 0 ifωi= 0. All in all, the constraintx∈X xes to zero the rate of generated information of every sleeping sensor, and each ow variable with a sleeping sensor as a source or origin.
A more elaborate formulation of the routing problem for networks with fast fading channels is derived in Example 2.8. The problem includes constraints on the channel capacities and power consumptions of the sensors. It is assumed that the gains of the channel and noise and interference levels can be regularly assessed by the sensors, and that energy costs of the packet transmissions can be estimated accordingly on a constant bit rate basis.
Example 2.8 (Fast fading) In this example, we suppose that the considered network undergoes fast fading, and that the channel gains are constantly varying in function of some parameter ω dened on an innite set Ω, which summarises the instant properties of all the communication channels of the network. As in Example 2.3, the ow policy x(ω) = (⃗x(ω), ⃗x(ω)) is composed of a vector⃗x(ω)withl `forward' ow variables and a vector ⃗x(ω)withl `backward' ow variables, which are all dependent on the parameterω. It is assumed that a model for the power consumptions of the sensors is known. A linear transmission model, for instance, is given, fori∈N and anyω∈Ω, by
pi(x(ω), ω) = ∑
j:(i,j)∈E
[¯eiij(ω)⃗xij(ω) + ¯eiji(ω) ji⃗x (ω)] + ∑
j:(j,i)∈E
[¯eiji(ω)⃗xji(ω) + ¯eiij(ω) ij⃗x (ω)], (2.19)
where ¯eiij(ω)and e¯jij(ω) denote the quantities of energy consumed under ω by the sensors i and j during the transmission of one packet fromitoj. These quantities may be estimated by the sensors for any ω ∈ Ω via channel state acquisition. Given uctuating network parameters such as the rate of generated informations(ω), and limitations on the channel capacitiesx(ω)¯ and on the total transmission powersp¯i(ω) of the sensors (i∈N), a stochastic version of the problem discussed in Example 2.3 takes the form
minimise
x∈X E[f(x(ω), ω)]
subject to E[A(⃗x(ω)− (⃗xω))−s(ω)] = 0 (2.20) where we deneX as
X(ω) ={y∈R2l: 0≤y≤x¯(ω), pi(y, ω)≤p¯i(ω)∀i∈N}, ∀ω∈Ω, (2.21) and the power consumption pi(·, ω) underω is specied by a model of the type (2.19).
The routing problem stated in (2.20) can be further rened by introduction of additional sources of power consumption, such as standby powers or the overhead due to channel sensing.
2.2.3 Distributed sensor networks
For several reasons which include casual failures of individual sensors and ob-structed communication channels, network architectures where the activity of