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Maximizing the Lifetime of Wireless Sensor Networks with Mobile Sink in Delay-Tolerant Applications
 

Maximizing the Lifetime of Wireless Sensor Networks with Mobile Sink in Delay-Tolerant Applications

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Maximizing the Lifetime of Wireless Sensor

Maximizing the Lifetime of Wireless Sensor
Networks with Mobile Sink in Delay-Tolerant
Applications

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    Maximizing the Lifetime of Wireless Sensor Networks with Mobile Sink in Delay-Tolerant Applications Maximizing the Lifetime of Wireless Sensor Networks with Mobile Sink in Delay-Tolerant Applications Document Transcript

    • Maximizing the Lifetime of Wireless Sensor Networks with Mobile Sink in Delay-Tolerant Applications YoungSang Yun and Ye Xia Computer and Information Science and Engineering Department, University of Florida, Gainesville, FL 30611-6120 email: {yyun, yx1}@cise.ufl.edu Abstract—This paper proposes a framework to maximize multi-hop routing is generally needed for distant sensor nodesthe lifetime the wireless sensor network (WSN) by using a from the sinks to save energy, the nodes near a sink can bemobile sink when the underlying applications tolerate delayed burdened with relaying a large amount of traffic from otherinformation delivery to the sink. Within a prescribed delaytolerance level, each node does not need to send the data nodes. This phenomenon is sometimes called the “crowdedimmediately as they become available. Instead, the node can center effect” [10] or the “energy hole problem” [11], [12],store the data temporarily and transmit them when the mobile [13]. It results in energy depletion at the nodes near the sinksink is at the most favorable location for achieving the longest too soon, leading to the separation of the sink from the rest ofWSN lifetime. To find the best solution within the proposed nodes that still have plenty of energy. However, by moving theframework, we formulate optimization problems that maximizethe lifetime of the WSN subject to the delay bound constraints, sink in the sensor field, one can avoid or mitigate the energynode energy constraints and flow conservation constraints. We hole problem and expect an increased network lifetime.conduct extensive computational experiments on the optimization This paper proposes a framework to maximize the lifetimeproblems and find that the lifetime can be increased significantly of the WSN by taking advantage of the mobile sink. Comparedas compared to not only the stationary sink model but also more with other mobile-sink proposals, the main novelty is that wetraditional mobile sink models. We also show that the delaytolerance level does not affect the maximum lifetime of the WSN. consider the case where the underlying applications tolerate delayed information delivery to the sink. In our proposal, within a prescribed delay tolerance level, each node does not I. I NTRODUCTION need to send the data immediately as they become available. A wireless sensor network (WSN) consists of sensor nodes Instead, the node can store the data temporarily and transmitcapable of collecting information from the environment and them when the mobile sink is at the stop most favorablecommunicating with each other via wireless transceivers. The for achieving the longest network lifetime. To find the bestcollected data will be delivered to one or more sinks, generally solution within the proposed framework, we formulate opti-via multi-hop communication. The sensor nodes are typically mization problems that maximize the lifetime of the WSNexpected to operate with batteries and are often deployed subject to the delay bound constraints, node energy constraintsto not-easily-accessible or hostile environment, sometimes in and flow conservation constraints. Another one of our contri-large quantities. It can be difficult or impossible to replace butions is that we compare our proposal with several otherthe batteries of the sensor nodes. On the other hand, the sink lifetime-maximization proposals and quantify the performanceis typically rich in energy. Since the sensor energy is the differences among them. Our computational experiments havemost precious resource in the WSN, efficient utilization of shown that our proposal increases the lifetime significantlythe energy to prolong the network lifetime has been the focus when compared to not only the stationary sink model but alsoof much of the research on the WSN. more traditional mobile sink models. We also show that the Although the lifetime of the WSN can be defined in many delay tolerance level does not affect the maximum lifetime ofways, we adopt the definition that it is the time until the first the WSN.node exhausts its energy, which is a widely used. Much work Our proposal is more sophisticated than most previoushas been done during recent years to increase the lifetime lifetime-improvement proposals that we know of. It integratesof the WSN. Among them, in spite of the difficulties in the following energy-saving techniques, multipath routing, arealization, taking advantage of the mobility in the WSN has mobile sink, delayed data delivery and active region control,attracted much interests from researchers [1], [2], [3], [4], [5], into a single optimization problem. Such sophistication comes[6], [7], [8], [9]. We can take the mobile sink as an example at a cost. Whether the proposal should be adopted in practiceof mobility in the WSN. The communications in the WSN has will depend on the tradeoff between the lifetime gain and thethe many-to-one property in that data from a large number of actual system cost. Even if the decision is not to adopt itsensor nodes tend to be concentrated into a few sinks. Since due to a high cost or high complexity, the framework in the
    • paper is still useful because it can supply the practitioners by optimizing not only the schedule of sink visits but also rout-with a performance benchmark, e.g., how much lifetime ing of the traffic. However, they did not consider applicationsimprovement opportunity there is. By also formulating the where delayed information delivery is allowed.optimization problems related to other proposals and providing The rest of the paper is organized as follow. Section IIcross comparison, the paper provides extra convenience for describes various related lifetime maximization problems thatcomparing and understanding different proposals. we will compare against. The mathematical formulations of the We now briefly review the most relevant work on how to models are provided for the purpose of comparison. In sectionexploit mobility to increase the network lifetime. In [4], the III, we propose two novel models with a mobile sink andauthors introduced the mobile agents called MULEs, which delayed information delivery. We show some nice propertiesmove around and collect data from nearby sensor nodes on that our models possess. In section IV, we compare our modelsbehalf of the immobile sink. When the mobile agents move to with others by simulation and numerical experiments. Thethe vicinity of the sink, they forward the collected data to the conclusions are given in Section V.sink. The mobile agents are assumed to have plenty of energy.The movement of each mobile agent is modeled as a random II. R ELATED L IFETIME M AXIMIZATION P ROBLEMSwalk. It was shown that the queues in the mobile agents and In this section, we discuss related lifetime maximizationthe sensor nodes are finite and the delay of the collected data problems that have been published in the literature. We willis bounded. However, the authors did not show the quantitative later compare their performance with our new proposal.improvement of the network lifetime by using mobile agents. First, we will describe the general assumptions about the In [1], the authors formulated a linear programming problem WSN models. Let the set of sensor nodes be denoted by N .to maximize the lifetime of a WSN where the sensor nodes For experimental convenience, we suppose they are uniformlyare deployed in a grid pattern and the sink can move to a randomly deployed into a circular area with radius R. Let thesubset of the grid points. When the location of the mobile center of the disk be the origin. Each node i is assumed tosink is known to the sensor nodes, each node can identify the generate data at a constant rate of di during its life span andminimum hop-count paths to the sink. The nodes distribute the initial energy of i is denoted by Ei . Furthermore, the nodesthe data evenly onto these paths. have the ability of adjusting their transmission power level to The authors of [3] showed that the network lifetime can match the transmission distance. Similar to [14], the energybe extended significantly if the mobile sink moves around required per unit of time to transmit data at the rate of xijthe periphery of the WSN. They assumed that, if the mobile from node i to j can be determined as follows.sink can balance the traffic load of the nodes, the lifetimeof the network can increase. Therefore, they proposed an t t Eij = Cij · xij , (1)optimization problem for choosing a mobility strategy that tminimizes the maximum traffic load of the nodes. However, where Cij is the required energy for transmitting one unit ofthey assumed the shortest path routing, which, in general, does data from node i to j and it can be modeled as follows [15].not produce the best lifetime. Cij = α + β · d(i, j)e , t (2) The problem of finding the sink movement path to optimizethe lifetime of the WSN, as in [5], is hard to solve. This type of where d(i, j) is the Euclidean distance between node i andproblems generally belong to the traveling salesman problem. j, α and β are nonnegative constants, and e is the path lossIn [9], the authors studied how to find the optimal sink stops exponent. Typically, e is in the range of 2 to 6, depending onand the schedule of visit to the each of the stops. If the can- the environment. Here, the energy cost per unit of data doesdidate locations for the stops are unconstrained, this problem not depend on the link rate, and this is valid for the low rateis also NP-hard. However, if the stops are constrained to be regime. Hence, we need to assume that the traffic rate xijselected from a finite set of known locations, the problem can is sufficiently small compared to the capacity of the wirelessbe easily formulated into linear programming. They proposed link.an approximation algorithms to the unconstrained problem by The energy consumed at node i per unit of time for receivingproperly dividing the whole sensor field into a finite number data from node k is given by [14]of disjoint small areas, and then, converted the unconstrained rproblem into a constraint problem. However, to obtain a good Eki = γ · xki , (3)approximation ratio, the number of small areas can potentiallybe very large, making the linear programming computation where γ is a given constant. Hence the total energy consump-time-consuming. Therefore, in this paper, we restrict the set tion per unit time at node i isof potential sink stops to be from a small number of given t r t Eij + Eki = Cij · xij + γ · xki . (4)locations rather than from arbitrary locations. j∈N k∈N j∈N k∈N The WSN model proposed in [7] is close to ours. Theauthors studied the maximum lifetime problem of the WSN We assume that each sensor node has the same transmissionwhere the mobile sink can visit only small number of loca- range. We define the neighbors of node i as N (i) = {j ∈tions. They showed that the lifetime can be further increased ¯ ¯ N |d(i, j) ≤ d}, when the transmission range is d.
    • A. Static Sink Model N1 L1 O L2 N2 In the static sink model (SSM), the sink is located at theorigin and remains stationary during the operation of the WSN. 1 1 1 1Data originated from the sensor nodes flow into the sink in amulti-hop fashion. As soon as the data becomes available at anode, it gets transmitted toward the sink. Typically, the rate atwhich each sensor node i harvests data from the outside worldis a constant. We denote it by di . The problem of maximizing Fig. 1. Examples of the static sink model (SSM), mobile sink model (MSM),the lifetime in this model is formulated as follows [16], [17]. and delay tolerant mobile sink model (DT-MSM) Problem : Staic Sink Model (SSM) maximization problem can be formulated as follows. max T Problem : Mobile Sink Model (MSM) s. t. xij − xki = di , i, j, k ∈ N (5) max T = z1 + z2 + · · · + z|L| j∈N (i) k:i∈N (k) (l) (l) s. t. xij − xki = di ,   t  Cij · xij + γ · xki  · T ≤ Ei , j∈N (i) k:i∈N (k) j∈N (i) k:i∈N (k) ˆ i, k ∈ N , j ∈ N , l ∈ L (9) i, j, k ∈ N   (6) |L| (l) (l) (l) xij ≥ 0, i, j ∈ N (7) zl  Cij · xij + γ · xki  ≤ Ei , l=1 j∈N (i) k:i∈N (k) T ≥ 0, (8) ˆ i, k ∈ N , j ∈ N , l ∈ L (10) (l) ˆThe constraint (5) is the “flow conservation constraint”, which xij ≥ 0, i ∈ N,j ∈ N,l ∈ L (11)states that, at a node i, the sum of all outgoing flows is equal zl ≥ 0, l∈L (12)to the sum of all incoming flows plus flows generated at nodei itself, or di . The inequality (6) is the energy constraint and The energy required for transmitting one unit of data whenit means that the total energy consumed by a node during the sink is at l can be expressed as,the lifetime (T ) cannot exceed the initial energy of the node.  tWith this formulation, the routing is dynamic and allows Cij  if j∈L / (l)multipath communications. There is no assumption on fixed- Cij = ∞ if j ∈ L and j = l (13)  t epath routing, such as the shortest path routing. The above  Cil = α + β · d(i, l) if j ∈ L and j = l,optimization problem can be easily converted into a linear tprogramming (LP) problem. where Cij is the same as in the SSM. Constraint (9) denotes the flow conservation for all nodes when the sink is at l. Constraint (10) says that the total energyB. Mobile Sink Model consumed at the node i can not exceed the initial energy (l) Ei . By multiplying (9) with zl and substituting xij · zl with (l) In the mobile sink model (MSM), we assume that the sink new variable yij , we have replace (9) with the following newcan move around within the sensor field and stop at certain constraint.locations to gather the data from the sensor nodes. We ignore (l) (l) yij − ˆ yki = zl · di , l ∈ L, i ∈ N , j ∈ N . (14)the traveling time of the sink between locations. Let L be the j∈N (i) k:i∈N (k)set of possible locations where the sink can stop. The sinkdoes not necessarily stop at (i.e., stays for a positive duration) Similarly, constraint (10) can be changed intoall locations in L in the interest of maximizing the network  lifetime [1], [9]. In this model, the order of visit to the stops |L| t(l) (l) (l)has no effect on the network lifetime and can be arbitrary. The  Cij · yij + γ · yki  ≤ Ei ,sink sojourn time at a location l ∈ L is denoted by zl ; it is the l=1 j∈N (i) k:i∈N(k)time that the sink spends at l to collect data from the sensor ˆ i ∈ N , j ∈ N , l ∈ L. (15)nodes. The overall network lifetime T = l∈L zl . To find theoptimal network lifetime, we need to consider the routing of With constraints (14), (15),(12), and non-negativity constraints (l)the traffic as well as the duration of stay by the sink at each of yij , the above optimization problem is converted into an (l) (l)stop [9], [2], [1], [7]. Let xij be the flow rate from node i to LP problem. Here, yij is interpreted as the total traffic volume ˆj while the sink is at stop l. Let N = N ∪ L. The lifetime for node i to send to node j while the sink stays at l.
    • III. L IFETIME M AXIMIZATION IN D ELAY T OLERANT 3.5 3 M OBILE S INK M ODEL Relative Lifetime 2.5 2 In this section , we consider how to maximize the network 1.5lifetime in applications that can tolerate a certain amount 1of delay. We call the resulting WSN model delay tolerant 0.5 MSM DT-MSMmobile sink model (DT-MSM). In this setting, each nodes can 0postpone the transmission of data until the sink is at the stop 10 12 14 16 18 20 22 24 26 Radius of the Coverage of the Sinkmost favorable for extending the network lifetime. This way,the nodes can collectively achieve a longer network lifetime. Fig. 2. Comparison of lifetimes of MSM and DT-MSM under the variousIn contrast, the SSM and MSM do not exploit this possibility. radii of coverage Let D be the maximum tolerable delay, or the delaytolerance level. We assume that the sink finishes one roundof visit to all the stops (where the sink stays for a positive in Rl , node i is said to be active at l ∈ L. Although we canduration to collect data) in D time units, and then, repeats construct Rl in many ways depending on the application ofwith another round again and again. Note that two consecutive interest, in this paper, a very simple method of constructingvisits to the same stop takes a time D. Rl is considered. Fix a positive number r. We call r the radius Let’s take an example to show how our framework can of coverage of the sink. For each l ∈ L, if d(i, l) ≤ r, whereoutperform other ones. Consider the two-node example shown i ∈ N , then i ∈ Rl . Here, the radius of coverage of the sinkin Figure 1. N1 and N2 are two sensor nodes and L1 and L2 (r) should be large enough so that every sensor node belongsare the candidate stops of the mobile sink. Suppose we ignore to at least one Rl .the receiving energy requirement and suppose the transmission In both SSM and MSM, the sink collects data from eachenergy per unit of data is equal to the square of the distance node i at the same rate at which node i generates the data.between the sender and the receiver. Both nodes N1 and N2 However, in the DT-MSM, the data transmission rate at node igenerate data at 1 bps and have 100 units of energy initially. during the collection time is no longer the same as the constantIf the sink is located at O in the SSM, both nodes spend 4 data generation rate di . When node i is not active (i.e., notunits of energy for sending a bit of data. It is obvious that the covered by the current sink location), it continues to gatheroptimal lifetime is 25 seconds. In the MSM with sink locations data and should store the newly generated data. Hence, data{L1 , L2 }, due to the symmetry of the structure, the sink stays buffering is required by our framework. Within a cycle of Dat both L1 and L2 for the same amount of time to achieve the time units, the total stored data at each node i is at mostmaximum lifetime. Each node spends 1 or 9 units of energy D · di . For ease of presentation, we assume the sink visits allfor sending 1 bit of data depending whether the sink is at locations in L in the order of 1 → 2 → · · · |L| → 1 · · · . TheL1 or L2 . The average energy consumption per bit is 5 units. sink may stay at some location for zero time. With slight abuseThus, the lifetime is 20 seconds. In the DT-MSM, we assume of terminology, we re-define the network lifetime T to be thethat the sink alternates between the two stops and stays for number of cycles made by the sink until the first node dies1 second at each stop in each cycle. Hence, D = 2 seconds. due to energy exhaustion. The actual lifetime is T ·D. We nextWhen the sink stays at L1 , only N1 sends 2 bits of data to describe two variants of the model: the sub-flow-based modelthe sink; when the sink moves to L2 , only N2 transmits 2 and the queue-based model.bits of data. Both nodes spend 1 unit of energy per secondon average. Thus, the lifetime is 100 seconds, a significant A. Sub-Flow-Based Modelincrease compared to the SSM and MSM. This is because, In the sub-flow-based model, the nodes in the current cover-in the DT-MSM, the nodes do not always participate in the age Rl are not allowed to buffer the relayed traffic from othercommunication for all the sink stops; they each wait until the nodes; as soon as a node in Rl receives the data from othersink’s location is most favorable for energy saving, and then nodes, it immediately forwards the data to its neighboringsend data at the higher rate. Recall that we have assumed that nodes. To model this constraint at each node i, we need tothe traffic rate is sufficiently small compared to the capacity of differentiate the data generated by node i itself and the datathe wireless link, and hence, sending data a higher rate does originally generated by other nodes but forwarded to node i.not alter the per-bit energy consumption. The data generated by the same source is sometimes called Unlike the MSM or SSM, the sink in the DT-MSM can (m,l) a commodity or a sub-flow [18], [19]. Let xij be the ratecollect data from only a subset of the set of all sensor nodes, assignment from node i to the node j, while the sink is at l,N , at each stop. Let Rl be the subset of N such that only for the traffic generated by node m (commodity m). Let xij (l)nodes in Rl can participate in the communication when the be the aggregated rate of traffic that needs to be forwarded tosink is at l ∈ L. We call Rl the coverage of the sink location node j from node i when the sink is at l. That is,l. Note that the union of Rl over l ∈ L must be the set ofall sensor nodes, N . In other words, any sensor node should (l) (m,l) xij = xij , i ∈ Rl , j ∈ Rl ∪ {l}, l ∈ L. (16)be covered by at least one sink location. When the node i is m∈Rl
    • Since at node i, the commodity or sub-flow of other nodes B. Queue-Based Modelm ∈ Rl , m = i must be forwarded as soon as it has been In the queue-based model, the sensor node can buffer thereceived, we must have (l) commodities or sub-flows of all nodes. Let qi be the queue (m,l) xki = xij (m,l) , length at node i just before the sink moves from location l to l + 1. Assume that each node i has D · di amount of k:i∈Nl (k) j∈Nl (i) (0) data at the beginning of a cycle, which is denoted by qi . i, m = i, k ∈ Rl , j ∈ Rl ∪ {l}, (17) When the sink finishes a cycle of visit, the queue at node i (|L|)where Nl (i) = Rl ∩ N (i). The flow conservation at node i must be cleared. Thus we have qi = 0. In this model, thecan be expressed as follows, which is the same as in the MSM flow conservation constraint has to do with the queue length,except that the net amounts of traffic originated from node i expressed as follows. (l)itself, (wi , l ∈ L, i ∈ N ), are now decision variables. zl (l) xij + qi (l−1) − zl (l) xij = qi , (l) k:i∈Nl (k) j∈Nl (i)   zl  (l) xij − (l) xki  = (l) wi . (18) ˆ i, k ∈ N ; j ∈ N ; l ∈ L. (30) j∈Nl (i) k:i∈Nl (k) The energy constraints can be expressed in the same way The data buffered during the previous sink-movement cycle as in the MSM or sub-flow based MSM. From the abovemust be cleared in the current cycle. This requirement can be discussion, we have the following optimization problem forwritten as maximizing the lifetime. (l) wi = D · di . (19) l:i∈Rl Problem : Queue-Based DT-MSM The following is the lifetime maximization problem in sub- max T (31)flow-based DT-MSM. (l) (l−1) (l) (l) s. t. zl xki + qi − zl xij = qi , Problem : Sub-Flow-Based DT-MSM k:i∈Nl (k) j∈Nl (i) max T (20) ˆ i, k ∈ N ; j ∈ N ; l ∈ L (32)   s. t. (m,l) xij (l) = xij , i ∈ Rl , j ∈ Rl ∪ {l}, l ∈ L (21)  |L| (l) m∈Rl zl  γ · xki +  (m,l) (m,l) l=1 k:i∈Nl (k) xki = xij ,  k:i∈Nl (k) j∈Nl (i)  (l) (l) + Cij · xij  · T ≤ Ei m = i; m, i, k ∈ Rl ; j ∈ Rl ∪ {l}; l ∈ L (22)    j∈Nl (i) (l) (l) (l) ˆ i, k ∈ N ; j ∈ N ; l ∈ L (33) zl  xij − xki  = wi , (l) ˆ j∈Nl (i) k:i∈Nl (k) xij ≥ 0, i ∈ N,j ∈ N,l ∈ L (34) (l) i, k ∈ Rl ; j ∈ Rl ∪ {l}; l ∈ L (23) qi ≥ 0, l ∈ L, i ∈ N (35)    |L| (0) (l) qi = D · di , i∈N (36) zl  γ · xki + (|L|)  l=1 k:i∈Nl (k) qi = 0, i∈N (37)   zl ≥ 0, l∈L (38) (l) (l) Cij · xij  · T ≤ Ei T ≥ 0. (39)  j∈Nl (i) The problem shown above can be converted into an LP ˆ i, k ∈ N ; j ∈ N ; l ∈ L (24) (l) (l) problem by substituting yij for zl · xij and introducing the (l) wi = D · di , i ∈ N (25) new variable u = 1/T . This linearization method can also be l:i∈Rl applied to the MSM. (m,l) xij ≥ 0, i, m ∈ Rl ; j ∈ Rl ∪ {l}; l ∈ L (26) C. Properties of Delay-Tolerant Mobile Sink Model (l) To facilitate later expositions, we define the inclusion re- wi ≥ 0, i ∈ Rl , l ∈ L (27) lationship between different optimization problems with the zl ≥ 0, l∈L (28) same decision variables x and the same objective function T ≥ 0. (29) f (x) but different constraint sets. Without loss of generality, we consider the minimization problem. In the sub-flow-based model, each node requires only D · diamount of buffer because there is no need to store other nodes’ Definition 1 (Inclusion). Consider two minimization problemssub-flows. P1 and P2 that have the same decision variables x and
    • the same objective function f of variables x, but different optimization problem P (N , L, r2 ).constraints set F1 and F2 , respectively. If F1 ⊆ F2 , we say  that P1 is said to be included in P2 and denote P1 ⊆ P2 . (l) (l) (l) zl  xij − xki  = wi . (40) j∈Nl (i,r2 ) k:i∈Nl (k,r2 ) If P1 ⊆ P2 , we can always obtain better optimal solutionfrom P2 than P1 . In general, we say that inclusion holds for We have the following by separating the neighbor sets intoa sequence of optimization problems {Pi }, where i is from ¯ ¯ A, A, B, and B.some linearly ordered index set, I, that is, Pa ⊆ Pb whenever  a < b, a, b ∈ I. (l) (l) (l) (l) (l) zl  xij + xij − xki − xki  = wi , In both sub-flow-based model and queue-based model, the j∈A ¯ j∈A k∈B ¯ k∈Bcoverage of the sink location is a very important factor for (41)the lifetime of the WSN. Consider the optimization problem for i, k ∈ N , j ∈ N ∪ {l}, l ∈ L. We set the variableformulated for the sub-flow-based model. Depending on the (l) (l) (l) ¯ xij = xij when j ∈ A, and xij = 0 when j ∈ A. In addition, ˆradius of coverage, we may obtain different instances of the (l) (l) (l) ¯ we set xki = xki when k ∈ B and xki = 0 when k ∈ B. For ˆoptimization problem. This observation is also valid in the (l) (l) the variables w and z , let w (i) = w (i) and zl = zl . Then, ˆ ˆqueue-based model. Thus, we can parameterize the optimiza- equation (41) is equivalent to the constraint (23) of the problemtion problems according to the radius of the coverage. Let x ˆ ˆ ˆ P (N , L, r1 ). Therefore, (ˆ, w, z , T ) satisfies the constraintP (N , L, r) be the optimization problem when the radius of (23) for the problem P (N , L, r2 ). For the energy constraintcoverage of the sinks is r, the set of sensor nodes is N , and the (24), we can also apply the similar procedures. Hence, we canset of sink locations is L. The value r must be large enough conclude that any feasible solution of the problem P (N , L, r1 )so that all sensor nodes can be covered by at least one sink is also a feasible solution for the problem P (N , L, r2 ). Hence,location and we denote this minimum radius of coverage for P (N , L, r1 ) ⊆ P (N , L, r2 ).connectivity by r0 . Under the same configuration with N and (l) As a consequence, we have the following theorem.L, different r values only affects Nl (i) and Cij . We will use (l)the notation Nl (i, r) and Cij (r) if it is necessary to specify Theorem 3. Consider the set of maximization problemthe radius of coverage. In the next theorem, we prove that the {P (N , L, r)}, r ∈ R. If r1 < r2 , then optimal objectivebigger the radius of coverage,the longer the optimal lifetime is. value for the problem P (N , L, r2 ) is greater than that forFor the proof of this theorem, we need the following lemma. the problem P (N , L, r1 ). In general, the queue-based model is less constraining thanLemma 2. The sequence {P (N , L, r)}, r ∈ R, satisfies the sub-flow-based model. Naturally, this results in lifetimeinclusion. That is, P (N , L, r1 ) ⊆ P (N , L, r2 ) for any pair gains in the former model. The following theorem formalizesof r1 , r2 such that r1 < r2 . the fact that the queue-based model always outperforms the sub-flow-based model. Proof: Let G = {N ∪ L, E} be a graph where E ={(i, j)|i ∈ N , j ∈ N ∪ L, d(i, j) ≤ c}, where c is the ˆ Theorem 4. Let T be the optimal objective value to problemtransmission range of the sensor nodes. The neighbor set of (20), and T ∗ be the optimal objective value to problem (31)node i is defined as N (i) = {j ∈ N ∪L|(i, j) ∈ E}. Let G(r) with the same configuration (N , L) and the same radius ofbe the graph for the DT-MSM model with the radius r and ˆ coverage r. Then T ≤ T ∗ . |L|G(r) = {N ∪ L, ∪l=1 E (l) (r)}, where E (l) (r) = {(i, j)|i ∈ (m,l) (l) ˆ ˆ ˆ Proof: Let (ˆij , wi , zk , T ) be the optimal solution to xRl , j ∈ Rl ∪ {l}, d(i, j) ≤ c}, and Rl = {i ∈ N |d(i, l) ≤ r}. problem (20). We will prove this theorem by constructing aThe neighbor set of node i when the sink is at l, Nl (i) is (m,l) (l) ˆ ˆ feasible solution to problem (31) with (ˆij , wi , Zk , T ), x ˆ{j ∈ Rl ∪ {l}|(i, j) ∈ E (l) (r)}. and showing that under this feasible solution, the objec- Consider the two optimization problems P (N , L, r1 ) and (l) tive value of problem (31) is the same. By (21), xij =P (N , L, r2 ) with r1 < r2 . It is obvious that Nl (i, r1 ) ∈ (m,l) (l) (l−1) (l) m∈Rl xij ˆ . In addition, let set qi = qi − wi forNl (i, r2 ) for all i ∈ N , l ∈ L by the definition of neighbor (0)set Nl (i, r). Therefore, we can split the larger set Nl (i, r2 ) all i ∈ N . Since qi = D · di by (36), we have the following ¯into two sets A and A, where A = Nl (i, r1 ), and A = ¯ sequence of assignments for the q(·)i .Nl (i, r2 ) − A. In a similar way, we can also split the incoming (1) qi = qi (0) − wi (1) = D · di − wi (1)neighbor set for the node i into B = {k ∈ N |(k, i) ∈ (2) (1) (2) ¯E (l) (r1 ) and (k, i) ∈ E (l) (r2 )} and B = {k ∈ N |(k, i) ∈ qi = qi − wi ¯ ¯E (l) (r2 )} − B. In other words, A and B are the extended . . .. . .outgoing and incoming neighbor sets for node i, respectively, (|L|) (|L|−1) (|L|)as the radius of coverage increases from r1 to r2 . qi = qi − wi . x ˆ ˆ ˆ Suppose that (ˆ, w, z , T ) is a feasible solution to the If we sum up above assignment for all l ∈ L, we have (|L|) |L| (l)problem P (N , L, r1 ). Now, consider equation (23) for the qi = D · di − l wi = D · di − D · di = 0 by (25)
    • TABLE Iand it exactly coincides with (37). Since the configuration E XPERIMENTAL PARAMETERS AND T HEIR VALUESand radius of coverage r for problem (20) are the same asproblem (31), Nl (i), i ∈ N , l ∈ L for (20) does not change the number of sensor nodes {100, 200} (l) (l−1) (l) the number of possible sink locations {5, 6, 7, 8, 9, 10, 15, 20, 30, 40}in (31). If we put wi = qi − qi into (23), we have path loss exponent (e) {2.0, 3.0}exactly the same constraint as (32). Energy constraint (24) is transmission range {5, 6, 7, 8, 9, 10, 15, 20, 30, 40, 50}equivalent to the (33). From the above arguments, we conclude α 10 pJ/bit/m2that for the optimal solution to problem (20), there exists a β 0.0013 pJ/bit/m4 Initial Energy (Ei ) 500 Jcorresponding feasible solution to problem (31) with the same Data generation rate (di ) 500 bpsobjective value. Hence, T ∗ ≥ T . ˆ In the following theorem, we show that the maximumlifetime of the system is the same for all values of D. Here, the ′ q( D ) = D′ di . Therefore we have Dmaximum lifetime of the system is expressed as the product ofthe D and the corresponding optimal objective value T ∗ (D). q = Ddi . (44)Theorem 5. Define P (D) as the lifetime optimization problem From above argument, we have shown that new solutionparameterized by the value D, for some fixed network con- (x, q, z, T ) is feasible to the problem P (D). According to thefiguration. Let T ∗ (D) and T ∗ (D′ ) be the optimal objective assumption, we know T ∗ (D′ ) > ( D′ )T ∗ (D). Therefore, we Dvalues for the problem P (D) and P (D′ ), respectively. Then, haveT ∗ (D) · D = T ∗ (D′ ) · D′ . D′ D′ D T = T ∗ (D′ ) > T ∗ (D) = T ∗ (D) (45) Proof: Consider the queue-based model. Let D D D′(x∗ (D), q ∗ (D), z ∗ (D), T ∗ (D)) be the optimal solution to the However, this is contradictory to the fact that T ∗ (D) is theproblem P (D), and let (x∗ (D′ ), q ∗ (D′ ), z ∗ (D′ ), T ∗ (D′ )) be optimal value for the problem P (D). Thus, it must be thatthe optimal solution to the problem P (D′ ). T ∗ (D)D ≥ T ∗ (D′ )D′ . First, suppose T ∗ (D) · D < T ∗ (D′ ) · D′ . We will show Similarly, we can also prove that T ∗ (D)D ≤ T ∗ (D′ )D′ .this contradicts the definition of the optimal value, T ∗ (D) Hence, T ∗ (D)D must equal to T ∗ (D′ )D′by constructing a feasible solution whose objective value isgreater than T ∗ (D) for the problem P (D). IV. E XPERIMENTAL R ESULTS Let x = ( D′ )x∗ (D′ ), q = ( D′ )q ∗ (D′ ), z = z ∗ (D′ ), D D In this section, we will present the results from numerical ′T = ( D )T ∗ (D′ ). We want to show that (x, q, z, T ) satisfies D experiments. In particular we have compared the networkthe constraints (32)-(39). Since it is obvious that the solution lifetimes of the following models.(x, q, z, T ) satisfies the constraints (34), (35), (37), (38), and • Static Sink Model (SSM): The stationary sink is located at(39), we focus here on constraints (32), (33), and (36) only. O. We take the performance of this model as the referenceSince the optimal solution (x∗ (D′ ), q ∗ (D′ ), z ∗ (D′ ), T ∗ (D′ )) for comparison.is feasible to the problem P (D′ ), it must satisfy constraint ′ ′ • Mobile Sink Model (MSM): The sink can move to(32). Next, let us plug ( D )x, z, and ( D )q into constraint (32) D D several locations to collect data. When the sink is at eachin the places for x∗ (D′ ), z ∗ (D′ ), and q ∗ (D′ ), respectively. location, all sensors participate in the communication,Then, we have sending and relaying traffic to the sink. (l) (l−1) (l) (l) • Delay-Tolerant Mobile Sink Model (DT-MSM): When zl xki + qi − zl xij = qi . (42) k:i∈Nl (k) j∈Nl (i) the mobile sink is at a stop, a subset of the sensor nodes ′ can participate in the communication. We use queue-If we put ( D )x, z, and ( D′ )T in the places for x∗ (D′ ), D D based variant of this model to evaluate the performance. ∗ ′ ∗ ′z (D ) and T (D ) on the left hand side of constraint (33), We have experimented with different parameters exten-we have sively, such as the number of nodes, the number of possible    |L| sink locations and the parameters for the energy consumption (l) D′ zl  γ · xki + model. Only a small subset of the results are reported here  D due to space limitation. In Table I, we provide the system l=1 k:i∈Nl (k)  (43) parameters and their values for the reported experiments in ′ (l) (l) D  D this paper. We adopt the data for the last four parameters from Cij · xij ·T D  D′ [20]. In all experiments, we use GLPK for solving the linear j∈Nl (i) programming problem.After canceling D and D′ , it is easy to see that the new solu- First, we would like to mention the impact of the radiustion (x, q, z, T ) satisfies the energy constraint of the problem of coverage of the sink on the performance of the DT-MSM.P (D). For this experiment, pointf for 100 nodes and 20 mobile-sink From the constraint (36) for the problem P (D′ ) , we have locations are randomly generated (|N | = 100, |L| = 20) in ∗(0) ∗(0)qi (D′ ) = D′ di . Since q = ( D′ )q ∗ (D′ ), qi (D′ ) = D a circular area with radius 25. We use a simple algorithm
    • Optimal Objective Value (=T) 12 12 1.6 3.5 10 10 1.4 3 Relative Lifetime Relative Lifetime Actual Lifetime 8 8 1.2 T 2.5 6 6 1 lifetime SSM SSM 2 4 MSM 4 MSM 0.8 DT-MSM DT-MSM 1.5 2 2 0.6 0.4 1 0 0 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 0.2 0.5 Number of Sink Location Number of Sink Location 0 0 0 2 4 6 8 10 (a) 100-node network (b) 200-node network Delay (=D)Fig. 3. Lifetime against the number of sink locations; maximum coverage;e = 2.0 Fig. 5. Lifetime/optimal objective value (T ) versus the delay tolerance level D 9 10 8 9 16 16 8 Relative Lifetime Relative Lifetime 7 14 14 6 7 6 12 SSM 12 SSM 5 SSM MSM MSM 10 10 Lifetime Lifetime SSM 5 SSM DT-MSM DT-MSM 4 MSM DT-MSM 4 8 8 3 3 DT-MSM 2 6 6 2 1 1 4 4 0 0 2 2 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 0 0 Number of Sink Locations Number of Sink Locations 5 10 15 20 25 30 35 40 45 50 5 10 15 20 25 30 35 40 45 50 Transmission Range Tarnsmission Range (a) 100-node network (b) 200-node network (a) Under minimum coverage (b) Under maximum coverageFig. 4. Lifetime against the number of sink locations; minimum coverage;e = 3.0 Fig. 6. Lifetimes versus transmission range: |N | = 200, |L| = 20, e = 2.0to find the minimum radius of coverage (denoted by r0 ): At locations are chosen as the true stops for the sink. However,each sink location, we increase the radius of coverage from in DT-MSM, the rate of lifetime increase is substantial as |L|0 simultaneously until the union of all coverages contains all increases. This is because each node can have better and bettersensor nodes. At that point, we have reached the minimum sink location as |L| increases, and it is not forced to participateradius of coverage required to cover all nodes. After that, we in the communication when the current coverage is not theincrease the radius of coverage in 0.1 increments. The result of most favorable for energy saving, even if the node may belongthe experiment is plotted in Figure 2. Note that, in the figure, to that coverage. This is not possible in the MSM because nothe lifetime is normalized to the optimal lifetime of the MSM. matter where the sink stops, every node must participate inAs shown in the figure, the lifetime of the DT-MSM increases the communication.as the radius of coverage increases, which is consistent with We conduct similar experiments with the same configurationTheorem 3. The increase is the sharpest when the radius just but minimum coverage. The result is shown in Figure 4.exceeds the minimum radius required to cover all nodes. After Although the slope of lifetime increase of the DT-MSM isthat, further increase of the radius has a negligible effect. lowered when compared to the maximum coverage case, theRecall that, when the mobile sink reaches one of the stops, increase pattern is similar. Although a larger set of sinksay l, only those sensor nodes in the coverage of l (i.e., Rl ) locations increases the network lifetime, it can be undesirablecan communicate. It is generally desirable for Rl to have as if the sink travel time cannot be ignored. The longer travelingfew nodes as possible, since this reduces the communication time may exceed the delay tolerance level D. Therefore, thereand coordination complexity. The aforementioned behavior of is a tradeoff between the gain from more sink locations andlifetime increase is desirable. the delay or other system costs. Next, we compare the lifetimes of models under various We also conducted experiments with different values for thenumbers of the sink locations. The number of nodes is set to delay tolerance level D. With 100 nodes and 20 sink locations,100 or 200, the path loss exponent e is 2.0, and the coverage we computed the optimal lifetime of the queue-based modelis as large as possible. We ran the experiment 100 times for as D varies from 1 to 10. In Figure 5, we plot the optimaleach configuration. The lifetimes of the MSM and DT-MSM objective values T (number of cycles) and the actual optimalare again normalized to the optimal lifetime of the SSM. As lifetime of the system. Note that the actual lifetime of theshown in Figure 3, the lifetime of the MSM is about 100% ∼ system is T × D. As claimed by Theorem 5, the optimal200% greater than that of the SSM. However, the DT-MSM lifetime is a constant regardless of the values of D.is 200% ∼ 1000% better than the SSM. Moreover, the curves In Figure 6, we show the lifetimes of the three models underall look linear; the performance gap can grow even larger with various values for the transmission range. The transmissionmore sink locations. range determines whether a link exists between a pair of nodes. Interestingly, the lifetime of the MSM increases very slowly For the minimum radius of coverage of the sink (See Figurewith the number of sink locations. As explained in [9], in the 6(a)), the effective transmission range limited by the dimensionoptimal solution, only a few locations from the set of sink of the coverage of the sink. In other words, a node cannot
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