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Pareto and Energy-Efficient Distributed Power Control With Feasibility Check in Wireless Networks Mehdi Rasti, Ahmad R. Sharafat, Senior Member, IEEE, and Jens Zander, Member, IEEE
Abstract—We formally define the gradual removal problem in wireless networks, where the smallest number of users should be removed due to infeasibility of the target-SIR requirements for all users, and present a distributed power-control algorithm with temporary removal and feasibility check (DFC) to address it. The basic idea is that any transmitting user whose required transmit power for reaching its target-SIR exceeds its maximum power is temporarily removed, but resumes its transmission if its required transmit power goes below a given threshold obtained in a distributed manner. This enables users to check the feasibility of system in a distributed manner. The existence of at least one fixed-point in DFC is guaranteed, and at each equilibrium, all transmitting users reach their target-SIRs consuming the minimum aggregate transmit power. Furthermore, in contrast to the existing algorithms, no user is unnecessarily removed by DFC, i.e., DFC is Pareto and energy-efficient. We also show that when target-SIRs are the same for all users, DFC minimizes the outage probability. Simulation results confirm our analytical developments and show that DFC significantly outperforms the existing schemes in addressing the gradual removal problem in terms of convergence, outage probability, and power consumption. Index Terms—Distributed feasibility check, gradual removal problem, Pareto and energy-efficient distributed power control, wireless networks.
I. INTRODUCTION
T
RANSMISSION power is a key resource in wireless networks, where transmit power control is used to maintain an acceptable QoS in terms of the bit-error-rate (BER) and/or transmission delay for the largest possible number of users by minimizing interference levels, and extending user’s battery life. A distributed scheme for power control is preferred to a centralized one, because in the former, transmit power level of a user is decided by that user by utilizing the locally available information and minimal feedback from the base station. In this way, available resources (including the frequency spectrum) are
Manuscript received July 20, 2008; revised October 14, 2009; accepted July 22, 2010. Date of current version December 27, 2010. This work was supported in part by Tarbiat Modares University (TMU), Tehran, Iran and in part by Wireless@KTH, Stockholm, Sweden, where the first author was a visiting researcher, and in part by Shiraz University of Technology, Shiraz, Iran. This paper was presented in part at IEEE PIMRC 2008, Cannes, France, September 2008. M. Rasti was with the Department of Electrical and Computer Engineering, Tarbiat Modares University, Tehran, Iran. He is now with the Department of Electrical Engineering, Shiraz University of Technology, Shiraz, Iran. A. R. Sharafat is with the Department of Electrical and Computer Engineering, Tarbiat Modares University, Tehran, Iran (e-mail:
[email protected]). J. Zander is with the Wireless@KTH, The Royal Institute of Technology, Electrum 229, Stockholm, Sweden. Communicated by A. Nosratinia, Associate Editor for Communication Networks. Digital Object Identifier 10.1109/TIT.2010.2090210
used more efficiently since the need for frequent power setting commands by the base station are avoided, and the processing capabilities at the base station needed to obtain instantaneous uplink power levels of all users (which may be quite significant) is substantially reduced. Since the early work on distributed power control in [1], designing distributed power control algorithms has received much attention. The target-SIR-tracking power control algorithm (TPC) was originally proposed in [2] and has been further studied in [3]–[5]. Under TPC, the information that each user needs to know, which is either locally available or is provided to each user through feedback from its corresponding base-station, is minimal. It is well known that in a feasible system (i.e., when there is a positive power vector that attains the target-SIR vector), the original TPC supports all active users, i.e., zero outage, consuming the minimum aggregate transmit power [2]. The feasibility of target-SIRs is an underlying assumption in TPC. In an infeasible system, some users may not reach their target-SIRs (called nonsupported users). Such users, due to the fact that a distributed mechanism in TPC cannot check the feasibility of target-SIRs, not only are not removed, but they transmit at their maximum power levels. This unnecessarily causes interference to others, and may also increase the number of nonsupported users. Obviously, this is not desirable, and a minimal number of users have to be removed to increase the number of supported users and to reduce the aggregate transmit power. A straightforward solution is that all nonsupported users simply stop transmitting. However it can be easily shown that although the aggregate transmit power is reduced, some users are unnecessarily removed. As an alternative, if a portion of nonsupported users are removed, the remaining users can be supported (the outage probability is reduced), and the aggregate transmit power required to support other users is also reduced. This is the gradual removal problem that we focus on in this paper. To solve the gradual removal problem, either a centralized, or a distributed approach can be considered. In the centralized approach, users basically apply TPC to update their transmit power levels, and the base station checks the feasibility of the system. If the system is infeasible, the base station identifies those users that must be removed, and orders them to switch off. This approach requires the base station to know an extensive set of global information, such as target-SIRs, path gains, and maximum transmit power levels for all users. As such information may not be readily available in practice, we do not consider the centralized approach in this paper. In contrast, in a fully distributed approach, each user sets its transmit power by utilizing only locally available information. In this scheme, in an infeasible system, some nonsupported users stop their transmissions in favor of others. We call this scheme a distributed constrained power control algorithm (DCPC) with the capability of gradual removal.
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Existing DCPCs with the capability of gradual removal have used either of the two different types of removal. In the first alternative, some users with poor channels permanently remove themselves as in [6] and [7]. In the second alternative, some users with poor channels temporarily remove themselves either by switching off as in [5], or by soft removal as in [8]. In the permanent-removal DCPC, each removed user either tries another channel (if one exists) or keeps disconnected, as opposed to the temporary-removal DCPC in which each removed user stays on the same channel and resumes its transmission (or increases its power) if its effective interference (the ratio of interference to the path-gain) becomes acceptable again. Generally, for the two algorithms with the same performance, the one with the capability of resuming transmission for removed users is preferred to the one with permanent removal. This is because the former adapts to interference changes and can either remove or resume users if conditions warrant. However, temporary removals may cause oscillations between removing and resuming states, and may suffer from possible inability of the algorithm to converge for infeasible systems. In [6] and [7], any user whose required transmit power for reaching its target-SIR exceeds its maximum transmit power is a candidate for permanent removal, and removes itself permanently (not resuming transmission on the same channel) with a given probability. The performance of this algorithm depends on the probabilities of permanent removals, which should be set in such a way to maximize the probability of exactly one removal at each step (power update iteration). As has been shown in [6] and [7], a lower probability results in a smaller outage ratio, but prolongs the convergence time. Irrespective of the case, if the effective interference becomes acceptable again for permanently removed users, they do not resume transmitting on the same channel. In some applications (especially the real-time ones) this is a drawback. In addition, unnecessary removals may ensue at the time of power updating (initial iterations prior to reaching the steady state). When a feasible system becomes periodically infeasible even for very short times, unnecessary permanent removals are caused. Due to such problems, in this paper we focus only on the DCPC with the capability of temporary removal. In [5], TPC was revised to temporarily switch off those users whose required transmit power for reaching their target-SIRs exceed their maximum transmit power. A switched-off user resumes transmission if its effective interference is reduced so that its required transmit power for reaching its target-SIR is below its maximum transmit power. If this algorithm converges, it reduces the outage probability as well as the total consumed power, as compared to TPC. However, the existence of a corresponding fixed-point, and consequently its convergence, cannot be guaranteed in an infeasible system, and some users may oscillate between switch-off and switch-on (target-SIR-tracking) modes. In addition, in contrast to TPC, which always has a fixedpoint (even in infeasible cases), there may exist no fixed-point for the power update function of DCPC in [5], implying that it may be impossible for the algorithm to converge. We wish to design a DCPC with the capability of temporary gradual removal and feasibility check, whose fixed-point can be guaranteed to exist, and at its equilibrium, all transmitting users attain their target-SIRs by consuming the minimum aggregate transmit power while no user is unnecessarily removed, which we call a Pareto and energy-efficient equilib-
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 1, JANUARY 2011
rium. In this algorithm, any transmitting user whose required transmit power for reaching its target-SIR exceeds its maximum power is temporarily removed, but resumes its transmission if its required transmit power for reaching its target-SIR is below a given threshold (lower than the maximum power, and adjusted in a distributed manner). This enables users to check the feasibility of the system in a distributed manner (which has been an open problem so far [9]), and if the system is recognized as infeasible, the least number of users are temporarily removed to make it feasible for the remaining users, meaning that TPC could be successfully employed by the remaining users. We will show that our proposed DCPC has at least one fixed-point, and its equilibrium is Pareto and energy-efficient. Furthermore, we show that when target-SIRs are the same for all users, our proposed distributed algorithm minimizes the outage probability. The rest of this paper is organized as follows. In Section II, we introduce the system model. In Section III, we describe TPC and the algorithm proposed in [5], and discuss their properties. A formal statement of the problem is presented in Section IV. In Section V, we define the efficient power control problem, present our proposed algorithm, discuss the intuition behind it, and show that it converges to an efficient equilibrium transmit power. Simulation results and conclusions are presented in Sections VI and VII, respectively. II. SYSTEM MODEL Consider a single cell in a wireless CDMA network with active users denoted by . Let be the , where is the upper transmit power of user and limit of the transmit power for user . The received power at the , where is the path gain base station of that cell is from user to the base station. The transmit power constraint for all imposes the received power to be bounded, i.e., , where is the upper bound on the received power. Noise is assumed to be additive white Gaussian whose power at the base station is . The receiver is assumed to be a conventional matched filter. Thus, for a given transmit power vector , the SIR of a user , denoted by is (1) where is the total interference caused in which is the intracell interto user , and ference. The effective interference, denoted by , and defined as the ratio of interference caused to each user to its path gain is (2) There is a one-to-one relation between a transmit power and the actual SIR vector vector [10], [11], which is (3) From this, the total received power plus noise at the base station, i.e., , is [10], [12] (4)
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Target-SIR of each user is denoted by , and is usually equivalent to a maximum tolerable BER, below which the user is not for each , i.e., when there is no satisfied. We assume interfering user, the target-SIR of user is reachable. Definition 1: Target-SIRs of users in a given subset are feasible if a power vector exists that satisfies implies target-SIRs of users in , where for all . We also say that the system is feasible if the ) is feasible, target-SIR vector for all users (i.e., when otherwise the system is called infeasible. , by using Given the target-SIR of each user in the set (3) we conclude that their target-SIRs are feasible if (5)
^( ~ =
III. BACKGROUND: EXISTING DISTRIBUTED POWER CONTROL ALGORITHMS In a DCPC, each user updates its transmit power by a power, that is update function (6) is the transmit power vector at time . The distributed where constrained power-update function is (7) The fixed-point of the power update function, denoted by , for the constrained and the unconstrained cases are obtained by and , respectively. If a solving DCPC converges to an equilibrium state, it will be a fixed-point of the corresponding power-update function. In this paper, we use equilibrium-point and fixed-point interchangeably. A. TPC In TPC, each user tries to maintain its SIR at the target level . The unconstrained power-update function in [2] is (8) where (9) is the effective interference experienced by user and at time . It was shown in [2] and [13] that if and only if the target-SIR vector is feasible, then the unconstrained-TPC converges either synchronously or asynchronously to a fixed point in which users attain their target-SIRs with the minimum aggregate transmit power, i.e., its fixed point solves the following optimization problem
subject to
= f (p ) = = ( ) = ^( + )=h . The ~ = f (~p ) and (0) ~ = (0)
Fig. 1. System with two users. The solid lines are p =h and p f p
p h
p h f ;p f , and p parameters are p f p . p
(10)
When the system is infeasible, the above problem has no solution, because there is no transmit power vector that can satisfy SIR constraints for all users. If the unconstrained TPC (which
+ ) ~ = (~ )
was originally designed to solve (10) in a distributed manner assuming the feasibility of system) is used in an infeasible system, since target-SIRs are rigidly tracked, all users increase their transmit power at each step and thus, the unconstrained TPC diverges. The algorithms developed in [3] and [4] assume that the transmit power is constrained to deal with the divergence in infeasible systems, that is Constrained-TPC:
(11)
Although the existence of a fixed-point and its convergence is guaranteed in the constrained TPC in both feasible and infeasible systems [4], it still suffers from a severe drawback in infeasible systems, where all nonsupported users (those who do not reach their target-SIRs) transmit at their maximum power. Thus, such users inefficiently consume their maximum power and cause unnecessary interference to others. This unnecessary interference may also unnecessarily increase the number of nonsupported users, which can be avoided if a portion of nonsupported users switch off. To show that the number of supported users in TPC can be increased if a few nonsupported users are removed, consider a system with two active users. Fig. 1 is a 2-D space whose x axis and y axis are the transmit power levels of users 1 and 2, respectively. The constrained power levels of the two users are upper bounded by the dash-lines. The and solid lines are , representing the transmit power levels required for reaching the target-SIRs of users 1 and 2, respectively, as a function of the other user’s power. If these two lines intersect, the intersection point A, i.e., is the fixed-point of the unconstrained-TPC. If this intersection point satisfies the power constraint, it is also the fixed-point of the power-update function of the constrained-TPC; otherwise, the system is infeasible and the fixed point for the constrained-TPC is . In and then the point B, i.e., this case, if is the fixed-point which is the worst case in the sense that both users transmit at their maximum power levels without reaching their target-SIRs. In this case, if a user is removed, the other user can reach its target-SIR by transmitting
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at a lower power level. In the rest of this paper, TPC implies (11) unless stated otherwise. B. TPC With Capability of Temporary Removal (TR) To deal with the stated drawback of TPC in infeasible cases, TPC is revised in [5] as if if
(12)
in which, a user with a poor channel refrains from transmitting at its maximum power level and removes itself temporarily. In contrast to TPC that has a fixed-point (even in infeasible cases), the existence of a fixed-point for the algorithm in (12) is not always guaranteed. If this algorithm converges, it reduces the outage probability and the total consumed power as compared to TPC, but the existence of its fixed-point, and consequently its convergence cannot be guaranteed in an infeasible system, and some users may oscillate between switch-off and switch-on (target-SIR-tracking) modes. The reason for such oscillations is that when a given user experiences an effective interference greater than a threshold, it switches itself off, thereby reducing its interference to others. This, in turn causes others to reduce their transmit power, resulting in a reduced interference to that user. If the effective interference is lower than the threshold value, the user resumes its transmission, and the same event is repeated. The possibility of power oscillations in (12) emanates from two problems. The first and the more important one is the possibility of nonexistence of a fixed-point for the power update function (12), implying that it is impossible for the algorithm to converge. The second problem that may cause oscillation, even if some fixed-points exist (it may have several fixed-points) is due to the initial transmit power. When there is at least one fixed-point for the power update function (12), depending on the initial transmit power, some users may oscillate between target-SIR-tracking and switch-off modes [5], specifically for the synchronous case, as we show below. To show these two drawbacks of algorithm (12), consider , again Fig. 1. When the system is infeasible, only or can be the fixed-point of (12), if , or , respectively ( and for are shown in Fig. 1). Otherwise (i.e., when and ), there is no fixed point for the power-update function (12), because and . It is obvious that when there is no fixed-point, the algorithm never converges. When there is at least one fixed-point, we note that both users, depending on their initial transmit power, may oscillate among a sequence of power vectors (e.g., ) including the no fixed-point, or converge to their fixed point. In summary, when there is no fixed-point, the power-updating algorithm (12) does not converge (there is no equilibrium). But when a fixed-point exists, depending on the initial value of the transmit power vector, the power-updating process may or may not converge (i.e., reaching an equilibrium depends on the initial value of the transmit power vector). In Section V, we propose a new DCPC with the capabilities of temporary removal and distributed feasibility check whose fixed point is guaranteed [i.e., it solves the first (the main) problem of (12)], and propose a heuristic solution to address the
second problem of (12). We also prove that our proposed algorithm has some desired properties, in contrast to TPC and (12). IV. PROBLEM STATEMENT: PARETO AND ENERGY-EFFICIENT POWER CONTROL In this section, motivated by the drawbacks of existing DCPCs, we formally define the gradual removal problem. In doing so, we introduce the concept of Pareto and energy efficiency by generalizing (reformulating) the minimum aggregate transmit power problem (10) to make it applicable to both feasible and infeasible systems. Under a Pareto and energy efficient power allocation, the aggregate transmit power to support a given subset of users whose target-SIRs are reachable is minimized, but no additional user (if one exists) can be supported at the same time. Given a transmit power vector , let us denote the sets of active users and supported users by and , respectively. Their and complementary sets are . Given a target-SIR vector and a transmit power vector , the resulting outage probability is , where and denote the total number of members in and , respectively. To compare the efficiency of the two transmit power allocations, the concept of Pareto dominance as defined below is used. Definition 2: A transmit power vector Pareto dominates . A power vector is Pareto another vector if efficient if there is no power vector that Pareto dominates . The problem of minimum-outage, or equivalently the problem of the minimum number of nonsupported users is (13) which is NP-complete, as was shown in [7]. It can be similarly shown that the problem of Pareto-efficient power control is also NP-complete. One can easily prove the following lemmas. Lemma 1: The transmit power vector is Pareto efficient if , target-SIRs of users in the set and only if for any are not feasible. Lemma 2: If the transmit power vector solves the minimum-outage problem (13), then it is Pareto efficient. The converse is not necessarily true, i.e., a Pareto-efficient transmit power vector does not necessarily minimize the outage. As implied by Lemma 2, the set of power vectors that minimizes the outage is a subset of all Pareto-efficient power vectors. The following example shows that there exists a Pareto-efficient power vector that does not minimize the outage. Consider an infeasible system with three users with the same target-SIR value . Suppose of 0.5 and the path-gain vector of the AWGN power is 0.0005 Watts and the maximum transmit power for each user is 0.1 Watts. It is straightforward to verify , and that target-SIRs of each set of users are feasible, and for any other set of users are infeasible. Thus , obtained by a transmit power vector the minimum outage is . Only the transmit power vecthat satisfies target-SIRs of or are Pareto effitors that satisfy target-SIRs of cient, among which the transmit power vectors that satisfy the does not minimize the outage. Furthermore, target-SIR of
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as implied by this example, the Pareto efficient transmit power vector(s) as well as the transmit power vector(s) that minimizes outage are not unique in general. Among the Pareto-efficient power vectors (and also among power vectors that minimize the outage), these vectors that minimize the aggregate transmit power required to provide the set of corresponding supported users with their target-SIRs are of most importance. Although the energy efficiency defined in (10) in terms of the minimum-aggregate-power is only applicable to feasible systems, in what follows, we use the concept of Pareto efficiency to generalize the optimization problem in (10) for making it applicable to both feasible and infeasible systems. Definition 3: Given the target-SIR vector , a transmit power vector is Pareto and energy efficient if is Pareto-efficient, and a) corresponds to the minimum aggregate transmit power b) needed to provide the supported users with their targetSIRs, i.e., is the solution to the following optimization problem:
subject to
(14)
If the transmit power vector in Definition 3 is also a solution to the minimum-outage problem (i.e., it minimizes the outage), we say it minimizes the outage probability in an energy-efficient manner. Lemma 3: If the transmit power vector is Pareto and energy , or equivalently, efficient, then , and . Using Lemmas 1 and 3, for a given target-SIR vector , the set of all Pareto and energy efficient transmit power vectors, de, can be obtained as follows. The set includes noted by that have the all constraint transmit power vectors following two properties. , reach a) All transmitting users, i.e., users in the set their target-SIRs with minimum aggregate transmit and . power, i.e., b) No user is unnecessarily removed, i.e., there exist no so that target-SIRs of users in are feasible. Under a Pareto and energy efficient transmit power allocation, all transmitting users are supported with their target-SIRs by consuming the minimum required power and no user is unnecessarily removed (i.e., no user (if one exists) can be even theoretically supported while the others are supported). In other words, the least number of users who cannot be supported along with the given corresponding supported users (due to the infeasibility of the system) are switched off. This policy has two main advantages. 1) All users save energy in the following manner. The removed users save their energies by switching-off, when they cannot reach their target-SIRs, hence the aggregate consumed power by transmitting users is minimized. 2) When some users are removed, no interference is introduced to others, which can increase the number of supported users. Remark 1: Note that a Pareto and energy efficient transmit power vector is not generally unique. When the system is feasible, the Pareto and energy efficient transmit power vector is unique and is the solution to (10), obtained by TPC in a dis-
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tributed manner. When the system is infeasible, at least one Pareto and energy efficient transmit power vector exists, but in general is not unique. These can be seen by considering again Fig. 1. When the system is feasible, the intersection of two solid-lines is uniquely Pareto and energy efficient. When the system is infeasible, there are two Pareto and energy efficient and . transmit power vectors When the system is feasible, all users can be supported, meaning that the minimum outage probability is zero and the transmit power vector is Pareto-efficient. In this case, TPC results in zero-outage (minimum-outage), consuming the minimum amount of aggregate transmit power. The problem that we focus on in this paper is how to identify and remove the least number of users in a distributed and Pareto-efficient manner (i.e., no user is unnecessarily removed) in infeasible systems. This would make target-SIRs of the remaining users feasible, and hence they can use TPC to attain their target-SIRs in an energy efficient manner. V. PROPOSED METHOD AND ITS PROPERTIES In this section we present our Pareto and energy-efficient distributed target-SIR-tracking power control algorithm with the capabilities of temporary removal and feasibility check (abbreviated as DFC), and discuss the intuition behind it. Then, we show that DFC has at least one fixed-point, and all of its fixed-points are Pareto and energy efficient. Furthermore, we show that when target-SIRs are the same for all users, our proposed distributed algorithm guarantees the minimum outage probability. A. Proposed Algorithm: DFC We first introduce the concept of feasibility of the system from a given user’s point of view and obtain a criterion to enable each user to check the feasibility of system in a distributed manner by using its pertinent information. We use this criterion to propose a new DCPC with the capability of temporary removal in which a user transmits only when the system is feasible from its own point of view, otherwise it switches off. Definition 4: Target-SIRs of users in are feasible from a , if the following ingiven user ’s point of view, where equality holds: (15)
Based on this definition, target-SIRs of users in are feasible (see Definition 1) if all users in consider these target-SIRs as feasible from their points of view using (15). We need a mechanism that enables each user to check the feasibility of the system from its own point of view in a distributed manner, when that user is active (transmitting), or when it has been temporarily removed. This issue is addressed in the following theorem. denote the fixed-point of the power-upTheorem 1: Let date function for TPC when only users in are active (transmitting), i.e., for all and for all . , if target-SIRs of users in are feasible a) Given a user from that user’s point of view, then (16)
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b) Given a new user , i.e., , if target-SIRs of users in are feasible from that user’s point of view, then (17) we know Proof: From for all . Now we prove the two parts of this theorem separately. a) Suppose target-SIRs of users in are feasible from the . Thus user ’s point of view, where
, if (17) holds, then target-SIRs of users b) For a user are feasible from that user’s point of view. in Proof: One can easily prove the first part. The second part can be proved by taking similar steps as in the proof of the second part in Theorem 1, but in reverse order, by noting that from the feasibility of target-SIRs of users in , we have for all . Based on the last two theorems, we propose a DFC in which each user either transmits according to TPC if from its point of view the system is feasible, or switches off. Define if if
(18) We show that this inequality is contradicted if does not hold. If , then from we know that and . Thus from (3) we have
(23)
where is a predefined threshold. This function is a generalized version of (12). DFC has the following distributed powerupdate function if if (24)
(19) and This strict inequality holds because for all , and contradicts (18). are feasible from b) Suppose target-SIRs of users in . Thus the user ’s point of view, where (20) which can be rewritten as (21) From this, and the fact that
and are the two thresholds utilized by user and where given by Theorem 1, as (25) (26) These two thresholds are determined by each user , in a distributed manner by using the minimal information pertinent to that user only. DFC can be interpreted as follows. There are two states for each user at any given time. If a user is transmitting (its transmit power is greater than zero), it operates in TPC as long as its required transmit power for reaching its target-SIR is less , otherwise it temporarily removes itself. A removed than user resumes its transmission if the required power to reach its (this threshold is lower than target-SIR becomes lower than by assuming ). B. Existence of Fixed-Point in DFC and Its Convergence
in which the equality is derived by using (4), and the infor all . We equality holds because conclude (22) where equivalent to (17).
which is
One can show that the converse of Theorem 1 is not always true. The following theorem states that the converse holds if we additionally assume that target-SIRs of users in are feasible . or equivalently, if (16) holds for all Theorem 2: If (16) holds for all , then a) Target-SIRs of users in are feasible.
In Theorems 3, we will show that by employing two different thresholds (25) and (26) by each user to decide between targetSIR-tracking or switching-off (in contrast to [5], where a single threshold is employed), the existence of at least one fixed-point is guaranteed. Before doing this, we present one lemma, and use it to propose an algorithm for obtaining an ordered set of users whose target-SIRs are Pareto-efficiently feasible, followed by one additional lemma, and use both lemmas to prove Theorem 3. , let be a Lemma 4: Given the set of total users are feasible subset of users. If target-SIRs of users in , then target-SIRs of from user ’s point of view, where are feasible from any user ’s point of view, users in where
and
.
Proof: Similar to (20) and (21), one can rewrite the feasi, where , bility condition of target-SIRs of users in from the user ’s point of view as (27)
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Thus, for a given from (27) we have
, if we have
then
(28) implying that target-SIRs of users in user ’s point of view.
are feasible from
Feasible-Set Algorithm: We now propose an algorithm for obtaining an ordered set of users whose target-SIRs are Paretobe efficiently feasible. Given the set of all users , let . Initialize . a subset of users and • Step 1: If target-SIRs of users in are feasible [it can be checked by using (5)], go to Step 2, otherwise let and repeat Step 1. and stop. • Step 2: Let Lemma 5: Let denote the set of users, obtained by the , targetproposed feasible-set algorithm. Given any user are not feasible from that user’s point SIRs of users in of view. Proof: If this is not true, then there exists at least one user , so that target-SIRs of users in are feasible from its point of view. Thus from Lemma 4, we conclude that are feasible from user ’s point target-SIRs of users in . This contradicts the of view, where fact that user is the last user removed from in the proposed feasible-set algorithm (as users in are removed in decreasing ). This completes the proof. order of Theorem 3: There exists at least one fixed-point for the power-update function of DFC. be the set of users obtained by the proposed Proof: Let feasible-set algorithm. It is obvious that target-SIRs of users in are feasible. We show that the transmit power vector obtained by if (29) is a fixed point of the proposed power-update function (24). To prove this, it is sufficient to show that (30) and (31) because,
if
the
former
holds then , and if the latter holds then . Thus the fixed-point constraint holds for all , by noting from for all and for all . (29) that are feasible, (30) is directly As target-SIRs of users in obtained from the first part of Theorem 1. Now we prove that (31). If (31) is not true, then there exists one . Thus from the second part of Theorem 2 we are feasible, conclude that target-SIRs of users in which is contradicted by Lemma 5.
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Remark 2: The fixed-point of the DFC’s power-update function is not generally unique. It is obvious that when there is no fixed-point, the algorithm never converges. If a distributed power update function has a fixed point, then that algorithm can potentially (not necessarily) converge. Based on the framework developed in [12] and [13], when there exists a fixed-point for a continuous power-update function, under certain conditions, the fixed-point is unique and the corresponding algorithm converges to the fixed-point for any initial transmit power vector. The frameworks in [12] and [13] are suitable for the algorithm without removal. However, since in DFC, some users switch off, the corresponding power-update function is not continuous, and thus these frameworks cannot be applied to study its convergence. In DFC, existence of at least one fixed-point is guaranteed. However, depending on the initial value of the transmit power vector, the power-updating process may or may not converge, i.e., reaching an equilibrium depends on the initial value of the transmit power vector. This problem, i.e., the existence of a fixed point and no convergence due to the initial value of transmit power vector can be resolved (as opposed to the case in which no fixed point exists), as explained in the following paragraph. For some initial transmit power vectors, some users oscillate among two modes of switch-off and target-SIR-tracking (due to discontinuity of the power-update function). This causes the algorithm to oscillate among a sequence of power vectors (including the no fixed-point), especially in synchronous cases. This possible oscillation is prevented if users avoid such sequences. To resolve this, we propose a heuristic solution as follows. Each user counts the number of times that it switches between the switch-off and transmitting states. When it exceeds a predefined threshold, it randomly and independently sets its transmit power level for the next iteration, resets the counter, and sets its transmit power according to DFC for the forthcoming iteration. This is equivalent to a new initial power vector that guarantees that the equilibrium (a fixed-point) is eventually reached. C. Pareto and Energy-Efficiency of DFC In Theorems 4 we show all fixed-points in DFC are Pareto and energy-efficient. Theorem 4: Any fixed-point for the power-update function of DFC is Pareto and energy-efficient. and Proof: It is obvious that . To prove the theorem, we only need to show so that target-SIRs of users in that there exists no are feasible. If this is not true, then there exists at so that target-SIRs of users in least one are feasible. Thus from the second part of Theorem 1, we which contradicts the fixedconclude that for . Thus is point constraint efficient. D. Outage-Minimization by DFC As stated earlier, in general the minimum-outage problem is NP-complete. In what follows, for the special case in which target-SIRs of all users are the same, we first find the solution to the minimum-outage problem for a single cell and then show that DFC minimizes the outage.1 1For the case when users do not have a common target-SIR, we simulate a multicell wireless network with different target-SIRs (see simulation results).
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Suppose all users have a common target-SIR, i.e., where is a positive constant. From (3) we conclude that targetare feasible if SIRs of users in (32)
that also satisfies the power constraint by noting (35)]. When , if the number of nonsupported users can be lower than , then at least target-SIRs of users in should be feasible. If this is true, then from (32) we have
When users are active and their SIRs are , the received . power at the base station for all of them is Theorem 5: Without loss of generality, suppose that users are indexed in increasing order of their max-received power, i.e., . Define (33) The minimum-outage, denoted by common target-SIR is if if if
for all
, which contradicts (36).
, Theorem 6: When target-SIRs of all users are the same any fixed-point of the power-update function of DFC minimizes , the outage probability, i.e., for any fixed-point where
for different values of the and
where (34)
Proof: Note that is the achieved SIR by each user inwhen users indexed from 1 to switch dexed from to off and users indexed from to transmit at a level so that their for received power at the base station is . We have . It can be easily shown that when , the all target-SIR is reachable by each user by transmitting at
meaning that the outage-ratio is zero. For , we have and thus for all , which means no user can reach its target-SIR and thus the minimum-outage-ratio is , then there exists a 1. If for which . Now we prove that in this case, the minimum number of removals (nonsupported users) is , meaning that the minimum-outage-ratio is . To prove this, we first demonstrate that by removing users, target-SIRs of all remaining users are reachable, and then show that the minimum-number of removals cannot be lower than . From the we have right- and the left-hand side of
is the minimum value of the outage given by (34). Proof: Similar to Theorem 5, suppose , and define as in (33). When the system is feasible and it can be easily seen that the fixed-point of TPC and of our algorithm are the same, implying that they minimize , the fixed-point for the the outage-ratio to zero. When power-update function of DFC is no transmission by each user (and thus the outage-ratio is 1), because in this case, we have for all , and consequently for all . Now we prove the theorem for the case of , i.e., we prove that for any given fixed-point for of DFC, we have the power-update function where for which . It . We now prove that is obvious that contradicts the following fixed-point constraint: (37) . As all users in and thus prove same target-SIR, their received power are the same
reach the
Thus we have for all and consequently, the fixed-point constraint (37) can be rewritten as
(35) (38) and (36)
If , then we have (38) we have
, and thus from
respectively. From (35) we conclude that (39)
for all . Therefore, by using (32), we are conclude that target-SIRs of users in feasible [i.e., they are supported with the common target-SIR, if users 1 to switch off and the transmit power for each remaining user is set at
For , we also conclude that the maximum-index is greater than , or equivalently there among users in for which . From this and exists a user from (39), we have (40) which contradicts
. This completes the proof.
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Fig. 4. Transmit power and SIRs (linear scale) of each user for DFC.
Fig. 2. Transmit power and SIRs of each user for TPC.
Fig. 3. Transmit power and SIRs (linear scale) of each user for TR.
VI. SIMULATION RESULTS Now we provide simulation results for our proposed algorithm to show that it outperforms other existing power control algorithms in terms of convergence, outage probability and energy efficiency. Similar to [13], the AWGN power at the reWatts. We adopt ceiver, i.e., , is assumed to be for the path-gain, a simple and well known model where is the distance between user and its base station, is the attenuation factor that represents power variations due to
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Fig. 5. Average outage probability for optimal solution of the minimum outage problem and for each algorithm versus common target-SIRs (in dB) for a singlecell network.
. The upper bound on the transmit path-loss, and take power for all users is 1 Watt. We consider a single-cell wireless network, first with fixed locations of users to study and track the performance of the algorithm in detail, and then proceed to different snapshots of users’ locations, to verify that the results do not depend on specific user-locations. We also consider a multi-cell environment where users have different target-SIRs. Consider 5 users indexed from 1 to 5 in a singlecell environment where their distance vector is m, in which each element is the distance of the corresponding user from its base station. The 2. One target-SIR vector is can easily verify that the system is infeasible. The system becomes feasible if user 5 switches off. Figs. 2–4 show the transmit-power and SIR versus each iteration for TPC, TR, and DFC, respectively. In TPC (Fig. 2), all nonsupported users, i.e., users 3–5 transmit at their maximum power. As we see in Fig. 3, TR does not converge, because in this example, its power-update function has no fixed-point. As we see in Fig. 4, in our proposed DFC, user 5 detects the infeasibility of the system and removes itself (in a distributed manner) which enables users 1–4 to reach their target-SIRs with minimum aggregate transmit power. It shows that our proposed algorithm outperforms TPC and TR with respect to the outage probability and power consumption. Furthermore, it outperforms TR with respect to the convergence. To show that the better performance of DFC is not dependent on locations of users, we compare the performances of TPC and DFC for different snapshots of users’ locations and for different values of target-SIRs3. In doing so, we consider a single-cell wireless network with a radius of 250 m and with 20 fixed users. Target-SIRs are considered the same for all users, to dB with a step size of 1 dB. For each ranging from value of target-SIR, we average the corresponding values of the outage probabilities, and average the total transmit power for
0 0
0
0 0
2This target-SIR vector is equivalent to ^ = [ 4; 5:2; 4:6; 6; 6] dB. Since under DFC and TR, some users may be removed (zero-transmission and thus zero-SIR), we use the linear scale for SIR in Figs. 2–4, and for the remaining simulations, we use the logarithmic scales, i.e., dB and dBm, for SIR and the aggregate transmit power, respectively. 3As TR may not converge for some snapshots, it is not included.
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Fig. 6. Average aggregate transmit power (dBm) for optimal solution of the minimum outage problem and for each algorithm versus common target-SIRs (in dB) for a single-cell network.
1500 independent snapshots for a uniform distribution of users’ locations within a single-cell. The initial transmit power for each for each snapshot. user is uniformly set from the interval Figs. 5 and 6 show the average outage probability and the average aggregate transmit power versus target-SIRs, respectively. The lower and the higher values of SIRs correspond to the feasible and the infeasible systems, respectively. In Figs. 5 and 6 we observe that DFC results in an optimum solution of the minimum outage probability problem as it was shown by Theorem 6. Note also that DFC outperforms TPC with respect to the outage probability and power consumption. For instance, for the dB, the outage probability of TPC and target-SIR value of DFC are 0.33 and 0.20, respectively, meaning that the number of supported users in our scheme is increased by 40% as compared to TPC. Similar improvements are achieved for other infeasible cases, to some extent. At the same time, as shown in Fig. 6, for infeasible cases, the aggregate transmit power consumed by all users in our scheme is significantly reduced as compared to dB, the avTPC. For instance, for a target-SIR value of erage aggregate transmit power of TPC and DFC are 39.7 and 13.4 dBm, respectively, which means that the aggregate power consumed by users in our scheme is decreased by 66% as compared to TPC. Now we consider a multicell wireless system in which users are distributed in a 1000 m 1000 m area covered by 4 base stations and have different target-SIRs. Each cell covers 500 m 500 m and each user is assigned to its nearest base station. Different total number of users are considered, ranging from 4 users (1 user/cell) to 28 users (7 users/cell) with step size of 4 users. To do so, for each total number of users, we consider 1500 independent snapshots with uniform distribution of users’ locations, and for each user, a target-SIR from the set dB is randomly assigned. For each snapshot, TPC and DFC are applied and their outage probability and aggregate transmit power are computed at the equilibrium. We also obtain a solution for the minimum outage problem with minimum aggregate transmit power by way of exhaustive search over all subsets of users for which target-SIRs are feasible. As several power vectors may exist, each resulting in the minimum outage, we select the one with minimum aggregate transmit power. For any given total number of users, we average the corresponding values of the outage probability
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Fig. 7. Average outage probability for the optimal solution of the minimum outage problem and for each algorithm versus the total number of users in a 4-cell network with different target-SIRs randomly assigned to users.
Fig. 8. Average aggregate transmit power (in dBm) for the optimal solution of the minimum outage problem and also for each algorithm versus the total number of users in a 4-cell network with different target-SIRs randomly assigned to users.
and the aggregate transmit power values for 1500 snapshots. Figs. 7 and 8 show the average outage probability and the average aggregate transmit power versus the total number of users, respectively, for TPC, DFC, and also for the optimum solution of the minimum outage probability problem. As can be seen, DFC outperforms TPC with respect to the outage probability and power consumption. For instance, for 16 users, the outage probability in TPC is 0.034, and in DFC is 0.019, which means that the number of supported users in our scheme is increased by 44% as compared to TPC. Better improvements are achieved for other infeasible cases. As shown in Fig. 8, the aggregate transmit power consumed by all users in our scheme is substantially lower as compared to TPC. For instance, for 16 users, the average aggregate transmit power in TPC and in DFC are 30 dBm and 12 dBm, respectively, which means a 60% reduction for the aggregate transmit power consumed by using DFC as compared to TPC. Note that while TPC significantly deviates from the optimum solution (especially for the aggregate consumed power), DFC closely follows the latter. The small difference between the outage probability in DFC and in the optimum solution is due to the fact
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that although all fixed-points of DFC are Pareto and energy efficient, some of them may not minimize the outage as explained by the example given after Lemma 2. This small difference indicates that the percentage of such fixed-points among all possible fixed-points in a general wireless network (multi-cell with different target-SIR values) is not significant.
VII. CONCLUSION We presented the concept of Pareto and energy efficient power control by reformulating the SIR constraint in the minimum aggregate transmit power problem, which is defined for and is applicable to both feasible and infeasible systems. We proposed a DFC in which any transmitting user whose required transmit power for reaching its target-SIR exceeds its maximum available power is temporarily removed. Each removed user resumes its transmission if the required transmit power for reaching its target-SIR goes below a given threshold (lower than the maximum power), determined in a distributed manner using locally available information pertinent only to that user and minimal feedback from the base station. This enables users to check the feasibility of system in a distributed manner. If the system is recognized as infeasible, a few users are temporarily removed. Specifically, we showed that the power-update function of our proposed algorithm has at least one-fixed point and at its equilibrium (where the algorithm converges), all transmitting users reach their target-SIRs by consuming the minimum aggregate transmit power vector, and no user is unnecessarily removed (i.e., it is Pareto efficient). We also showed that when target-SIRs are the same for all users, DFC minimizes the outage probability. Simulation results confirmed our analytical developments and showed that our scheme outperforms the existing DCPC algorithms in addressing the gradual removal problem, in terms of convergence, outage probability (more than 40% reduction in infeasible cases on the average), and power consumption (more than 60% reduction in infeasible cases on the average).
REFERENCES [1] J. Zander, “Distributed cochannel interference control in cellular radio systems,” IEEE Trans. Veh. Technol., vol. 41, no. 3, pp. 305–311, Aug. 1992. [2] G. J. Foschini and Z. Milijanic, “A simple distributed autonomous power control algorithm and its convergence,” IEEE Trans. Veh. Technol., vol. 42, pp. 641–646, Nov. 1993. [3] S. A. Grandhi, R. Vijaya, and D. J. Goodman, “Distributed power control in cellular radio systems,” IEEE Trans. Commun., vol. 42, pp. 226–228, 1994. [4] S. A. Grandhi and J. Zander, “Constrained power control in cellular radio system,” in Proc. IEEE Veh. Technol. Conf., Stockholm, Sweden, June 1994, vol. 2, pp. 824–828. [5] F. Berggren, R. Jantti, and S. Kim, “A generalized algorithm for constrained power control with capability of temporary removal,” IEEE Trans. Veh. Technol., vol. 50, no. 6, pp. 1604–1612, Nov. 2001. [6] M. Andersin, “Power control and admission control in cellular radio systems,” Ph.D. dissertation, Royal Inst. Technol. (KTH), Stockholm, Sweden, 1996.
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[7] M. Andersin, Z. Rosberg, and J. Zander, “Gradual removals in cellular PCS with constrained power control and noise,” ACM/Baltzar Wireless Netw. J., vol. 2, no. 1, pp. 27–43, 1996. [8] M. Rasti and A. R. Sharafat, “Distributed power control with soft removal in wireless networks,” IEEE Trans. Commun., 2009, submitted for publication. [9] P. Hande, S. Rangan, M. Chiang, and X. Wu, “Distributed uplink power control for optimal SIR assignement in cellular data networks,” IEEE/ACM Transactions on Networking, To appear in. [10] C. W. Sung and W. S. Wong, “A noncooperative power control game for multirate CDMA data networks,” IEEE Trans. Wireless Commun., vol. 2, no. 1, pp. 186–194, 2003. [11] M. Rasti, A. R. Sharafat, and B. Seyfe, “Pareto efficient and goal driven power control in wireless networks: A game theoretic approach with a novel pricing scheme,” IEEE/ACM Trans. Netw., vol. 17, no. 2, pp. 556–569, Apr. 2009. [12] C. W. Sung and K. Leung, “A generalized framework for distributed power control in wireless networks,” IEEE Trans. Inf. Theory, vol. 51, no. 7, pp. 2625–2635, Jul. 2005. [13] R. D. Yates, “A framework for uplink power control in cellular radio systems,” IEEE Journal on Sel. Areas Commun., vol. 13, no. 7, pp. 1341–1347, 1995. [14] C. U. Saraydar, N. B. Mandayam, and D. J. Goodman, “Efficient power control via pricing in wireless data networks,” IEEE Trans. Commun., vol. 50, no. 2, pp. 291–303, Feb. 2002. Mehdi Rasti received the B.Sc. degree from Shiraz University, Shiraz, Iran, and the M.Sc. and Ph.D. degrees from Tarbiat Modares University, Tehran, Iran, all in electrical engineering in 2001, 2003, and 2009, respectively. From November 2007 to November 2008, he was a Visiting Researcher with the Wireless@KTH, Royal Institute of Technology, Stockholm, Sweden. He is now with Shiraz University of Technology, Shiraz. His current research interests include resource allocation in wireless networks, and application of game theory and pricing in wireless networks.
Ahmad R. Sharafat (S’75–M’81–SM’94) received the B.Sc. degree from Sharif University of Technology, Tehran, Iran, and the M.Sc. and Ph.D. degrees from Stanford University, Stanford, CA, all in electrical engineering in 1975, 1976, and 1981, respectively. He is currently a Professor of electrical and computer engineering at Tarbiat Modares University, Tehran. His research interests are advanced signal processing techniques, and communications systems and networks. Dr. Sharafat is a Senior Member of Sigma Xi.
Jens Zander (S’82–M’85) received the M.S. degree in electrical engineering and the Ph.D. Degree from Linköping University, Sweden, in 1979 and 1985, respectively. From 1985 to 1989, he was a partner of SECTRA, a high-tech company in telecommunications systems and applications. In 1989, he was appointed Professor and Head of the Radio Communication Systems Laboratory, Royal Institute of Technology, Stockholm, Sweden. Since 1992, he also serves as Senior Scientific Advisor to the Swedish National Defence Research Institute (FOI). Between 2001–2002, he was Scientific Director and since 2003, the Director of the Center for Wireless Systems (Wireless@KTH), Royal Institute of Technology. He has published numerous papers in the field of radio communication, in particular on resource management aspects of personal communication systems. He has also coauthored four textbooks in radio communication systems, including the English textbooks Principles of Wireless Communications and Radio Resource Management for Wireless Networks. His current research interests include future wireless infrastructures and in particular related resource allocation and economic issues. Dr. Zander was the recipient of the IEEE Vehicular Technology Society “Jack Neubauer Award” for best systems paper in 1992. He is a member of the Royal Academy of Engineering Sciences. He is the chairman of the IEEE VT/COM Swedish Chapter. He is an Associate Editor of the ACM Wireless Networks Journal and Area Editor of Wireless Personal Communications.
