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Transient Analysis for Wireless Power Control
Maryam Fazel 1 Dennice Maynard Gayme 1 Mung Chiang 2
1 Control and Dynamical Systems, Caltech. 2 Electrical Engineering, Princeton University.
Abstract — Power control mitigates interference and main- tains required QoS levels in cellular wireless networks. An im- portant class of distributed power control (DPC) was proposed by Foschini and Miljanic in 1993, with many variants developed since. Almost all related work focuses on the equilibrium and asymptotic convergence properties. However, for many appli- cations transient behavior is more important. If a link’s SIR drops below a critical threshold for too long, the connections over this link will be dropped, rendering the entire concept of equilibrium resource allocation meaningless. This paper proposes a systematic approach to the analysis of transient properties of DPC algorithms, in particular Foschini-Miljanic, based on tools from control theory. Analytically, we present a sufficient condition to ensure that after links reach their minimum SIR levels, their SIR requirements can be guaranteed for future time steps. Computationally, we pose this problem as checking the invariance of certain regions in the SIR space, which for the basic DPC algorithm can be cast as a Linear Program (LP). Furthermore, using insights gained from the analysis, we propose a preliminary design framework for new iterative resource allocation schemes. Keywords: Invariant sets, Lyapunov functions, Power con- trol, Wireless network.
I. INTRODUCTION Power control in both voice and data cellular wireless networks is an important interference mitigation mechanism that has been extensively studied since the early 1990s. Many iterative [5], [3] and distributed [5], [6], [9] algorithms have been proposed; some of these converge to a globally optimal power allocation, while others converge to a Nash equilibrium. In all cases, efficiency and fairness of power allocation and Signal-Interference-Ratio (SIR) configuration at equilibrium have been the focus. In this paper, we shift attention to the transients of power control. Indeed, while equilibrium properties are often more analytically tractable, they do not capture some of the more important issues in practical network operations. At the application connection level, dipping below a minimum SIR threshold during the transient phase of an iterative algorithm could cause the connection to disappear, thus rendering the whole concept of equilibrium SIR meaningless. Furthermore, the transient phase dominates the entire operating range when network dynamics vary at a similar time scale as the algorithm convergence speed. As a starting point for a systematic study of transient behaviors in wireless power control, we consider the cel- ebrated iterative algorithm of Foschini and Miljanic [5]. There are many transient properties worth examining, this paper mainly focuses on providing a guarantee that once a link is active, it retains the SIR level required to keep it
active. Active link protection (and providing SIR guarantees in general) can be thought of in terms of invariance of regions in the SIR space. We use tools from control theory to obtain an analytic condition for providing SIR guarantees across all users. We also present a computational approach to verifying the invariance of the same SIR region via Linear Programming (LP). Then, turning from analysis to design, we present preliminary results on how to modify the basic algorithm to improve its transient behavior. In general, we believe that a similar shift of research focus from asymptotic convergence to transient properties , including invariance , is desirable in many other research areas involving iterative resource allocation algorithms, such as congestion control, medium access, and buffer allocation. This paper is an initial step along this research direction. Often in current practice unpredictable and undesirable tran- sient behaviors are treated through ad hoc means, and one of our goals is to illustrate how tools from control theory can help provide a systematic approach. The paper is organized as follows; the following section introduces the problem setting, and provides a review of the Foschini-Miljanic algorithm. In section III the analytical results are presented along with a discussion of what they imply for link SIR guarantees. In section IV, we express the problem of checking the invariance of certain SIR regions as an LP, which can then be readily solved. The analytic results are verified by simulation in section V. New design of power control algorithm based on these insights is outlined in section VI, before we discuss some immediate steps in future research in section VII. Notation. Vectors are represented by letters in boldface, matrices are represented by capital letters and the ith^ entry of x is denoted xi.
II. BACKGROUND AND RELATED WORK We consider a network of n interfering links, each link consisting of a logical transmitter-receiver pair. This could, for example, model n uplinks in a cellular network. Signal- to-Interference-Ratio (SIR) of the ith^ link is denoted by
ri =
Giipi ∑ j 6 =i Gij^ pj^ +^ ηi
, i = 1,... , n (1)
where Gij > 0 is the power gain (path loss) from the transmitter of the jth^ link to the receiver of the ith^ one, pi is the power of the transmitter of the ith^ link, ηi is the thermal noise power at its receiver. Each link has a threshold SIR requirement αi to maintain its connection through a minimum level of proper decoding, i.e., the link
disappears if ri < αi. A target SIR level of γi ≥ αi can be assigned through QoS provisioning to provide a ‘safety margin’. Using (1) the inequalities ri ≥ γi, ∀i (i.e., that all links meet or exceed their target SIR requirements) can be written in matrix form as
(I − Dγ F )p ≥ Dγ u, p > 0 , (2)
where p ∈ Rn^ is the power vector, u ∈ Rn^ is the (normalized) noise vector, i.e., ui = (^) Gηiii , Dγ is a diagonal matrix with γis on its diagonal, and F ∈ Rn×n^ is the matrix of cross-link power gains,
Fij =
0 , i = j, Gij Gii ,^ i^6 =^ j,
where i, j = 1,... , n. The following is a standard result on whether a given set of target SIRs are feasible. Fact 1: Existence of a feasible power vector [3], [5]. Let P = {p | p > 0 , (I − Dγ F )p ≥ Dγ u}. The following statements are equivalent:
(i) There exists p ∈ P, (ii) ρ(Dγ F ) < 1 , where ρ denotes the maximum modulus eigenvalue, (iii) (I − Dγ F )−^1 exists and is positive componentwise. Based on the above, the iteration
p(k + 1) = Dγ F p(k) + Dγ u, (3)
introduced in [5] by Foschini and Miljanic, converges to p∗^ = (I − Dγ F )−^1 Dγ u when p∗^ exists. This iteration can alternatively be expressed as
pi(k + 1) =
γi ri(k)
pi(k), i = 1,... , n, (4)
where only local SIR measurements ri(k) are needed for the update. Since the update decision at each link is per- formed independently, based on information collected on it exclusively, the algorithm is referred to as a Distributed Power Control algorithm (DPC). Many other variants of this DPC have been proposed over the years. In particular, this algorithm has been extended for other network models, e.g., [7], [8], [9] examined asynchronous implementation, bursty transmissions, and multiclass traffic; [13], [14] considered joint power and base station assignment, and [1], [2] stud- ied admission control with power control. As a key paper that started to examine non-equilibrium properties of DPC algorithms, [3] tackled the issue of protecting active links while new links are introduced, by modifying the Foschini- Miljanic algorithm to include two different update rules for active and inactive links. The present work further shifts the attention to what happens to DPC before reaching the equilibrium. We use tools from control theory to study aspects of the dynamics and the evolution of the SIRs in the Foschini-Miljanic algorithm. Using these tools we not only provide analytical results on some SIR invariant regions for the Foschini- Miljanic algorithm but also provide computational methods to evaluate invariance for a particular problem setup. Such analysis not only provides a deeper understanding of DPCs,
but also helps with designing new power control algorithms with better SIR guarantees and transient properties. This type of design approach is more systematic than the current practice of treating undesirable transient behaviors through ad hoc means (e.g., raising target SIR levels and hoping for fast convergence or averaging effects of network dynamics). We set up and discuss a general framework for such designs in the last section of this paper. III. INVARIANT REGIONS IN SIR SPACE Consider an autonomous discrete-time dynamical system x(k + 1) = f (x(k)), y(k) = g(x(k)), with initial state x(0). The state of the system and the output at time k are x(k) ∈ Rn^ and y(k) ∈ Rm^ respectively. A subset of the output space is called invariant if once the output enters this set, it remains there for all future time steps [12], i.e. y(k) ∈ S ⇒ y(k + 1) ∈ S, ∀k. For the Foschini-Miljanic algorithm we call the power pi(k) the state of each link i at time k and refer to the corre- sponding SIR ri(k) as the output, so the system is described by equations (3) and (1). We are interested in the transient behavior of the output (SIR’s) of this system. To study this problem we begin by addressing questions such as: if all links achieve their minimum SIR, will they continually stay above this level for all future time? Also, if an individual link achieves its minimum SIR, under what conditions will it remain above this level? Both of these questions can be addressed though determining if there are corresponding invariant regions in the SIR space. Throughout this section we assume that ρ(Dγ F ) < 1 , so the γi are always feasible SIR targets. A. SIR guarantees We begin with the first question posed in the previous section: Assuming all links have already achieved their minimum SIR level, under what conditions can we guarantee that they retain this SIR level for all subsequent time steps? This type of guarantee could be sought for example when all the links are powering up at the same time.
Proposition 1: Common ratio condition. A sufficient con- dition for ri(k) ≥ αi, ∀i =⇒ ri(k + 1) ≥ αi, ∀i is that there is a constant δ > 0 such that γi αi
= δ, ∀i. (5)
Proof: The invariance relation can be written using (2), (I − DαF )p(k) − Dαu ≥ 0 =⇒ (6) (I − DαF )p(k + 1) − Dαu ≥ 0. (7)
Substituting for p(k + 1) in (7) yields (I − DαF )Dγ F p(k) + [(I − DαF )Dγ − Dα]u ≥ 0. (8) Noting Dγ = δDα ⇒ DαF Dγ = Dγ F Dα and rearranging (8) δDαF [(I − DαF )p(k) − Dαu] + (δ − 1)Dαu ≥ 0.
δ
γ 1
δ
γ (^2)
γ 2 ∆ SIR
r( k )
δ V ( r)= 1 −^1
r 2
γ 1 r 1
Fig. 2. The level sets of the Lyapunov function
this region. Therefore, drops in the SIR of each link, i.e., components of vector r(k) − r(k + 1), can by no larger than the componentwise distance of r(k) to the line r γii = δ; this distance is labeled ∆SIR in figure 2.
C. The case where γ 6 = δα (links may have different ‘safety margins’)
Even when the condition in Proposition 1 is not satisfied, that is the SIR ‘safety margin’ is not the same for all links, the result in Lemma 1 provides some conditions on the link SIR levels. Figure 3 illustrates a situation where the condition
J 1
J 2
r 1
r 2
D 1
D 2
D^ ~ 1
D^ ~ 2
N
Fig. 3. The case where γ 6 = δα.
from Proposition 1 is not satisfied (i.e. the vectors γ and α are not aligned). If we let α˜ 1 = γ γ^12 α 2 , we know from Proposition 1 that the cone K with its vertex at (˜α 1 , α 2 ) is invariant. So once the SIR for link 2 (r 2 ) enters this cone at time K, r 2 (k) ≥ α 2 (k) ∀ k > K. Thus, to keep link 2 active, link 1 need only achieve an SIR of α˜ 1 , which is lower than its SIR threshold (α 1 ). This property extends to n links as follows.
Proposition 2: If ri(k) ≥ αi and rj (k) ≥ γ γji αi for all j 6 = i, then ri(k + 1) ≥ αi (i.e., link i is protected).
Proof: The proof follows immediately from the fact that γj α ˜j =^
γi αi =^ δi, for all^ j^6 =^ i, so^ γ^ and^ α˜^ satisfy the common ratio condition.
Note that in figure 3 the cone that guarantees both SIRs are above their thresholds has its vertex at (α 1 , α˜ 2 ), where α˜ 2 ≥ α 2. This means that, if we raise the SIR thresholds of other users, invariance is guaranteed (at the cost of consuming more power).
IV. INVARIANCE ANALYSIS: LINEAR PROGRAMMING
APPROACH
In this section we discuss how to computationally verify that all links remain active after they all achieve their minimum SIRs. Note that, here and in the sequel, p(k) is replaced by p for simplicity in notation. Proposition 3: Given the system described by (3) and (1), the following conditions are equivalent (i) ri(k) ≥ αi ⇒ ri(k + 1) ≥ αi, ∀i (ii) The set { p| p ≥ 0 , (I − DαF )p ≥ Dαu, (I − DαF )Dγ F p < [Dα − (I − DαF )Dγ ]u } (10)
is empty. (iii) ∃ λ 0 such that AT^ λ = 0 , λT^ c < 0 where
A = −
I − DαF [DαF − I] Dγ F I
(^) , and
c = −
Dαu [I − DαF ] Dγ u − Dαu + ε 0
and finding this λ is equivalent to solving the dual of the LP feasibility problem^2 : find p such that Ap c. Proof: (ii) ⇔ (i): Emptiness of the set (10) implies ∀ p ≥ 0 the region defined by (I − DαF )p ≥ Dαu (at time k) and the complement of this region at time (k + 1) do not intersect; i.e., no vector exists that satisfies all conditions in (10) simultaneously. This means starting in the region guarantees remaining there for all future time. (ii) ⇔ (iii): Farkas’ Lemma or LP duality [4] states that the LP: find p such that Ap c is infeasible by if and only if the dual problem: find λ 0 such that AT^ λ = 0 , λT^ c < 0 is feasible. So, the solution to this problem provides a certificate of infeasibility for the primal.
A. Numerical example We consider a case with four links, a power gain matrix
G =
2 6 4
- 730 0. 241 0. 407 0. 316
- 263 7. 883 0. 165 0. 247
- 219 0. 224 7. 939 0. 146
- 184 0. 0498 0. 117 7. 373
3 7 5 ,^ (11)
ηi = 10 −^2 , and ε = 10 −^10. Selecting α = [9. 27 , 7. 92 , 8. 75 , 5 .83] and γ = [10, 9 , 8. 5 , 9 .5] generates the vector of ratios (^) αγii = [1. 187 , 1. 2 , 1. 2 , 1 .2], (which does not satisfy the common ratio condition). To solve the LP we used, among the many existing solvers, the software package SeDuMi [11]. Solving the dual problem yields Lagrange multipliers λ 0 that satisfy λT^ (c−Ap) ≈ − 1 and provide a certificate of infeasibility for the LP. Recall that the feasibility problem is finding p such that c − Ap 0. Multiplying this it by λ 0 would yield a positive number. So the equation λT^ (c−Ap) ≈ − 1 provides (^2) The ε in the c vector is to deal with the fact that in (10) there is a strict inequality whereas the standard form of an LP has a nonstrict inequality.
(^22 4 6 8 )
4
6
8
10
12
14
16
18
Time step
SIR
target= target= target=8. target=9.
Fig. 4. Power Up Phase for 4 links
(^45 10 15 )
5
6
7
8
9
10
11
Time Step
SIR target= target= target=8. target=9. NewUser
Fig. 5. Adding a new user (link)
a contradiction that proves the LP is infeasible and thus the region {r(k) | r(k) ≥ α} is invariant. This proof method is a special case of a more general proof method in [10]. This example shows that the common ratio condition from Proposition 1 is not necessary in general; that is, invariance can hold without it.
V. NUMERICAL RESULTS In this section we give a numerical example that demon- strates that Proposition 1 is valid both during startup and when a new user is added. These are the conditions that induce a disturbance on the system and thus the most likely to cause a link’s SIR to drop. For the startup case the G matrix and ηi’s are the same as in the example IV- A. The target SIR’s for each link are given in the vector γ = [10, 9 , 8. 5 , 9 .5] as shown in figure 4, the minimum SIR for each link is set to α = [8. 43 , 7. 5 , 7. 08 , 7 .92] which means that δ = 1. 2 for this example. The initial power was set to p(0) = 10−^2 [6. 83 , 2. 13 , 8. 39 , 6 .29]. After the four links had reached and settled at their target SIR’s we introduced a fifth link with an SIR target of 10. 5 and corresponding Gij ’s that keep the system feasible (i.e ρ(Dγ F ) < 1 ). The new link has a higher target SIR than all of the old links because this was thought to lead to a larger disturbance for the other links. The new power gain matrix is
G =
- 730 0. 241 0. 407 0. 316 0. 106
- 263 7. 883 0. 165 0. 247 0. 151
- 219 0. 224 7. 939 0. 146 0. 259
- 184 0. 0498 0. 117 7. 373 0. 184
- 155 0. 0968 0. 299 0. 237 7. 982
Figure 5 shows the effect of adding a new link. The new five link system has the minimum SIR vector α = [8. 43 , 7. 5 , 7. 08 , 7. 92 , 8 .75]. It is clear from the figure that once all of the links surpass their minimum values they remain above them for all time. In fact all of the links eventually converge to their target values.
VI. DESIGN FOR TRANSIENTS Given that we can compute conditions to quantify the transient behavior, an interesting question now is how we can use that knowledge to design a power control algorithm that limits undesirable behavior. This includes slow convergence during the startup phase and links dropping below their target SIR’s when the network changes due to an external disturbance (e.g., the addition of a new link, or a change in
the target SIR level of an active link). In general, control design (synthesis) is a difficult problem. However, one can reformulate the Foschini-Miljanic DPC into a more general framework (12) (also see ?? ), which allows one to adjust the update rule based on the current state of the system. Here we present a preliminary study and an example of how this framework can be used. Systematic control design for this problem is a topic for future work. Consider a generalization Foschini-Miljanic update (4) where the target SIR is time varying,
pi(k + 1) =
γi(k) ri(k)
pi(k). (12)
This provides the following framework for general γ updates
γi(k + 1) = f (ri(k), pi(k), {γi(1),... , γi(k)}), (13)
which allows many choices for γi(k + 1) with each choice leading to a different algorithm. For example, the DPC/ALP algorithm of Bambos et al [3], which involves two separate modes for active and inactive links, can be viewed as a case of (12), with γi(k) given by the piecewise linear function
γi(k) =
δri(k) for ri(k) < γ∗ i , δγi, for ri(k) ≥ γ∗ i , where γ∗^ is the target SIR. In this case there is feedback based on ri(k) that allows each link to determine which mode they are in and perform a mode switch when it is appropriate. In many cases it may not be desirable for each link to have to do an additional calculation to determine which mode they are operating in or to keep track of more than one update rule. There are many reasons that a user may need to switch from one mode to the other such as variations in network conditions, changes in target SIR’s or the addition of new users in a cooperative network. In (14) a simple alternative that does not require switching between two operation modes is proposed, where γi is updated based on how far it is from its target value. γi(k + 1) = γ i∗ − κ|γ i∗ − γi(k)|, (14)
where κ is an arbitrary constant that affects the rate of convergence and the sensitivity of the system to disturbances. A low κ results in a disturbance having a larger effect but a faster convergence rate whereas a larger κ makes the SIR levels less likely to drop when a new user is added. This tradeoff between sensitivity to disturbances and performance is a common tradeoff in control design. Figures 6 and 7 illustrate the behavior of this algorithm for under the exact same conditions of figures 4 and 5 with κ = 0. 675. If one assumes a δ = 1. 175 then the algorithm allows all of the links to maintain their minimum SIR’s with the addition of the fifth user. In both the basic Foschini-Miljanic algorithm and the DPC/ALP of Bambos et al there is a single gain parameter (γ) associated both the transient properties and the equilib- rium properties (target SIR). A more general approach is to introduce some means to ‘decouple’ the transients from
