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International Journal of Distributed Sensor Networks
Volume 2012 (2012), Article ID 523787, 14 pages
Robust Maximum Lifetime Routing and Energy Allocation in Wireless Sensor Networks
1Division of Systems Engineering, Department of Electrical and Computer Engineering, Center for Information and Systems Engineering, Boston University, 8 St. Mary's Street, Boston, MA 02215, USA
2Center for Information and Systems Engineering, Boston University, Boston, MA 02215, USA
Received 28 November 2011; Accepted 10 May 2012
Academic Editor: Pedro Ruiz
Copyright © 2012 Ioannis Ch. Paschalidis and Ruomin Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider the maximum lifetime routing problem in wireless sensor networks in two settings: (a) when nodes####^~^~^~^~^~^####x2019; initial energy is given and (b) when it is subject to optimization. The optimal solution and objective value provide optimal flows and the corresponding predicted lifetime, respectively. We stipulate that there is uncertainty in various network parameters (available energy and energy depletion rates). In setting (a) we show that for specific, yet typical, network topologies, the actual network lifetime will reach the predicted value with a probability that converges to zero as the number of nodes grows large. In setting (b) the same result holds for all topologies. We develop a series of robust problem formulations, ranging from pessimistic to optimistic. A set of parameters enable the tuning of the conservatism of the formulation to obtain network flows with a desirably high probability that the corresponding lifetime prediction is achieved. We establish a number of properties for the robust network flows and energy allocations and provide numerical results to highlight the tradeoff between predicted lifetime and the probability achieved. Further, we analyze an interesting limiting regime of massively deployed sensor networks and essentially solve a continuous version of the problem.
Wireless sensor networks (WSNETs) have emerged as an exciting new paradigm of inexpensive, easily deployable, completely untethered device networks that enable the automated and intelligent monitoring and control of physical systems. WSNET nodes can be equipped with a variety of sensors, have a built-in radio to communicate with each other, are powered by batteries, and have limited information storage and processing capabilities. WSNETs can be useful in a plethora of applications including industrial and building automation, health monitoring, wildlife monitoring, and asset and personnel tracking . Battery technology, however, remains a critical bottleneck. In many applications one would like to use the WSNET for long periods, often years, without changing batteries. As a result, energy conservation is a primary concern and aggressive optimization becomes indispensable.
In this paper, we focus on the problem of selecting an optimal strategy for routing packets from data-collecting sensor nodes to a set of gateways (or sinks) in order to minimize the rate at which energy is consumed or, equivalently, to maximize the lifetime of the network. We consider two situations: (i) when the initial energy of every node is given and (ii) when it is also subject to optimization given an overall energy budget. Routing, of course, has received quite a bit of attention in WSNETs. Various aspects of the problem have been considered in [2####^~^~^~^~^~^####x2013;11], which mostly focus on finding a single path from origin to destination. A more static view is adopted in , followed by , and , which provide a linear programming formulation for optimizing average flows between nodes.
Our starting point is the flow optimizing formulation of [12, 14]. A different but equivalent formulation using optimal control ideas is in . Key data to solve this problem include the total available energy at the nodes and the energy consumption rates. These quantities are hardly known with any degree of certainty or accuracy. Yet, they affect both the optimal flows and the corresponding optimal objective value, that is, the predicted network lifetime. The latter value will in fact be equal to the actual network lifetime if all problem data are known with certainty. We note that both these quantities are quite important for the network designer. The predicted network lifetime is useful for planning purposes, and the optimal flows indicate how routing should be done to achieve such a lifetime.
Uncertainty, though, renders the predicted lifetime overly optimistic. For the case without energy allocation, we show that for specific, yet typical, topologies including linear and two-dimensional grid-like networks, the actual lifetime will reach the predicted value with a probability that converges to zero as the number of nodes grows large. This suggests that the predicted network lifetime is not a particularly useful estimate under uncertainty.
For the energy allocation case, we show the same result without any topological assumptions. We also find that uncertainty impacts the optimal policy as well, and one needs to use a different set of ####^~^~^~^~^~^####x201c;robust####^~^~^~^~^~^####x2019;####^~^~^~^~^~^####x2019; flows to protect against uncertainty. To that end, we develop a series of alternative robust problem formulations, ranging from pessimistic to optimistic. A set of parameters enable the tuning of the conservatism of the formulation with a desirably high probability that the corresponding lifetime prediction will be achieved####^~^~^~^~^~^####x2014;a lifetime guarantee probability. Our robust formulations are based on recent work in robust linear programming in [16, 17]. However, the problem we consider has special structure which we exploit to establish a number of interesting properties. Robust optimization has in general received a lot of attention lately and has found applications in many areas. It started with  with more recent contributions in [16, 19].
To gain more insight, we consider maximum lifetime routing with energy allocation in a continuous setting of massively dense WSNETs. Related limiting regimes have previously been considered in [8, 20, 21]. For a single point source and a single point sink, we show that the optimal route is a straight line from the source to the sink. For multiple sources and sinks, we show that sources send their flows to the closest sink, again over a straight line.
The rest of the paper is organized as follows. In Section 2, we tackle the maximum lifetime routing problem without energy allocation, introducing robust formulations and characterizing their solutions. Section 3 incorporates the energy allocation into the problem. In Section 4, we develop the continuous version of the problem with energy allocation. Numerical examples are in Section 5. Conclusions are in Section 6.
2. Maximum Lifetime Routing without Node Energy Allocation
We represent a WSNET as a directed graph , where is the node set and is the set of directed links with . Link exists if and only if , where is the set of nodes that can be reached by . Each node has an initial battery energy of and consumes per data unit to transmit to , while consumes per data unit to receive from . We assume that the nodes are able to relay packets and to adjust the transmit power level to the minimum required in order to reach the intended receiver. Origin nodes (or sources) include all with a positive (constant) information generation rate . is the set of sink nodes (or sinks) responsible for collecting all data. Assume ; we refer to nodes in simply as sensor nodes.
Every source node seeks to send its data to one of the sinks, not necessarily the same one for each data unit generated. To that end, node may use multiple other nodes as relays. Let be the information transmission rate from to . We write for the vector of all ####^~^~^~^~^~^####x2019;s. (We use bold letters to denote vectors and all vectors are assumed to be column vectors unless explicitly stated otherwise.) Note that routing and power control are intrinsically coupled since the power level is adjusted depending on the choice of the next hop.
In the sequel, we only consider the energy spent for communications since this is the dominant energy consumption term in WSNETs (see ). Additional energy consumption terms could be incorporated into . For example, a sensing/processing energy cost at transmissions or receptions per data unit can be incorporated into and . We also assume that is monotonically increasing with the distance between and . Finally, sink nodes are assumed to be powered by line power.
The lifetime of a sensor node under a given set of flows is given by Define the network lifetime under flow as the minimum lifetime over all nodes, that is, The network lifetime is equivalent to the earliest time, a sensor node runs out of energy.
2.1. Problem Formulations
The maximum lifetime routing problem without node energy allocation is the problem of selecting flows to maximize . Letting denote the amount of information transmitted from to over the lifetime ,  formulated the problem as a linear program as follows: where the decision variables are and the ####^~^~^~^~^~^####x2019;s. On a notational remark, we will use to denote flow over the lifetime and to denote flow per unit of time. Thus, when we refer to an optimal solution (resp. ) of (3) we mean optimal flow per unit of time (resp. over the lifetime). The first set of constraints correspond to flow conservation and the second set of constraints follows from the definition of lifetime. We note that this formulation can also account for the energy consumed while the node####^~^~^~^~^~^####x2019;s radio is listening. Specifically, we can add to the lefthand side of (5), where is the energy consumption rate by the radio while listening, is the fraction of time node and is ####^~^~^~^~^~^####x201c;awake####^~^~^~^~^~^####x2019;####^~^~^~^~^~^####x2019; and listening. We refer to (3) as the nominal problem. Note that it is always feasible if for every sensor node there exists a path to a sink node. We assume that this will always be the case. We note that problem (3) can be solved in a distributed manner using subgradient optimization techniques for the dual . This is appealing for WSNET applications. Here, however, we concentrate on the impact of uncertainty and do not focus on distributed solution approaches. It can be also argued that in several application contexts a distributed approach is not critical since (3) is solved during a planning/deployment stage of the WSNET.
The data for the nominal problem are , , and and these affect both the optimal solution and the optimal value. As these may be uncertain, we model them as symmetrically bounded nonnegative random variables (r.v.####^~^~^~^~^~^####x2019;s) with ranges given by: , , and . We will call , , and the nominal values and assume that they are the means of the corresponding r.v.####^~^~^~^~^~^####x2019;s. The values , , and represent the maximum deviations from the mean which are assumed to be identical left and right from the mean (hence, the term symmetrically bounded r.v.####^~^~^~^~^~^####x2019;s). These deviations are defined so that all r.v.####^~^~^~^~^~^####x2019;s have positive support. We also define the uncertainty sets and , for all .
Due to data uncertainty, the optimal solution of (3) may not be feasible. It can be easily seen that the following worst-case formulation guarantees feasibility for any realization of the following data: We refer to the above as the fat problem. By construction, its optimal solution is feasible for any data realization but it may be overly conservative. Intuitively, the probability that all parameters take their ####^~^~^~^~^~^####x201c;extreme####^~^~^~^~^~^####x2019;####^~^~^~^~^~^####x2019; value should be small, thus, motivating a less conservative formulation.
We introduce the uncertainty budget for every sensor node and define the restricted uncertainty set as We view the uncertainty budget as an -norm constraint for the vector Similarly, let be the uncertainty budget for , namely, . The following robust maximum lifetime routing problem is formulated so that we can guarantee feasibility for all data realizations in the following restricted uncertainty sets: In the Appendix, we show that the above is equivalent to a linear programming problem.
Theorem 1. The robust problem (11) is equivalent to the linear following programming formulation: Furthermore, solving (13) one obtains an optimal solution so that is feasible for (11) and is equal to the optimal value of (11).
2.2. Properties of Optimal Solutions
Next, we study the relationships between the three formulations and establish properties of the optimal solutions. We also introduce a metric####^~^~^~^~^~^####x2014;the lifetime guarantee probability####^~^~^~^~^~^####x2014;to quantify how likely it is for the predicted lifetime to be achieved.
2.2.1. Optimal Lifetime
Let , , denote the optimal values of the nominal, fat, and robust problems, respectively. Let and . Note that depends on and . To express this dependence, we write . The following proposition is almost immediate. It simply states that by adjusting the uncertainty budgets one can generate a continuum of formulations whose predicted lifetime ranges from the fat to the nominal.
Proposition 2. is a nonincreasing function of both and . Furthermore, .
Proof. Fix so that , and . It follows that , for all . Let be an optimal flow for the robust routing problem under . For all , we have
which suggests that is a feasible flow vector for the robust routing problem under . It follows that .
Next notice that when , , the uncertainty set becomes and the robust routing problem (11) reduces to the nominal routing problem (3), that is, .
When and for all the uncertainty sets becomes and for all , which implies that the robust routing problem (11) reduces to the fat one (7).
Standard sensitivity analysis results from linear programming yield the following corollary.
Corollary 3. is a concave function of .
Observe now that at optimality at least one of the energy constraints (5), (8), and (12) will be active. This is stated in the following proposition. We will call dead the nodes that correspond to active constraints at optimality. The lifetime of a dead node equals the lifetime of the network.
2.2.2. Optimal Flows
Consider an optimal flow vector obtained by solving one of the three formulations. Recall that denotes total flow over the lifetime and flow per unit of time. We associate a directed graph (subgraph of ) to , where contains all with . We say that a flow is acyclic(resp., cyclic) if contains no cycles (resp., otherwise).
Proof. Let be an optimal solution with forming a cycle in . Let . Subtract from all the flows on the cycle. At least one of becomes zero and all other flows remain nonnegative. Because both the inflow and outflow at each node is reduced by the same amount, the flow conservation condition for all the nodes still holds. Since the above operation only reduces flows, all the energy constraints remain satisfied. Hence, the reduced flows remain optimal. We can repeat the same process to eliminate any other cycle.
Since is a trivial cycle, we obtain the following corollary.
Proof. Let be an acyclic optimal flow (cf. Theorem 5). Suppose there are sinks with positive flows emanating from them. Let such that is not empty. For let . We reduce to zero by proportionally allocating this flow reduction to all outflows from node . To be specific, for all we set the new reduced flow as which maintains the nonnegativity of the resulting flow. The flow reduction can be propagated to the node downstream from in a similar way. Since is acyclic and the network is finite, propagating the flow reduction as described above terminates at some other sink nodes. During this process, flow conservation and energy constraints are maintained. This yields a new optimal flow vector with no flows out of sinks.
2.2.3. Lifetime Guarantee Probability
Consider one of the three formulations (3), (7), and (11) and let be an optimal solution. We will refer to the probability evaluated under the distributions of the r.v.####^~^~^~^~^~^####x2019;s , , , as the lifetime guarantee probability. This is the probability that the actual lifetime obtained by applying the optimal flow achieves the predicted optimal lifetime. We denote by the lifetime guarantee probabilities for the nominal (3), fat (7), and robust (11) formulations, respectively. By design, the fat formulation provides an ####^~^~^~^~^~^####x201c;absolute####^~^~^~^~^~^####x2019;####^~^~^~^~^~^####x2019; guarantee; we omit the proof.
Theorem 8. It holds that .
The straightforward observation is that when , , for all , then ; while when , , for all , then .
Now let be the set of nodes having active energy constraints at optimality in the nominal formulation (3). For any random variable with mean and support in , we say that it is symmetrically distributed if for all , where is the cumulative distribution function of .
Theorem 9. If are independent symmetrically distributed r.v.####^~^~^~^~^~^####x2019;s, then .
Proof. Let be an optimal solution to the nominal problem (3). We have For and because is feasible for the nominal problem it holds . Since , , are independent symmetrically distributed r.v.####^~^~^~^~^~^####x2019;s with means , , , respectively, it follows that By independence, we have .
2.3. Linear and Square Arrays
In this section, we study two regular network topologies: linear and square arrays. Linear arrays appear, for instance, in pipeline monitoring applications and square arrays are applicable in environmental monitoring applications.
2.3.1. Linear Arrays
We consider a linear array segment where one sink node is at the center and an equal number of sensor nodes are aligned one by one on both sides of the sink. The distance between neighboring nodes is . The radio range is in , that is, every node can only communicate with its very next neighbors. Lining up such multiple segments, we can build a linear array network. We grow the network in this manner since one would need a sink per given number of sensor nodes. We assume that all sensor nodes have identical characteristics, that is, has the same distribution for all , and have the same distribution among equidistant nodes, and the information generation rate is identical for all . The network we described is motivated by oil or gas pipeline monitoring applications. The following theorem establishes a decomposition property.
Theorem 10. The maximum lifetime routing problem under either the nominal (3), fat (7), or robust formulation (11) for a linear array network described above can be decomposed into the corresponding subproblems for each one of its segments.
Proof. Without loss of generality, consider a linear array network denoted by consisting of two segments and . Consider any of the three routing formulations and let , , be the optimal values for networks , , and, , respectively. Clearly, since by combining the optimal flow vectors for and we obtain a feasible flow vector for .
Due to homogeneity and symmetry in , there exists an optimal flow vector which is symmetric about the center of . Flows in the interface between the two segments and can fall into one out of two possible cases shown in Figure 1 (top). In each case, we can reconstruct the optimal flows between nodes and of and nodes and of as shown in Figure 1 (bottom). This flow reconstruction process maintains feasibility and eliminates any communication between segments and . Then . Together with our earlier observation it follows , which establishes the result.
The following theorem establishes that the nominal formulation (3) is not particularly useful since its predicted lifetime will be achieved with a diminishing probability as the size of the network increases.
Theorem 11. Consider a WSNET formed by aligning linear arrays as described before. Assume , , are i.i.d. and nondegenerate r.v.####^~^~^~^~^~^####x2019;s (i.e., not equal to a constant). Then, as , .
Proof. By applying Theorem 10####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009; times, we decompose the network into identical segments. With this decomposition, we have identical optimal flows in all linear segments. As we have seen before, each segment has at least one node with a binding energy constraint. Let denote a set which contains one node from each segment with a binding energy constraint. It follows that where the last equality follows from the fact that every corresponds to a binding energy constraint. Notice that for nondegenerate r.v.####^~^~^~^~^~^####x2019;s and that . Hence, as , .
2.3.2. Square Arrays
A square array network consists of square array segments. Each segment is a two-dimensional (square) grid of a given dimension with a node at each point in the grid and a sink node located at the center point of the grid. The vertical and horizontal distance between neighboring nodes is and we assume that the radio range is slightly less than . As with linear arrays, we assume that all sensor nodes have identical characteristics, that is, has the same distribution for all , and have the same distribution among equidistant nodes, and the information generation rate is identical for all . We grow a square network in both dimensions by stitching together segments. As an example, a network with four segments can be formed by placing segment in the northeast orthant, segment in the southeast orthant, in the southwest orthant, and in the northwest orthant. The following result is analogous to Theorem 10.
Theorem 12. Consider a network consisting of 4 segments as outlined above. The maximum lifetime routing problem under either formulation ((3), (7), or (11)) for can be decomposed into the corresponding problems for .
Proof. Fix a particular formulation, fat, nominal, or robust. Let , be the optimal values for network , , and , respectively. As in the proof of Theorem 10####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009; for all .
Due to the homogeneity and symmetry of , there exists an optimal flow vector for with no flows out of sinks which is symmetric about the vertical line that separates and . As in Theorem 10, we consider all possible cases and reconstruct the optimal flow as shown in Figure 2 (right), resulting in the new flow with no communication between and . A similar flow reconstruction process can result in a flow with no communication between and . These flow reconstruction steps maintain flow conservation and do not violate the energy constraints, so the resulting flow is optimal. It follows that for all which concludes the proof.
Analogous to the linear array case, we can now show that the nominal formulation does not provide a useful lifetime prediction. We omit the proof as it is similar to the proof of Theorem 11.
Theorem 13. Let a square network be constructed by repeating times the process of constructing from . Assume , , are i.i.d. and nondegenerate r.v.####^~^~^~^~^~^####x2019;s (i.e., not equal to a constant). Then, as , .
2.4. Uncertainty Only in
Here we focus on the case where uncertainty appears only in the initial available energy . Namely, for all results in this subsection we assume that ####^~^~^~^~^~^####x2019;s and ####^~^~^~^~^~^####x2019;s are known with certainty.
We define a global robustness budget and incorporate the allocation of to individual into the following robust formulation: where the decision variables are , the ####^~^~^~^~^~^####x2019;s, and the ####^~^~^~^~^~^####x2019;s. The following monotonicity property is immediate. Concavity follows from the fact that (19) maximizes a concave (linear) objective over linear constraints and appears in the right hand side of these constraints.
Proposition 14. The optimal value of (19) is monotonically nonincreasing and concave as a function of the global robustness budget .
2.4.1. Optimizing P over the Optimal Flows q####^~^~^~^~^~^####x2a;
When the uncertainty is only in ####^~^~^~^~^~^####x2019;s, we can maximize the lifetime guarantee probability over the set of optimal flows while guaranteeing that we achieve the corresponding predicted lifetime. One can think of this optimization as maximizing ####^~^~^~^~^~^####x201c;robustness####^~^~^~^~^~^####x2019;####^~^~^~^~^~^####x2019; while guaranteeing the same objective (predicted lifetime). We next show that this problem is a well-structured concave optimization problem. We only treat the robust case. For the fat case we have already shown that and the nominal case is similar to the robust.
Assume that only ####^~^~^~^~^~^####x2019;s are uncertain, and let , , form an optimal solution of the robust formulation (19), where denotes the vector of slack variables corresponding to the energy constraints. Suppose all ####^~^~^~^~^~^####x2019;s are independent, then Taking the ####^~^~^~^~^~^####x2019;s to be uniformly distributed in : and , where we defined .
To maximize while achieving the optimal lifetime , we can equivalently maximize which yields the following concave optimization problem:
3. Maximum Lifetime Routing with Energy Allocation
In this section, we consider the problem of maximizing the WSNET lifetime by jointly optimizing the routing decisions and the initial energy allocated to the nodes. Suppose is the total available energy for a WSNET. Similar to formulation (3) we have the nominal problem:
Here the ####^~^~^~^~^~^####x2019;s (appearing in (5) and above) are decision variables. The corresponding fat and robust formulations, respectively, are As before, the robust problem (26) can be shown to be equivalent to a linear programming problem; we omit the details for brevity. From the structure of the formulation with energy allocation, we have the following result.
Proposition 15. At optimality, all the energy constraints for nonsink nodes are active and the total energy constraint is also binding. This holds for all three formulations.
Proof . Consider first the robust problem (26). We will use contradiction. Assume that at optimality, the energy constraint (27) for some nonsink node is not active. Notice that we can decrease and increase all the other while maintaining their sum. This improves the lifetime which contradicts optimality. Similarly, the total energy constraint is also binding at optimality. If not, we can increase all to achieve a better lifetime, which again contradicts optimality. The nominal and fat cases are almost identical.
3.1. Properties of Optimal Solutions
As before, let , , denote the optimal values of the nominal, fat, and robust routing problems, respectively. Let . Note that depends on . To express this dependence, we write . The following result is similar to Proposition 2.
Proposition 16. is a nonincreasing function of####^~^~^~^~^~^####x2009;####^~^~^~^~^~^####x2009; and .
As in Section 2.2, one associates a directed graph to a feasible flow vector , where contains all with . Recall that we name as acyclic when contains no cycles. The following results are similar to Theorem 5 and Corollary 7; we omit the proofs.
Proposition 18. For all three formulations the optimal flows satisfy , for all .
3.2. Lifetime Guarantee Probability
The development in this section is similar to Section 2. We have the following results; we omit the details in the interest of brevity.
Proposition 19. It holds .
Note that when , for all , then ; while when , for all , then .
Proposition 20. If are independent symmetrically distributed r.v.####^~^~^~^~^~^####x2019;s, then .
It follows that as we have , and this now holds for all topologies.
4. Routing and Energy Allocation in Massively Dense WSNETs
It is straightforward that the joint problem of routing and energy allocation (24) is equivalent to finding paths from sources to sinks with lowest energy consumption rate. If we consider the energy consumed by both the sender and the receiver over a link as the cost (or length) of the link, the problem is reduced to finding shortest paths between sources and sinks. Imagine now that the WSNET is scaled by uniformly deploying an increasing number of nodes while decreasing their radio range in order to maintain a fixed density of one-hop-reachable neighbors. Although the approach we developed so far scales well since we are dealing with linear programming problems, it is of interest to consider whether the scaled problem exhibits, in the limit, a structure that simplifies its solution and deepens our understanding. In particular, we will consider a limiting regime of massively dense WSNETs and study maximum lifetime routing formulations with energy allocation. Such WSNETs can only be described by macroscopic parameters, such as the information generation and energy distribution densities.
4.1. Problem Formulation
Let be the planar area where a massively dense WSNET is deployed. Mathematically, is a convex set in . We assume that the WSNET is uniformly deployed over .
Let represent the information generation density function defined on whose units are . We assume is known. Denote by the information consumption density function defined on whose units are . In the next subsection we will consider the special cases of ####^~^~^~^~^~^####x201c;point####^~^~^~^~^~^####x2019;####^~^~^~^~^~^####x2019; sources and sinks where and become Dirac functions on the plane. Let be the energy density function defined on whose units are . The energy density function characterizes the distribution of the globally available energy over . Define the information traffic flow function as . The interpretation of is as follows: is the rate at which information crosses a linear segment of infinitesimal length which is centered on and perpendicular to (see Figure 3). The units of are .
The continuous maximum life routing problem with energy allocation can be formulated as: where , , , and are decision functions and variables. Using an argument in , (29) states that the divergence of the traffic flow function measures the degree with which the traffic increases or decreases; we can think of this as a detailed flow conservation equation. (31) is a global energy constraint while (32) can be seen as a global flow conservation constraint. As for (30), consider a point and let denote an infinitesimal square centered at with a side length equal to and one of its sides parallel to . Let (in ) be a constant indicating how much energy is consumed per unit of transmitted information per second. Then, (30) expresses the fact that the total energy consumed when the traffic flow passes through during a period of time should be no more than the total energy available in this area.
In this section, we are only interested in the structure of the optimal solutions to (28), hence we only consider the nominal version of the problem. Uncertainty in can be easily incorporated as we have done with the discrete instances. This will only change the right hand side of the total energy constraint and would not affect the optimal solution structure. Uncertainty in can also be incorporated but that is beyond the main focus of this section.
From the structure of (28), we have the following results. The proof is immediate as whenever and we can reduce to zero while maintaining feasibility. The energy savings can be allocated to other points resulting in a potential increase of the lifetime.
Proposition 21. (28) has optimal solutions such that whenever .
Similar to Proposition 15 we can showthe following.
4.2. Single Point Source and Sink
In this subsection, we focus on the scenario where there is a single point source and a single point sink in a massively dense WSNET. We start with the definition of a point source/sink.
Definition 23 (point source on ). Let be the location of the source on and denote by its information generation rate. The point information generation density function satisfies
Similarly, we define the information consumption density function for a point sink at with a sink rate equal to . These are Dirac impulse functions on .
In the single point source and single point sink case, let and be the source and sink locations, respectively. Denote by and the corresponding information generation/consumption density function with rates and , respectively. It follows from the global flow balance equation (32) that . We next define the notion of a marginal density function; its units are .
Definition 24 (marginal energy density function on curve ). Let be a continuous curve connecting two points and , and let denote an -tube around as shown in Figure 4. The marginal energy density function on curve satisfies
4.2.1. Properties of Optimal Solutions
Let and be the source and sink positions, respectively. Information generated at the source (with rate ) gets consumed at the sink (with rate , where it follows that ).
Consider an arbitrary set of paths (see Figure 5) traversed by the traffic as it flows from to . Specifically, the traffic flow first follows curve then forks into branches which merge into . Denote by the traffic flows on , respectively, where . We have already established that at optimality we have an energy density function that is nonzero only on the curves and all energy constraints are active. The problem reduces to where denotes the length of the curve . Note this is a linear program. Given the allocation of into an optimal lifetime is It follows that to maximize , the branches should all be straight lines (minimal length). Furthermore, the best can be achieved by a straight line from to . The result is summarized below.
Proposition 25. The path that maximizes the network####^~^~^~^~^~^####x2019;s lifetime is the straight line from to . The corresponding energy distribution function is nonzero only on this line with a uniform marginal energy density function.
One notes that the argument above can be extended to handle an infinite number of (forked and merged) paths. The key idea is the same, that is, one can show that any solution using an infinite number of paths is no better than the straight line connecting with . one will omit the details to avoid obfuscating the discussion.
4.3. Multiple Point Sources and Sinks
The result in the previous subsection readily generalizes to the situation where we have point sources, say , and sinks, denoted by . The result is provided in the following proposition; we omit the details because it follows the same line of reasoning.
Proposition 26. For problem (28) with multiple point sources and sinks, there exist an optimal solution such that every source sends its information to its nearest sink along the straight line segment connecting them, and the corresponding marginal energy density function on the line segment is uniform.
The result implies that sinks generate a Voronoi tessellation of the deployment area, and the sources send their flows over straight lines to the sink in the cell they reside in, thus, resulting in a star-like network within each cell.
5. Numerical Experiments
In this section, we present a set of numerical examples. For all examples we adopt the communication energy consumption model from .
Let be the transmission range of each node. Then if and only if , where is the distance between nodes and . The energy expenditure per data unit transmitted from to satisfies , , where nJ/bit and nJ/bit denote the energy consumed in the transceiver circuitry at the transmitter and the receiver, respectively, and pJ/bit/ is the energy consumed at the output transmitter antenna for transmitting a bit over one meter. The receiver circuitry is in general more complex and consumes more energy than the transmitter circuitry within the same order of magnitude. The path loss exponent of four is chosen to account for multipath reflections. In all the numerical experiments is estimated by Monte-Carlo simulation with samples, thus is accurate with a error and confidence (by Chebyshev####^~^~^~^~^~^####x2019;s inequality).
5.1. A 4-Node WSNET
We start with a toy example to give some intuition on the routing policies produced by each formulation. The WSNET consists of one origin node , two relay nodes, and , and one sink node , where bits/sec and the radio range is m. The origin node has to use relays and to reach the sink . Further, . First, we consider the case without energy allocation.
5.1.1. Routing without Energy Allocation
All , , are uniformly distributed with ####^~^~^~^~^~^####x2009;J where ####^~^~^~^~^~^####x2009;J, ####^~^~^~^~^~^####x2009;J where J, ####^~^~^~^~^~^####x2009;J with J, , and for all other appropriate and . for all appropriate and . Note that , , , and for all . Take , , and for all .
In Figure 6(a), the red (dot-dash), black (dash), and green (solid-star) lines with arrows represent the nominal, fat, and robust optimal flows, respectively. Note the difference in the selected routes: the nominal picks the shorter path , the fat picks the more ####^~^~^~^~^~^####x201c;stable####^~^~^~^~^~^####x2019;####^~^~^~^~^~^####x2019; but a little longer path , while the robust balances the two to maintain a relatively high lifetime guarantee probability while not suffering too much from the low predicted lifetime.
As we adjust , and will change accordingly. The solid blue curve in Figure 6(b) describes the relationships between and (the percentage predicted lifetime gain of the robust formulation over the fat). It can be seen that there is significant predicted lifetime gain (e.g., ####^~^~^~^~^~^####x25;) while the lifetime guarantee probability remains high (e.g., close to ). The red dash curve represents the relationship versus . It can be seen that as we protect more against the randomness, the predicted lifetime goes down and the lifetime guarantee probability gets enhanced. The two extreme cases of no protection and full protection correspond to the nominal and fat situations.
To gain further insight on the impact of uncertainty on the nominal formulation, consider the probability distribution of the actual lifetime achieved by applying the nominal optimal policy to random instances (where , , and are randomly selected). Figure 7 shows the histogram of generated from a million instances. We can see that can be substantially smaller than and in fact most of the probability mass corresponds to such ####^~^~^~^~^~^####x2019;s. The nominal lifetime guarantee probability would be fairly low but that does not capture how far from the actual lifetime can be.
5.1.2. Routing with Energy Allocation
If energy allocation is an option, set the global available energy J. As before, , and for all other appropriate and . for all appropriate and . Note that , , . Take , .
Figure 8(a) presents the nominal, fat, and robust optimal flows and energy allocation. The situation is very similar as before but energy allocation improves the predicted lifetime since no energy is wasted. Optimal values in a number of nominal, fat, and robust cases with and without energy allocation are listed in Table 1.
5.2. A Randomly Deployed WSNET
In this case, we have 20 nodes (4 sinks, 10 origins, 6 relays) uniformly deployed on a square. m. bits/sec, for all . All , , are uniformly distributed and J, is uniformly sampled from . , , and are uniformly sampled from .
5.2.1. Routing without Energy Allocation
We use and solve the three routing problems without energy allocation. The policies are quite different since we compute and .
5.2.2. Routing with Energy Allocation
Let now . Solving the problems with energy allocation we obtain and . The results for both cases are presented in Table 2.
Again adjusting or , respectively, for the two cases, changes and accordingly (see Figures 9(a) and 9(b)). It can be seen that as we protect more against the randomness, the predicted lifetime goes down and the lifetime guarantee probability gets enhanced. For energy allocation problems, since at optimality all energy constraints are active, the lifetime guarantee probability gets reduced but still the gain over the fat formulation is nonnegligible.
As we did in the 4-node example, we plot in Figure 10 the histogram of achieved by computed from a million random instances of the problem (without energy allocation). It is clear that as the number of nodes grows the probability mass for shifts away from and the actual is typically substantially smaller than . This is consistent with our result that .
We presented a new framework to accommodate uncertainty in designing maximum lifetime routing policies for WSNETs. We considered two scenarios####^~^~^~^~^~^####x2014;one (Scenario A) assuming that energy is already allocated to various nodes and the other (Scenario B) where such allocation is also subject to optimization. We formulated a worst case (fat) problem and compared it with the nominal problem that makes certainty equivalence assumptions and ignores uncertainty. As a compromise between the two, we also devised a robust formulation. We established, analytically and numerically, that the nominal solutions are always too optimistic. Specifically, for common Scenario A topologies (like regular linear arrays and grid-like WSNETs) the nominal formulation predicts a lifetime that is (almost) never achieved in the presence of uncertainty. In Scenario B, the same result holds for all topologies. The robust solutions, on the other hand, provide a useful and practical way to tradeoff performance versus robustness. We extended our analysis to massively dense WSNETs and characterized optimal solutions of the routing problems.
Proof of Theorem 1
Let and be optimal solutions of (13) and (11), respectively. We will show that is a feasible solution of (11) with . For any , the maximization problem in the energy constraint for node in the robust problem (11) is Note that (A.1) is equivalent to the following linear optimization problem: where are the decision variables. Then the dual of (A.2) is: where are the dual variables. Fix in (A.2) and (A.3), and let be an optimal solution of (A.2). Note that is feasible for (A.3). For all we have where the first equation is due to the equivalence of (A.1) and (A.2), the following inequality is due to the weak duality between (A.2) and (A.3), and the second inequality is due to the feasibility of in (13). This shows that is feasible to (11), implying that .
Next, set in (A.2) and (A.3), and let be an optimal solution to (A.2). By strong duality, there exists a feasible to (A.3) such that for all we have Thus, satisfies the second set of constraints of (13). Since the remaining constraints are also satisfied, is a feasible solution of (13), hence, .
Research was partially supported by the NSF under Grant no. EFRI-0735974, by the DOE under Grant no. DE-FG52-06NA27490, by the ARO under Grant no. W911NF-09-1-0492, and by the ODDR####^~^~^~^~^~^####x26;E MURI10 program under Grant no. N00014-10-1-0952.
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