Joint Data Rate and Power Allocation for Lifetime ... - IEEE Xplore

1 downloads 0 Views 429KB Size Report
for Lifetime Maximization in. Interference Limited Ad Hoc Networks. Riku Jäntti, Member, IEEE, and Seong-Lyun Kim, Member, IEEE. Abstract— In this paper, we ...
1086

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 5, MAY 2006

Joint Data Rate and Power Allocation for Lifetime Maximization in Interference Limited Ad Hoc Networks Riku J¨antti, Member, IEEE, and Seong-Lyun Kim, Member, IEEE

Abstract— In this paper, we consider the following problem in the wireless ad hoc network: Given a set of paths between source and destination, how to divide the data flow among the paths and how to control the transmission rates, times, and powers of the individual links in order to maximize the operation time of the worst network node. If all nodes are of equal importance, the operation time of the worst node is also the lifetime of the network. By solving the problem, our aim is to investigate how the network lifetime is affected by link conditions such as the maximum transmission power of a node and the peak data rate of a link. For the purpose, we start from a system model that incorporates the carrier to interference ratio (CIR) into a variable data rate of a link. With this, we can develop an iterative algorithm for the lifetime maximization, which resembles to the distributed power control in cellular systems. Numerical examples on the iterative algorithm are included to illustrate the network lifetime as a function of the maximum transmission power and the peak data rate. Index Terms— Lifetime, power control, rate control, routing, wireless ad hoc network.

I. I NTRODUCTION

T

HE wireless ad hoc network is an autonomous system that does not have any fixed network infrastructure, where the topology of a network dynamically changes, as nodes are free to move around [1]. For ad hoc networks, the issue of routing packets between any pair of nodes is not a trivial task because of mobility of the nodes. A path that was considered to be optimal at a given instant might not work at all a few moments later. Moreover, variations of wireless channels as well as interference coming from other transmissions nearby add uncertainty to the routing. The lifetime of an ad hoc network depends mainly on each mobile node’s battery capacity. With this constraint, routing algorithms must provide energy-efficient route discovery and maintenance mechanisms (see [2]–[7]) and literature therein). Even when all the traffic is routed through minimum energy paths to the destination, the lifetime of the network may not be

Manuscript received January 27, 2004; revised December 7, 2004; accepted March 23, 2005. The associate editor coordinating the review of this paper and approving it for publication was H. Yanikomeroglu. This paper was presented in part at IEEE Vehicular Technology Conference Spring, Jeju, Korea, April, 2003. This work is supported jointly by the Academy of Finland (grant no. 74585) and BROMA-ITRC, Ministry of Information and Communications, Korea. Riku J¨antti is with the University of Vaasa, P. O. Box 700, Vaasa, FIN-5101, Finland (e-mail: [email protected]). Seong-Lyun Kim is with the School of Electronic Engineering, Yonsei University, Seoul, 120-749, Korea (email: [email protected]). Digital Object Identifier 10.1109/TWC.2006.05016

maximized. This is because some nodes may belong to several routes at the same time, quickly exhausting their batteries and shortening the lifetime of the network. An interesting issue is to investigate the lifetime of an ad hoc network and to see how it is affected by other factors. For example, the power increase of a node would accelerate energy consumption and thus might shorten the lifetime. However, a node can send packets with a higher transmission rate due to increased power, and thereby it would also be possible to prolong the lifetime by reducing the interference duration to the other nodes. Thus it is not obvious whether the power increase will affect the lifetime of an ad hoc network positively or negatively. To the best of our knowledge, there has not been any theoretic framework to analyze the issue, although there are numerous papers dealing with energyefficient routing algorithms. Main purpose of this paper is to figure out relationships between the network lifetime and the link conditions such as the maximum transmission power of a node and the peak data rate of a link. We consider the case, in which all nodes are of equal importance. This could be the case for instance in a peer-topeer network, where all the transceivers are user devices. In such case, the lifetime of the network is equal to the battery operation time of the worst node in the network. A similar definition of the lifetime can be found, for example in [3]. An alternative definition for the lifetime would be the time until there does not exist a route between a given source destination pair. This definition of the lifetime would be rather difficult to theoretically handle, and it is out of the scope of this paper. However, we believe that our analysis and results in this paper will be applicable to maximizing the lifetime of a network, under any relevant definition of the network lifetime. Most of the previous works on ad hoc networks utilize very simple link models, in which the power consumption in a link is assumed to depend only on the distance between transmitter and receiver. This model is valid, if only a single link is active on a given frequency band at the time. However, if the frequency is reused, simultaneous transmissions will cause interference that affects the link performance negatively. In this paper, we consider a system utilizing the direct sequence spread spectrum (DSSS) technique to share the spectrum. We assume that the instantaneous data rate of a link can be controlled by varying the spreading factor. The achievable data rate of a link is a function of the receiver carrier-to-interference ratio (CIR). Perhaps the closest system model to ours can be

c 2006 IEEE 1536-1276/06$20.00 

¨ JANTTI AND KIM: JOINT DATA RATE AND POWER ALLOCATION FOR LIFETIME MAXIMIZATION IN INTERFERENCE LIMITED AD HOC NETWORKS

found in [8], [9], and [10] which consider the CIR level of each link into their analysis on power control [8], capacity issues [9], and routing [10] of an ad hoc network. With the system model, we consider the following problem: Given a set of paths between source and destination, how to divide the data flow among the paths and how to control the transmission rates, times, and powers of the individual links, in order to maximize the operation time of the worst network node. If all nodes are of equal importance, the operation time of the worst node is also the lifetime of the network. By solving the problem, our aim is to investigate how the network lifetime is affected by link conditions such as the maximum transmission power of a node and the peak data rate of a link. We develop an iterative algorithm that maximizes the lifetime of the ad hoc network of our interest. The iterative algorithm resembles to the distributed power control in cellular systems [11]–[13]. Using the proposed algorithm, we can investigate how the optimal routing would change, for example, as node’s transmission power varies and how it affects the lifetime. Our results suggest that by increasing the transmission power, the data rates of the links can be increased and transmission times can be shortened, mitigating the interference in the system. Thus, the overall effect is that by using higher transmission power, less energy is needed to maintain the average data rates in the network. When there is a strict peak data rate constraint on each link, there will be some power threshold in each link, after which incremental power affects the lifetime of the network negatively. In this case, we can improve the situation by introducing power control in order to decrease power consumption of the links. The rest of this paper is to describe how we obtained our results. In the next section, we will provide our system model. In Section III, we will define a maximum lifetime routing problem for jointly determining routing paths, mean rates, transmission power and transmission time of each node. An iterative algorithm that exactly solves the problem is described in Section IV. Section V contains simulation results that provide meaningful insights on the lifetime of an ad hoc network. Finally Section VI concludes the paper. II. S YSTEM M ODEL Consider an ad hoc network consisted of nodes that transmit data and signalling packets asynchronously employing direct sequence spread spectrum (DSSS) waveforms. A node i having a packet to send to node j transmits it at a given instant with a given probability φij . In other words, φij ≥ 0 denotes the average fraction of time in the routing (scheduling) interval T that node i transmits to node j. This can be interpreted as a form of stochastic routing resembling to the concept suggested in [7]. We assume that there is only a single frequency band available. Hence a node must divide its time for transmitting and receiving. Consequently, we require that  (φij + φji ) ≤ 1, (1) j=i

for each node i.

1087

Let (i, j) denote a link between nodes i and j, where node i is a transmitter and node j is a receiver. Further, the instantaneous data rate rij (t) of a link (i, j) is a monotonous function of the instantaneous received carrier-to-interference ratio (CIR): γij (t) =





gij (t)pij (t)

⎣gkj (t)

k=i k=j



l=k l=i l=j



,

(2)

pkl (t)⎦ + νj

where pij (t) denotes the transmitter power allocated to link (i, j) at time t, gij (t) is the link gain between transmitter i and receiver j, and νj denotes the noise power at the receiver j. We assume that a node, if it transmits, will transmit with a fixed power value pij . Hence, pij (t) = pij with probability φij and pij (t) = 0 with probability 1 − φij . The link gains, gij (t) are assumed to be mutually independent stationary (ergodic) stochastic processes with mean values gij . Hence, also the resulting process rij (t) is a stationary process. In order to achieve the instantaneous rate with tolerable bit error probability, we assume that the bit-energy-to-interference+noisepower-density Eb /(I0 + N0 ) of the link must be greater than or equal to some predefined value Γij . Let r˜ij (Φ, P ) denote the expected data of the link (i, j) conditioned on pij (t) = pij . The expected rate is a function of the activity factors {φij } denoted by the vector Φ and transmit power levels {pij } denoted by the vector P . It follows that the expected rate can be written as: r¯ij = r˜ij (Φ, P )φij

(3)

If the link adaptation is very fast, i.e., the data rate of the link is changed immediately when the channel conditions change, then the instantaneous data rate of a link is a function of the instantaneous CIR, rij (t) = rij (γij (t)) while the mean data rate of a link would follow the expected value of the rate r¯ij = E{rij (γij (t))}. On the other hand, if the transmission time interval (TTI), during which the modulation and coding is kept fixed, is longer than the coherence time of the channel, then the data rate becomes a function of the average carrier power to the average interference ratio averaged over the TTI: γ¯ij (Φ, P ) =



k=i k=j

⎡ ⎣gkj

gij pij 

l=k l=i l=j



(4)

φkl pkl ⎦ + νj

The latter model constitutes the lower bound for the previous case. This follows directly form the Jensen inequality. For simplicity, we will focus on the case, in which the relationship between rate and CIR is linear and the average rate follows the Jensen bound. Furthermore, we assume that there is some level of coordination (medium access control) between the nodes such that no collision can happen between adjacent links sharing a common source or a destination node. Under these simplifying assumptions, the average rate of link (i, j) conditioned on the link activity would be equal to:  W γ¯ij (Φ, P ) (5) r˜ij (Φ, P ) = min rmax , Γij

1088

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 5, MAY 2006

This model holds approximately well for DSSS systems, in which different rates are obtained by changing the spreading factor, i.e., chip rate per bit rate W/rij (t), and rate adaptation frequency is slower than the rate at which the channel conditions change. The spreading factor must be greater than one. Hence we bound the rate to some maximum value rmax < W . It should be noted that (1) alone is not enough to guarantee the existence of collision free schedule. The problem of finding such a schedule is equivalent to the edge coloring problem of a graph and the existence of a solution is very much dependent on the network (graph) topology. If there is no collision avoidance mechanism, i.e., an ALOHA type of medium access control is used, then the effect of collisions could be taken into account by scaling the average rate of a link (i, j) down by the collision probability which is a function of the activity factors of links adjacent to link (i, j). Although our model might seem to be very restrictive, we note that it is straightforward to extend our results to more general forms of r˜ij (Φ, P ) as long as the mapping rij (Φ, P ) is contraction with respect to Φ in the domain r¯ij /˜ P ≥ 0 and Φ ≥ 0 for sufficiently small r¯ij . The model considered in this paper was chosen due to its simplicity in order to keep the derivations short and readable. It also allows us to make comparisons with the power control literature instead of providing lengthy convergence proofs. We believe that even though our model is simple, the obtained results give good insight to the lifetime of ad hoc networks. Assume that we can find all possible paths for any sourcedestination pair using, e.g., some on-demand routing algorithms like those suggested in [14] and [15]. Let P (k,l) denote the set of nonrecurrent paths between nodes k and l and Pij , the set of paths using link (i, j). From the total usage of link (p) (i, j), i.e., φij , a partition φij is used by a path p. Therefore, we must have  (p) φij (6) φij = p∈Pij

We assume that node k wishes to send packets to node l at the average rate of R(k,l) . Let r(p) denote the average data rate path p. If the rate requests are feasible, then  used by (p) r ≥ R(k,l) . The rate of an individual link, must (k,l) p∈P  fulfill r¯ij = p∈Pij r(p) , where r¯ij of link (i, j) denotes the average data rate of the link given by r¯ij = r˜ij (Φ, P )φij . Furthermore, we assume that all the links on a path p have the same Eb /(I0 + N0 ) target, Γ(p) . It thus follows from (5) and (6) that r(p) Γ(p) (p) φ (7) φij =  (q) Γ(q) ij q∈Pij r Solving φij from (3) and (5) indicates that φij ≥

r¯ij def min = φij rmax

(8)

Hence, we can take the peak data rate into account by imposing a minimum transmission time constraint. That is, in order to the time fraction φij should be at least φmin ij achieve the average data rate r¯ij = r˜ij (Φ, P )φij when the the peak data rate is bounded above by rmax . Clearly, a necessary condition for the system to be feasible, i.e., for a solution to exist, is that (1) holds for φij = φmin ij .

p1 p2 p3 p4

3

1

Fig. 1.

2

A three node network.

Example 1. To clarify our notation, consider a fully connected three node network illustrated in Fig. 1. Consider the case, in which nodes 1 and 2 are communicating with node 3 with data rates R(1,3) and R(2,3) , respectively. There are two alternative paths between node 1 and 3: The one hop path {(1, 3)} denoted by p1 and the two hop path {(1, 2), (2, 3)} denoted by p2 . Similarly, there are two paths from node 2 to node 3: The one hop path {(2, 3)} denoted by p3 and the two hop path {(2, 1), (1, 3)} denoted by p4 . Hence, we have P (1,3) = {p1 , p2 } and P (2,3) = {p3 , p4 }. The data rate of the route (1, 3) is divided among the two possible paths such that r(p1 ) is transmitted using path p1 and r(p2 ) is transmitted using path p2 , R(1,3) = r(p1 ) + r(p2 ) . Now let us consider the link (1, 3). This link is utilized both paths p1 and p4 . Thus P13 = {p1 , p4 }. Let us consider node 1. There are two links originating from that node, (1, 2) and (1, 3), and one link terminating to it (2, 1). Hence the constraint (1) becomes φ12 + φ13 + φ21 ≤ 1. Since the link (1, 3) is shared by the (p ) (p ) paths p1 and p4 we have φ13 = φ131 + φ134 . III. M AXIMUM L IFETIME ROUTING P ROBLEM The expected lifetime of a node i is given by τi = 

Ei  j=i φij (Pt + pij ) + j=i φji Pr + Ps

(9)

where Ei is the energy left in the battery of the node i in the beginning of the routing interval, Ps denotes the power consumption of the node without transceiver, Pt is the power consumed by the baseband part of the transmitter and Pr denote the power consumption of the receiver. If all nodes are of equal importance, the expected lifetime of the network is as large as the expected lifetime of the worst node. Hence, the objective of the maximum lifetime routing problem can be written as max{mini {τi }} or equivalently as min{maxi {τi−1 }}. With this objective function and the constraints discussed in previous section, we have the following problem to solve. Maximum Lifetime Routing Problem (MLRP) Take (10), subject to (p)

r(p) = φij r˜ij (Φ, P ),

(11)

¨ JANTTI AND KIM: JOINT DATA RATE AND POWER ALLOCATION FOR LIFETIME MAXIMIZATION IN INTERFERENCE LIMITED AD HOC NETWORKS

⎧ ⎛ ⎛ ⎞ ⎞⎫ ⎬ ⎨1    (p) (p) ⎝ ⎝ min max φij (Pt + pij ) + φji Pr ⎠ + Ps ⎠ (p) ⎭ φij ,pij i ⎩ Ei j=i

for every link (i, j) on a given path p,  r(p) ≥ R(k,l) ,

p∈Pij

(12)

Let xij = φij pij denote the energy consumption of the link in unit time interval. By summing over p ∈ Pij and multiplying pij , we can write (17) as follows: ⎛

for every source destination pair (k, l), ≥ 0,

for every link (i, j) on a given path p,  (p) φij φij =

 (13)



(φij + φji ) ≤ 1,

(14)

(15)

j=i

for every node i, and 0 ≤ pij ≤ pmax

xij =



⎜ ⎟ r(p) Γ(p) ⎜  gkj νj ⎟ ⎜ ⎟, x + kl ⎜ W gij ⎟ ⎝ k=i l=k gij ⎠

p∈Pij

(18)

k=j l=i l=j

p∈Pij

for every link (i, j),

(10)

p∈Pji

p∈P (k,l)

(p) φij

1089

(16)

for every link (i, j). The problem is to find such a set of transmission times (p) {φij } and powers {pij } that the data rates of the paths r(p) belonging to a route between nodes k and l meet the average rate requirement of the route given by (12) while fulfilling the transmission time constraints (13)-(15) and the maximum transmission power constraint (16). Our goal is to investigate the sensitivity of the optimal objective function value (10) with respect to changes in pmax and rmax . Unfortunately, however, MLRP is a nonlinear optimization (p) problem with respect to φij and pij and as such nontrivial to solve directly. To solve the problem, we first define the transmission energy per unit scheduling interval T , φij · pij by xij . Then, we solve the problem in three steps: First, we consider the problem, in which the rate to path allocations r(p) are known and we solve the optimal energy allocation xij . Secondly, with the obtained xij , we optimally divide it into φij and pij . Finally, we update the rate to path allocations r(p) . The three steps are repeated until r(p) , φij and pij converge. Once the time fraction φij is obtained, fractions (p) {φij } allocated to different paths p can be obtained from (7). More detailed description of our solution method and its convergence is given in the next section.

for every link (i, j). The maximum rate limit imposes lower bound xij ≥ φmin ij pij . However, since pij is bounded below only by zero, we can ignore this bound. Let us define an ordered set of links L, in which we denote the ith link as (ai , bi ). Let H and η be a |L|× |L| matrix and a |L| × 1 vector, respectively (|L| denotes the cardinality of the ga b set L). The elements of H = [hij ] are defined as hij = j i gai bi νbi and the elements of η = (ηi ) are defined as ηi = . gai ,bi Furthermore, let us define a diagonal matrix Ξ(R) = diag{ξi }, where  (p) (p) Γ p∈Pai bi r (19) ξi (R) = W and R = (r(p) ) is a vector of data rates. It is well known that the linear equation system (18) has a nonnegative solution with respect to xij if and only if the spectral radius, the largest eigenvalue in modulus, of the matrix Ξ(R)H is less than one, i.e., ρ(Ξ(R)H) < 1 [16]. Hereafter, for a given rate to path allocation r(p) , we say that the problem (18) is feasible when ρ(Ξ(R)H) < 1. The problem is equivalent to the power control problem in cellular radio systems [11]–[13]. The similarity in the power control between ad hoc and cellular networks is also discussed in [8]. Now, let us denote the time fraction φai bi by φi , transmission power pai bi by pi , and transmission energy per unit time xai bi by xi . Let Φ = (φi ), P = (pi ) and X = (xi ) denote the corresponding vectors. In [12], it has been shown that, if the problem (18) is feasible, then it can be solved using the Jacobi iteration [16] of the following form: ⎞ ⎛  (n+1) (n) xi = ξi (R) ⎝ hij xj + ηi ⎠ j

def

IV. A N I TERATIVE A LGORITHM FOR THE M AXIMUM L IFETIME ROUTING P ROBLEM A. Transmission Energy Allocation Assume now that r˜ij is given by (5). It follows from (11) that r(p) (p) (17) φij = r˜ij (Φ, P )

=

  Ii R, X (n)

(20)

for all i and n = 0, 1, 2, · · · , with any nonnegative initial value (0) (0) x0i = φi pi . Note that Ii (R, X) is equal to the mean of received interference power divided by the mean link gain. Hence, it can determined locally in each link. Let I (R, X) denote a vector valued function of R and X.  This allows us to write the above as X (n+1) = I R, X (n) .

1090

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 5, MAY 2006

B. Transmission Time and Power Allocation Let us switch to the problem of dividing the energy xij between transmission time φij and power pij . With the help of xij , we can rewrite the lifetime (9) in terms of φij : Ei (φ P + φ ij t ji Pr + xij ) + Ps j=i

τi = 

(21)

We have no control over xij as it solely depends on the utilized rate to path allocation R. Consequently, in order to maximize the lifetime of each node, the transmission time φij should be as short as possible. This also maximizes the slack in constraint (15). We note that for given xij we can express the power constraint (16) in terms of φij as follows: xij (22) φij ≥ max p Recall that the average rate of a link r¯ij can be found from (11) and (14). Hence, we can now use (8) to express the peak data rate constraint as follows:  (p) r¯ij p∈Pij r = = φmin (23) φij ≥ ij rmax rmax Thus φij is bounded below by (22) and (23). Hence, we can conclude that the transmission time should be the largest of these two minimum bounds. Considering this, we suggest the following iterative algorithm to obtain the optimal values for energy, time and power. Joint Energy, Time, and Power Allocation (JETP) Algorithm (n+1)

xi

(n+1)

φi

(n+1)

pi

= Ii (R, X (n) )   (n+1) xi min = max , φi pmax

(24) (25)

(n+1)

=

xi

(n+1)

Positivity Ii (R, X) ≥ 0 for all X ≥ 0 Monotonicity X  ≥ X implies Ii (R, X) ≥ Ii (R, X). • Scalability αIi (R, X) > Ii (R, X) for all α > 1. (n) Therefore, Theorem 4 in [12] guarantees that xij converges even if the the transmission energies of the links are updated in an asynchronous fashion as long as each component is updated (n) asymptotically infinitely many times. Clearly, if xij converge (n) (n) so must φij and pij , since they are directly determined by (n) the value of xij . In each link, the lifetime is maximized, if the transmission time is minimized. The transmission time of a link φi is bounded below by (22) and (23). Hence, the best choice for φi is to take it to be as small as possible while xi should be solved from (18).  •

(26)

φi

For a given rate allocation that satisfies (12), the iteration (24) follows from the constraint on average data rate (11). Iteration (25) is needed to keep the rate below the maximum peak data rate. It also takes care of the constraints (13), (14) and (16). Iteration (26) is straightforward from the definition of xij . The constraint (15) depends on the selected rate to path allocation and cannot be guaranteed by JETP alone. Since JETP seeks to find the smallest (magnitude) Φ vector that would satisfy the constraints (11)-(16), it does its best to meet (p) the constraint. The obtained φai bi is mapped to φai bi using (7). Proposition 1. JETP converges for a fixed rate allocation R = (r(p) ), starting from any nonnegative initial values X (0) , Φ(0) , P (0) , to a solution that jointly maximizes (21) for all i, if such solution exists. Furthermore, the iteration converges even if the updates are done in an asynchronous fashion. Proof. The energy control mapping Ii (R, X) is a standard interference function with respect to X in the sense of Yates [12]. That is, it fulfills:



C. Transmission Rate Allocation There still remains the problem of allocating rates to paths, R = (r(p) ). For this, let us consider the inverse of the lifetime of the worst node written as: F (X, P ) = ⎧ ⎛ ⎞⎫   ⎬ ⎨ 1   P P t r ⎝ max xji + Ps ⎠ (27) 1+ xij + i ⎩ Ei ⎭ pij pij j=i

We note that for a given R, the optimal X can be obtained from the mapping X = Ii (R, X), which is a linear function with respect to R. Hence, with the help of this mapping, we can rewrite the objective function F (X, P ) in terms of R for fixed P : F (R, X, P ) = F (I(R, X), P ). In order to write the constraint (15) with respect to R, we will write φi and pi in terms of xi and then replace xi by Ii (R, X). In addition to that, we need to consider the mean rate constraint (12) and require that r(p) ≥ 0. The rest of the constraints are assumed to hold for the given X. We will denote this set of constraints for R by CR . Now we are ready to solve MLRP of (10)-(16) in terms of R, assuming Φ and P (and consequently X) in (11) are fixed. This can be done by using the linear programming technique. Then for the obtained rate R, we can solve X, Φ, and P by applying the JETP algorithm described in the previous section. Finally, we have the following algorithm for our maximum lifetime problem. Iterative Routing (IR) Algorithm 1) Start with any feasible (X (0) , Φ(0) , P (0) , R(0) ). 2) Solve the following linear programming problem    R(n+1) = argminR∈CR F I(R, X (n) ), P (n) 3) For the obtained R(n+1) , solve (n+1) (n+1) (n+1) ,Φ ,P ) by applying JETP. (X 4) if ||X (n+1) − X (n) || < , then stop; otherwise, set n = n + 1 and goto Step 2 Proposition 2. Starting with any feasible (X (0) , Φ(0) , P (0) , R(0) ), the IR algorithm will converge to (X ∗ , Φ∗ , P ∗ , R∗ ) that will minimize (10) under the constrains (11)-(16), if such a (X ∗ , Φ∗ , P ∗ , R∗ ) exits.

¨ JANTTI AND KIM: JOINT DATA RATE AND POWER ALLOCATION FOR LIFETIME MAXIMIZATION IN INTERFERENCE LIMITED AD HOC NETWORKS

Proof. Let us denote the set of feasible (X, Φ, P, R) that fulfill (11)-(16) by C. Now consider an iterative algorithm described by   (28) R(n+1) = argminR∈C F R, X (n) , P (n) X (n+1)

=

  argminX∈C F R(n+1) , X, P (n)

2

1091

3

1.5

1

4

0.5

0

1

2

5

8

9

(29) −0.5

(30)

where P (X, Φ) denotes a power vector that is written in terms xi of X and Φ by noting that pi = . The above algorithm φi is equivalent to the IR algorithm, since iteration (28) equals to Step 2 and iterations (29) and (30) can be shown to be equivalent to the JETP algorithm. It is notable that the mapping F is strictly concave with respect one of the variables R, X, and Φ at the time. Hence iterations (28)-(30) can be interpreted as a nonlinear GaussSeidel algorithm for solving the MLRP. Proposition 3.9 in [17] (p. 219) implies that the algorithm converges to the optimal  solution of MLRP, (X ∗ , Φ∗ , P ∗ , R∗ ). In order to solve the linear programming problem associated with rate to path allocation, we need to determine the network topology, the activity factors, and transmit powers used in all the links. It is noteworthy that the optimization problem does not scale well as the dimension of the network grows. The larger the network, the more paths there are between source and destination and thus the larger the dimension of the optimization problem. Hence, in practice, the rate to path allocation must be limited to a subset of possible paths. It is noteworthy that the JETP algorithm is totally distributed. So given the rate to path allocations, the activity factors and transmit powers can be solved locally in each link. Thus JETP could serve as a building block of more practical routing scheme, in which some heuristics are used to determine the rate to path allocations. We leave the development of such heuristics as further study. In case of mobile nodes, the topology of the network is changing and thus the optimal rate to path allocation is changing. If the rate at which the parameters (activity factors, power, and rates) are controlled is fast compared to mobile speed, then the IR algorithm can converge before the topology changes considerably. In that case, the scheme always selects the best parameters for the current network topology from the lifetime point of view. Of course, if the mobiles move very fast, then IR is not able to track the topology changes. V. N UMERICAL E XAMPLE As mentioned in Introduction, it is not clear if the transmitter power increase of a node will either positively or negatively affects the lifetime of an ad hoc network. In this section, we apply the IR algorithm to a sample network, in order to get insights on how the maximum transmitting power of a node and the peak rate of a link affect the lifetime of the network. For the purpose, we consider a nine node symmetric network shown in Fig. 2. The nodes are 1 km apart from their nearest neighbors. The distance based attenuation was assumed to be

6

−1

−1.5

7

−2 −2

−1.5

Fig. 2.

−1

−0.5

0

0.5

1

1.5

2

Positioning of the nodes (kilometers).

2

10

0

10

−2

10

−4

10 ||X(n+1)−X(n)||

Φ(n+1) =   (n+1)  (n+1) (n+1) ,X ,P X ,Φ arg minΦ∈C F R

−6

10

−8

10

−10

10

−12

10

−14

10

0

5

10

15 Iteration

20

25

30

Fig. 3. Convergence of X as a function of iterations when pmax = 1000 mW and the middle node has 10% less energy.

proportional to the fourth power of the distance. Both shadow and fast fading were neglected from the analysis. The power consumption of the radio independent part of the nodes Ps was assumed to be 50 mW. The power consumption of the transmitter electronics Pt was assumed to be 800 mW and the corresponding value at the receiver Pr was assumed to be 400 mW. Actual radiated air-interface power pij was assumed to vary from 2 mW up to 1000 mW. Receiver noise power density was assumed to be -174 dBm/Hz. The bandwidth W was set to 1 MHz and the (Eb /I0 )-target, Γ(p) was set to 5 dB for all paths p. We considered two routes in Fig. 2: (1, 9) and (7, 3). For each route there were three possible paths: The one (path 1) passing trough the middle node (node 5) and two (path 2 and path 3) circumventing it; via node 6 or 4 for the first route and via node 2 or 8 for the second route. All the nodes, except the middle node, had initially the same amount of energy in their battery while the middle node had 10% or 90% less. The average bit rate requirement for each route was set to 30 kbit/s. In addition, it was assumed that each link had 1 kbit/s signaling connection for link state and routing information exchange. Figs. 3 and 4 illustrate the convergence of X and R as a

1092

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 5, MAY 2006

0.4

30 Path 1 of route 1 Path 2 of route 1 Path 3 of route 1

r max =500 kbps with power control 0.35

25 r max =500 kbps without power control

0.3

Normalized lifetime

Data rate (kbit/s)

20

15

0.25 r max =250 kbps with power control 0.2 r max =250 kbps without power control

10

0.15

0.1

5

0.05

0

0

5

10

15 Iteration

20

25

0

5

10 15 20 Maximum transmit power (dBm)

25

30

30

Fig. 4. Rate per path allocation as a function of iterations when pmax = 1000 mW and the middle node has 10% less energy.

Fig. 6. energy.

Normalized network lifetime when the middle node has 10% less

0.9 25

r

max

=500 kbps

r

max

=250 kbps

0.8

0.7 20

0.6 Normalized lifetime

Data rate (kbit/s)

Path 3 of route 1

15 Path 2 of route 1

10

0.5

0.4

0.3

0.2 5

0.1

Path 1 of route 1 0

0

5

10 15 20 Maximum transmit power (dBm)

0 25

30

0

5

10 15 20 Maximum transmit power (dBm)

25

30

Fig. 5. Rate per path allocation when the middle node has 90% less energy.

Fig. 7. energy.

Normalized network lifetime when the middle node has 90% less

function of iterations (Proposition 2) when pmax = 1000 mW and node five had 10% less energy than the rest of the nodes. In this simple example, the routing scheme seemed to require only ten iterations to find the final R vector. Since our test network is symmetric, the found rate per path allocation for the two routes are the same. The solution is to let most of the data to pass trough the shortest path and divide the rest of the data between the two longer paths. Since the destination nodes 3 and 9 only transmit at the signaling rate, the interference power in the network is not distributed evenly between the nodes. Consequently, the two paths circumventing the middle node experience uneven load. It is notable, however, that the rate per path allocation is very much dependent on the battery states of the nodes. If, for example, the middle node would have 90% less energy than the rest of the nodes, then the solution would be to allocate all the rates to the two circumventing paths. This is illustrated in Fig. 5 which shows how the rate behaves as a function of maximum power. In the figure, continuous lines correspond to 500 kbit/s peak data rate and dotted lines to 250 kbit/s peak data rate. With low power values the achievable

data rates of the circumventing paths are small and thus node five is forced to carry most of the traffic. As we increase the maximum power, also the achievable data rates of the circumventing paths increase and more data can be routed through them. In this extreme case, where the node five has much less energy, the path using it is finally shutdown. In general case, the rate of the first path would converge to some nonnegative value. Now let us investigate how the lifetime would be affected by the maximum transmission power of a node and the peak rate of a link. Figs. 6 and 7 illustrate the lifetime of the worst node as a function of maximum air-interface power when the middle node has 10% and 90% less energy than the rest, respectively. We consider two types of link adaptation. In the first case, transmission power is fixed and the link adaptation is done by varying the processing gain and transmission time. That is, (26) is replaced by pij = pmax . In the second case, we also control the power to avoid excessive interference using (26). Two different peak data rates were considered, 500 kbps and 250 kbps.

¨ JANTTI AND KIM: JOINT DATA RATE AND POWER ALLOCATION FOR LIFETIME MAXIMIZATION IN INTERFERENCE LIMITED AD HOC NETWORKS

1093

In the case the middle node had 10% less energy than the rest, the lifetime with 500 kbps is about 37% and with 250 kbps is about 22% of mini {Ei }/Ps . The figure also suggests that there is a threshold power after which there is no gain in increasing the maximum transmitter power pmax . The threshold is dependent on the network configuration and the peak data rate. For 250 kbs peak rate, the threshold is 12 dBm and for 500 kbps, it is 3 dB higher. If the energy is controlled by keeping the transmitter power fixed and only controlling the time, then due to the peak rate constraint, there will be another threshold value, which is lower than the previous one, after which incremental power affects the lifetime negatively. The lifetime with power control seems to be at least two times longer than without power control. In case the middle node had 90% less than the rest, the effect of power control and peak data rate is less dramatic. As can bee seen from Fig. 5, the solution quickly converges to rate to path allocation, in which the node five does not carry any traffic. In this case, the relative lifetime is 0.84 for 500 kbps peak rate and 0.72 for 250 kbs peak rate. Hence, the lifetime of the network is close to the lifetime the bottleneck node in idle mode. This is due to the fact that with the power above 10 dBm, the bottleneck node does not participate in relaying data traffic (see Fig. 5) and thus its battery is exhausted by the idle state power consumption and signaling traffic. By increasing the peak data rate, the energy consumption of the signaling traffic can be decreased a little. In case the middle node had only 10% less energy, significant portion of the traffic is still routed through it. Hence, the energy saving due to high peak rate and short transmission time is higher.

lifetime routing, however, is topology dependent. In a sparse network, there might be bottleneck nodes that have to carry most of the traffic in the network. In such a case, all routing schemes perform more or less the same, since the set of available paths is very small. In dense networks, the traffic can be divided into several paths and thus the load carried by low battery nodes can be decreased affecting their lifetime positively. In this case, there is gain in using maximum lifetime routing instead of minimum-hop routing or minimum energy routing. To solve the maximum lifetime problem, we have developed an iterative routing algorithm (IR) that consists of two parts. First the joint energy, time, and power control algorithm (JETP) is used to solve these parameters for given rate to path allocation. The JETP algorithm is fully distributed and can be locally executed in each link. In the second phase, linear programming is applied to find the optimal data rates. This procedure is iterated until the parameters converge. The rate control part requires full knowledge of the link states and thus requires a lot of signaling. We leave the development of a practical maximum lifetime routing scheme, in which some heuristics are utilized to solve the rate allocation, as further study. We note that our results are valid for networks having relatively stable topology; either fixed or very slowly moving mobiles. We expect the gain of using maximum lifetime routing to diminish as the mobility increases due to the large protocol overhead caused by constant rerouting. This overhead could be decreased by considering only a subset of paths that are relatively stable.

VI. C ONCLUDING R EMARKS In this paper, we have considered joint routing and rate allocation that maximizes the lifetime of an ad hoc network supporting variable-rate transmissions (link adaptation). Our analysis suggests that if the rate is a linear function of the instantaneous CIR and the peak rate is bounded above, then the lifetime of the network is a nondecreasing function of the maximum air-interface power pmax and peak data rate rmax . This is due to two aspects: First, by increasing power, a node can decrease the interference duration to the other nodes, which results in power-saving in the other nodes. Secondly, the increased power will give a node more freedom in selecting its route as higher data rates can be used in long links. Due to the peak rate constant, there will be a threshold power value, after which the lifetime of the worst node cannot be prolonged by increasing the maximum transmission power. If the transmission powers of the links are not controlled, then excess-transmission power affects the network negatively. Although we provided numerical results only for a simple symmetric network configuration, the behavior of a network having an arbitrary topology is expected to be similar. That is, the lifetime of the network is a nondecreasing function of the transmit power. In concurrent radio technology, the airinterface, i.e., radiated, power constitutes only a small portion of the total energy consumption of a radio link. By decreasing the transmission time by using high transmit power and high peak data rate, we can decrease the power consumed by the baseband processing. The benefit of using the maximum

R EFERENCES [1] M. Frodigh, P. Johansson, and P. Larsson, “Wireless ad hoc networkingThe art of networking without a network,” Ericsson Review, no. 4, pp. 248-263, 2000. [2] S. Singh, M. Woo, and C. S. Raghavendra, “Power-aware routing in mobile ad hoc networks,” in Proc. Conf. Mobile Computing, 1998, pp. 181-190. [3] J.-H. Chang and L. Tassiulas, “Maximum lifetime routing in wireless sensor networks,” IEEE/ACM Trans. Networking, vol. 12, no. 4, pp. 609-619, Aug. 2004. [4] C.-K. Toh, H. Cobb, and D. A. Scott, “Performance evaluation of battery-life-aware routing schemes for wireless ad hoc networks,” in Proc. IEEE Int. Conf. Commun., vol. 9, June 2001, pp. 2824-2829. [5] S. Banerjee and A. Misra, “Minimum energy paths for reliable communication in multi-hop wireless networks,” in Proc. ACM Int. Symposium Mobile Ad Hoc Networking & Computing, 2002, pp. 146156. [6] W. Cho and S.-L. Kim, “A fully distributed routing algorithm for maximizing lifetime of a wireless ad hoc network,” in Proc. IEEE 4th Int. Workshop-Mobile & Wireless Commun. Network, Sep. 2002, pp. 670-674. [7] P. Floreen, P. Kaski, J. Kohonen, and P. Orponen, “Lifetime maximization for multicasting in energy-constrained wireless networks,” IEEE J. Select. Areas Commun., vol. 23, no. 1, pp. 117-126, Jan. 2005. [8] T. ElBatt and A. Ephremides, “Joint scheduling and power control for wireless ad-hoc networks,” IEEE Trans. Wireless Commun., vol. 3, no. 1, pp. 74-85, Jan. 2004. [9] M. Grossglauser and D. N. C. Tse, “Mobility increses the capacity of ad hoc wireless networks,” IEEE/ACM Trans. Networking, vol. 10, no. 4, pp. 477-486, Aug. 2002. [10] C. Comaniciu and H. V. Poor “QoS provisioning for wireless ad hoc networks,” in Proc. 42nd IEEE Conference on Decision and Control, Dec. 2003, vol. 1, pp. 92-97. [11] G. J. Foschini and Z. Miljanic, “A simple distributed autonomous power control algorithm and its convergence,” IEEE Trans. Veh. Technol., vol. 42, no. 4, pp. 641-646, Nov. 1993.

1094

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 5, MAY 2006

[12] R. Yates, “A framework for uplink power control in cellular radio systems” IEEE J. Select. Areas Commun., vol. 13, no. 7, pp. 1341-1347, Sep. 1995. [13] R. J¨antti and S.-L. Kim, “Second-order power control with asymptotically fast convergence,” IEEE J. Select. Areas Commun., vol. 18, no. 3, pp. 447-457, Mar. 2000. [14] D. B. Johnson, D. A. Maltz, Y.-C. Hu, and J. G. Jetcheva, “The dynamic source routing protocol for mobile ad hoc networks,” IETF Internet Draft, Mobile Ad-hoc Networking Group (IETF), July 2004. [Online] Available: http://www1.ietf.org/internet-drafts/draft-ietf-manetdsr-10.txt [15] C. E. Perkins, E. M. Royer, and S. Das, “Ad hoc on demand distance vector (AODV) routing,” IETF Internet Draft, Mobile Ad-hoc Networking Group (IETF), July 2003. [Online] Available: http://www1.ietf.org/rfc/rfc3561.txt [16] R. S. Varga, Matrix Iterative Analysis. Englewood Cliffs, NJ: PrenticeHall, 1962. [17] D. P. Bertsekas and J. N. Tsitsiklis, Distributed Computation: Numerical Methods. Belmont, MA: Athena Scientific, 1997. Riku J¨antti (M’02) is an acting professor and docent (adjunct professor) of Telecommunication Engineering at University of Vaasa, Finland. He is also a docent (adjunct professor) of Control Engineering at Helsinki University of Technology (TKK), Finland. From 1998 to 1999, he was with the Radio Communication Systems Laboratory (now Wireless@KTH), Royal Institute of Technology (KTH), Stockholm, Sweden, as a visiting scholar. He has also been a visiting professor at Radio Resource Management & Optimization Laboratory, Information and Communications University (ICU), Taejon, Korea. His current research interests include various control and resource management problems of wireless radio networks and their applications to automation systems.

Seong-Lyun Kim (S’93-A’95-M’01) is an associate professor at the School of Electrical and Electronic Engineering, Yonsei University, Seoul, Korea. Prior to joining Yonsei, he was an associate professor at Information & Communications University (ICU), Taejon, Korea and an assistant professor at Radio Communication Systems Group (now Wireless@KTH) , Department of Signals, Sensors and Systems, Royal Institute of Technology (KTH), Stockholm, Sweden. He was also a visiting professor at Control Group, Helsinki University of Technology (HUT), Espoo, Finland. His research interest includes radio resource management in wireless networks and economics of wireless multimedia. He co-authored a book with Prof. Jens Zander, Radio Resource Management in Wireless Networks (Artech House, Boston & London, 2001). He is currently serving as an associate editor for the IEEE T RANSACTIONS ON V EHICULAR T ECHNOLOGY and an editor for the Journal of Communications and Networks.