Lifetime maximization under reliability constraint via cross ... - MWNL

Lifetime Maximization under Reliability Constraint via Cross-Layer Strategy in Wireless Sensor Networks Hojoong Kwon, Tae Hyun Kim, Sunghyun Choi, and Byeong Gi Lee School of Electrical Engineering Seoul National University, Seoul, 151-744, Korea Email: [email protected]

Abstract— In this paper, we investigate the problem of the lifetime maximization in a wireless sensor network under the constraint of the target end-to-end transmission success probability, by adopting a cross-layer strategy that considers both physical layer (i.e., power control) and network layer (i.e., routing protocol) jointly. We first present the optimal power allocation algorithm for a given routing path. This allocation determines per-hop success probability for each link along the path in order to minimize the total energy consumption while guaranteeing the reliability constraint. Then, we develop an optimal routing algorithm that maximizes the network lifetime and a heuristic algorithm that has lower and tractable complexity. Simulation results reveal that a trade-off relation exists between the network lifetime maximization and the reliability constraint, and that the proposed routing algorithm combined with the optimal power allocation algorithm increases the network lifetime significantly.

I. I NTRODUCTION Recently, wireless sensor networks have become an area of attractive research interest [1]. Sensor networks, in most applications, are required to have a long lifetime in the order of months to years, but the constituent sensor nodes have limited battery power. Therefore, it is one of the major challenges of sensor networks to maximize the network lifetime under power constraint. There have been reported many routing algorithms for the network lifetime maximization [2, 3], which dealt mainly with the energy efficiency. In certain sensor networks, reliability becomes a critical factor when the information collected by sensor nodes needs to be conveyed reliably to the sink node. Though most of the previous works on reliability in sensor networks focused on designing new MAC or transport layer [4, 5], the authors in [6] presented a routing protocol to reduce energy consumption for reliable communications. They proposed a heuristic routing algorithm based on the link cost that incorporates the link error rate in order to reflect the potential retransmission cost for successful packet delivery. Conventional energy-efficient routing protocols preferred the multi-hop routes with short hop-to-hop distances to singlehop routes with long distances [2, 3]. The reason is that the transmission power required to support a constant transmission success probability over one hop is proportional to the κth power of the distance between two nodes1 . In practice, 1 The parameter κ is typically 2 for short distances and omni-directional antennas, and is about 4 for longer distances.

IEEE Communications Society / WCNC 2005

however, reliability in sensor networks should be supported such that the success probability of end-to-end (or sensor-tosink) transmission, rather than hop-by-hop transmission, meet the requirement specified by the given applications. The endto-end success probability is the product of multiple per-hop success probabilities. Accordingly, the per-hop success probability required for the target end-to-end success probability increases as the number of hops increases, thereby requiring a higher power over each hop. This demonstrates that choosing a path consisting of a large number of short hops is not always optimal for energy saving. Furthermore, for such a multi-hop path, it is not necessarily optimal to make the per-hop success probability equal for every link, as the distances may differ one another. Therefore, in this paper, we investigate the problem of maximizing the network lifetime under the reliability constraint which is given in terms of the end-to-end success probability. We divide the problem into two constrained optimization subproblems: The first sub-problem is to minimize the energy consumption for guaranteeing the reliability constraint by controlling the per-hop success probability of each link along a given routing path, which can be done by controlling the transmission power allocation to each link. The second subproblem is to maximize the network lifetime by controlling the time fraction of using each routing path in consideration of the energy consumption determined by the above power allocation algorithm. Out of the two sub-problems, we can get an optimal power allocation algorithm and an optimal routing algorithm, respectively. However, the resulting optimal routing algorithm has high implementation complexity, and hence we devise a heuristic algorithm that has a lower and tractable complexity. We demonstrate by simulations that a trade-off relation exists between the network lifetime maximization and the reliability constraint, and that the proposed routing algorithm combined with the optimal power allocation increases the network lifetime significantly. II. S YSTEM M ODEL We consider a network composed of multiple sensor nodes and one sink node. Each sensor node periodically sends its acquired data to the sink node located at the center of the sensor field. We assume that a time division multiple access (TDMA)-based medium access control (MAC) protocol

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is employed. Time is divided into periodic fixed-size MAC frames, and each MAC frame comprises multiple time slots. Each sensor node is allowed to transmit packets in its own allocated time slots in each MAC frame so that no collision can occur. The sink node broadcasts a control packet that contains the time slot information and the routing information at the beginning of each MAC frame. We assume that the control packet can be received by all the sensor nodes in a single hop thanks to a larger transmission power used by the sink node. In each time slot, only the sending and receiving nodes are kept awake while others stay in the sleeping (or doze) state, which minimizes energy consumption. Since there exists no interference from other nodes’ transmissions, the transmission failure can occur only due to channel errors, which depends on the transmission power, channel gain, and receiver noise condition. The channel gain, normalized by the noise power, between two nodes is given by c (1) g = κ, d where d is the distance between two nodes, κ the path loss exponent, and c a constant, respectively. We define the probability of successful packet delivery (i.e., per-hop success probability) as a function of transmission power P tx , which is given by f (gP tx ). We assume that f (x) is a monotonically increasing and concave function of x, which is consistent with the practical observations [10, 11]. Each sensor node can adapt its transmission power up to Pmax . The transmission power used by a sensor node in a given time slot is determined by the sink node, and also announced via the control packet transmitted at the beginning of each MAC frame. The energy consumption for the transmission is assumed to be given by E =E tx

elec

+ P /R, tx

III. O PTIMAL P OWER A LLOCATION A LGORITHM In this section, we first assume that a routing path is given, and develop the optimal power control schemes that allocate the transmission power to each sensor over the path in order to minimize the total energy consumption while meeting the reliability constraint. We first consider the system in which sensor nodes can control the transmission power in continuous level. However, in practice, sensor nodes may not have the capability of controlling power continuously, but rather support only a finite set of power levels. Accordingly, we also consider the system which allows only discrete power levels. A. Continuous Power Level We first consider the continuous power level case. Let gi and Pi denote the channel gain normalized by the noise power and the transmission power at the i-th link, respectively. Then, the end-to-end success probability is given by H

f (gi Pi ),

(3)

i=1

where H denotes the number of links over the path. The problem of the total energy consumption minimization under the reliability constraint is formulated as follows: min

(2)

where E elec denotes the energy consumption of the electronic circuitry and R is the transmission rate in packets per second. We further assume that the energy consumption for packet reception is fixed at E rx . We ignore the energy consumption in the sleeping state, assuming that it is much smaller than the transmission and reception energy consumption. The sink node should know the channel gains of all the links in order to determine the transmission power and select the routing path for each sensor node. We assume that each sensor node periodically reports to the sink node the information about channel gains between itself and all the neighboring nodes. Noting that sensor networks are usually stationary, we can make the reporting period sufficiently large to keep the required signaling overhead minimal. Reliable packet delivery can be also achieved through retransmission schemes [6]. In this case, the probability of successful packet delivery depends also on the number of the retransmissions. However, the time slots secured for retransmissions in the case of the TDMA system is likely to cause network resource waste. Since the time slots are allocated at the beginning of a MAC frame, the number of allocated time slots should be made according to the maximum allowed


retransmissions, but they would be wasted if there are not that many retransmissions. Therefore, we limit the scope of our discussion to the system having no retransmission in this paper.

subject to

H i=1 H

(E elec + Pi /R) + (H − 1) · E rx

(4)

f (gi Pi ) ≥ Qsuccess ,

i=1

0 ≤ Pi ≤ Pmax ,

where Qsuccess denotes the target end-to-end success probability. Note that the energy consumed by the sink node for reception is excluded in Eq. (4), as the sink node is usually mains-powered, and hence its energy consumption is not of our concern. If an optimization problem is transformed into the standard convex optimization problem, it is not difficult to solve the problem because there are many efficient algorithms available for solving convex optimization problems. Fortunately, the problem in (4) is a convex optimization problem, as hi (Pi ) ≡ H log f (gi Pi ) is a concave function and hence − i=1 hi (Pi ) is convex in vector P = (P1 , · · · , PH ). If we apply the the Karush-Kuhn-Tucker (KKT) conditions [7] to the problem, then the optimal power Pi∗ satisfies the following relation:   < 0 , if Pi∗ = 0 ∂hi (Pi ) 1 = 0 , if Pi∗ ∈ (0, Pmax ) , = − ∂Pi Pi =P ∗ λR  > 0 , if Pi∗ = Pmax i (5)

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where λ is the Lagrangian multiplier chosen to meet the reliability constraint. We can find the optimal solution of the problem in (4) using the primal-dual interior point method [8]. Note that the problem in (4) may not have any feasible solution, since there may exist no power allocation that simultaneously meets all the constraints if the reliability constraint is too strict. We can exclude such paths when considering the routing path selection in the next section.

algorithm is optimal to the problem of maximizing the endto-end success probability under the constraint of the sum of power levels as follows: H

max subject to

i=1 H

hi (li )

(8)

li ≤ N,

i=1

li ∈ {1, · · · , L},

B. Discrete Power Level We now consider the discrete power level case. The power set that a sensor node can support is given by {∆P , 2∆P , · · · , L∆P }, where L is the number of the power levels, and ∆P ≡ Pmax /L. Let li denote the power level over the i-th link. Under the discrete power level arrangement, the problem in (4) is reformulated as follows: H

min

(E elec + li ∆P /R) + (H − 1) · E rx (6)

i=1

subject to

H

where hi (x) ≡ log f (gi ∆p x) and N is a positive integer. Let li∗ (n) and li∗ (n+1) denote the optimal solution to the problem in (8) for N = n and for N = n + 1, respectively. We first prove the following relation. For all i, li∗ (n + 1) ≥ li∗ (n),

where li∗ (0) = 0. If there exist p and q such that lp∗ (n + 1) < lp∗ (n) and lq∗ (n + 1) > lq∗ (n), then the optimality of li∗ (n) yields the following inequality

f (gi li ∆P ) ≥ Qsuccess ,

H

i=1

li ∈ {1, 2, · · · , L}.

f (gi min(li + 1, L)∆P ) . φi = f (gi li ∆P )

(7)

Then, we consider a greedy algorithm that increases the power level of the link of which the increment is the highest, as outlined below: 1) Allocate the lowest level to all links, i.e., li ← 1 for i = 1, 2, · · · , H. 2) Increase by one the power level of the link that has the highest value of φi , i = 1, 2, · · · , H, i.e., li∗ ← li∗ + 1 for i∗ = arg maxi φi . 3) Repeat the above process until the end-to-end success probability becomes equal to or larger than the target value. It may happen that a path does not attain the target endto-end success probability threshold even if all the links are allocated with the maximum power level. Then, we exclude that particular path from the routing path selection. Theorem 1: The greedy algorithm is the optimal solution to the problem in (6). Proof: The problem in (6) is equivalent to minimizing the sum of power levels of each link under the reliability constraint. For the proof, it suffices to show that the greedy


hi (li∗ (n)) > hq (lq∗ (n + 1) − 1) +

i=1

This problem cannot be solved through the standard convex optimization because it is an integer programming problem. So, instead, we first define an incremental gain that can be obtained by increasing power level, and devise a new algorithm based on it. Specifically, we introduce the incremental ratio φi of the per-hop success probability contributed by the power level increment at the i-th hop, i.e.,

(9)

H

hi (li∗ (n + 1)).

i=1,i=q

(10) Since hi (x) is a concave function, the following inequality holds: hq (lq∗ (n + 1) − 1) − hq (lq∗ (n))

≥ hq (lq∗ (n + 1)) − hq (lq∗ (n) + 1),

(11)

where the equality holds when lq∗ (n + 1) − lq∗ (n) = 1. We combine the two inequalities (10) and (11) to obtain the following inequality hq (lq∗ (n) + 1) +

H

hi (li∗ (n)) >

H

hi (li∗ (n + 1)), (12)

i=1

i=1,i=q

which contradicts the assumption that li∗ (n + 1) is the optimal solution for N = n + 1. This prove that the optimal solution satisfies the relation (9). H Since l∗ (n + 1) ≥ li∗ (n) for all i and i=1 li∗ (n + 1) − H ∗ i ∗ i=1 li (n) = 1, there exists only one i such that li (n + 1) = ∗ li (n) + 1 while the others remain intact. Therefore we get H i=1

hi (li∗ (n + 1)) = max{hj (lj∗ (n) + 1) − hj (lj∗ (n))} j

+

H

hi (li∗ (n)).

(13)

i=1

Accordingly it is optimal to allocate the additional unit power ∆p to the link with the highest increment of hi (li ).

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TABLE I N OTATION D EFINITIONS N Ei Gi Ls usp Pspr

gspr

the transmission power of Pspr , which is determined by (2). Then, the time duration for the battery of node i to drain out is given by

the total number of sensor nodes the initial energy of node i the packet generation rate at node i the number of distinct paths from node s to the sink the time fraction that the p-th path of node s is used the transmission power of node r when the p-th path of node s is used, which is determined through the optimal power allocation algorithm the channel gain at the link originated from node r on the p-th path of node s

s=1

rx + srx spi E )usp

.

(15)

Now, we define the lifetime of the network to be the minimum of the lifetimes of all sensor nodes, i.e., T = min Ti .

(16)

i

A sensor node uses multiple candidate paths in a mixed order and frequency during the network lifetime, meeting the constraint: Ls usp = 1. (17)

IV. ROUTING A LGORITHM In general, there exist multiple routing paths between a particular sensor node and the sink node. For each path, it is possible to determine the energy consumption required to support the target reliability using the optimal power allocation algorithm presented in the previous section. If all the packets were routed through the path with the minimum energy consumption, the batteries of the sensor nodes along that path would be drained out quickly while other sensor nodes would have some residual energy. In order to maximize the network lifetime, it is necessary to route the packets such that the energy consumption is balanced among the multiple paths. So, in this section, we present two routing algorithms that determine how to distribute the traffic among the multiple paths: one is the optimal routing algorithm based on a linear programming approach, and the other is the minimum-cost path routing algorithm with lower and tractable complexity. A. Optimal Routing Algorithm We define some notations as summarized in Table I for the problem formulation. If node r is on a path from sensor s to the sink, it can relay the packets originated from node s only when they are received successfully. Let hspr denote the number of hops from node s to node r on the p-th path of node s. If node r does not exist on the p-th path of node s, the value of hspr is set to ∞. Then, the probability stx spr that node r transmits a packet from node s is given by   k∈Kspr f (gspk Pspk ), if 0 < hspr < ∞ stx , 1, if hspr = 0 spr =  0, if hspr = ∞ (14) where Kspr ≡ {k | 0 ≤ hspk < hspr }. Similarly, let srx spr denote the probability that node r receives a packet from node s, which is equal to the probability that the preceding node tx transmits the packet to node r, given by srx spr = sspk , where k is the index of the preceding node (i.e., hspk = hspr − 1). The rate of energy consumption by sensor node i for transmitting packets and relaying others’ packets is given N Lits s tx rx rx tx Gs (stx by s=1 p=1 spi Espi +sspi E )usp , where Espr is the energy consumption by node r for transmitting packets with


Ei tx tx p=1 Gs (sspi Espi

Ti = N Ls

p=1

Then, the problem of maximizing the network lifetime is formulated as follows: max

T

subject to T

(18) Ls N

tx rx rx Gs (stx spi Espi + sspi E )usp

s=1 p=1

≤ Ei , i = 1, 2, · · · , N, Ls

usp = 1, s = 1, 2, · · · , N,

p=1

usp ≥ 0, s = 1, 2, · · · , N. The problem in (19) can be easily reformulated into a standard linear programming (LP) problem [9]. We can obtain the optimal value of usp by solving the LP problem. Note that sensor node s is supposed to use path p, p = 1, 2, · · · , Ls , in proportional to the determined optimal value usp . B. Minimum-Cost Path Routing Algorithm The optimal routing algorithm derived above involves very high implemental complexity, thus making it impractical. So, in this subsection, we develop a suboptimal algorithm, called minimum-cost path routing (MCPR) algorithm, that can possibly perform close to the optimal algorithm but has lower and tractable complexity. We first define the link cost function as the ratio of the required transmission energy for a packet transmission to the remaining energy, similar to the case in [2]. Specifically, we define the cost of the link originated from node i on the p-th path of node s as follows: Cspi =

tx Espi + E rx , (E i /Ei )w

(19)

where E i denotes residual energy, and w is a weighting exponent on the remaining energy. Note that the residual energy is normalized by the initial energy because sensor nodes can have different initial energy levels. The path cost is calculated as the sum of the costs of all the links along the path. Then, the MCPR algorithm selects

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3600

the path with the least cost. The path selection requires an additional knowledge on the residual energy of each sensor node. In order to support this, we may arrange all sensor nodes to report the residual energy periodically, but it results in a large transmission overhead. Instead, we take an alternative method that the sink node estimates the residual energy of all sensor nodes based on the energy consumption determined by the power allocation algorithm. As mentioned earlier, the sink node periodically broadcasts the routing information and the allocated time slot information at the beginning of each MAC frame.

Network Lifetime (frame)

3400

2600

2400

0

50

100

150 Weighting factor

200

250

300

Fig. 1. Network lifetime of various combinations of algorithms with respect to weighting factor w. 3600

3400


We conduct computer simulations in order to evaluate the performance of the proposed algorithms. For the performance comparison, we consider three different combinations: 1) nonoptimal power allocation and MCPR (namely, NP-MCPR), 2) optimal power allocation and MCPR (namely, OP-MCPR), and 3) optimal power allocation and optimal routing (namely, OPOR). In the case of non-optimal power allocation, we take the following arrangement: the transmission power is allocated such that all links have the same per-hop success probability, which is set to the value that can guarantee the target end-toend success probability for the path with the maximum number of hops. We randomly place 50 sensor nodes in a 100 m x 100 m sensor field and put the sink node at the center. For simulations, we use the success probability function of f (γ) = exp(−1/F ), where F = F0 γ, representing the fading margin for a SNR γ and a parameter F0 , which is set to 7.5 [11, 12]. We set κ = 2, c = 2.6 × 105 m2 /W , R = 10 packets per second and E elec = E rx = 500 µJ. These parameters are adopted from [13], assuming that the per-hop success probability is 95% and the packet length is 10000 bits. Pmax is set to the value for which the success probability of 99% can be achieved over the distance of 20 m. Each sensor node is supplied with an initial energy level of 50 J and generates one packet per frame. The target end-to-end success probability is set to 90%. We first determine the value of the weighting factor w which optimally makes balance between the energy consumption and the residual energy in the link cost for the MCPR algorithm. Fig. 1 shows the performance of the three combinations of algorithms with respect to the weighting factor w. We observe that the performance of the MCPR algorithm becomes close to that of the optimal routing algorithm when w lies between 50 and 150, regardless of the power allocation algorithms. Hence, we set the value of w to 100 in the subsequent simulations. We examine the influence of the route update period of the MCPR algorithm to the network lifetime. Fig. 2 depicts the network lifetime of various different settings with (‘w’) and without (‘w/o’) signaling overhead with respect to the route update period. The signaling overhead includes only the power consumption of the sensor nodes for receiving the control packet as we assume that the power level with

OP−OR OP−MCPR NP−MCPR

3000

2800

V. S IMULATION R ESULTS


3200

3200 OP−OR OP−MCPR w/o overhead OP−MCPR w/ overhead NP−MCPR w/o overhead NP−MCPR w/ overhead

3000

2800

2600

2400

0

20

40

60

80 100 120 Route update period (frame)

140

160

180

200

Fig. 2. Network lifetime of various different settings with respect to route update period (w = 100).

which sink node transmits control packets is large enough to deliver packets to all sensor nodes in one hop successfully. The signaling overhead is not taken into consideration in the case of the optimal routing as the frequency of the route update is usually hard to be determined. We observe from the figure that the network lifetime increases as the update period decreases, but this requires more signaling overhead caused by frequent routing updates. Moreover, we observe that the effect of signaling overhead is negligible when the update period is larger than 50 frames, and the margin between the MCPR and the optimal routing algorithm is less than 3% when the update period is less than 150 frames. Accordingly, we set the update period to 50 frames in the subsequent simulations. We compare the performances of the continuous and the discrete power level cases. Fig. 3 plots the network lifetime of both cases as the different number of power levels varies. (Note that the case of the continuous power level is represented by straight lines). We observe, as expected, that the performance improves as the number of power levels increases, but it requires higher implementation cost. Note that the

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3600

50 to 80. We may interpret this phenomenon as follows: the more sensor nodes lies in the field, the more packets the sensor nodes near the sink node have to relay, thus causing higher energy consumption.

3400

Network lifetime (frame)

3200

OP−OR (con.) OP−OR (dis.) OP−MCPR (con.) OP−MCPR (dis.) NP−MCPR (con.) NP−MCPR (dis.)

3000

2800

2600

2400

2200

4

6

8

10 Number of power level

12

14

16

Fig. 3. Network lifetime of various combinations with respect to the number of power levels (w = 100, update period = 50 frames). 3600 OP−OR (N=50) OP−MCPR (N=50) NP−MCPR (N=50) OP−OR (N=80) OP−MCPR (N=80) NP−MCPR (N=80)

3400 3200


3000 2800 2600 2400 2200 2000

VI. C ONCLUSIONS In this paper, we have investigated how to maximize the network lifetime while guaranteeing the end-to-end success probability in a wireless sensor network. The resulting optimal power allocation and the optimal routing algorithms form as a cross-layer strategy in that the former is a physical layer processing and the latter is a network layer processing. The optimal power allocation algorithm was designed to minimize the total energy consumption required to support the reliability constraint for each path. In contrast, the optimal routing algorithm was to maximize the network lifetime by taking balanced energy consumption among the constituent paths. The combined optimal algorithms can balance the trade-off between the network lifetime and the reliability constraint. However, since the complexity of the optimal routing algorithm is high, we also proposed a suboptimal algorithm called MCPR instead. We demonstrated through simulations that the MCPR algorithm performs very closely to the optimal algorithm. We have also confirmed through simulations that the combined power allocation and routing optimization can increase the network lifetime remarkably over the non-optimal algorithms.

1800

R EFERENCES

1600 1400 0.9

0.905

0.91

0.915

0.92 0.925 0.93 Reliability threshold

0.935

0.94

0.945

Fig. 4. Network lifetime of various combinations with respect to target end-to-end succuss probability (w = 100, update period = 50 frames).

performance of the discrete power level case approaches that of the continuous power level case by 6% when the number of power levels is 6 or larger. Fig. 4 shows the network lifetime as the target ene-toend success probability varies. We observe that the network lifetime of each combination decreases as the threshold increases. This implies that a trade-off relation exists between the network lifetime and the reliability constraint. We also observe that the OP-MCPR combination outperforms the NPMCPR combination by about 35% in terms of the network lifetime. In order to examine the performance of the MCPR algorithm in comparison with a random routing algorithm, we have additionally conducted simulations on the combination of non-optimal power allocation and random routing, from which we confirmed that the OP-MCPR combination outperforms this NP-RR combination by about 95%. (Note that we omit the NP-RR combination plot in the figure as it lies far below all the other curves). In addition, we observe from Fig. 4 that the network lifetime decreases when the number of sensor nodes increases from


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