IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 3, MARCH 2011
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Distributed Uplink Power Control with Soft Removal for Wireless Networks Mehdi Rasti and Ahmad R. Sharafat, Senior Member, IEEE
Abstract—In the well-known distributed target-SIR-tracking power control algorithm (TPC), when the system is infeasible (a constrained power vector does not exist to attain target-SIRs), all non-supported users (those who cannot obtain their targetSIRs) transmit at their maximum power. Such users inefficiently consume their energy, and cause interference to others, which increases the number of non-supported users. To deal with this, some non-supported users should decrease their transmit power (the gradual removal problem). We present a distributed power control scheme with gradual soft removal, by which either TPC or OPC (opportunistic power control) is used, depending on whether the ratio of interference-to-path-gain is below or above a threshold that is chosen by each user in a distributed manner. We show that our algorithm converges to a unique fixedpoint in both feasible and infeasible systems, and that when the system is infeasible, it results in less outage with significantly less consumed power, as compared to TPC. We also provide a game theoretic analysis of our algorithm by introducing a new pricing when users are selfish. As our algorithm is fully distributed and requires only local information, it can be applied to both cellular and ad hoc networks. Simulation results confirm our analysis. Index Terms—Distributed power control, gradual removal problem, selective target-SIR-tracking or opportunistic power control algorithm, wireless networks.
I. I NTRODUCTION
P
OWER control in a wireless network is an important and challenging issue as the network’s capacity depends highly on how interference and fading are managed. In addition to capacity enhancement, power control is also used to extend user’s battery life, and to maintain an acceptable quality of service (QoS) in terms of bit-error-rate (BER) and/or transmission delay for all users by minimizing interference to others. A power control algorithm should work well for both data and voice services. Data services require a higher signal-to-interference ratio (SIR) as compared to the voice service, because the former is less tolerant to bit-errors. Also, in contrast to a non-real-time data service, the voice service is highly sensitive to transmission delays. Since the early work on distributed power control in [1], designing distributed power control algorithms has received
Paper approved by K. K. Leung, the Editor for Wireless Network Access and Performance of the IEEE Communications Society. Manuscript received November 19, 2009; revised May 10, 2010 and September 4, 2010. This work was supported in part by Tarbiat Modares University, Tehran, Iran, and in part by Shiraz University of Technology, Shiraz, Iran. M. Rasti is with the Department of Electrical Engineering, Shiraz University of Technology, Shiraz, Iran. Prior to this, he was with the Department of Electrical and Computer Engineering, Tarbiat Modares Univesity, Tehran, Iran. A. R. Sharafat (corresponding author) is with the Dept. of Electrical and Computer Engineering, Tarbiat Modares Univesity, P.O. Box 14155-4838, Tehran, Iran (e-mail:
[email protected]). Digital Object Identifier 10.1109/TCOMM.2011.122110.090711
much attention. In distributed algorithms, the information that each user globally needs to know is minimal. In doing so, two approaches have been considered, namely the target-SIRtracking scheme that was originally proposed in [2] and has been further studied in [3]–[8], and the opportunistic approach that was proposed in [9] and [10]. They are different in the way interference (traffic load) and deep fading (low path-gain) are managed. In the target-SIR-tracking power control algorithm (TPC), each user is tracking its own predefined target SIR. This causes a user to increase its transmit power when the effective interference (i.e., the ratio of interference to path-gain) is increased because of heavy traffic and/or deep fading and/or far distance. In contrast to TPC, in an opportunistic power control scheme (OPC), each user increases its transmit power as the effective interference decreases [9], [10]. The OPC significantly increases the aggregate throughput (in terms of the aggregate capacity) as compared to TPC by allocating more power to users with good channels, and very low power to users with poor channels (softly removing users with poor channels). We bring this latter property of OPC (soft removal) into TPC to improve its performance in infeasible systems. It is well known that in a feasible system, TPC (both constrained and unconstrained cases) provides all active users with their target-SIRs, i.e., zero outage, with minimum power consumption [2]. When the system is infeasible, it was shown in [3], [4] and [11] that constrained TPC converges to a unique fixed point, as opposed to unconstrained TPC that does not converge. As TPC was originally designed to provide all users with their target-SIRs in feasible systems, in constrained TPC, when the system is infeasible, all non-supported users (those who cannot obtain their target-SIRs) transmit at their maximum power (without reaching their target-SIRs), and cause interference to others as well. Obviously, this is not desirable, and can be easily shown that it increases the number of non-supported users, which can be avoided if only some non-supported users reduce their transmit power. In other words, by gradually removing some non-supported users, the remaining users can be supported and thus, the outage ratio (the ratio of the number of non-supported users to the total number of users) as well as the transmit power consumed by each user are reduced. This is the gradual removal problem that we focus on in this paper. In this paper, we address the gradual removal problem where the outage ratio is minimized subject to power constraint, by removing a minimal number of users in a distributed manner in an infeasible system. It has been shown in [4] that minimizing the outage ratio is NP-hard. To address the gradual removal problem, in [5] and [12]
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TPC was modified to temporarily remove users with a poor channel, but its convergence is not guaranteed [5], and power oscillations may ensue [12]. In [13], it was proposed that users with a poor channel remove themselves permanently with a given probability. If the effective interference becomes acceptable again for such users, they do not resume transmission on the same channel, as opposed to a temporarily removed user as in [5] and [12]. However, temporary removal may result in power oscillations [5], [12]. In real-time applications, permanent removal is a severe drawback, particularly because unnecessary removals may ensue during initial power updating iterations (before reaching the steady state), even in feasible cases. Due to these problems, we focus on power control algorithms with the capability of temporary removal. In order to address the gradual removal problem in a distributed manner that requires minimal information by users, we present a selective target-SIR-tracking or opportunistic power control algorithm (TOPC). In TOPC, when the effective interference is below a threshold (i.e., a good channel), the user updates its transmit power using TPC, and when it is above the threshold (poor channel), OPC is used. Under TOPC, in an infeasible system, a portion of non-supported users are identified as candidates for soft removal. For each of such users, if its interference level is above the threshold, the user would operate according to OPC, i.e., instead of transmitting at its maximum power as in TPC, or instead of switching off as in [5] and [12] that causes power oscillations, it gradually reduces its transmit power, i.e., the user is softly removed. In other words, in TOPC, OPC is applied only when the channel is poor, resulting in soft removal in order to reduce interference, and consequently increase the number of supported users. Note that the objective of employing OPC in TOPC is completely different from the objective of the original OPC that was basically applied in both poor and good channels. We will prove that the proposed algorithm is convergent (in contrast to the algorithms in [5] and [12]), and satisfies the objective of gradual removal, i.e., it reduces outage ratio and power consumption (a given user’s energy consumption as well as the aggregate transmit power consumed by all users) as compared to TPC. We also provide a game theoretic analysis of our TOPC via a non-cooperative power control game with a new pricing function, when users are assumed to be selfish, and show that it results in a best response function for each user, which is the same as our proposed TOPC’s power update function. Simulation results confirm our analysis. Since our proposed TOPC is fully distributed and always convergent, it can be applied to cellular as well as to ad hoc networks. The rest of this paper is organized as follows. In Section II, we introduce the system model and present the background. In Section III, we state the gradual removal problem, present our proposed TOPC, and prove its convergence. In Section IV, we describe how to choose the TOPC parameters, and observe that TOPC reduces both the outage ratio and power consumptions by users as compared to TPC. A game theoretic analysis of TOPC is presented in Section V, where the power control problem is formulated as a non-cooperative game. Simulation results and conclusions are presented in Sections VI and VII, respectively.
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 3, MARCH 2011
II. S YSTEM M ODEL AND BACKGROUND A. CDMA Wireless Network Models and Notations Consider a multi-cell wireless CDMA network with 𝐾 base stations (cells) and 𝑀 active users denoted by 𝒦 = {1, 2, ⋅ ⋅ ⋅ , 𝐾} and ℳ = {1, 2, ⋅ ⋅ ⋅ , 𝑀 }, respectively. Let 𝑝𝑖 be the transmit power of user 𝑖 and 0 ≤ 𝑝𝑖 ≤ 𝑝𝑖 , where 𝑝𝑖 is the upper limit of the transmit power for user 𝑖. The path gain from user 𝑖 to the base station 𝑘 is denoted by ℎ𝑘𝑖 . Noise is assumed to be additive white Gaussian (AWGN) whose power is 𝜎𝑘2 at the receiver of the base station 𝑘. Let 𝑠𝑖 denote the base station to which user 𝑖 is assigned. The receiver is assumed to be a conventional matched filter. Thus, for a given transmit power vector p = [𝑝1 , 𝑝2 , ⋅ ⋅ ⋅ , 𝑝𝑀 ]T , the SIR of user 𝑖, denoted by 𝛾𝑖 is 𝛾𝑖 (p) =
𝑔𝑖 ℎ𝑠𝑖 𝑖 𝑝𝑖 , 𝐼𝑖 (p)
(1)
where 𝑔𝑖 is the processing gain for user 𝑖 (defined as the ratio of chip rate (or spreading bandwidth) to transmit data rate), and ∑ 𝐼𝑖 (p) = ℎ𝑠𝑖 𝑗 𝑝𝑗 + 𝜎𝑠2𝑖 (2) 𝑗∕=𝑖
is the interference to user 𝑖’s base station. We also define the effective interference as the ratio of interference to the product of processing gain and path-gain for a user 𝑖 (as in [9], [12]), denoted by 𝑅𝑖 , 𝐼𝑖 (p) . (3) 𝑅𝑖 (p) = 𝑔𝑖 ℎ𝑠𝑖 𝑖 The value of 𝑅𝑖 represents the channel status for user 𝑖, i.e., given 𝑔𝑖 (assumed to be fixed for all users), a higher interference and a lower path gain results in a higher 𝑅𝑖 . The terms poor channel or good channel for user 𝑖 imply that the value of 𝑅𝑖 is high or low, respectively. Definition 1: A SIR vector denoted by 𝜸 = [𝛾1 , 𝛾2 , ⋅ ⋅ ⋅ , 𝛾𝑀 ]T is feasible if a feasible power vector 0 ≤ p ≤ p exists that corresponds to the SIR vector, where the vector inequality 0 ≤ p ≤ p implies 0 ≤ 𝑝𝑖 ≤ 𝑝𝑖 for ˆ , we also all 𝑖. For a given target-SIR vector denoted by 𝜸 ˆ is feasible, otherwise the say that the system is feasible if 𝜸 system is called infeasible. B. Existing Distributed Power Control Algorithms In TPC, each user 𝑖 tries to maintain its SIR at a target level denoted by 𝛾 ˆ𝑖 . The unconstrained power-update function in [2] is ˆ𝑖 𝑅𝑖 (p(𝑡)) (4) 𝑝𝑖 (𝑡 + 1) = 𝛾 where the target-SIR 𝛾 ˆ𝑖 is fixed and 𝑅𝑖 (p(𝑡)) is the effective interference to user 𝑖’s base station at time 𝑡. It was shown in ˆ is feasi[2] and [8] that if and only if the target-SIR vector 𝜸 ble, then unconstrained-TPC converges either synchronously or asynchronously to a fixed point at which users attain their target-SIRs with minimal aggregate transmit power, i.e., its fixed point solves the following optimization problem ∑ 𝑝𝑖 min p
𝑖∈ℳ
subject to 𝛾𝑖 (p) ≥ 𝛾 ˆ𝑖 , ∀𝑖 ∈ ℳ.
(5)
RASTI and SHARAFAT: DISTRIBUTED UPLINK POWER CONTROL WITH SOFT REMOVAL FOR WIRELESS NETWORKS
When the system is infeasible, the above problem has no solution, because there is no transmit power vector that can satisfy SIR requirements for all users. When the unconstrained TPC (that was originally designed to solve (5) in a distributed manner assuming feasibility of the system) is used in infeasible systems, since the target-SIRs are rigidly tracked, all users increase their transmit power at each step and thus, the unconstrained TPC diverges. Algorithms developed in [3] and [4] assume constrained transmit power to deal with divergence in infeasible systems, that is 𝛾𝑖 𝑅𝑖 (p(𝑡))} . 𝑝𝑖 (𝑡 + 1) = min {𝑝𝑖 , ˆ
(6)
Although convergence to a fixed point is guaranteed for constrained TPC in both feasible and infeasible systems, it suffers from a severe drawback in infeasible systems (high outage and high power consumption) as described in the following section where the gradual removal problem is formally stated. In the rest of this paper, TPC implies (6) unless stated otherwise. Convergence of the unconstrained TPC is facilitated by a dynamic target-SIR, denoted by 𝑇𝑖 (𝑅𝑖 ), which is a decreasing function of 𝑅𝑖 and is chosen so that 𝑇𝑖 (𝑅𝑖 ) 𝑅𝑖 is an increasing function of 𝑅𝑖 [12]. The power update function for a dynamic target-SIR-tracking power control (DTPC) is { 𝑇𝑖 (𝑅𝑖 (p(𝑡))) 𝑅𝑖 (p(𝑡)), if 𝑇𝑖 (𝑅𝑖 (p(𝑡))) ≥ 𝛾 𝑖 𝑝𝑖 (𝑡 + 1)= 0, if 𝑇𝑖 (𝑅𝑖 (p(𝑡))) < 𝛾 𝑖 (7) where 𝛾 𝑖 is the threshold SIR for user 𝑖. The updated transmit power for user 𝑖 is 𝑇𝑖 (𝑅𝑖 ) 𝑅𝑖 so far as 𝑇𝑖 (𝑅𝑖 ) ≥ 𝛾 𝑖 , and is 0 (switch-off) otherwise. This algorithm dynamically adjusts the target-SIR according to the channel condition. If the channel is poor, the target-SIR is decreased and may even vanish (i.e., the user is switched off). This facilitates the convergence [12], in the sense that as users are not rigidly tracking their fixed target SIRs, some users may switch off at high traffic loads to make the system feasible. However, in infeasible systems, some users may switch off when 𝑇𝑖 (𝑅𝑖 ) < 𝛾 𝑖 , and start transmitting again when 𝑇𝑖 (𝑅𝑖 ) ≥ 𝛾 𝑖 , which means power oscillations [12] and possible inability of the algorithm to converge. The reason is that the power-update function is neither standard nor continuous over the entire 𝑇𝑖 (𝑅𝑖 ). Another drawback of the algorithm in [12] is that no upper limit for transmit power is assumed. In OPC, the transmit power is updated in a manner opposite to TPC. The opportunistic power update function proposed in [10] is 𝜂𝑖 𝑝𝑖 (𝑡 + 1) = (8) 𝑅𝑖 (p(𝑡)) where 𝜂𝑖 is a constant for user 𝑖. As (8) implies, the transmit power is increased when the channel is good and is decreased when the channel is poor. This algorithm is always convergent, and significantly increases the aggregate throughput by transmitting more power by users with good channels, and transmitting less power by users with bad channels, but leads to unfairness as well. In Section III, we employ OPC to reduce the transmit power of users only when their channel is poor (i.e., soft removal), as their effective interference levels are increased (in contrast to TPC, in which higher transmit power levels are set for poor channels).
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III. P ROBLEM S TATEMENT AND P ROPOSED A LGORITHM A. Problem Statement and Motivation TPC is suitable for attaining target-SIRs of users in feasible systems. Using TPC in infeasible systems causes all nonsupported users to transmit at their maximum power because all users (including the non-supported ones) rigidly track their respective SIRs. This unnecessarily drains the batteries of nonsupported users and increases interference to others. As we will see in the following example and in simulation results, this also increases the number of non-supported users, which can be avoided if some non-supported users reduce their transmit power. As an example, consider an infeasible system with four users whose path gain and target-SIR vectors are [0.15, 0.40, 0.50, 0.60], and [50, 40, 50, 40], respectively. Suppose AWGN power is 0.1 Watts, processing gain is 100, and the maximum transmit power for each user is 1 Watt. The fixed-point power-vector of the constrained TPC is [1, 1, 1, 0.767] Watts and thus, their achieved SIRs at the equilibrium are [10, 33, 45, 40], meaning that only one user (user 4) attains its target-SIR using TPC. One can easily see that the transmit power vector [0.22, 1, 0.93, 0.67] Watts results in the target-SIR vector [2.4, 40, 50, 40] which means that users 2, 3, and 4 can be supported and their required transmit power to attain their target-SIRs are also reduced if user 1 refrains from transmitting at maximum power and instead, transmits at 0.22 Watts or less. This shows that in an infeasible system, the policy of reducing transmit power by some non-supported users has at least two advantages: 1) the remaining ones can be supported, and consequently the number of supported users is increased, and 2) the aggregate transmit power is reduced as well. For the DTPC and the algorithm proposed in [5], although a surge in the transmit power of non-supported users is avoided, they nevertheless suffer from power oscillations [12] (see also simulation results). Motivated by the stated drawbacks of TPC and the advantages of softly removing a portion of non-supported users in infeasible systems, we now formally state the gradual removal problem. Given a transmit power vector p, we denote the set of supported users by 𝒮(p) = {𝑗 ∈ ℳ∣𝛾𝑗 (p) ≥ ˆ 𝛾𝑗 }, and the total number of its members by ∣𝒮(p)∣. The outage ratio is (∣ℳ∣ − ∣𝒮(p)∣)/∣ℳ∣. We define the problem of minimum-outage-based removal, or equivalently the problem of maximum number of supported users by max ∣𝒮(p)∣,
0≤p≤p
(9)
which is applicable to both feasible and infeasible systems. When the system is feasible, all users can be supported (i.e., the minimum outage ratio is zero) and its solution is given by TPC consuming minimal transmit power. When the system is infeasible, a minimal number of users should be removed, which is a NP-hard problem [4]. In this case, as stated earlier, TPC results in some avoidable non-supported users. In what follows, we present a distributed power control algorithm for addressing the gradual removal problem, i.e., reducing the outage that not only prevents transmission at maximum power in a poor channel (instead, reduces the transmit power), but also is convergent and stable.
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C. Convergence of TOPC The work presented in [8] provides a framework to examine the convergence of TPC. This framework is generalized to a new framework in [9] applicable to a wider range of distributed power control algorithms including TPC and OPC. Our proposed TOPC falls into this generalized framework. We will show that TOPC’s power update function has a unique fixed point to which it converges. To show this, we first define two-sided scalable functions as in [9] and prove that TOPC’s power update function (10) is two-sided scalable. Definition 3: A power update function f (p) = T [𝑓1 (p), 𝑓2 (p), ⋅ ⋅ ⋅ , 𝑓𝑀 (p)] is two-sided scalable if for all 𝑎 > 1, 𝑎1 p ≤ p′ ≤ 𝑎p implies Fig. 1. The proposed TOPC algorithm. The regions corresponding to th 𝑅𝑖 < 𝑅th 𝑖 or to 𝑅𝑖 > 𝑅𝑖 are called TPC mode or OPC mode of TOPC, respectively.
B. Proposed Algorithm: Selective Target-SIR-Tracking or Opportunistic Power Control Algorithm (TOPC) The observations made in Section II-B on TPC and OPC motivate us to design a distributed power control algorithm that adjusts the transmit power according to either TPC or OPC, depending on the channel conditions to address the gradual removal objectives. We call this strategy TOPC, as defined and formulated below. Definition 2: A power control algorithm is TOPC if the power-update function 𝑓𝑖 (p) is continuous and is either an increasing function of 𝑅𝑖 for 𝑅𝑖 values below a given threshold, or a decreasing function of 𝑅𝑖 for 𝑅𝑖 values above that threshold (see Fig. 1) for all 𝑖. In TOPC, when a given user 𝑖 operates in the target-SIRtracking mode, an increase in 𝑅𝑖 would increase its transmit power, until the user’s threshold for 𝑅𝑖 is reached, upon which further increases in 𝑅𝑖 would reduce the user’s transmit power (opportunistic mode). We propose TOPC’s power-update function as { 𝑝𝑖 (𝑡+1) =
( ) 𝑓𝑖 C (p(𝑡))
≜
𝛾 ˆ𝑖 𝑅𝑖 (p(𝑡)), 𝜂𝑖 , 𝑅𝑖 (p(𝑡))
if 𝑅𝑖 (p(𝑡)) ≤ 𝑅𝑖th if 𝑅𝑖 (p(𝑡)) ≥ 𝑅𝑖th
(10) where 𝑅𝑖th is the effective interference threshold, 𝜂𝑖 is a constant, and 𝛾 ˆ𝑖 is the target-SIR for user 𝑖 (see Fig. 1). These (C) three parameters of TOPC are adjusted so that 𝑓𝑖 (p(𝑡)) is continuous, i.e., 𝛾𝑖 (𝑅𝑖th )2 . 𝜂𝑖 = ˆ
(11)
Obviously, the performance of TOPC highly depends on the threshold values of the effective interference set by users. An important question is how each user 𝑖 should adjust its 𝑅𝑖th in a distributed manner to satisfy the stated objectives. In Section IV we will propose a distributed method for choosing the effective interference thresholds to reduce both the outage ratio and the power consumption as compared to TPC. In what follows, for a given 𝑅𝑖th for each user 𝑖, we show that TOPC converges to a unique fixed point in both feasible and infeasible systems.
1 𝑓𝑖 (p) ≤ 𝑓𝑖 (p′ ) ≤ 𝑎𝑓𝑖 (p) for all 𝑖 ∈ ℳ. 𝑎
(12)
Theorem 1: TOPC’s power update function f (C) (p) in (10) is two-sided scalable. Proof: If both 𝑅𝑖 (p) and 𝑅𝑖 (p′ ) are either smaller or greater than 𝑅𝑖th , then for p and p′ , a user 𝑖 operates in either the target-SIR-tracking mode or the opportunistic mode, respectively. This means that for user 𝑖, (12) holds, because the TPC’s and the OPC’s power update functions are twosided scalable [9]. If 𝑎1 p ≤ p′ ≤ 𝑎p for a given 𝑎 > 1, then by simple mathematical manipulations, it is easy to show that
and
1 2 ′ 𝑅 (p ) ≤ 𝑅𝑖 (p′ )𝑅𝑖 (p) ≤ 𝑎𝑅𝑖2 (p), 𝑎 𝑖
(13)
1 2 𝑅 (p) ≤ 𝑅𝑖 (p′ )𝑅𝑖 (p) ≤ 𝑎𝑅𝑖2 (p′ ). 𝑎 𝑖
(14)
If 𝑅𝑖 (p) ≤ 𝑅𝑖th ≤ 𝑅𝑖 (p′ ), then from (13) we have ( )2 1 ( th )2 ≤ 𝑅𝑖 (p′ )𝑅𝑖 (p) ≤ 𝑎 𝑅𝑖th , 𝑅𝑖 𝑎 or equivalently 𝑎 1 1 ≤ ( )2 . ( th )2 ≤ ′ 𝑅𝑖 (p )𝑅𝑖 (p) 𝑅𝑖th 𝑎 𝑅𝑖
(15)
(16)
We substitute (11) into (16), multiply (16) by 𝑅𝑖 (p), and get 1 𝜂𝑖 𝛾𝑖 𝑅𝑖 (p) ≤ ˆ ≤ 𝑎ˆ 𝛾𝑖 𝑅𝑖 (p) 𝑎 𝑅𝑖 (p′ )
(17)
which is equivalent to (12). If 𝑅𝑖 (p′ ) ≤ 𝑅𝑖th ≤ 𝑅𝑖 (p), then from (14) we have )2 ( 1 ( th )2 𝑅𝑖 ≤ 𝑅𝑖 (p′ )𝑅𝑖 (p) ≤ 𝑎 𝑅𝑖th . (18) 𝑎 We substitute (11) into (18), divide (18) by 𝑅𝑖 (p), and get 1 𝜂𝑖 𝜂𝑖 ≤𝛾 ˆ𝑖 𝑅𝑖 (p′ ) ≤ 𝑎 , 𝑎 𝑅𝑖 (p) 𝑅𝑖 (p)
(19)
which is equivalent to (12). Theorem 2: a) TOPC’s power update function f (C) (p) has a fixed point, i.e., there exists a transmit power vector p∗ such that p∗ = f (C) (p∗ ). Besides, the fixed point is unique. b) For any given initial power vector, TOPC converges to the fixed point of f (C) (p) in both synchronous and asynchronous cases.
RASTI and SHARAFAT: DISTRIBUTED UPLINK POWER CONTROL WITH SOFT REMOVAL FOR WIRELESS NETWORKS
Proof: The followings have been proved in [9]. 1. If a given f (p) is two-sided scalable and there exists a fixed point p∗ so that p∗ = f (p∗ ), then – the fixed point is unique, and – for any initial power vector, the power control algorithm p(𝑡 + 1) = f (p(𝑡)) converges to p∗ . 2. If f (p) is two-sided scalable, continuous, and f (p) ≤ u for all p (i.e., f (p) is upper bounded) for a u > 0, then a fixed point exists. From the above, we conclude that if f (p) is two-sided scalable and continuous, and there is a u > 0 such that f (p) ≤ u for all p, then the fixed point p∗ = f (p∗ ) exists and is unique. Also, for any initial power vector, the power control algorithm p(𝑡 + 1) = f (p(𝑡)) converges to p∗ . Thus, since f (C) (p) is continuous (with constraint (11)) and is upper bounded ( ) ˆ𝑖 𝑅𝑖th for all 𝑖), and since in Theorem 1, it (𝑓𝑖 C (p) ≤ 𝛾 was shown that f (C) p) is two-sided scalable, this theorem is proved. IV. TOPC PARAMETERS So far we have shown that, given 𝑅𝑖th for each user 𝑖, TOPC converges to a unique fixed point. We now show how users can choose their TOPC’s parameters in a distributed manner to reduce both the outage ratio and power consumption as compared to TPC. There are three parameters in TOPC given by (10), namely 𝛾 ˆ𝑖 , 𝑅𝑖th and 𝜂𝑖 . The values of these parameters depend on one another via (11). We choose the values of ˆ𝑖 for 𝛾 ˆ𝑖 and 𝑅𝑖th , and obtain 𝜂𝑖 from (11). The value of 𝛾 each user 𝑖 is simply decided by its target-SIR. Thus we need only to propose a distributed method for choosing the effective interference threshold 𝑅𝑖th . In what follows we explain how to choose the effective interference thresholds in the proposed algorithm to reduce the outage ratio and power consumption as compared to TPC. Assume that each user 𝑖 with the target-SIR of 𝛾 ˆ𝑖 , chooses its 𝑅𝑖th by 𝑝 𝑅𝑖th = 𝑖 . (20) 𝛾 ˆ𝑖 𝑝𝑖 is the maximum value of 𝑅𝑖 for which the target𝛾 ˆ𝑖 SIR for user 𝑖 is achievable (by transmitting at its maximum power). Note that if 𝑅𝑖th is chosen at a value higher than the above, it may result in maximum transmit power without reaching target-SIRs for some users when the system is infeasible (similar to TPC), which is not desirable, and if it is chosen at a value lower than the above, it may result in not reaching target-SIRs even when the system is feasible (because the user may go to the OPC mode without being 𝑝2 required). By substituting (11) into (20), we get 𝜂𝑖 = 𝑖 . 𝛾 ˆ𝑖 In the following two theorems, we will show that by using TOPC (as compared to TPC), each user consumes less energy, resulting in a reduced total power consumption, and a lower outage ratio. Theorem 3: Given 𝛾 ˆ𝑖 for all 𝑖, let p∗(T) and p∗(C) as the fixed points of TPC and TOPC for which the effective interference threshold is chosen by (20), respectively. where
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ˆ) a) If and only if the system (i.e., the target-SIR vector 𝜸 ∗(C) ∗(T) =p . is feasible, then p ∗(C) b) If and only if the system is infeasible, < p∗(T) ∑ ∑ then p and consequently 𝑖 𝑝𝑖 ∗(C) < 𝑖 𝑝𝑖 ∗(T) . Proof: The first part can be easily proved, so we only prove the second part. We know that if p(C) (𝑡) ≤ p(T) (𝑡) for a given 𝑡, then from (3) we have 𝑅𝑖 (p(C) (𝑡)) ≤ 𝑅𝑖 (p(T) (𝑡)) for all 𝑖 and thus p(C) (𝑡 + 1) ≤ p(T) (𝑡 + 1) obtained by comparing TOPC’s power-update function (10) for which the effective interference threshold is cho(C) sen according as 𝑝𝑖 (𝑡 + } { to (20) (which can be re-written 𝜂𝑖 ) with TPC’s power1) = min 𝛾 ˆ𝑖 𝑅𝑖 (p(C) (𝑡)), 𝑅𝑖 (p(C) (𝑡)) update function (6). Thus, if p(C) (𝑡0 ) ≤ p(T) (𝑡0 ) for a given time 𝑡0 , then p(C) (𝑡) ≤ p(T) (𝑡) for all 𝑡 ≥ 𝑡0 . Since for any given initial transmit power p(0), TPC and TOPC converge to their own unique fixed point [(3], [4] and Theorem 2), we assume the initial transmit power p(0) is the same for both. Therefore, we have p(C) (𝑡) ≤ p(T) (𝑡) for all 𝑡 ≥ 0 and consequently p∗(C) ≤ p∗(T) . In what follows, we prove that for an infeasible system, the strict inequality holds, i.e., p∗(C) < p∗(T) . As a direct result of the first part of this theorem, and from p∗(C) ≤ p∗(T) , there is at least one 𝑗 for which 𝑝𝑗 ∗(C) < 𝑝𝑗 ∗(T) in an infeasible case, because otherwise, it contradicts the first part. Now we prove that if there is at least one 𝑗 for which 𝑝𝑗 ∗(C) < 𝑝𝑗 ∗(T) , then 𝑝𝑖 ∗(C) < 𝑝𝑖 ∗(T) for all 𝑖. This is true because from 𝑝𝑗 ∗(C) < 𝑝𝑗 ∗(T) and p∗(C) ≤ p∗(T) , we conclude that for all 𝑖 ∕= 𝑗, 𝑅𝑖 (p∗(C) ) < 𝑅𝑖 (p∗(T) ), which implies 𝑝𝑖 ∗(C) < 𝑝𝑖 ∗(T) . Thus, if the system is infeasible, then p∗(C) < p∗(T) . Also from this and from the first part, we conclude that if p∗(C) < p∗(T) , then the system is infeasible. The two latter statements complete the proof of the second part of this theorem. Remark 1: Theorem 3 states that in a feasible system, the performance of TOPC for which the effective interference threshold is chosen according to (20) is the same as that of TPC. This implies that under TOPC, the outage ratio in a feasible system is zero (i.e., target-SIRs are reached and all users operate in the target-SIR-tracking mode), and the minimum transmit power vector required to satisfy the users’ target-SIRs is obtained, i.e., both optimization problems in (5) and (9) are solved by TOPC and TPC in a similar manner. Remark 2: As we see in the above theorem and in its proof, when the system is infeasible, for any initial power vector, in addition to the equilibrium (the fixed-point), the instantaneous power consumed by TOPC for each user and consequently the instantaneous total power for all users are strictly less than those of TPC. In simulation results we show that this energy efficiency is significant. Remark 3: In TOPC for which the effective interference threshold is chosen according to (20), if the system is infeasible, then at the fixed point, there exists at least one user operating in the opportunistic mode and vice versa. This is because, at the fixed point of TOPC in an infeasible system, if all users operate in TPC, Theorem 3 is contradicted. In this case, the users that operate in the opportunistic mode are not supported (do not reach their target SIRs). Such users transmit at less than their maximum power, as opposed to TPC where all non-supported users transmit at their maximum power.
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Theorem 4: Given 𝛾 ˆ𝑖 for all 𝑖, 𝒮(p∗(T) ) ⊆ 𝒮(p∗(C) ), ∗(T) which implies ∣𝒮(p )∣ ≤ ∣𝒮(p∗(C) )∣, where p∗(T) and ∗(C) p are the fixed points of TPC and TOPC (for which the effective interference threshold is chosen according to (20)), respectively. In other words, if a user reaches its target-SIR using TPC, then that user also reaches its target-SIR using TOPC. The inverse is not necessarily true, i.e., there may exist a user that does not reach its target-SIR by using TPC, although its target-SIR is reachable by using TOPC. ∗(C) ≤ p∗(T) and Proof: ( ∗(From ) Theorem ( ∗(T) 3) we have p ) C ≤ 𝑅𝑖 p for all 𝑖. Suppose thus 𝑅𝑖 p ( a given ) user 𝑖 reaches its target-SIR using TPC, i.e., 𝛾𝑖 p∗(T) = 𝛾 ˆ𝑖 , ( ∗(T) ) 𝑝𝑖 ≤ . In this case, user 𝑖 reaches which implies 𝑅𝑖 p 𝛾 ˆ𝑖 ( ) ˆ𝑖 . its target-SIR when TOPC used, i.e., 𝛾𝑖 p∗(C) = 𝛾 ( ∗( is ) This is because if 𝛾𝑖 p C) < 𝛾 ˆ𝑖 , then we must have ( ) ( ) ( ) 𝑝 𝑅𝑖 p∗(C) > 𝑖 , and thus 𝑅𝑖 p∗(T) < 𝑅𝑖 p∗(C) , which ˆ𝑖 (𝛾 ) ( ) contradicts 𝑅𝑖 p∗(C) ≤ 𝑅𝑖 p∗(T) . The following example shows that the inverse is not true. Consider again the example in Section III.A for an infeasible system with four users. At the fixed point of TPC, only one user is supported. But in TOPC, one can easily verify that at the fixed point, the users’ transmit power vector is [0.22, 1, 0.93, 0.67] Watts and their achieved SIR vector is [2.4, 40, 50, 40], respectively. This means that user 1 (the non-supported user with the worst channel) senses that the system is infeasible and reduces its transmit power, instead of transmitting at its maximum power, and thus users 2 and 3 can attain their target-SIRs in addition to user 4. Besides, the transmit power consumed by each user is reduced as well (see also simulation results). Employing TOPC in an infeasible system (in contrast to employing TPC) causes some users to operate in the opportunistic mode (Remark 3 above), meaning that they start reducing their transmit power as the effective interference increases. This makes more resources available to other users to reach their target-SIRs. Hence, the number of users that reach their target-SIRs using TOPC is always equal to or higher than that of using TPC. Equivalently, the outage ratio in TOPC for which the effective interference threshold is chosen according to (20) is less than that of TPC (see also simulation results in Section VI). Thus, although in general, TOPC does not guarantee the minimum-outage ratio defined in (9), but it has a unique equilibrium to which the algorithm converges, which is closer to the optimum solution of (9) (in the sense that the outage ratio as well as the power consumption are less) as compared to TPC’s equilibrium. Now we discuss which users in an infeasible system may operate in the opportunistic mode in favor of others by adjusting 𝑅𝑖th according to (20). Consider the case that 𝑝𝑖 and ˆ 𝛾𝑖 are the same for all users and the system is infeasible. In this case, 𝑅𝑖th is the same for all users. Thus those users that operate in the opportunistic mode have high values of 𝑅𝑖 (have low path-gains and/or high interference levels). This means that when the system is infeasible, users with a poor channel operate in the opportunistic mode (they start reducing their transmit power levels as their effective interference is increased), thus favoring users with a good channel, which is
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 3, MARCH 2011
desirable in this case. Similarly, for the case that target-SIRs are not the same for different users, in an infeasible system, those users with low path-gains and/or with high values for target-SIRs may operate in the opportunistic mode. V. G AME -T HEORETIC A NALYSIS OF TOPC Recently, the non-cooperative game theoretic analysis is widely applied to power control in wireless networks [12], [14]–[18] when users are assumed to be selfish (in contrast to what we have assumed so far). In what follows, assuming selfish and non-cooperative users, we investigate how and with what pricing scheme, at the Nash equilibrium (NE) of the power control game, the gradual removal problem is addressed similar to TOPC. In a game theoretic view of the power control problem, each user 𝑖 ∈ ℳ chooses its transmit power level from its strategy space 𝑃𝑖 = [0, 𝑝𝑖 ] in a selfish manner to maximize its own utility denoted by 𝑢𝑖 (𝑝𝑖 , p−𝑖 ) in which p−𝑖 is the transmit power vector for all users except user 𝑖. The utility function is defined for each user and represents the QoS offerings to that user, as well as its associated costs. When a user transmits in a shared medium, that user should pay a price (cost) for receiving the service and for causing interference to others. A non-cooperative power control game (NPCG) is denoted by 𝐺 = ⟨ℳ, (𝑃𝑖 ), (𝑢𝑖 )⟩ and is formally stated [19] by max 𝑢𝑖 (𝑝𝑖 , p−𝑖 ) for all 𝑖 ∈ ℳ.
(21)
𝑝𝑖 ∈𝑃𝑖
The commonly used concept in solving game theoretic problems is the Nash equilibrium (NE) at which no user can improve its utility by unilaterally changing its transmit power. Definition 4: A transmit power vector p∗ is the NE point for NPCG 𝐺 = ⟨ℳ, (𝑃𝑖 ), (𝑢𝑖 )⟩ if for every user 𝑖, 𝑢𝑖 (𝑝∗𝑖 , p∗−𝑖 ) ≥ 𝑢𝑖 (𝑝𝑖 , p∗−𝑖 ), ∀𝑝𝑖 ∈ 𝑃𝑖 . Another commonly used concept in game theory is the best response function for each player. Formally, the user 𝑖’s best response function 𝑏𝑖 : 𝑃−𝑖 → 𝑃𝑖 , where 𝑃∏ −𝑖 is the 𝑃𝑗 ), is Cartesian product of 𝑃𝑗 for 𝑗 ∕= 𝑖 (i.e., 𝑃−𝑖 = 𝑗∕=𝑖
a set-valued function that assigns the set of best power levels in the utility sense to each power vector p−𝑖 ∈ 𝑃−𝑖 , that is 𝑏𝑖 (p−𝑖 ) = {𝑝𝑖 ∈ 𝑃𝑖 ∣∀𝑝′𝑖 ∈ 𝑃𝑖 : 𝑢𝑖 (𝑝𝑖 , p−𝑖 ) ≥ 𝑢𝑖 (𝑝′𝑖 , p−𝑖 ) }. The NE is the fixed point of the best response function set, that is p = b(p), where b(p) = [𝑏1 (p), 𝑏2 (p), ⋅ ⋅ ⋅ , 𝑏𝑀 (p)]T . Note that 𝑏𝑖 (p) and 𝑏𝑖 (p−𝑖 ) are equivalent. A pricing-based utility function in many NPCGs is 𝑈𝑖 (𝑝𝑖 , p−i ) = 𝑞𝑖 (𝑝𝑖 , p−i ) − 𝑐𝑖 (𝑝𝑖 , p−i ),
(22)
where 𝑞𝑖 is the function representing the QoS that user 𝑖 receives, and 𝑐𝑖 is the pricing function for user 𝑖. It is well established that in contrast to the case in which no pricing is applied, the pricing scheme could affect the individual user’s decision in such a way that the efficiency of NE from a given goal’s point of view (e.g., from fairness’s or system’s point of view) is improved. In [18], for a QoS function which is concave and increasing with respect to SIR, we have shown that SIR-based pricing can be utilized to satisfy specific goals (such as fairness, aggregate throughput optimization, or trading off between these two goals) at the NE. In [9], [10] it
RASTI and SHARAFAT: DISTRIBUTED UPLINK POWER CONTROL WITH SOFT REMOVAL FOR WIRELESS NETWORKS
was shown that the best response function for utility function √ 𝑈𝑖 (𝑝𝑖 , p−i ) = 𝛾𝑖 − 𝛼𝑖 𝑝𝑖 corresponds to the opportunistic power control scheme. Note that the QoS function defined in [9], [10] has no physical meaning. Similar to [16]–[18] and [20], we use an information theoretic approach to define a meaningful QoS function in terms of channel capacity as the highest rate at which user 𝑖’s information can be sent with an arbitrary low probability of error [21]. We do not restrict ourselves to a specific channel model, modulation, or coding scheme. For a given SIR experienced by user 𝑖 at its base-station, we denote the corresponding QoS function by 𝑞(𝛾𝑖 ). Generally, 𝑞(𝛾𝑖 ) is an increasing and concave function of 𝛾𝑖 for every channel model with average power constraint [22]; thus the following two properties hold. Property 1: 𝑞 ′ (𝛾𝑖 ) > 0 for all 𝛾𝑖 Property 2: 𝑞 ′′ (𝛾𝑖 ) < 0 for all 𝛾𝑖 , where 𝑞 ′ (𝛾𝑖 ) and 𝑞 ′′ (𝛾𝑖 ) are the first and the second derivatives of 𝑞(𝛾𝑖 ) with respect to 𝛾𝑖 , respectively. We rely only on these two general properties, and our developments hold for any other QoS function that satisfies Properties 1-2 above. As two examples for the QoS function, consider a logarithmic function of SIR, defined in [16] and [20] denoted by 𝑞G (𝛾), and channel capacity for a binary symmetric channel (BSC) as in [17] denoted by 𝑞BSC (𝛾), and write 𝑞G (𝛾) = and
𝑏 log2 (1 + 𝛾) 2
bits/second,
(23)
( 𝑞BSC (𝛾)=𝑏 1 + 𝑝e (𝛾) log2 𝑝e (𝛾)
) +(1 − 𝑝e (𝛾)) log2 (1 − 𝑝e (𝛾)) bits/second,(24)
where 𝑏 is the data rate, and 𝑝e (𝛾) is the cross error probability defined in [21]. Note that these two examples for QoS function satisfy Properties 1 and 2 above. The logarithmic function (4) is the capacity of a Gaussian channel, provided that noise plus interference for each user is Gaussian [16]. In what follows, we set up a NPCG with a pricing scheme that is a function of SIR, and analytically show that with a proper choice of pricing units, its outcome is the same as that of our proposed TOPC, implying that at the NE, the gradual removal problem is addressed. Let 𝑐𝑖 (𝛾𝑖 ) be the pricing function of user 𝑖 for 𝛾𝑖 at the base station, and the pricing-based utility function for user 𝑖 be 𝑈𝑖 (𝑝𝑖 , p−𝑖 ) = 𝑞(𝛾𝑖 (𝑝𝑖 , p−i )) − 𝛼𝑖 𝛾𝑖 (𝑝𝑖 , p−i ),
(25)
where 𝛼𝑖 ≥ 0 is the price per unit of the actual SIR at the base station for user 𝑖. We will show that this pricing scheme enables us to adequately influence the best response function of each user to address the gradual removal (similar to TOPC) by a proper choice of pricing units introduced in the following theorem. Theorem 5: In NPCG 𝐺 = ⟨ℳ, (𝑃𝑖 ), (𝑈𝑖 )⟩ in which 𝑈𝑖 is (25), the best response of user 𝑖 ∈ ℳ to a given power vector p−𝑖 is the same as our proposed power update function of TOPC in (10) with the given parameters of 𝛾 ˆ𝑖 and 𝑅𝑖th if the
pricing units in (26) for each user 𝑖 is set to ⎧ if 𝑅𝑖 ≤ 𝑅𝑖th ⎨𝑞 ′ (𝛾ˆ𝑖 ), th 𝛼𝑖 = 𝑅 ⎩𝑞 ′ (( 𝑖 )2 𝛾ˆ𝑖 ), if 𝑅𝑖 > 𝑅𝑖th . 𝑅𝑖
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(26)
Thus, a unique Nash equilibrium for this game exists, and is the fixed point of TOPC, i.e., the fixed-point of TOPC denoted by p∗ is the solution to max 𝑞(𝛾𝑖 (𝑝𝑖 , p∗−i )) − 𝛼𝑖 𝛾𝑖 (𝑝𝑖 , p∗−i ), for all 𝑖 ∈ ℳ.
𝑝𝑖 ∈𝑃𝑖
(27)
Proof: See Appendix I. Note that as the QoS function is concave, its first derivative, i.e., 𝑞 ′ (⋅), is a decreasing function. Thus the proposed pricing in (26) implies that when the effective interference for a given user is below the threshold, the pricing unit for that user is fixed at a value given by 𝑞 ′ (𝛾ˆ𝑖 ), and when the effective interference exceeds the threshold, its pricing unit is increased as its effective interference is increased in order to discourage selfish users from transmitting with high power. Another important property of the proposed pricing scheme is that the pricing unit in (26) can be obtained by each user in a distributed manner, because it depends on the information pertinent to that user only. VI. S IMULATION R ESULTS Now we show that TOPC outperforms other existing power control algorithms in terms of convergence, outage ratio, and energy efficiency1 . The outage ratio in one iteration (one power update step) is the ratio of the number of non-supported users to the total number of active users in that iteration, as defined in Section III. A given user 𝑖 with target-SIR ˆ 𝛾𝑖 is 𝛾𝑖 , where 0 < 𝛿 ≤ 1 is the utilized fade supported if 𝛾𝑖 ≥ 𝛿ˆ margin. We use 𝛿 = 0.88 that corresponds to -0.5 dB, and assume a data rate of 104 bits/second and a chip rate of 106 bits/second for each user as in [15], which means that the processing gain is 100. The AWGN power at the receiver, i.e., 𝜎 2 , is assumed to be 5 × 10−15 Watts as in [15]. We adopt a simple and well known model ℎ𝑠𝑖 𝑖 = 𝑘𝑑−4 𝑠𝑖 𝑖 for the path-gain as in [23] where 𝑑𝑠𝑖 𝑖 is the distance between user 𝑖 and its base station 𝑠𝑖 , and 𝑘 is the attenuation factor that represents power variations due to the shadowing effect, assumed to be 𝑘 = 0.09 as in [15]. The upper bound on the transmit power for all users is 1 Watt. We first consider a single cell wireless network, and then proceed to a multi-cell one. A. Single Cell Networks We consider a single-cell wireless network, first with fixed locations of users to study and track the performance of the algorithms in detail, and then proceed to different snapshots of users’ locations, to verify that the results do not depend on specific user-locations. Consider 6 users indexed from 1 to 6 in a single cell environment where their distance vector is d = 1 Comparing TPC, DTPC, and TOPC with OPC with respect to the stated performance measures does not make much sense, because their objectives are different. However to show how OPC works as compared to TOPC, we have included it in our simulation results.
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Fig. 2. Transmit power and SIR of each user vs. iteration number, using TPC. Users 5 and 6 become active after iterations 30 and 60, respectively.
Fig. 4. Transmit power and SIR of each user vs. iteration number, using OPC. Users 5 and 6 become active after iterations 30 and 60, respectively.
Fig. 3. Transmit power and SIR of each user vs. iteration number, using DTPC. Users 5 and 6 become active after iterations 30 and 60, respectively.
Fig. 5. Transmit power and SIR of each user vs. iteration number, using TOPC with the effective interference threshold of (20). Users 5 and 6 become active after iteration 30 and 60, respectively.
T
[200, 320, 460, 530, 550, 800] m, in which each element is the distance of the corresponding user from the base station. ˆ = [24, 20, 30, 28, 24, 30]T . One The target-SIR vector is 𝜸 can easily verify that the system is infeasible. The system becomes feasible if either user 6 or users 5 and 4 switch off. Without loss of generality, we assume that initially, users 1 to 4 are active (from the first iteration), and then user 5 followed by user 6 become active after 30 and 60 iterations, respectively. The parameter values for TPC, DTPC, OPC, and TOPC = 𝛾 ˆ𝑖 and 𝑇𝑖 (𝑅𝑖 ) )) = 𝛾 ˆ𝑖 − are as(follows. For 𝑖 ( DTPC (7), 𝛾√ ( )2 𝑎𝑖 1 𝛾𝑖 − 1) 2𝛼𝑎𝑖𝑖𝑅𝑖 − 1 − −1 as 𝑎𝑖 ln (𝑎𝑖 ˆ 2𝛼𝑖 𝑅𝑖 − 1 in [12], where 𝑎𝑖 = 200, 𝛼𝑖 = 𝛼ℎ𝑠𝑖 𝑖 for all 𝑖 and 𝛼 = 3000. In TOPC (10), for a given target-SIR for each user 𝑖, the value of 𝑅𝑖th is set according to (20) and 𝜂𝑖 is obtained from (11). For OPC, the value of 𝜂𝑖 is set to 10 for all 𝑖. The transmit power and SIR for each user versus the iteration number are shown in Figs. 2 to 5 for TPC, DTPC, OPC, and TOPC, respectively. As expected, since OPC aims to increase the aggregate throughput, it sacrifices fairness and those users with a good channel transmit at high power (Fig. 4). As seen in Figs. 2, 3 and 5 for a feasible system (when both users 5 and 6 are switched off, i.e., before iteration 30), TPC and TOPC work well, and DTPC oscillates (albeit this oscillation can be avoided by increasing the pricing coefficient
𝛼 for oscillatory users, meaning a centralized decision making that uses a trial and error approach, which contradicts the notion of a distributed power control algorithm). When user 5 becomes active but user 6 is still switched off (i.e., from iteration 30 to iteration 60), TPC and TOPC work well and all users reach their target-SIRs, and DTPC becomes stable but users 3 and 4 do not reach their targetSIRs. When user 6 starts transmitting (after iteration 60), the system becomes infeasible (as shown in Fig. 2). As user 6 has a poor channel, under TPC, it transmits at its maximum power but does not reach its target-SIR. This increases interference to other users, resulting in users 4 and 5 not reaching their target-SIRs although they transmit at their maximum power, and users 1 to 3 have to consume more power to reach their target-SIRs. In this case, if user 6 reduces its transmit power, the interference to other users is decreased, resulting in users 4 and 5 to reach their target-SIRs and users 1 to 3 to attain their target-SIRs at lower transmit power. This is achieved if TOPC is applied as shown in Fig. 5. In an infeasible system, in contrast to TPC and DTPC, TOPC for which the effective interference threshold is chosen according to (20) causes those users with a poor channel to operate in the opportunistic mode of TOPC, where the transmission power is reduced with an increase in the effective interference levels (transmitting at very low power and thus experiencing a lower SIR than its
RASTI and SHARAFAT: DISTRIBUTED UPLINK POWER CONTROL WITH SOFT REMOVAL FOR WIRELESS NETWORKS
Fig. 6. Average outage ratio vs. target-SIR for TPC, for TOPC with the effective interference threshold chosen by (20) and for the optimum solution obtained via exhaustive search. The outage ratio in TOPC is closer to the optimum solution as compared to TPC.
target value to make target-SIR requirements for the remaining users achievable), while other users operate in the target-SIRtracking mode of TOPC and attain their target-SIRs. In summary, the outage ratio for OPC is the worst one, as we expected. The outage ratio for DTPC increases from 0 to 0.4 and 0.5 when users 5 and 6 are switched on, respectively. We see that the outage ratio for both TOPC and TPC is zero when the system is feasible, but after both users 5 and 6 are switched on (i.e., after iteration 60), it reaches 0.16 (only user 6 does not reach its target-SIR) in TOPC, and gets to 0.5 in TPC, indicating that the outage ratio for TOPC is less than those of other algorithms. This outage reduction is achieved while each user consumes less power and thus the total power consumption is significantly reduced by using TOPC as compared to using TPC (as shown in Figs. 2 and 5, and in Theorem 3). To show that the better performance of TOPC as compared to TPC in reducing the outage is not dependent on specific locations and target-SIRs of users, we compare the performances of TOPC with both TPC and the optimum solution obtained by exhaustive search for different snapshots of users’ locations and for different values of target-SIRs2 . To do so, we consider a single cell network with a radius of 1 Km and with 12 fixed users. To compare the performances of TOPC with that of TPC and the minimum outage, the target-SIRs are considered the same for all users, ranging from 5 to 25 with step size 5. The lower and higher values of SIRs correspond to the feasible and the infeasible systems, respectively. For each target-SIR, we average the corresponding values of outage ratio (for TPC, TOPC and also for the optimum one), and average the transmit power consumed by each user (the average total transmit power over the entire snapshots divided by the total number of users) for 1000 independent snapshots for a uniform distribution of users’ locations within a single-cell. The initial transmit power for each user is uniformly set from the interval [0, 1] for each snapshot. Figs. 6 and 7 show the average outage ratio and the average total transmit power versus target-SIR, respectively. Note that TOPC outperforms TPC with respect to 2 As
DTPC may not converge for some snapshots, it is not included.
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Fig. 7. Average transmit power vs. target-SIR for TPC and for TOPC with the effective interference threshold chosen by (20). The average transmit power consumed by users in TOPC is significantly lower as compared to TPC.
Fig. 8. Users and base stations locations. Users are marked “×” with their index next to them and base stations are marked “△”.
the outage ratio and the consumed power and its performance is closely following that of the optimum one obtained via exhaustive search. For instance, for the target-SIR of 15, by using TOPC, the outage ratio is reduced by 13% and the average transmit-power is significantly reduced by 54%, as compared to using TPC. As seen in Figs. 6 and 7 for other target-SIRs that make the system infeasible, either similar or better improvements are achieved by using TOPC as compared to using TPC. B. Multi-Cell Networks Now we consider a multi-cell wireless system with 20 users distributed in an area covered by 4 base stations. The locations of users for a uniformly generated set marked by “×” and the base stations marked by “Δ” are shown in Fig. 8, where each user is assigned to its nearest base station. Each cell covers 500 m × 500 m. Assume that for each user, the target-SIR is 24 and the value of 𝑅𝑖th is obtained from (20). To show how TOPC works when users move in a multi-cell environment, we assume that initially, user 20 is at its starting point in cell No. 4, and after 30 iterations3 , it moves towards cell No. 3 along 3 User 20 is assumed to be fixed in the initial 30 iterations to show infeasibility of the system when it stays at the starting point in Fig. 8.
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 3, MARCH 2011
target-SIR, thus the outage ratio for cell No. 4 is 0.2. However, when user 20 gets closer to base station 3 and far from base station 4, since its interference to cell No. 4 is reduced, user 19 can also be supported, thus the outage ratio for cell No. 3 is reduced to zero. These numbers show that TOPC also works well when users move, and supports more users than TPC by causing some users (those that cause bottlenecks) to operate in the opportunistic mode. VII. C ONCLUSIONS
Fig. 9. Transmit power and SIR of remotely located users vs. iteration number, using TOPC. User 20 after iteration 30 starts moving from the start point and at uniform speed reach the end point at iteration 250.
We proposed TOPC for distributed power control in wireless networks. The algorithm was proved to converge to a unique fixed point (as opposed to DTPC), and its power consumption is significantly less than that of TPC for individual users and for the aggregate of all users. This significant energy saving is achieved while our scheme reduces the outage ratio as well, compared to existing algorithms. Thus TOPC outperforms TPC with respect to addressing the minimum-outage problem (less outage) by consuming less power. For the case that users are selfish, we also provided a game theoretic analysis of our proposed algorithm by presenting a new pricing scheme. Due to its fully distributed power updating scheme and its guaranteed convergence, TOPC can be used in both cellular and ad hoc networks. A PPENDIX A P ROOF OF T HEOREM 5
Fig. 10. Outage ratio for each cell in TOPC vs. iteration number. The reason for the brief transition in the outage ratio for cell 3 is the hand-over of user 20 from cell 4 to cell 3.
the line in Fig. 8 at a uniform speed, so that its movement from the start point to the end point takes 220 iterations (equivalent to a speed of 38.21 m/s and transmit power updating every 50 ms (20 Hz)). When user 20 enters cell No. 3 coverage area, base station 3 is assigned to it. This system is infeasible when user 20 is at its starting point, but becomes feasible when it reaches towards the end point. Note that (ignoring user 20), user 5 in cell No. 1, user 10 in cell No. 2, user 14 in cell No. 3, and user 19 in cell No. 4 are the farthest away users in each cell from their respective base stations. The transmit power and SIRs for these users and for the moving user 20 are shown in Fig. 9. The outage ratio for each cell versus the iteration number are shown in Fig. 10. As we see in Fig. 9, prior to start of moving by user 20 (i.e., before iteration 30), only users 10, 19 and 20 fail to reach their target-SIRs and thus the outage ratio for cell Nos. 1 to 4 are 0, 0.2, 0.33, and 0, respectively. When user 20 starts moving (at iteration 30) and gets far away from cell No. 2, as its interference to cell No. 2 is reduced, user 10 in cell No. 2 switches to target-SIR-tracking mode, and reaches its target SIR, and thus the outage ratio for cell No. 2 becomes zero. As user 20 moves farther and is handed over from cell No. 4 to cell No. 3, it switches to TPC and reaches its target-SIR, but user 19 still fails to reach its
To obtain the best response function denoted by 𝑏𝑖 (p−𝑖 ), we use the first and the second derivatives of the pricing-based utility with respect to 𝑝𝑖 1 ′ ∂𝑈𝑖 = (𝑞 (𝛾𝑖 ) − 𝛼𝑖 ) , ∂𝑝𝑖 𝑅𝑖 ( )2 1 ∂ 2 𝑈𝑖 = 𝑞 ′′ (𝛾𝑖 ). ∂𝑝2𝑖 𝑅𝑖
(28) (29)
For a given 𝑅𝑖 , we note from (28) that ∂𝑈𝑖 /∂𝑝𝑖 = 0 has ˆ𝑖 if 𝑅𝑖 ≤ 𝑅𝑖th a unique root 𝛾𝑖 = 𝑞 ′−1 (𝛼𝑖 ), that is 𝛾𝑖 = 𝛾 th 𝑅 and 𝛾𝑖 = ( 𝑖 )2 𝛾ˆ𝑖 if 𝑅𝑖 > 𝑅𝑖th . From Property II in Section 𝑅𝑖 V, we have 𝑞 ′′ (𝛾𝑖 ) < 0 for all 𝛾𝑖 , thus ∂ 2 𝑈𝑖 /∂𝑝2𝑖 < 0 which means the unique root of ∂𝑈𝑖 /∂𝑝𝑖 = 0 globally maximizes 𝑈𝑖 . For a given effective interference 𝑅𝑖 , a one-to-one relation exists between SIR and the transmit power, and thus the best transmit power in response to p−𝑖 that maximizes 𝑈𝑖 is also unique and is equal to 𝑝𝑖 = 𝑞 ′−1 (𝛼𝑖 )𝑅𝑖 , i.e., the best response function is ⎧ for 𝑅𝑖 ≤ 𝑅𝑖th ⎨𝛾ˆ𝑖 𝑅𝑖 , ′−1 th 2 (30) 𝑏𝑖 (p) = 𝑞 (𝛼𝑖 )𝑅𝑖 = (𝑅𝑖 ) ⎩ 𝛾ˆ𝑖 , for 𝑅𝑖 > 𝑅𝑖th 𝑅𝑖 which is the same as in TOPC and its fixed point is the Nash equilibrium for 𝐺 = ⟨ℳ, (𝑃𝑖 ), (𝑈𝑖 )⟩. As there exists a unique fixed point in TOPC, the Nash equilibrium exists and is unique. Thus (27) holds.
RASTI and SHARAFAT: DISTRIBUTED UPLINK POWER CONTROL WITH SOFT REMOVAL FOR WIRELESS NETWORKS
R EFERENCES [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., vol. 2, Stockholm, Sweden, June 1994, 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] S. C. Chen, N. Bambos, and G. J. Pottie, “Admission control schemes for wireless communication network with adjustable transmitter powers," in Proc. IEEE INFOCOM, Toronto, Canada, June 1994, pp. 21-28. [7] N. Bambos, S. C. Chen, and G. J. Pottie, “Channel access algorithms with active link protection for wireless communication networks with power control," IEEE/ACM Trans. Networking, vol. 8, no. 5, pp. 583597, Oct. 2000. [8] R. D. Yates, “A framework for uplink power control in cellular radio systems," IEEE J. Sel. Areas Commun., vol. 13, no. 7, pp. 1341-1347, 1995. [9] 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, July 2005. [10] K. Leung and C. W. Sung, “An opportunistic power control algorithm for cellular network," IEEE/ACM Trans. Networking, vol. 14, no. 3, pp. 470-478, June 2006. [11] S. A. Grandhi, R. Yates, and J. Zander, “Constrained power control," Wireless Commun., vol. 2, no. 3, Aug. 1995. [12] M. Xiao, N. B. Shroff, and E. K. P. Chong, “A utility-based powercontrol scheme in wireless cellular systems," IEEE/ACM Trans. Networking, vol. 11, no. 2, pp. 210-221, Apr. 2003. [13] 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. [14] P. Liu, P. Zhang, S. Jordan, and L. Honig, “Single-cell forward link power allocation using pricing in wireless networks," IEEE Trans. Wireless Commun., vol. 3, no. 2, pp. 533-543, Mar. 2004. [15] 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.
843
[16] T. Alpcan, T. Basar, and R. Srikant, “CDMA uplink power control as a noncooperative game," Wireless Netw., vol. 8, pp. 659-670, 2002. [17] 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. [18] 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. Networking, vol. 17, no. 2, pp. 556-569, Apr. 2009. [19] D. Fudenberg and J. Tirole, Game Theory. Cambridge, MA: MIT Press, 1995. [20] A. R. Fattahi and F. Paganini, “New economic perspectives for resource allocation in wireless networks," in Proc. American Control Conf., Portland, OR, USA, June 2005, pp. 3690-3695. [21] T. M. Cover and J. Thomas, Elements of Information Theory. Wiley, 1991. [22] M. C. Gursoy, H. V. Poor, and S. Verdu, “Noncoherent rician fading channel–part II: spectral efficiency in the low-power regime," IEEE Trans. Wireless Commun., vol. 4, no. 5, pp. 2207-2221, Sep. 2005. [23] A. J. Viterbi, CDMA: Principle of Spread Spectrum Communication. Addison-Wesley, 1995.
in wireless networks.
Mehdi Rasti received his B.Sc. degree from Shiraz University, Shiraz, Iran, and the M.Sc. and Ph.D. degrees both 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 at the Wireless@KTH, Royal Institute of Technology, Stockholm, Sweden. He is now with Shiraz University of Technology, Shiraz, Iran. His current research interests include resource allocation in wireless networks, and application of game theory and pricing
Ahmad R. Sharafat (S’75-M’81-SM’94) is a professor of Electrical and Computer Engineering at Tarbiat Modares University, Tehran, Iran. He received his B.Sc. degree from Sharif University of Technology, Tehran, Iran, and his M.Sc. and his Ph.D. degrees both from Stanford University, Stanford, California, all in Electrical Engineering in 1975, 1976, and 1981, respectively. His research interests are advanced signal processing techniques, and communications systems and networks. He is a Senior Member of the IEEE and Sigma Xi.
