74

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 1, JANUARY 2004

Joint Scheduling and Power Control for Wireless Ad Hoc Networks Tamer ElBatt, Member, IEEE, and Anthony Ephremides, Fellow, IEEE

Abstract—In this paper, we introduce a cross-layer design framework to the multiple access problem in contention-based wireless ad hoc networks. The motivation for this study is twofold, limiting multiuser interference to increase single-hop throughput and reducing power consumption to prolong battery life. We focus on next neighbor transmissions where nodes are required to send information packets to their respective receivers subject to a constraint on the signal-to-interference-and-noise ratio. The multiple access problem is solved via two alternating phases, namely scheduling and power control. The scheduling algorithm is essential to coordinate the transmissions of independent users in order to eliminate strong levels of interference (e.g., self-interference) that cannot be overcome by power control. On the other hand, power control is executed in a distributed fashion to determine the admissible power vector, if one exists, that can be used by the scheduled users to satisfy their single-hop transmission requirements. This is done for two types of networks, namely time-division multiple-access (TDMA) and TDMA/code-division multiple-access wireless ad hoc networks. Index Terms—Code-division multiple-access (CDMA), crosslayer protocol design, multiple access, power control, scheduling, time-division multiple-access (TDMA), wireless ad hoc networks.

I. INTRODUCTION

I

T is well known that power is a precious resource in wireless networks due to the limited battery life. This is further aggravated in ad hoc networks since all nodes are mobile terminals of limited weight and size. In addition, power control is of paramount importance to limit multiuser interference and, hence, maximize the spatial reuse of resources [1]. Power control has been studied extensively in the context of channelized cellular systems [2], [4], code-division multiple-access (CDMA)-based systems [7], and in a general framework [8]. Distributed iterative power control algorithms have been introduced for cellular systems and convergence results have been established [2], [4], [8]. More recently, there has been some focus on formulating the distributed power control problem as a noncooperative game [9]–[12]. In [12], the authors modified

Manuscript received August 1, 2000; revised December 1, 2001; accepted February 1, 2003. The editor coordinating the review of this paper and approving it for publication is S. Tekinav. This work was supported by the Center for Satellite & Hybrid Communication Networks, a NASA Commercial Space Center (CSC) at the University of Maryland, under a NASA Cooperative Agreement NCC3-528. T. ElBatt is with the Information Sciences Lab, HRL Laboratories, LLC, Malibu, CA 90265 USA (e-mail: [email protected]). A. Ephremides is with the Electrical and Computer Engineering Department, University of Maryland, College Park, MD 20742 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2003.819032

the power control problem formulation to incorporate the notions of utility and cost which are shown to improve the convergence characteristics of the algorithm. Our main objective in this paper is to develop a power control-based multiple access algorithm for contention-based wireless ad hoc networks. This is done via investigating the similarities and differences of this problem from the problem solved earlier for cellular networks. The concept of controlling the transmission radii in multihop packet radio networks was first introduced in [15]. They determined the optimal transmission radius (that maximizes the packet forward progress toward destination) under the constraint that the transmission powers for all nodes are the same. In [16], the authors developed a model for analyzing the throughput and forward progress where each mobile node may have a variable and different transmission range. Recently, the work in [17] employed transmission power as the link metric for shortest path routing algorithms in an attempt to realize the minimum-power routing algorithm discussed in [13]. However, the congestion caused by multiuser interference was not represented in the link metric. In [18], the authors employed transmission power adjustment in order to control the topology of wireless ad hoc networks. Unlike these studies, our work employs power control as part of the multiple access algorithm. Although the authors in [19] introduced a power control-based multiple access protocol, it was limited only to the class of carrier sense multiple access with collision avoidance (CSMA/CA) protocols. In this study, we introduce the notion of power control as part of a contention-based multiple access protocol that characterizes successful transmissions depending on a set of signal-to-interference-and-noise ratio (SINR) constraints (which directly translates to quality of service (QoS) constraints on the bit-error rate (BER) at individual receivers). Moreover, there are no guarantees in [19] that the computed powers are minimum, while in our study we determine the minimum power vector subject to SINR constraints. On the other hand, the problem of scheduling nonconflicting transmissions, in order to achieve efficient spatial reuse, in TDMA multihop packet radio networks has received considerable attention in the literature [20]–[27]. Two versions of this problem have been addressed, namely, broadcast scheduling [20], [23], [25], [26] and unicast scheduling [21], [23], [24]. However, the common limitation among these studies is the limited transmission range assumption where no interference is caused beyond that range. Accordingly, nodes that are more than two hops away are assumed to be conflict-free (i.e., do not cause interference to each other at their respective receivers). This, in turn, led to the strong connection between the transmission scheduling problem and graph-theoretic problems,

1536-1276/04$20.00 © 2004 IEEE

ELBATT AND EPHREMIDES: JOINT SCHEDULING AND POWER CONTROL

such as graph coloring. In reality, transmitting antennas may cause negligible amounts of interference individually at a far receiver; however, the aggregate interference of a large number of far transmitters could be significant and may cause the SINR to fall below the threshold and, hence, disrupt an ongoing communication. Therefore, in this paper we embrace a ”node-centric” approach where the wireless link is treated as a relative notion that is controllable via the transmitter and/or receiver design. More specifically, the existence of a wireless link is characterized by the condition for successful reception , where is a prespecified threshold defined as that depends on the acceptable BER, detector structure, modulation/demodulation scheme, and channel coding/decoding algorithm. On the other hand, the SINR depends on the channel, multiuser interference, antenna gain, transmission power, and transmission rate [28]. Our main contribution in this work is to introduce a cross-layer design framework to the multiple access problem in contention-based wireless ad hoc networks. We attempt to tie two research areas in wireless networks, that have been largely studied in isolation, namely, power control (physical layer problem) and time-division multiple-access (TDMA) scheduling (multiple access problem). The motivation behind this coupling is twofold: 1) accounting for the wireless link volatility in the design of higher layer protocols, e.g., link scheduling and 2) introducing more realistic and accurate characterization of nonconflicting transmissions as those satisfying a prespecified SINR constraint. Thus, we propose to solve the multiple access problem via two alternating phases that search for an admissible set of users along with their transmission powers. In the first phase, the scheduling algorithm is responsible for coordinating independent users’ transmissions to eliminate strong levels of interference inherent to wireless ad hoc networks (e.g., self-interference caused by a node simultaneously transmitting and receiving). Self-interference, along with other types of interference described later, cannot be overcome by computationally expensive power control algorithms and has to be eliminated first via scheduling. In the second phase, power control is executed in a distributed fashion, to determine the ”admissible” set of powers that could be used by the scheduled nodes, if one exists. If no set of positive powers can be found, control is transferred again to the scheduling phase to reduce interference via deferring the transmissions of one or more users participating in this scenario. The paper is organized as follows. In Section II, the assumptions and definitions necessary for formulating the problem are presented. This is followed by a detailed description of the joint scheduling-power control algorithm in Section III. Section IV is devoted to describing the distributed power control algorithm. The simulation results and discussion are given in Section V. Finally, the conclusions are drawn in Section VI.

II. ASSUMPTIONS AND DEFINITIONS In this section, we provide the assumptions underlying this study and introduce appropriate notation.

75

1) Consider a wireless ad hoc network consisting of nodes. There is no wireline infrastructure to interconnect the nodes, i.e., they can communicate only via the wireless medium. 2) Each node is supported by an omni-directional antenna. 3) Each node knows the geographical location of all other nodes via location discovery schemes [29]–[31]. This information is necessary for the receivers to feedback their SINR measurements to their respective transmitters. 4) Routing is not considered in this study. We focus on next neighbor transmissions since the multiple access problem depends solely on the next neighbor transmission requirements. However, interference-induced congestion may lead to a conflict between the routing and multiple access decisions. Balancing this trade-off via joint routing, scheduling and power control design is out of the scope of this paper and is a subject of ongoing research. 5) The effect of users’ mobility is not considered in this study. However, this assumption can be relaxed to the case of low users’ mobility (typically pedestrians) where the link gain matrix is expected to change slowly compared to the dynamics of the iterative power control algorithm. 6) Assume that all nodes share the same frequency band, and time is divided into equal size slots that are grouped into frames. Thus, part of the study is conducted in the context of TDMA. In addition, we investigate the implications and tradeoffs in the case where all nodes have unique quasi-orthogonal spreading codes on top of the TDMA scheme, i.e., TDMA/CDMA. 7) The slot duration is assumed to be larger than the packet duration by an interval called a “guard band.” These bands are necessary to compensate for arbitrary delays incurred by transmitted packets due to signal propagation delays and/or clock drifts. 8) In this study, we assume that the frame length (or transmission schedule length) is fixed throughout system operation. It is chosen, heuristically, depending on the number of nodes, network load, and quality-of-service constraints. However, there is an inherent tradeoff between the choice of the frame length and the convergence of the power control algorithm as illustrated in the following extreme cases: short frame lengths lead to packing excessive number of transmissions in each slot and thus make it impossible for the power control algorithm to experience convergence in many slots. On the other hand, long frame lengths make it easier for the power control algorithm to converge at the expense of wasting system resources since most slots will be underutilized. Therefore, we envision room for balancing this tradeoff at the expense of adapting the frame length dynamically depending on the number of required transmissions at the beginning of each frame and their spatial separation. More precisely, the objective would be to find, on a frame-by-frame basis, the minimum frame length that guarantees convergence of the power control algorithm in all slots. This tradeoff

76

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 1, JANUARY 2004

9)

10)

11)

12)

13)

14)

15)

16)

falls out of the scope of this paper and is a subject of future research. The complexity of this problem stems from its combinatorial nature which renders heuristic techniques unavoidable. Each node generates information packets (e.g., data packets) of fixed length, destined to all other nodes, according to a Poisson distribution with aggregate rate packets/second. We assume that each generated packet is intended for a single neighbor only, i.e., the cases of broadcasting and multicasting are out of the scope of this work. , that We assume a maximum power level, denoted a node can use for transmission. This is enforced by the limited weight and size of the wireless terminal. The interference model adopted assumes that each node in the area could potentially cause interference at any receiving node, even if it is too far. We consider this model more realistic than the models introduced in the literature (e.g., IEEE 802.11) which assume that the transmission range of any node is limited (typically circular) and beyond that range no interference is caused [15], [16], [18]. The reason behind this is that a very large number of far interferers might cause negligible amounts of interference individually, but their aggregate effect could bring the SINR below the threshold and, hence, may disrupt an on-going transmission. The power decay law is assumed to be inversely proportional to the fourth order of the distance between the transmitter and the receiver. Accordingly, the link gain matrix is assumed to be constant throughout this study. We assume the existence of a separate feedback channel that enables receivers to send their SINR measurements to their respective transmitters in a contention-free manner. This can be justified based on the fact that feedback messages are typically smaller than data packets and, hence, the scarce wireless bandwidth will not be wasted. We assume the existence of a central controller responsible for executing the scheduling algorithms presented in the next section. In this paper, we limit our attention to centralized scheduling algorithms in order to: 1) avoid complexity in the first step of the study and 2) focus on the cross-layer protocol design concept. Introducing distributed scheduling algorithms within the proposed framework is out of the scope of this paper and is a subject of future research. On the other hand, computationally intensive power control is to be executed in a distributed fashion in order to reduce the communication overhead. Define the average slot throughout as the long-run average of the percentage of packets successfully received by single-hop neighbors in each time slot. III. CROSS-LAYER PROTOCOL DESIGN

Future wireless networks are expected to accommodate a wide variety of nodes with different power constraints, bandwidth capabilities, and vastly different QoS requirements

as dictated by a large set of applications. One of the main hurdles toward designing protocols that fully exploit the scarce power and bandwidth resources is the dynamic and volatile nature of the underlying wireless link which is characterized by time-varying quality due to fading, shadowing, in addition to multiuser interference. A major shortcoming in the design of wireless ad hoc networks is that research efforts has been devoted to introducing protocols in a specific layer of the International Standards Organization (ISO) Open System Interconnection (OSI) protocol stack. Moreover, these protocols has been designed independently without considering their interaction or impact on the design choices at other layers of the stack. Recent work has provided overwhelming evidence of the need to couple physical layer design with other aspects of the system design in the link and network layers. For instance, adapting the transmission rate based on link state information passed from the physical to the MAC layer was shown to improve the performance of the IEEE 802.11 protocol [33]. Moreover, the role of power control in improving the CSMA/CA throughput performance has been studied in [19]. The goal of this paper is to explore another dimension of the cross-layer design concept, namely the joint design of scheduling (link layer task) and power control (physical layer task) in the context of contention-based wireless ad hoc networks. A. Algorithm Description In this section, we present the joint scheduling-power control algorithm. This algorithm is to be executed at the beginning of each time slot in order to cope with excessive interference levels that might be developed in some slots. The proposed algorithm determines the admissible set of users that can safely transmit in the current slot without disrupting each other’s transmission. Accordingly, the objective is twofold: first, to determine the set of users who can attempt transmission simultaneously in a given slot and second to specify the set of powers needed in order to satisfy SINR constraints at their respective receivers. This is done via two alternating phases, namely scheduling and power control. The following two definitions are instrumental in illustrating the problem since they are related to the scheduling and power control phases, respectively. Definition 1: In TDMA wireless ad hoc networks, a transmission scenario is valid iff it satisfies the following three conditions. 1) A node is not allowed to transmit and receive simultaneously. 2) A node cannot receive from more than one neighbor at the same time. 3) A node receiving from a neighbor should be spatially separated from any other transmitter by at least a distance . However, if nodes use unique signature sequences (i.e., joint TDMA/CDMA scheme), then the second and third conditions can be dropped, and the first condition only characterizes a valid transmission scenario. The purpose of the third condition above is to enforce spatial separation among simultaneous transmissions in order to reduce the amount of interference induced at nonintended receivers before executing computationally intensive power control algorithms. The choice of the parameter

ELBATT AND EPHREMIDES: JOINT SCHEDULING AND POWER CONTROL

77

Fig. 1. Flowchart of the joint scheduling-power control algorithm.

affects the amount of interference eliminated via scheduling. is too small, no spatial separation between simultaneous If transmissions is guaranteed and most of the interference will be is passed to the power control phase. On the other hand, if large, considerable amounts of interference are eliminated via the scheduling phase. For example, to limit multiuser interference to levels comparable to those in channelized cellular systems, the parameter should be equal to the well-known “frequency reuse distance” parameter [37]. The choice of the parameter generally depends on the minimum acceptable SINR levels. links is Definition 2: A transmission scenario involving , admissible iff there is a set of transmission powers, which solves the following minimization problem: (1) s.t.

The key observation that led to the development of the proposed two-phase solution is twofold: first, examining the “validity” constraints of a given transmission scenario is much easier and computationally more efficient than examining the “admissibility” conditions [which involves solving the optimization problem in (1)]. Second, eliminating strong levels of interference (indicated in Definition 1) in the scheduling phase is essential since they cannot be overcome by power control alone. In addition, employing a scheduling algorithm first makes the structure of the power control problem in wireless ad hoc networks exactly similar to the structure of the power control problem in cellular networks. This interesting observation has led to the applicability of existing power control algorithms to emerging wireless ad hoc networks as shown in Section IV. Fig. 1 shows a flowchart that demonstrates the operation and interaction of the scheduling and power control portions of the algorithm. Given the transmission schedule specified at the beginning of each frame, the scheduling phase is responsible for checking whether the scenario in the current slot is valid or not.

78

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 1, JANUARY 2004

If valid, it proceeds to the power control phase. Otherwise, it searches for a valid subset of users via deferring the transmissions of some of the users causing high interference to the next slot in the frame. The power control phase is responsible for investigating the power admissibility of the valid scenario specified in the scheduling phase. If it turns out to be power admissible, the specified nodes start transmission in the current slot using the determined set of transmission powers. Otherwise, control is transferred again to the scheduling phase where a search algorithm is activated to find the optimum subset of users who are admissible. B. Scheduling Policies It is evident from the proposed algorithm that the objective is to pack the maximum number of transmissions that can be successfully detected at their respective receivers in each slot. This, in turn, maximizes the spatial reuse of system resources. According to the previous section, the scheduling algorithm is invoked in one of the following two cases: 1) invalid transmission scenario denoted (INV) or 2) valid, yet inadmissible, transmission scenario denoted (INA). Thus, the scheduling algorithm is responsible for solving two optimization problems, namely “valid scenario optimization” and “admissible scenario optimization”. Our objective in the first problem is to determine the transmission scenario that solves the following constrained optimization problem

various scheduling policies and rank them for deciding optimality. Although DES would be practically infeasible, due to the real-time nature of the algorithm, it is quite insightful and serves as a benchmark for gauging the performance of candidate heuristic policies. Finally, we examine two simple heuristic algorithms and show their performance compared to the aforementioned optimum. For the valid scenario search problem, a simple algorithm is to examine the set of valid scenario constraints sequentially and defer users’ transmissions accordingly to resolve the conflicts. It is evident that this algorithm is suboptimal in the sense that it could lead to deferring more transmissions than needed in order to reach a valid scenario. On the other hand, for the admissible scenario search problem, we examine a heuristic policy introduced earlier by Zander [2]. It suggests deferring the user with minimum SINR as an attempt to lower the level of multiaccess interference. This might allow other users to converge to the optimum power vector quite fast. It is worth noting that the latter algorithm lends itself to distributed implementation if the SINR measurement at each receiver is fed back to all transmitters via efficient information dissemination algorithms [31]. IV. DISTRIBUTED POWER CONTROL In this section, we formulate the power control problem and introduce possible distributed implementations within the context of TDMA and TDMA/CDMA wireless ad hoc networks. A. TDMA Wireless Ad Hoc Networks

s.t.

is a valid scenario where cardinality of the transmission scenario (i.e., number of links participating in scenario ). On the other hand, our objective in the second problem is to determine the transmission scenario that solves the following optimization problem

s.t.

In this section, we assume that all nodes share the same frequency band and those scheduled will attempt transmission to their respective neighbors in the assigned time slot. Prior experience, from the context of cochannel interference control in channelized (FDMA or TDMA) cellular systems [2], [4], shows the existence of distributed power control algorithms which converge exponentially fast to the optimal (minimum) power vector, if one exists. The main result of this section indicates that under some transmission constraints, the structure of the power control problem at hand is similar to the problem formulated and solved earlier for channelized cellular systems. According to [4], the uplink distributed power control algorithm executed by node follows the following iteration: (2)

is an admissible scenario. The complexity of the previous two problems is attributed to their combinatorial nature since the size of the search space grows in an exponential manner with the number of links in the transmission scenario. This, in turn, constitutes a major hurdle toward finding the optimum schedule and renders heuristic solutions unavoidable. This is further aggravated by two factors: 1) the two problems are not amenable to analytical optimization techniques due to the lack of a tractable mathematical structure and 2) the two problems should be solved on a slot-by-slot basis. Therefore, simulation-based discrete exhaustive search (DES) [34] is a candidate technique to measure the performance of

where power transmitted by node to its base station (BS); signal-to-interference-and-noise ratio at BS ; iteration number. Theorem 1: For valid transmission scenarios in TDMA wireless ad hoc networks, the distributed power control algorithm in (2) converges exponentially fast to the minimum power vector, if one exists. Proof: Our approach to prove the above assertion is to show the similarity of the problem structure at hand to the power control problem in channelized cellular systems. Once we achieve this, it is straightforward to establish the proof since

ELBATT AND EPHREMIDES: JOINT SCHEDULING AND POWER CONTROL

79

validity constraint. The parameter introduced in Definition 1 is chosen to be equal to the distance between the interferer and its intended receiver (i.e., ) as an example of spatially separating transmissions prior to examining the iterative power control algorithm. Accordingly, the constraints in (3) can be written in the form (4) On the other hand, it is shown in [4] and [13] that the SINR constraints in the uplink power control problem formulated for channelized cellular systems can be reduced to

m

Fig. 2. Example of a valid transmission scenario ( = 3 links) {1 3 4, 5 6} for a TDMA wireless ad hoc network of = 6 nodes.

!

!

n

! 2,

convergence results are already available for the iterative algorithm in (2) in the context of channelized cellular systems [4]. Accordingly, we compare the mathematical structure of the two problems and show their similarity under the aforementioned set of valid scenario constraints. We compare the structure of the power control problem for a valid scenario that has links (shown in Fig. 2), where , to that of a channelized cellular system having users in different cells reusing the same frequency channel. The objective in both problems is to minimize the total power transmitted by participating nodes subject to a constraint on the SINR at their receivers. The formulation of the power control problem for a valid scenario in TDMA wireless ad hoc networks is given by (3) s.t.

where

is the power transmitted from node to node and

where link gain from node to node ( is the distance between nodes and ); receiver thermal noise power; interference power at node from transmitters other than node . It is given by and It is worth noting that the receiver in the above expression depends on the specific transmission scenario under investigain tion. For valid transmission scenarios, the constraint the above expression satisfies the first condition in Definition 1, i.e., a node cannot transmit and receive simultaneously. In adis necessary to satisfy the second dition, the constraint condition (no common receivers among the links). Finally, guarantees the satisfaction of the third the constraint

(5) where is the power transmitted from node to BS and link gain from node to BS . It is straightforward to notice that for channelized cellular systems, the first, second, and third conditions of the valid scenario constraints are inherently satisfied by the different uplink and downlink frequencies, the system’s cellular structure (i.e., mobile users communicate with their closest BS’s only), and the frequency reuse constraints, respectively. From (4) and (5), it is evident that the power control problem formulated for a specific valid scenario in TDMA wireless ad hoc networks has exactly the same structure as the power control problem for channelized cellular systems. They are both characterized as eigenvalue problems for nonnegative matrices [35]. Therefore, for a transmission scenario involving links, whether in channelized cellular or TDMA wireless ad hoc networks, there different link gains between all transmitters and rewill be ceivers. The only difference between the two cases is that in cellular systems, wireless terminals are restricted to communicate only with their assigned BSs, whereas in ad hoc networks a wireless terminal can potentially establish communication with any neighbor. Accordingly, the distributed power control algorithm in (2) and its convergence properties turn out to be directly applicable to TDMA wireless ad hoc networks. Based on Theorem 1, the results of power control with maximum power constraint for channelized cellular systems [5] are also directly extendable to TDMA wireless ad hoc networks. In this case, the iterative power control algorithm in (2) is modified to take the following form: (6) B. TDMA/CDMA Wireless Ad Hoc Networks In this section, we assume that, on top of the TDMA scheme, each node has a unique preassigned signature sequence that it can use to encode the transmitted symbols. Again, our main objective is to show the applicability of the distributed power control algorithm in [7] and [8], introduced for cellular CDMA systems, to contention-based wireless ad hoc networks. First, we introduce the physical layer assumptions underlying the system. We adopt a simple signaling structure with BPSK modulation. The symbol stream is assumed to be i.i.d. and the 1 symbols are assumed to be equally probable. The

80

Fig. 3.

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 1, JANUARY 2004

Asynchronous CDMA system.

noise is assumed to be independent of the symbols and has variance . Users are assumed to have preassigned unique signature sequences which they use to modulate their information which bits. The signature sequence of user is denoted is nonzero only in the bit interval [0, ] and is normalized to . The receiver is assumed unit energy, i.e., to be a conventional single-user detector, namely a bank of filters matched to the signature waveforms of various users [36]. For each user , we assume that all other users create interference asynchronously. The relative delays of the users, which can have any value not exceeding the bit duration , do not change with time and are assumed to have a uniform distribution. For the th bit of a given user , an interfering user creates ) and or bits and ( ), deinterference by either bits ( pending on whether the interfering user has a positive or negative delay relative to the user of interest. In Fig. 3, two possible cases are depicted. The delay of user relative to the matched . In Fig. 3, user has a positive filter of user is denoted delay relative to user and creates interference to the th bit of ) and . On the other hand, user has user with bits ( a negative relative delay with respect to user and creates in). Acterference to the th bit of user with bits and ( cordingly, three types of cross correlations between the signature sequences of any two users and can be defined. They and represent the cross correlaare denoted as , , and tions between the symbol of interest in one hand and the previous symbol, current symbol, and next symbol of an interferer, respectively. Theorem 2: For valid transmission scenarios in TDMA/ CDMA wireless ad hoc networks, the distributed power control algorithm in (2) converges exponentially fast to the minimum power vector, if one exists. Proof: Again, our approach is to show the similarity of the problem structure at hand to the power control problem in cellular CDMA systems. Given that, it is straightforward to establish the proof since convergence results are available for cellular CDMA uplink power control [7], [8]. We compare the structure of the power control problem for a valid scenario that has links, as shown in Fig. 4, to that of a multicell CDMA system having users, as shown in Fig. 5. The power control problem for a valid transmission scenario in TDMA/CDMA wireless ad hoc networks would have a formulation similar to (3).

m

Fig. 4. Example of a valid transmission scenario ( = 4 links) {1 3 4, 5 2, 6 4} in a TDMA/CDMA wireless ad hoc network of nodes.

!

Fig. 5.

!

!

! 2,

n=6

Cellular CDMA system with four users located in two cells.

In this case, the interference power is given by

Under the CDMA assumption, the constraint is sufficient to characterize a valid scenario. It represents that each

ELBATT AND EPHREMIDES: JOINT SCHEDULING AND POWER CONTROL

81

TABLE I SIMULATION PARAMETERS

node is not allowed to transmit and receive simultaneously. Accordingly, the SINR constraints can be written in the form

(a)

(7) For the uplink power control problem in cellular CDMA systems, it is straightforward to show that the SINR constraints are given by (7), where BS represents the closest BS to node . In addition, the aforementioned valid scenario constraint is inherently satisfied by the different uplink and downlink frequency bands. Based on the above observation, we conclude that the power control problem formulated for a specific valid scenario in TDMA/CDMA wireless ad hoc systems has exactly the same structure as the power control problem formulated in [7] and [8] for minimizing multiuser interference in cellular CDMA systems. Again, the power control problem is characterized as an eigenvalue problem for nonnegative matrices. In both systems, for a scenario consisting of physical links, there will be “effective” link gains, due to the cross correlations between the spreading sequences of various transmitters, irrespective of the number of receivers involved. Accordingly, the distributed power control algorithm in (2) and its convergence properties turn out to be directly applicable to TDMA/CDMA wireless ad hoc networks. Finally, it is straightforward to show that the constrained distributed power control algorithm in (6) is directly applicable to TDMA/CDMA wireless ad hoc networks.

(b)

V. PERFORMANCE EVALUATION In this section, we show the behavior of the joint scheduling and power control algorithm and its convergence properties for admissible transmission scenarios. In addition, we compare the long-run average slot throughput and average transmission power per slot throughput for TDMA and TDMA/CDMA wireless ad hoc networks. The simulations were carried using the numerical parameters given in Table I. We limit our attention to a small system consisting of nodes since it adequately captures the tradeoffs addressed in this paper and provides valuable insights about the joint algorithm behavior under various interference conditions.

(c)

m

Fig. 6. (a) Example of an Inadmissible scenario with = 5 links in a TDMA/CDMA wireless ad hoc network (b) Inadmissible subscenario with = 4 links. (c) Admissible subscenario with = 3 links.

m

m

82

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 1, JANUARY 2004

Fig. 7. Average slot throughput of the optimum valid and admissible scenario policies.

Fig. 8.

Average slot throughput of the optimum and heuristic scheduling policies for TDMA/CDMA wireless ad hoc networks.

First, we verify via simulations, the applicability of the distributed iterative power control algorithm in (6) to wireless ad hoc networks. For TDMA/CDMA wireless ad hoc networks, Fig. 6 shows the behavior of the power control algorithm applied to a valid scenario that involves five links. In

Fig. 6(a), it can be noticed that the algorithm fails to converge due to the inadmissibility of this scenario. By deferring the user with minimum SINR the power control algorithm fails, again, to converge as shown in Fig. 6(b). Finally, by deferring another transmission according to the same heuristic policy, the

ELBATT AND EPHREMIDES: JOINT SCHEDULING AND POWER CONTROL

Fig. 9.

83

Average normalized power of the optimum scheduling policies.

transmission scenario having links becomes admissible as demonstrated in Fig. 6(c). Next, we show the average slot throughput for the optimum valid and admissible scenarios under light and heavy load conditions. In Fig. 7, we notice that the average slot throughput for a TDMA/CDMA wireless ad hoc network outperforms that of a TDMA wireless ad hoc network by a factor that varies from approximately twice the throughput at heavy loads to 17% at light loads. This, in turn, emphasizes the benefits of deploying CDMA at the expense of the computational complexity associated with determining the cross correlations at various receivers. In Fig. 8, we compare the slot throughput performance of the optimum valid and admissible scenarios to their heuristic counterparts described in Section III—B. It can be easily noticed that the optimum policy significantly outperforms the heuristic policy by a factor of 57% at heavy loads. This performance gain gradually diminishes as load decreases. For larger systems, we expect the gap in performance to be even larger, especially at heavy loads. Therefore, we envision more room for developing computationally efficient heuristic scheduling policies that achieve performance levels close to the optimum and at the same time guarantee fairness among various users. It is evident that the proposed algorithm does not guarantee any fairness among various users, since the same transmitter may be frequently penalized (i.e., deferred to transmit in the next slot) due to the high interference it causes at neighboring receivers. However, there might be a tradeoff between fairness and slot utilization since the schedule that guarantees fairness may not necessarily be the one that maximizes the slot throughput. Addressing this tradeoff is out of the scope of this paper and is a subject of future research.

Finally, we demonstrate the behavior of the average power transmitted in a slot, normalized by the slot throughput, as the system load varies for both TDMA and TDMA/CDMA systems. As expected, Fig. 9 shows that the normalized transmitted power monotonically increases with the system load. Moreover, the average normalized power consumption per slot for a TDMA/CDMA system is almost half that of a TDMA system. This implies that the CDMA system saves transmission power, at the expense of the power consumed in the computations associated with determining the cross correlation coefficients at various receivers. Hence, there is a fundamental tradeoff between transmission power and computation power that needs to be studied carefully throughout the design phase of power-controlled multiple access algorithms. In Fig. 10, we compare the normalized power consumption of the optimum and heuristic scheduling policies for TDMA/CDMA wireless ad hoc networks. We notice that the normalized power consumption of the optimum policy is considerably less than the heuristic policy, especially at heavy loads.

VI. CONCLUSION In this paper, we presented a cross-layer design framework for the multiple access problem in contention-based wireless ad hoc networks. We focused on next neighbor transmissions where nodes are to send packets to their respective receivers while, at the same time, satisfy a set of SINR constraints. Our main contribution in this paper is to solve the problem via two alternating phases until an admissible set of users, along with their transmission powers, is reached. This, in turn, reduces the

84

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 1, JANUARY 2004

Fig. 10.

Average normalized power of the optimum and heuristic scheduling policies for TDMA/CDMA wireless ad hoc networks.

computational overhead significantly and simplifies the structure of the power control problem. In the first phase, a simple scheduling algorithm coordinates independent users’ transmissions to eliminate strong levels of interference that cannot be overcome by power control. In the second phase, a distributed power control algorithm determines the set of powers that could be used by the scheduled users to satisfy their transmissions, if one exists. We showed that distributed power control algorithms introduced earlier for cellular networks are directly applicable to emerging wireless ad hoc networks. Furthermore, we conducted a simulation study that verifies the theoretical convergence results of the proposed algorithm. This was done, first, under the assumption of a TDMA scheme and later for TDMA/CDMA ad hoc networks. It was shown that CDMA, on top of TDMA, improves the single-hop throughput and reduces transmission power consumption at the expense of excessive processing power. Thus, there is a tradeoff between transmission power and processing power consumption that needs further investigation. Furthermore, we showed the performance of the optimum scheduling policies compared to simple heuristic policies under light and heavy load conditions. We conclude that there is room for performance improvement via introducing distributed, heuristic, and fair scheduling policies within the proposed framework. REFERENCES [1] P. Gupta and P. R. Kumar, “The capacity of wireless networks,” IEEE Trans. Inform. Theory, vol. 46, pp. 388–404, Mar. 2000. [2] J. Zander, “Distributed cochannel interference control in cellular radio systems,” IEEE Trans. Veh. Technol., vol. 41, pp. 305–311, Aug. 1992.

[3] S. Grandhi, R. Vijayan, D. Goodman, and J. Zander, “Centralized power control in cellular radio systems,” IEEE Trans. Veh. Technol., vol. 42, pp. 466–468, Nov. 1993. [4] G. Foschini and Z. Miljanic, “A simple distributed autonomous power control algorithm and its convergence,” IEEE Trans. Veh. Technol., vol. 42, pp. 641–646, Nov. 1993. [5] S. Grandhi, J. Zander, and R. Yates, “Constrained power control,” Int. J. Wireless Personal Commun., vol. 1, no. 4, Apr. 1995. [6] N. Bambos, S. Chen, and G. Pottie, “Radio link admission algorithms for wireless networks with power control and active link quality protection,” Proc. IEEE INFOCOM’95, pp. 97–104, 1995. [7] S. Ulukus and R. Yates, “Stochastic power control for cellular radio systems,” IEEE Trans. Commun., vol. 46, pp. 784–798, June 1998. [8] R. Yates, “A framework for uplink power control in cellular radio systems,” IEEE J. Select. Areas Commun., vol. 13, pp. 1341–1348, Sept. 1995. [9] D. Goodman and N. Mandayam, “Power control for wireless data,” IEEE Personal Commun. Mag., vol. 7, pp. 48–54, Apr. 2000. [10] C. Saraydar, N. Mandayam, and D. Goodman, “Pricing and power control in a multi-cell wireless data network,” IEEE J. Select. Areas Commun., vol. 19, pp. 1883–1892, Oct. 2001. , “Efficient power control via pricing in wireless data networks,” [11] IEEE Trans. Commun., vol. 50, pp. 291–303, Feb. 2002. [12] M. Xiao, N. Shroff, and E. Chong, “Utility-based power control in cellular wireless systems,” Proc. IEEE INFOCOM, Apr. 2001. [13] N. Bambos, “Toward power-sensitive network architectures in wireless communications: Concepts, issues, and design aspects,” IEEE Personal Commun. Mag., pp. 50–59, June 1998. [14] Z. Rosberg and J. Zander, “Toward a framework for power control in cellular systems,” Wireless Networks, vol. 4, pp. 215–222, 1998. [15] L. Kleinrock and J. Silvester, “Optimum transmission radii packet radio networks or why six is a magic number,” Proc. IEEE Nat. Telecommun. Conf., pp. 4.3.1–4.3.6, Dec. 1978. [16] T. Hou and V. Li, “Transmission range control in multihop packet radio networks,” IEEE Trans. Commun., vol. 34, pp. 38–44, Jan. 1986. [17] T. ElBatt, S. Krishnamurthy, D. Connors, and S. Dao, “Power management for throughput enhancement in wireless ad hoc networks,” Proc. IEEE ICC, 2000.

ELBATT AND EPHREMIDES: JOINT SCHEDULING AND POWER CONTROL

[18] R. Ramanathan and R. Rosales-Hain, “Topology control of multihop wireless networks using transmit power adjustment,” Proc. IEEE INFOCOM, 2000. [19] J. Monks, V. Bharghavan, and W. Hwu, “A power controlled multiple access protocol for wireless packet networks,” Proc. IEEE INFOCOM, Apr. 2001. [20] I. Chlamtac and S. Kutten, “A spatial reuse TDMA/FDMA for mobile multi-hop radio network,” Proc. IEEE INFOCOM, 1985. [21] A. Ephremides, J. Wieselthier, and D. Baker, “A design concept for reliable mobile radio networks with frequency hopping signaling,” Proc. IEEE, vol. 75, pp. 56–73, Jan. 1987. [22] I. Chlamtac and S. Pinter, “Distributed nodes organization algorithm for channel access in a multi-hop dynamic radio network,” IEEE Trans. Comput., vol. 36, pp. 728–737, 1987. [23] I. Cidon and M. Sidi, “Distributed assignment algorithms for multihop packet radio networks,” IEEE Trans. Comput., vol. 38, pp. 1353–1361, Oct. 1989. [24] L. Pond and V. Li, “A distributed time-slot assignment protocol for mobile multi-hop broadcast packet radio networks,” Proc. IEEE MILCOM, 1989. [25] R. Ramaswami and K. Parhi, “Distributed scheduling of broadcasts in a radio network,” Proc. IEEE INFOCOM, 1989. [26] A. Ephremides and T. Truong, “Scheduling broadcasts in multihop radio networks,” IEEE Trans. Commun., vol. 38, pp. 456–460, Apr. 1990. [27] S. Ramanathan and E. Lloyd, “Scheduling algorithms for multihop radio networks,” IEEE/ACM Trans. Networking, vol. 1, no. 2, pp. 166–177, 1993. [28] A. Ephremides, “Energy concerns in wireless networks,” IEEE Wireless Commun. Mag., pp. 48–59, Aug. 2002. [29] L. Williams, “Technology advances from small unit operations situation awareness system development,” IEEE Personal Commun. Mag., pp. 30–33, Feb. 2001. [30] K. Chadha, “The global positioning system: Challenges in bringing GPS to mainstream consumers,” in Proc. ISSCC’98, Feb. 1998. [31] M. Mauve, J. Widmer, and H. Hartenstein, “A survey on position-based routing in mobile ad hoc networks,” IEEE Networks, pp. 30–39, Nov./Dec. 2001. [32] C. Huang and R. Yates, “Rate of convergence for minimum power assignment algorithms in cellular radio systems,” Wireless Networks, vol. 4, pp. 223–231, 1998. [33] G. Holland and N. Vaidya, “A rate adaptive MAC protocol for multi-hop wireless networks,” in Proc. ACM MOBICOM, 2001. [34] B. Gottfried and J. Weisman, Introduction to Optimization Theory. Englewood Cliffs, NJ: Prentice-Hall, 1973. [35] E. Seneta, Non-Negative Matrices. New York: Wiley, 1973. [36] J. Proakis, Digital Communications, 3rd ed. New York: McGraw-Hill, 1983. [37] W. C. Y. Lee, Mobile Cellular Telecommunication Systems. New York: McGraw-Hill, 1989.

Tamer ElBatt (S’98–M’01) received the B.S. degree with honors and the M.S. degree from Cairo University, Egypt, in 1993 and 1996, respectively, and the Ph.D. degree from the University of Maryland, College Park, in 2000, all in electrical engineering. From 1993 to 1996, he served as a Teaching Assistant in the Department of Electronics and Communications Engineering, Cairo University, Egypt. From June 1995 to July 1996, he was a part-time Software Development Engineer at the International Computer and Communication Consultants (ICCC), Cairo, Egypt. From 1996 to 2000, he was a Research Assistant in the Department of Electrical and Computer Engineering and Center for Satellite and Hybrid Communication Networks at the University of Maryland, College Park. In September 2000, he joined the Information Sciences Lab, HRL Laboratories, LLC, Malibu, CA, where he is currently a Research Staff Member. His research interests include multiple access in wireless ad hoc networks, cross-layer protocol design, sensor networks, satellite-terrestrial hybrid wireless networks, queuing systems, and optical wireless networks. Dr. ElBatt received a technology achievement award from HRL in 2002.

85

Anthony Ephremides (S’68–M’71–SM’77–F’84) received the B.S. degree from the National Technical University of Athens (1967), Greece, and the M.S. and Ph.D. degrees from Princeton University, in 1969 and 1971, respectively, all in electrical engineering. He has been at the University of Maryland since 1971 and currently holds a joint appointment as Professor in the Electrical and Computer Engineering Department and in the Institute of Systems Research (ISR), of which he is a founding member. He is co-founder of the NASA Center for Commercial Development of Space on Hybrid and Satellite Communications Networks established in 1991 as an off-shoot of the ISR. He served as Co-Director of that Center from 1991 to 1994. He was a Visiting Professor in 1978 at the National Technical University, Athens, Greece, and in 1979 at the EECS Department of the University of California, Berkeley, and at INRIA, France. During 1985–1986 he was on leave at the Massachusetts Institute of Technology, Cambridge, and ETH, Zurich, Switzerland. He has authored or co-authored over 100 technical journal papers and 300 technical conference presentations. He has also contributed chapters to several books and edited numerous special issues of scientific journals. He has been the Dissertation Supervisor of over 20 Ph.D. students who now hold prominent positions in academia, industry, and research labs. He is the founder and President of Pontos, Inc., a MD company that provides technical consulting services, since 1980. His research interests include the areas of communication theory, communication systems and networks, queuing systems, signal processing, and satellite communications. Dr. Ephremides was the General Chairman of the 1986 IEEE Conference on Decision and Control in Athens, Greece, and of the 1991 IEEE International Symposium on Information Theory in Budapest, Hungary. He also organized workshops on Information theory in 1984 (Hot Springs, VA) and in 1999 (Metsovo, Greece). He was the Technical Program Co-Chair of the IEEE INFOCOM in New York City, in 1999, and of the IEEE International Symposium on Information theory in Sorrento, Italy, in 2000. He has also been the Director of the Fairchild Scholars and Doctoral Fellows Program, an academic and research partnership program in Satellite Communications between Fairchild Industries and the University of Maryland. He won the IEEE Donald E. Fink Prize Paper Award (1992) and he was the first recipient of the Sigmobile Award of the ACM (Association of Computer Machinery) in 1997 for contributions to wireless communications. He has been the President of the Information Theory Society of the IEEE (1987) and has served on its Board of Governors almost continuously since 1981. He was elected to the Board of Directors of the IEEE in 1989 and 1990. He has also won awards from the Maryland Office of Technology Liaison for the commercialization of products and ideas stemming from his research. He has served on the Editorial Boards of the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, IEEE TRANSACTIONS ON INFORMATION THEORY, the Journal of Wireless Networks, and the International Journal of Satellite Communications.