Autonomous Dynamic Power Control for Wireless Networks: User ...

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 3, MARCH 2008

Autonomous Dynamic Power Control for Wireless Networks: User-Centric and Network-Centric Consideration Siamak Sorooshyari and Zoran Gajic, Senior Member, IEEE

Abstract— Dynamic adaptation of transmission power has been researched as a technique for improving the performance and capacity of wireless networks. In this paper an estimatorbased algorithm is presented for distributed power control. The proposed power control policy is optimal with respect to users dynamically allocating transmit power so as to minimize an objective function consisting of the user’s performance degradation and the network interference. The policy enables a user to address various user-centric and network-centric objectives by adapting power in either a greedy or energy efficient manner. The algorithm is predictive, with a user performing autonomous interference estimation and prediction prior to adapting transmit power. Also, closed-loop implementation of the algorithm is of reasonable complexity thus allowing for distributed online operation. Subsequently, the robustness of the algorithm to stochastic detriments such as a time varying channel and noisy measurements is investigated. Index Terms— Distributed algorithms, H∞ estimation, Kalman filtering, nonlinear programming, power control, radio communication.

I. I NTRODUCTION DAPTIVE allocation of transmit power has been shown to be an efficient and effective means of improving the performance and capacity of wireless networks. The importance of distributed methods for power control has traditionally been fueled by the application to the uplink power control problem in cellular systems. More recently, attention has been focused on the application of autonomous power control to ad hoc wireless networks which lack infrastructure. Within the context of distributed power allocation policies, perhaps the most prominent algorithm was derived by Foschini and Miljanic [1]. The Foschini-Miljanic algorithm has been studied and expanded upon in a number of works including [2] [3] [4] [5]. The aforementioned works consist of power control algorithms which may be deemed greedy in that a user adapts power with the sole purpose of maintaining a target QoS metric during communication.

A

Manuscript received September 20, 2006; revised December 30, 2006; accepted February 18, 2007. The associate editor coordinating the review of this paper and approving it for publication was E. Serpedin. This work was supported by NSF grant ANIR-0106857. This work was presented in part at the 2006 IEEE International Conference on Communications (ICC 2006), Istanbul, Turkey, June 2006. S. Sorooshyari is with Alcatel-Lucent, Whippany, NJ 07981 USA (e-mail: [email protected]). Z. Gajic is with WINLAB, Rutgers University, Piscataway, NJ 08854 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2008.060731.

Although achieving a specific performance is important to users, depending on the application, a user may find it detrimental to attain such performance at arbitrarily high energy expenditure. Furthermore, a network may strongly oppose of such user transmitting with arbitrarily high power and disproportionately interfering with the remaining users. More recent works such as [6] [7] have advocated utility-based power control as devising policies which allow users to consider multiple objectives rather than solely instantaneous performance. In this presentation the terms user-centric, network-centric, greedy, and energy efficient are defined within the context of power control as we present an autonomous policy which allows a user to address various user-centric and network-centric objectives by adapting power in either a greedy or energyefficient manner. The works of [1] [3] [4] [6] [7] are deterministic in the sense that noiseless feedback and static channels are presumed between transmitters and receivers. In this work, noisy feedback and the time varying nature of the link gains are considered in the design and evaluation of a closed-loop power control algorithm. The policy is predictive with a user performing interference estimation and prediction prior to power adaptation. Within the context of power control, the benefits of predictive policies as far as providing improved robustness and convergence speed have been most notably motivated by [2] [5]. The presented estimator-based power control algorithm is optimal with respect to users adapting power so as to minimize an objective function consisting of QoS degradation and the network interference. Due to its predictive nature, with sufficiently good estimates, the algorithm has the potential of achieving rapid convergence. Essential implementational aspects of power control are addressed since the presented algorithm is decentralized, robust, and able to rapidly converge to a globally optimal solution. The remainder of this paper is organized as follows. Section II presents the derivation of a policy which is optimal in dynamically minimizing an objective function consisting of a user’s user-centric and network-centric objectives. The analysis is expanded upon with the solution of a constrained nonlinear programming problem pertaining to the policy under power restrictions. Additional insight is provided in Section III by quantifying the tradeoff incurred in a user’s QoS when the user adapts transmit power so as to conserve energy and reduce network interference. Attention is also given to the implementation of autonomous interference estimation and its

c 2008 IEEE 1536-1276/08$25.00 

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TABLE I W ITHIN THE CONTEXT OF POWER ALLOCATION THE FOLLOWING OBJECTIVES ARE CATEGORIZED AS EITHER BENEFITING AN INDIVIDUAL USER ( USER - CENTRIC ) OR THE ENTIRE NETWORK ( NETWORK - CENTRIC ). User-Centric Objective:

Network-Centric Objective:

satisfaction of SIR or delay constraint

reduction of network interference

increased throughput

reduced power dissipation by network users

conservation of transmit power

increased network lifetime

prolonged battery life

increased network capacity

value with respect to the presented algorithm. Section IV provides simulation results illustrating the utility and robustness of the power control algorithm when considered for the uplink of a CDMA system with users having heterogeneous service requirements. II. S YSTEM M ODEL AND P OWER A LLOCATION P OLICY A multiple access wireless network is frequently modeled as a collection of radio links separating transmitters and receivers. Each user is characterized as having an intended receiver, with its transmission causing interference to the signals of the other transmitting users. The dispersive nature of the wireless channel is modeled by the multiplicative link gains {Gij (k)}, with Gij (k) denoting the attenuation from the jth user’s transmitted signal to the ith user’s intended receiver. Although not crucial for our formulation; the link gains will be assumed as being fixed for the duration of the convergence of the power control algorithm. This indicates that the fading rate of the channel is slow in comparison to the rate at which power updates are performed. We shall consider the stochastic nature of the link gains in Section IV. Within a wireless network, the signal-to-interference ratio (SIR) determines a user’s QoS for a given transmission rate and bandwidth. Considering a power control policy with N users of power levels {Pi (k)}, at time k, the SIR of the ith user is defined as SIRi (k) = 

Pi (k)Gii Pj (k)Gij + ηi

=

Pi (k)Gii I−i (k)

(1)

j=i

with the constant ηi denoting the thermal noise power at the ith user’s intended receiver. The ith user’s perceived interference and aggregate interference are defined by  I−i (k) = Pj (k)Gij + ηi (2) j=i

and Ii (k) =



Pj (k)Gij + ηi = I−i (k) + Pi (k)Gii ,

(3)

j

respectively, with the subscript “−i” denoting the exclusion of the ith user’s signal. Even with no constraints on transmit power the issue still remains of whether there exists a set of power levels {Pi } such that the N users can achieve application-specific target SIR values. The so-called feasibility issue has been addressed in [4]. Within the context of distributed power control we deem a user’s user-centric objectives as benefits that a user may seek with disregard to other network users. Conversely, a user’s network-centric objectives

will be characterized as benefits seen by other network users, or the network as a whole, due to the user’s actions [6]. In Table I we have categorized several user-centric and networkcentric objectives that a user may strive for by dynamically adapting transmit power. With the user-centric and networkcentric concepts defined within the realm of power control, we conjecture that depending on a user’s application and the network dynamics, a power control policy will follow one of two strategies: • Greedy approach: Adaptation of transmit power in response to channel impairments with the goal of maintaining a target SIR threshold. A user may be best suited to use such a strategy when supporting an application which has a continuous stream of delay sensitive traffic. • Energy efficient approach: Opportunistic allocation of transmit power relative to the channel state. In other words, a user would increase transmit power during good channel conditions (i.e. low I−i (k)), decrease power for poor channel conditions, and transmit with minimal power once its channel quality falls below a certain threshold. Such an approach is typically referred to as "water-filling" [8], and is beneficial to an energyconstrained user with an application that consists of bursty traffic which is delay insensitive. The greedy approach leads to a user increasing transmit power when witnessing increasing levels of interference, whereas the energy efficient approach would see the user lowering power or possibly ceasing transmission. After all, the greedy user is continuously aiming for a target SIR, but is certainly not energy efficient in doing so. Conversely, an energy efficient user is willing to sacrifice instantaneous QoS by lowering transmit power during poor channel conditions and halting transmission after a certain point. Naturally, it would be advantageous for an energy efficient user to either decrease its transmission rate or increase forward error correction (FEC) when lowering transmit power during unfavorable channel states. With transmitter power being an indispensable commodity in reliable wireless communication, its regulation and the subsequent conservation of energy is imperative. From a user-centric perspective, energy efficiency is important in lessening power dissipation and prolonging battery life. From a network-centric perspective, energy efficiency reduces the aggregate network interference. This results in increased network capacity, admission of additional users, and prolonged lifetime of the network. The following power allocation policy is derived with the intent of enabling a user to address various user-centric and network-centric objectives by adapting power in either a greedy or energy efficient manner.

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^I (k) -i Compute

Estimator Filter

α*i(k ) +

α*i (k )

P(k+1) i

Pi (k)

-1

Z

^I (k+1) -i Yi(k)

I-i (k)

+ vi(k)

Intended Receiver of ith User

wi (k)

SIR i (k)

Feedback Channel

Fig. 1. Distributed closed-loop power control between the ith user and its intended receiver. The estimator filter is used to obtain a local estimate of the current and next-step values of the user’s perceived interference. Subsequently, the optimal power update gain with respect to the minimization of the ith user’s objective function is computed. The exogenous disturbances vi (k) and wi (k) denote the ith user’s measurement noise and process noise, respectively.

A fundamental criterion in the determination of a user’s QoS is the attained SIR during communication. The ith user’s SIR deviation − SIRi (k) Ei (k) = SIRtar i

(4)

may be viewed as a user-centric metric since it represents and the deviation between the user’s target QoS of SIRtar i received QoS at time k. A user’s adaptation of transmit power so as to minimize its SIR deviation is perceived as an action which only benefits that user. Contrarily, a user’s aggregate interference may be viewed as a network-centric metric since it indicates the amount of interference introduced into the network by that user. Thus, a user’s adaptation of transmit power so as to reduce interference shall be interpreted as an action which ultimately benefits other network users, or equivalently, the network. Accordingly, in determining a user’s distributed power allocation within the network we propose the convex cost function Ji (k) = ρi1 Ei2 (k + 1) + ρi2 Ii2 (k + 1)

(5)

as being representative of both user-centric and networkcentric objectives. For user i, the positive weights ρi1 and ρi2 dictate the priority given to the fulfillment of a QoS requirement and controlling the level of network interference, respectively. The motivation behind (5) stems from an autonomous user adapting power so as to minimize cost. The SIR deviation alone is a cost which may be minimized by not transmitting with more power than that necessary for meeting the target SIR. The interference term constitutes an additional cost, or penalty, meant to inhibit the user from achieving a desired QoS with arbitrarily high energy expenditure. For the duration of the paper we shall maintain a level of abstraction with respect to the manner by which the weights {ρi1 , ρi2 } are specified. In practice, the weights may be assigned to a user by the network based on network congestion, the user’s location, or other global information. Alternatively, the weights may be selected by a user in ad hoc manner based on the user’s application, channel state, delay constraint, or other local information. Although it shall be assumed from hereon, the weights need not remain static for the duration of the power control algorithm. It will be shown in our first theorem that, for a given user, only the ratio of the weights (ρi2 /ρi1 ) will be of relevance rather than the specific values of ρi1 and ρi2 .

In correspondence to an interference-based power control ideology we propose that the ith user dynamically adapt power in response to its perceived interference via an assignment of the form Pi (k + 1) = Pi (k) + f {I−i (k)}. The power update Pi (k + 1) = Pi (k) + αi (k)I−i (k)

(6)

supports such a notion with the gain αi (k) parameterizing the increase/decrease in transmit power at each time instant. Our first result is given in the following theorem. Theorem 1: At each power control iteration, the power update gain given by (7) with ρi  ρi2 /ρi1 is optimal with respect to the joint minimization of the ith user’s SIR deviation and aggregate interference in accordance with the objective function of (5). Proof: Due to page restrictions we shall outline this proof while omitting the algebra. The optimal gain in (6) is αi∗ (k) = arg minαi (k) {ρi1 Ei2 (k + 1) + ρi2 Ii2 (k + 1)}. We note that  Pj (k + 1)Gij + ηi Ii (k + 1) = j

= Ii (k) +



αj (k)I−j (k)Gij

j=i

+ αi (k)I−i (k)Gii Ei (k + 1)

= =

SIRtar − SIRi (k + 1) i I−i (k) (SIRi (k) + αi (k)Gii ) SIRtar − i I−i (k + 1)

from which Ii2 (k + 1) and Ei2 (k + 1) can be readily obtained. Evaluation of a partial derivative followed by solving ∂Ji (k)/∂αi (k) = 0 for αi (k) yields (7). The globality of the minimum is confirmed via ∂ 2 Ji (k)/∂αi2 (k) = 2 2 2 (k)G2ii + 2ρi1 G2ii I−i (k)/I−i (k + 1) > 0. 2ρi2 I−i  With the optimal gain derived, the optimal power update Pi∗ (k + 1) = Pi (k) + αi∗ (k)I−i (k) can be performed in distributed manner as shown in Fig. 1. It is imperative to note that the computation of αi∗ (k) requires knowledge of the current (estimated) and the next-step (predicted) values for the perceived interference. In general, a user may devise any estimator to autonomously calculate such quantities when provided with feedback from its intended receiver. Interference estimation and prediction will be addressed in the following section. The decentralized nature of the policy is indicated by the fact that, in performing a power update, a user only

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TABLE II C LOSED - LOOP AUTONOMOUS INTERFERENCE - AWARE POWER CONTROL (AIPC) ALGORITHM FOR THE iTH USER . T HE PARAMETERS {ρi ≥ 0, Pimin , Pimax } ARE EITHER SELECTED BY THE iTH USER , OR ASSIGNED TO THE USER BY THE NETWORK . % time = k measure interference: Yi (k) = I−i (k) + vi (k); estimate Iˆ−i (k) and predict Iˆ−i (k + 1); determine ρi ; if ρi = 0 { % user i is energy efficient compute α∗i (k) via equation (7);





Pi (k + 1) = max min Pi (k) + α∗i (k)Iˆ−i (k), Pimax , Pimin };

else { % user i is greedy





Pi (k + 1) = max min SIRitar Iˆ−i (k + 1)/Gii , Pimax , Pimin };

αi∗ (k) =

2 (k + 1) (−Pi (k)Gii − I−i (k + 1)) + SIRtar ρi I−i i I−i (k + 1) − SIRi (k)I−i (k) 2 I−i (k)Gii (1 + ρi I−i (k + 1))

requires local information pertaining to its perceived interference and the link gain to its intended receiver. The presented power control policy was derived with the intent of enabling a user the capability to address various usercentric and network-centric objectives via either an energy efficient or greedy approach to power allocation. In fact, the power control policy allows a user to be either greedy or energy efficient through the assignment of ρi . A greedy user is solely concerned with its user-centric performance and, with an assignment of ρi = 0, will adapt power so as to maintain a target QoS irrespective of energy expenditure or interference production. This is verified by noting the absence of the aggregate interference in the objective function (5) when ρi = 0. Conversely, an energy efficient user, with an assignment of ρi > 0, addresses its networkcentric performance by regulating the amount of interference it introduces into the network, and addresses its user-centric performance by conserving transmit power and battery life via opportunistic power adaptation with respect to channel state. It’s possible for a user to waver between the greedy and energy-efficient approaches by adapting ρi in correspondence with its application, traffic, or channel condition. Alternately, ρi may be statically assigned to a user by the network for the duration of the user’s lifetime. Realistically, network users will have hard constraints on their dynamic range of transmit power. Thus, we restrict the ith user to a minimum and maximum power level of Pimin and Pimax , respectively. This translates to user i dynamically adapting transmit power in order to achieve minimize subject to

ρi1 Ei2 (k + 1) + ρi2 Ii2 (k + 1) Pimin ≤ Pi (k + 1) ≤ Pimax .

(8)

The solution of the constrained nonlinear programming problem is given as follows. Corollary 1: For a user having the power constraint of (8), the distributed power update given by (9) is optimal with respect to the minimization of the objective function pertaining to the user’s SIR deviation and aggregate interference.

(7)

Proof: The substitution of (6) into the box constraints of (8) allows the power constraints to be alternatively expressed as (Pi (k) − Pimax )/I−i (k) + αi (k) ≤ 0 and (Pimin − Pi (k))/I−i (k) − αi (k) ≤ 0. The constrained nonlinear programming problem of (8) may now be solved via the KarushKuhn-Tucker conditions [9]. We introduce the scalars u1 ≥ 0 and u2 ≥ 0 so as to solve   ∂Ji (k) u1 Pi (k) − Pimax + + αi (k) ∂αi (k) αi (k) I−i (k)   min Pi − Pi (k) u2 − αi (k) = 0 + αi (k) I−i (k)   Pi (k) − Pimax + αi (k) = 0 u1 I−i (k)   min Pi − Pi (k) − αi (k) = 0 (10) u2 I−i (k) for the optimal gain αi∗ (k). Evaluation of (10) for {u1 = 0, u2 = 0}, {u1 = 0, u2 > 0}, {u1 > 0, u2 = 0}, and {u1 > 0, u2 > 0} yields the power update of (9) as the solution to the optimization problem. The convexity of (8) justifies the global optimality of the solution.  A description of the presented power control policy which we deem autonomous interference-aware power control (AIPC) is given in Table II. It is noteworthy that, alternatively, a centralized power allocation problem may be formulated as minimize

N 

ρi1 Ei2 (k + 1) + ρi2 Ii2 (k + 1)

(11)

i=1

subject to Pimin ≤ Pi (k + 1) ≤ Pimax ,

i = 1, . . . , N

with the network allocating power to the users so as to minimize a network-wide objective function. It can be verified that (11) is non-convex with respect to the variables {αi (k)}. Also, the solution to (11) will generally differ from the solution obtained with each user adopting the policy in (8). III. C LOSED -L OOP P OWER C ONTROL A LGORITHM The application of interference reduction to distributed power control has thus far been viewed from the perspective of

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⎧ P min −P (k) ⎪ Pimin if I−i (k) > i α∗ (k)i and αi∗ (k) < 0 ⎪ ⎪ i ⎪ ⎪ ⎨ P max −P (k) Pi (k + 1) = Pimax if I−i (k) > i α∗ (k)i and αi∗ (k) > 0 ⎪ i ⎪ ⎪ ⎪ ⎪ ⎩ otherwise Pi (k) + αi∗ (k)I−i (k)

TIˆ(Pi (k)) 

(9)



2 ˆ ˆ ˆ I I SIRtar (k + 1) − ρ (k + 1) P (k) G + I (k + 1) −i i −i i ii −i i

the minimization of (5). Analysis of the power control policy presented in the previous section would involve comprehension of the dynamics of a user’s power adaptation with respect to its perceived interference. This section aims to provide such analysis and to quantify the tradeoff in a user’s QoS when a user adapts transmit power so as to address its networkcentric performance. Autonomous interference estimation and closed-loop implementation of the presented policy will also be addressed. A. Dynamics of AIPC The distributed power allocation policy of (9) allows a user the flexibility of adapting power in either a greedy or energyefficient manner. This can be seen by examination of a user’s power adaptation with respect to its perceived interference. We note through (7) that the condition αi∗ (k) ≶ 0 in (9) can be alternatively expressed as Iˆ−i (k) ≷ TIˆ(Pi (k)) via (12). Assuming that Pimin < Pi (k) < Pimax and   TI (Pi (k)) = TIˆ(Pi (k)) Iˆ−i (k)=I−i (k), Iˆ−i (k+1)=I−i (k+1) via perfect interference estimation, user i will subsequently transmit with minimum power iff I−i (k) > min

P −P (k) max i α∗ (k)i , TI (Pi (k)) . This indicates that a user i who experiences a “bad channel” in terms of its perceived interference exceeding a certain threshold will autonomously opt-out and transmit with minimum power. Inspection of (12) reveals the threshold by which a bad channel is defined as being heavily dependent on the value of ρi . This is explained by the fact that with the power updates performed according to (9); a user that experiences a high level of interference (i.e., a bad channel) will increase power and introduce a disproportionately large amount of additional interference into the network. While rectifying the user’s performance during unfavorable channel conditions, such user would incur a large dissipation of transmit power and the network would witness a correspondingly large amount of interference. Thus, in such scenario an energy efficient user would find it beneficial to opt-out, or according to [3] autonomously perform voluntary drop-out (VDO). In fact, the decrease of the “bad-channel threshold” value with increasing ρi indicates a lower (interference) tolerance level before the user decides to opt-out. Alternately, this may be viewed as a variant of distributed interactive admission control [10] where an active user is not allowed to transmit unless its measured interference remains below a threshold. It should be clear that an increase in ρi

(12)

SIRi (k)

and the resultant energy-efficiency would favor the ith user’s network-centric performance since the user would regulate the amount of interference which he introduces into the network. The dynamics of a user’s power evolution as a function of perceived interference is shown in Fig. 2, with Pimax denoting the state at which the user transmits with maximum power, Pi∗ indicating a non-extremum power level, and Pimin denoting the opt-out state. We now wish to quantify the implications of interference reduction on a user’s QoS. The classical greedy approach of Foschini-Miljanic to distributed power control may be presented as minimize

N 

Pi

(13)

i=1

≤  subject to SIRtar i

Pi Gii

,

i = 1, . . . , N

Pj Gij + ηi

j=i

and represented in matrix form via1 (I − A)p ≥ b where the vector p = [P1 , P2 , . . . , PN ]T denotes the converged powers of the network users. The entries of the matrix A are specified as  0 if i = j (14) A(i, j) = SIRtar Gij i otherwise Gii with i, j ∈ {1, 2, . . . , N }, and  T SIRtar SIRtar SIRtar 1 η1 2 η2 N ηN b= , , ..., . G11 G22 GN N

(15)

When (I − A)−1 > 0 exists, the problem given by (13) is said to be feasible with the power vector p∗ = (I − A)−1 b being the globally optimal solution such that for any power vector p satisfying (13), p∗ ≤ p . Given feasibility of the power control problem, p∗ may be iteratively reached via [1] p(k + 1) = Ap(k) + b, whereas each user may perform this power update in distributed fashion according to Pi (k + 1) = SIRtar i Pi (k)/SIRi (k). Reverting attention to the power control algorithm presented in Section II; the following result provides a connection between the globally optimal solution of the presented policy and that of the Foschini-Miljanic scheme reviewed above. Theorem 2: At the convergence of the proposed power control algorithm the ith user will have achieved a modified 1 We adopt the convention that the matrix inequality X ≥ X or the 1 2 vector inequality x1 ≥ x2 denotes inequality in all components.

SOROOSHYARI AND GAJIC: AUTONOMOUS DYNAMIC POWER CONTROL FOR WIRELESS NETWORKS

− Pi α*i(k)

max

max

{

min

I−i(k) > max{ Pi

1009

, TI (Pimax )

min

− Pi (k) , Pi − Pi (k) α*i(k) α*i(k)

min

− Pi (k) , TI (Pi(k)) α*i(k)

I−i(k) > max{ Pi

min

{

I−i (k) > TI (Pimin )

{

I−i(k) < max{ Pi

T I (Pi max) < I−i(k)
0 being equivalent to the condition ρ(A) < 1. The fact that ˜ ≤ ρ(A) and p ˜ ∗ ≤ p∗ are readily verified by noting ρ(A) ˜ ≤ b with equality holding for the case ˜ ≤ A and b that A of βi = 1 ∀ i, which would occur only if ρi = 0 ∀ i. 

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ρ =0

~ i1 UU

UU

i

-

max i2

ρ >0 i

0 < ρi < ρ =

ρ =0

i

-

SIR

SIR

~ tari

min i2

i

-

UU

-

min i1

-

UU

-

max i1

User utility 2

User utility 1

UU

tar i

E

min i

E

max i

Energy expenditure

Signal to interference ratio

Fig. 3. Illustration of competing utilities obtained by the ith user with the presented power control policy. Naturally, a tradeoff is inherent between a utility which is a monotonically increasing function of SIR (U Ui1 ) and a utility which is a monotonically decreasing function of energy expenditure (U Ui2 ).

ρ =0 ∀ i i

ρ = i

max NU 2

{ ρi } : ∃ i , 0 < ρi < i

-

min

NU 2 S

-

min

S

max

Sum throughput

-

1

ρ =0 ∀ i i

-

min

-

NU

∀ i

{ ρi } : ∃ i , 0 < ρi
0 ∀ i. The users will feasible solution p  i ≤ SIRi ∀ i. achieve modified target SIR values SIR Proof: We note from the proof of Theorem 2 that βi ∈ [0, 1] is monotonically decreasing with the ith user’s interference. If all users are energy efficient, each users’ modified tartar i } get SIR will decrease with I−i (k) so that eventually {SIR will be low enough so as to be feasible. The corresponding ˜ ∗ will be unique since the problem is still conpower vector p vex.  It should be noted that the condition set forth in the above corollary is sufficient rather than necessary. For instance, consider a specific infeasible realization of (13). For the AIPC policy to converge to a unique feasible solution, it may be enough that only a few users be energy efficient. Nevertheless, the condition ρi > 0 ∀ i is required to ensure feasibility of our policy for all infeasible realizations of (13). From a user’s

perspective, energy efficient adaptation of transmit power with the intent of conserving battery life and addressing networkcentric objectives leads to the sacrifice of striving for a reduced target SIR. B. Utility-Based Power Control Within the framework of utility-based power control [6] [7], Fig. 3 depicts the ith user’s utility (U Ui ) with the presented policy. Naturally, there is a tradeoff between utility obtained from QoS (U Ui1 ) and utility obtained from energy conservation (U Ui2 ). The presented policy is versatile in allowing a max min , U Ui2 } or user to operate at the extremum points {U Ui1 min max {U Ui1 , U Ui2 } by being either greedy or overly energy efficient2 , respectively. Alternatively, a user may be energy efficient via 0 < ρi < ∞ and balance the utility gained by QoS versus the utility gained by energy conservation so as to maximize an aggregate utility of U Ui = U Ui1 + U Ui2 . It should be noted that in Fig. 3 we have arbitrarily depicted 2 A user with ρ = ∞ is deemed as being overly energy efficient. It can be i verified from (7) and (9) that such user will transmit with minimum allowable min power Pi irrespective of its channel state I−i (k).

SOROOSHYARI AND GAJIC: AUTONOMOUS DYNAMIC POWER CONTROL FOR WIRELESS NETWORKS

With the presented power control algorithm each user’s controller is parameterized by a gain which effectively adapts its transmit power. Computation of the optimal gain is dependent upon a user’s estimate of its interference for the current time instant and the predicted interference of the subsequent time instant. The estimated and predicted interference values Iˆ−i (k) and Iˆ−i (k + 1) are shown in Fig. 5 with Ki (k) denoting the filter gain. Once calculated the interference estimates are used in computing the optimal controller gain as shown in Fig. 1. Traditionally, the Kalman filter has been the preferred filter for predictive power control algorithms [2] [5]. The Kalman filter, however, presupposes a Gaussian assumption on the stochastic disturbances. The H∞ estimator shares a similar structure to the Kalman filter while not requiring an assumption on the statistics of the disturbances except that they be bounded. The H∞ filter was developed within the framework of H∞ optimization [11]. The dynamics of the ith user’s interference are given by I−i (k + 1) Yi (k)

= I−i (k) + wi (k) = I−i (k) + vi (k)

(18)

where wi (k) represents the driving disturbance and the measurement noise is denoted by vi (k). With respect to the scalar interference estimation problem modeled by the discrete-time system of (18), the performance measure of H∞ filtering

I-i (k)

I-i (k+1)

+ +

Z

-1

I-i (k)

wi (k)

+

C. Autonomous Interference Estimation

Intended receiver of user i

+ +

Z

K i (k)

vi (k)

Yi (k)

Estimator of user i ^I (k+1) -i

Feedback channel +

the user utilities U Ui1 and U Ui2 as being sigmoid-like for 0 < ρi < ∞, while our analysis generalizes to user utility functions U Ui1 and U Ui2 that are strictly non-decreasing and non-increasing, respectively. Furthermore, for U Ui1 the abscissa may be any monotonically increasing function of SIR such as throughput, and for U Ui2 the abscissa may be any monotonically increasing function of energy expenditure such as battery drain. In the scenario of {ρi } being assigned to the users by the network, Fig. 4 depicts the network’s utility (N U ) with users allocating power according to the presented policy. An apparent tradeoff exists between the utility the network would obtain from the sum throughput of the supported users (N U1 ) and the utility obtained from network lifetime (N U2 ). The extremum points {N U1max , N U2min } and {N U1min , N U2max } correspond to the network requesting that all N users be greedy or overly energy efficient, respectively. The interval in between the extremum points correspond to all possible assignments of {ρi } so that there exists at least one user with 0 < ρi < ∞. It would seem sensible that the network would assign {ρi } so as to maximize the network utility N U = N U1 +N U2 . It should be apparent that for N U1 the abscissa may be any monotonically increasing function of the users’ SIRs, and for N U2 the abscissa may be any monotonically decreasing function of the users’ energy expenditure. In Fig. 4, we have arbitrarily depicted the network’s utilities N U1 and N U2 as being sigmoid-like for the interval {ρi } : ∃ i, 0 < ρi < ∞. Nevertheless, our comments apply to network utility functions N U1 and N U2 that are strictly non-decreasing.

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e i (k)

+ -

^I (k) -i

-1

Fig. 5. Autonomous interference estimation and prediction at the ith user. The measurement Yi (k) may contain noise due to stochastic impairments in the feedback channel between the ith user and its intended receiver.

defined by  2 k ei (k) = −1 −1  −1  2 2 2 Bi (0)ei (0) + Wi k wi (k) + Vi k vi (k) (19) reveals the level of the uniform estimation error ei (k) = I−i (k) − Iˆ−i (k). The positive scalar constants Vi , Wi , and Qi must be specified by the designer and will dictate the robustness of the estimator. For convenience and tractability, the suboptimal formulation requiring sup{JiH∞ } < 1/ci , for a constant ci > 0, will be used in specifying the H∞ filter of the ith user. Subsequently, the gain of the suboptimal finite horizon H∞ filter is given as Ki (k) = Bi (k)/ (Vi − ci Vi Qi Bi (k) + Bi (k)). As shown in Fig. 5, interference predictions will be made via

JiH∞

Qi

Iˆ−i (k + 1) = Iˆ−i (k) + Ki (k)(Yi (k) − Iˆ−i (k)) .

(20)

The existence of the H∞ filter for the considered dynamic system is justified by the fact that the discrete-time scalar Riccati equation Bi (k + 1) = Bi (k)(1 − ci Qi Bi (k) + Bi (k)Vi−1 )−1 + Wi , with initial condition Bi (0) > 0, will always have a stabilizing solution Bi (k) > 0 ∀ k [11]. It is noteworthy that the practical importance behind the consideration of stochastic measurements in autonomous power control was initially motivated by the authors in [12]. The stochastic disturbance vi (k) in (18) is representative of an unavoidable measurement noise present in the ith user’s estimate of its perceived interference. As discussed, the local estimates are computed on-line so as to allow for decentralized real-time implementation. The distributed power control policy presented in this paper is fundamentally different than that of [12] in the deployment of an estimator, and in the allocation of power so as to jointly address the user-centric and network-centric objectives of Table I. In the following section we initially examine the dynamics of the users’ transmit power and SIR evolution in the absence of exogenous disturbances, and subsequently assess the performance of the proposed power control algorithm with stochastic detriments.

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0 0

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Fig. 6. Dynamics of the transmit power and SIR of N = 20 users with AIPC. A target SIR of SIRtar = 5 was specified for each user. The two users i with indices 3 and 8 were characterized as being greedy during 0 < k ≤ 100, and energy efficient afterwards via ρ3 = 10, ρ8 = 10 for 100 < k ≤ 200 and ρ3 = 100, ρ8 = 10 for 200 < k ≤ 300. The remaining 18 users were characterized as being greedy (i.e. ρi = 0 for i = 3, 8) for the duration of the simulation.

IV. S IMULATION R ESULTS AND D ISCUSSION In this section the performance of the AIPC algorithm is evaluated by considering a network consisting of N users. Each user’s transmitter updates power as shown in Fig. 1. In doing so, users will autonomously perform interference estimation and prediction at each iteration according to Fig. 5. We consider the uplink of a CDMA system with the users communicating with a single base station. The use of a linear receiver allows the ith user’s SIR to be defined as  2 Pi (k)hi cTi (k)si (k) SIRi (k) =   2 Pj (k)hj cTi (k)sj (k) + cTi (k)ci (k)ηi j=i

(21) with si (k) ∈ RL and ci (k) ∈ RL denoting the user’s codeword and receive vector, respectively. The constant L denotes the processing gain and the gains {hj } represent the pathloss. The signature sequence si = √1L [si1 , si2 , . . . , siL ]T is fixed for the duration of convergence of the power control algorithm, and a matched filter receiver (i.e., ci = si ) will be used for demodulation. We note that the excellent crosscorrelation properties of Hadamard and Gold sequences [14] will be mitigated by the asynchronous nature of the CDMA uplink. Thus, in correspondence with works such as [12][15] we consider randomly generated signature sequences with sij ∈ {−1, 1}. Comparison of (21) with (1) reveals that the link gains may be represented as  hi if i = j (22) Gij = hj (sTi sj )2 if i = j . A frequently used path loss model for cellular radio comn munication is given by hi = PR (dR /di ) = A/dni where dR is a reference distance, PR is the received power at the reference distance, di is the distance between the ith user and the base station, and n denotes the path loss exponent. We shall assume a path loss exponent of n = 4 and assign A = 10−4 in correspondence to a path gain of −40 dB at a reference

distance of 1 km with a 1.9 GHz carrier frequency [13]. A receiver noise power of η = 10−3 mW will be assumed along with a single circular cell with a coverage range of radius r = 1 km. Within the cell the N users locations’ will be generated uniformly on the interval of (0, r]. We shall consider = 5 for each user. A processing N = 20 users with SIRtar i gain of L = 128 will be allocated to each user along with a maximum power level of Pimax = 500 mW. Initially each user will transmit with a power level of Pi (0) = Pimin = 0.0 mW. The H∞ filter optimization parameters are selected as Vi = 1.0, Wi = 0.1, Qi = 20.0, and ci = 0.001 ∀ i. With the empirical system outlined above, we investigate the dynamics of the presented policy as far as enabling a user to jointly address the user-centric and network-centric objectives of Table I by adopting either a greedy or energy efficient approach to power adaptation. Subsequently, the robustness of the distributed power control algorithm shall be investigated within a stochastic setting. For lucidness in presentation, in examining the dynamics of the power control policy we shall ignore the stochastic detriments giving rise to the measurement noise and the process disturbance. The power and SIR evolution of the 20 users is shown in Fig. 6 with each user adapting power in a greedy manner for the first one-hundred power updates via the assignment ρi = 0 ∀ i. Due to the feasibility of the target SIRs all users attain their target SIR value upon convergence of the transmit powers. In fact, the converged power and SIR values of the users correspond to those obtained with classical greedy algorithms such as [1]. The two users dissipating the most power (users 3 and 8) autonomously choose3 to be energy efficient by selecting ρ3 = 10 and ρ8 = 10 for 3 We use the word choose with caution since ρ = 10 and ρ = 10 3 8 may have been dynamically assigned to user 3 and user 8 by the network due to congestion, a change in the users’ application, or as a way of prioritizing the QoS of various users. Alternatively, ρ3 and ρ8 may have been dynamically adapted by user 3 and user 8, respectively, during transmission in accordance with changes in application, traffic, or channel state. We maintain this abstraction so as to sustain generality.

SOROOSHYARI AND GAJIC: AUTONOMOUS DYNAMIC POWER CONTROL FOR WIRELESS NETWORKS

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Fig. 7. Stochastic power and SIR evolution of N = 20 users with AIPC. Each user was greedy via ρi = 0 with a target QoS of SIRtar = 5. Deployment i of the H∞ filter allows for robust and autonomous interference estimation while alleviating the need for tracking the statistics of the exogenous disturbances.

100 < k ≤ 200. A resultant savings of 27 percent and 47 percent in transmission power is witnessed by user 3 and 8, respectively. Within the user-centric paradigm, the two users are sacrificing instantaneous SIR in favor of energy conservation and prolonged battery life. A less obvious occurrence is the collective power conservation of the greedy users which comes at no sacrifice to their QoS. Specifically, a mean power savings of 17 percent is experienced by the 18 greedy users. This is due to users 3 and 8 addressing their networkcentric objectives of decreasing the interference which they introduce into the network, and reducing the power dissipation of the other network users. We make the following point in retrospect to users 3 and 8 achieving degraded SIR values during the interval 100 < k ≤ 200. An energy efficient user may still fulfill its applicationwitnessing SIRi (k) < SIRtar i specific QoS requirement at a sacrifice in delay. Such a user would require excessive retransmissions at the link layer, or increased redundancy at the physical layer to compensate for its degraded SIR. This further justifies the utility of the opportunistic energy efficient approach for applications with bursty and delay-insensitive traffic. Lastly, user 3 chooses to further address energy efficiency via an assignment of ρ3 = 100 during the interval 200 < k ≤ 300. This leads to the user opting out via a converged transmit power of P3 (k) = P3min = 0 mW for 200 < k ≤ 300. As illustrated in Fig. 2, user 3 will only commence transmission in response to an improved channel state, or equivalently, a sufficiently smaller value of I−3 (k) for k > 300. In response to the increased network-centric performance of user 3 an additional mean power savings of 23 percent is witnessed by the 19 transmitting users at no loss to their SIR performance. In addition to addressing network-centric objectives, user 3 may seek to optimize its throughput by resuming transmission with a higher data rate for a favorable channel state after having remained dormant during such an unfavorable transmission scenario. In light of the example, we comment on the selection or assignment of ρi within a practical setting. The 18 users with ρi = 0 ∀ k are best suited as having had a real-time

application (i.e. voice) which can not gracefully cope with outage or delay. A value of ρ8 = 10 for k > 100 would correspond to user 8 having a data-based application that can withstand delay due to retransmission. Finally, a value of ρ3 = 100 for user 3 with k > 200 would be appropriate for a user wishing to transmit in opportunistic fashion due to a non-rechargeable battery and a delay insensitive application. We now wish to investigate the robustness of the predictive power control algorithm to stochastic impairments. In a wireless channel the fading process and the mobility of the users render the channel response as a stochastic process. The time varying nature of the channel shall be depicted by representing each link gain by a first-order Gauss-Markov model Gij (k + 1) = Gij (k) + gij

(23)

with gij ∼ N (0, Var[gij ]). At time k, the link gain ˜ ij (k) Gij (k) = Gij + G

(24)

shall consist of a deterministic component Gij = E[Gij (k)] ˜ ij (k) ∼ given by (22) and a stochastic component G ˜ ij (k)]) denoting fluctuations brought on by N (0, Var[G small-scale effects such as user mobility and multipath fading. Since 0 < Gij (k) ≤ 1.0, the stochastic perturbations shall ˜ ij (k) ∈ (−Gij , 1 − Gij ]. We be limited to the interval G shall model the variance of the perturbations in (24) as ˜ ij (k)]) = μ1 Gij with μ1 < 1, and model the variance Var[G of the sequence of random variates {gij } in (23) as Var[gij ] = μ2 Gij with μ2 < 1. The power and SIR evolution of each user The variance is shown in Fig. 7 for μ = μ1 = μ2 = 1/10.  of the measurement noise Vi (k) = E vi2 (k) was specified as Vi (k) = ηv I−i (k) with ηv = 1/5 and vi (k) ∼ N (0, Vi (k)). All users were designated as being greedy via the assignment = 5. ρi = 0 ∀ i, and each user had a target QoS of SIRtar i We define a user’s outage rate as the percentage of time during which a user’s attained SIR is below 95 percent of the target value. In Fig. 7 the stochastic convergence of the transmit powers, defined by |αi∗ (k)| ≤ 0.01 ∀ i, is seen after approximately 90 iterations with an average outage rate of

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Fig. 8. Comparison of the mean outage rate obtained by 20 network users with differing degrees of variation in the stochastic link gains and the measurement noise. Each user was greedy via ρi = 0 with a target QoS of SIRtar = 5. In the three scenarios above the outage rates of the 20 users were averaged in i order to obtain the mean outage rate.

nearly 3 percent after convergence. It is imperative to note that the users were able to obtain such performance in autonomous fashion with no knowledge pertaining to the statistics of the disturbances. We now aim to compare the robustness of the power control algorithm with users using the H∞ filter with that of users using a Kalman filter for interference estimation and prediction. The Kalman filter gain is specified as Ki (k) = Bi (k)/(Bi (k) + Vi (k)) with the corresponding Riccati equation Bi (k + 1) = Bi (k) − Bi2 (k)(Bi (k) + Vi (k))−1 + Wi (k)   where Wi (k) = E wi2 (k) . Deployment of the Kalman filter requires a user to have knowledge of the statistics of the disturbances in (18). The measurement noise is locally impingent upon the received feedback of the ith user, and hence its statistics are assumed to be known a priori as vi (k) ∼ N (0, Vi (k)) with Vi (k) = ηv I−i (k). We note that the dynamics of the ith user’s next-step perceived interference can be expressed as  Pj (k + 1)Gij (k + 1) + ηi I−i (k + 1) = j=i

=



 Pj (k) + αj∗ (k)I−j (k) (Gij (k) + gij )

j=i

+ ηi = I−i (k) + wi (k)

(25)

with the stochastic process  wi (k) = αj∗ (k)I−j (k)Gij (k)+Pj (k)gij +αj∗ (k)I−j (k)gij j=i

(26)

denoting the driving disturbance acting upon the perceived interference of the ith user. Inspection of wi (k) reveals that the distribution of the driving disturbance may not be very instructive to derive since it is a function of parameters which are not locally known by the ith user. More specifically, an arbitrary user would not be aware of the transmit power of other users nor have statistical information pertaining to the stochastic link gains of the other transmitters. With {Gij (k)} and {gij } being normal we invoke a Gaussian assumption on the process noise by assuming wi (k) ∼ N (bi (k), Wi (k)). From the state equation in (18) it follows that bi (k) = E[I−i (k +1)−I−i (k)]. Therefore, we designate the sequences ˆbi (k) ˆ i (k) W

=

=

1 K 1 K

k 

Iˆ−i (n + 1) − Iˆ−i (n)

(27)

n=k−K+1 k 



2 Iˆ−i (n + 1) − Iˆ−i (n) − ˆb2i (k)

n=k−K+1

as approximations to the maximum-likelihood (ML) estimates of the mean and variance of the driving disturbance wi (k), respectively. The deviation between the two approximations above and the ML estimates is dependent upon the accuracy of the approximation I−i (k+1)−I−i (k) ∼ = Iˆ−i (k+1)−Iˆ−i (k). A window size of K = 100 samples will be used in empirically obtaining the statistics of the driving disturbance. Fig. 8 illustrates that the performance of the H∞ filter implementation of the power control algorithm is comparatively robust to the performance attained with the Kalman filter. It should be stressed that the Kalman filter is the optimal estimator in this case due to the Gaussian nature of the disturbances. The H∞ filter, however, provides comparable performance

SOROOSHYARI AND GAJIC: AUTONOMOUS DYNAMIC POWER CONTROL FOR WIRELESS NETWORKS

while alleviating the need for tracking the statistics of the disturbances. Established power control works such as [1][7][12] presume deterministic channels between transmitters and receivers, while [1] [3] [4] [6] [7] also assume noiseless feedback from the receivers to the transmitters. In Fig. 8, the notable discrepancy between the outage rates of users performing estimator-based power control and those using the conventional Foschini-Miljanic algorithm strongly advocates the necessity of an estimator for robust power control over a time varying channel with noisy measurements. Examination of (26) reveals the absence of process noise for a deterministic channel since wi (k) = 0 upon convergence of the powers so long as gij = 0 ∀ j = i. We observe from Fig. 8 that the degradation in performance is more pronounced with increasing variation in the link gains (increasing μ) than with increasing variation in the measurement noise (increasing ηv ). This gives insight into the importance of modeling and accounting for the stochastic nature of the wireless channel between transmitters and receivers when devising a closedloop power control algorithm. V. C ONCLUSION A predictive algorithm has been presented for distributed power control in wireless networks. Each user dynamically adapts power so as to minimize an objective function consisting of the user’s QoS degradation and aggregate interference. The predictive nature of the controller allows for rapid convergence with a user adapting power in accordance with interference measurements. A critical feature of the algorithm is the capability of a user to allocate power so as to address various user-centric and network-centric objectives by being either greedy or energy efficient. It was observed that an energy efficient user shall adapt power so as to address user-centric and network-centric objectives; while a greedy user will solely address user-centric objectives. Subsequently, closed-loop implementation of the distributed algorithm with autonomous interference estimation and no a priori knowledge of the exogenous disturbances was proposed. The versatility of the power control policy was shown via simulation with users adapting transmit power in either greedy or energy efficient manner. Simulation results demonstrate superb performance with respect to robustness to stochastic detriments caused by a time varying channel and noisy measurements. ACKNOWLEDGMENT The authors wish to thank the anonymous reviewers for their constructive comments which have improved the quality of the paper. We also thank Roy Yates for several helpful discussions at an early stage in this work. R EFERENCES [1] G. J. 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.

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[2] K. K. Leung, “Power control by interference prediction for broadband wireless packet networks,” IEEE Trans. Wireless Commun., vol. 1, pp. 256-265, Apr. 2002. [3] N. Bambos, S. C. Chen, and G. Pottie, “Channel access algorithms with active link protection for wireless communication networks with power control,” IEEE/ACM Trans. Networking, vol. 8, pp. 583-597, Oct. 2000. [4] R. D. Yates, “A framework for uplink power control in cellular radio systems,” IEEE J. Select. Areas Commun., vol. 13, pp. 1341-1348, July 1995. [5] K. Shoarinejad, J. Speyer, and G. Pottie, “Integrated predictive power control and dynamic channel assignment in mobile radio systems,” IEEE Trans. Wireless Commun., vol. 2, pp. 976-988, Sep. 2003. [6] N. Feng, S. Mau, and N. B. Mandayam, “Pricing and power control for joint network-centric and user-centric radio resource management,” IEEE Trans. Commun., vol. 52, pp. 1547-1557, Sep. 2004. [7] M. Xiao, N. B. Shroff, and E. Chong, “A utility-based power-control scheme in wireless cellular systems,” IEEE/ACM Trans. Networking, vol. 11, pp. 210-221, Apr. 2000. [8] A. J. Goldsmith and S. G. Chua, “Variable-rate variable-power MQAM for fading channels,” IEEE Trans. Commun., vol. 45, pp. 1218-1230, Oct. 1997. [9] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2003. [10] M. Andersin, Z. Rosberg, and J. Zander, “Soft and safe admission control in cellular networks,” IEEE/ACM Trans. Networking, vol. 5, pp. 255-265, Apr. 1997. [11] K. Zhou and J. C. Doyle, Essentials of Robust Control. Upper Saddle River, NJ: Prentice Hall, 1997. [12] S. Ulukus and R. D. Yates, “Stochastic power control for cellular radio systems,” IEEE Trans. Commun., vol. 46, pp. 784-798, June 1998. [13] M. J. Feuerstein, K. L. Blackard, T. S. Rappaport, S. Y. Seidel, and H. H. Xia, “Path loss, delay spread, and outage models as functions of antenna height for microcellular system design,” IEEE Trans. Veh. Technol., vol. 43, pp. 487-498, Aug. 1994. [14] G. Stuber, Principles of Mobile Communication. Norwell, MA: Kluwer Academic Publishers, 1996. [15] S. Hanly and D. Tse, “Power control and capacity of spread-spectrum wireless networks,” Automatica, vol. 35, pp. 1987-2012, Dec. 1999. Siamak Sorooshyari received his B.S. and M.S. degrees in electrical engineering from Rutgers University in 2000 and 2003, respectively. He is currently a Member of the Technical Staff at Bell Laboratories Alcatel-Lucent where he is involved in the development of physical layer and link layer algorithms for next-generation wireless data networks. His current research has been focused on resource allocation for wireless networks, optimization and stochastic control of communication networks, and distributed algorithms for resource allocation. Zoran Gajic (S’81-M’83-SM’90) received the Dipl. Ing. and Mgr. Sci. degrees in electrical engineering from the University of Belgrade, Belgrade, Yugoslavia, and the M.S. degree in applied mathematics and the Ph.D. degree in systems science engineering from Michigan State University, East Lansing, MI. He is a member of the Wireless Information Networks Laboratory (WINLAB) and a Professor of electrical and computer engineering at Rutgers University, Piscataway, NJ, where he has been teaching courses on electrical circuits, linear systems and signals, controls, and networking since 1984. His research interests are in control systems, wireless communications, and networking. He is the author or coauthor of more than sixty journal papers and seven books in the fields of linear systems and linear and bilinear control systems. His textbook, Linear Dynamic Systems and Signals (Upper Saddle River, NJ: Prentice-Hall, 2003), has been translated into the Chinese Simplified language. He has delivered two plenary lectures at international conferences and presented more than one hundred conference papers. Prof. Gajic serves on the editorial board of the journal Dynamics of Continuous, Discrete, and Impulsive Systems and was the Guest Editor of a special issue of that journal, on Singularly Perturbed Dynamic Systems in Control Technology.