# Power Control is Not Required for Wireless Networks in the Linear Regime

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7 We can similarly define the set of average end-to-end rates, link rates and power allocations T = T (P) that is achievable with power allocations belonging to P, as the set of all (f, x, p) that are achievable using power allocation P. Thus, sets T and T (P E ) represent the sets of all possible average end-to-end rates, link rates and power consumptions with an arbitrary and with 0 P MAX power control, respectively. When we consider rate maximization under constraints on average consumed power, we are interested only in the set of feasible rates. If the average consumed power is limited by P MAX, then the set of feasible rates is F = {f (f, x, p) T, p P MAX }. Similarly, with 0 P MAX power control, the set of feasible rate is F E = {f (f, x, p) T (P E ), p P MAX }. For notational conveience, we analogly define X = { x (f, x, p) T, p P MAX } and X E = { x (f, x, p) T (P E ), p P MAX }. Similarly, when considering power minimization, we focus on the set of feasible average consumed powers. If the average end-to-end flow rate is lower-bounded by F MIN, then the set of feasible average consumed powers, under arbitrary power control, is P = { p (f, x, p) T, f F MIN }. Similarly, with 0 P MAX power control, the set of feasible rate is P E = { p (f, x, p) T (P E ), f F MIN }. Performance Objectives: Finally, we formally define notion of Pareto efficiency that was introduced in Section 2.5. Rate vector f F is Pareto efficient on F if there exist no other vector f F such that for all i, f i f i and for some j, f j > f j. Average power dissipation vector p P is Pareto efficient on P if there exists no other vector p P such that for all i, p i p i and for some j, p j < p j. 4 Main Findings 4.1 Rate Maximization In this section we show that any rate allocation that is feasible with an arbitrary power control and under some average power constraint, is also achievable with 0 P MAX power control. Moreover, if we consider a scenario without average power constraints, then 0 P MAX is the only optimal power control. We clearly have F E F, and we want to show that every feasible flow rate allocation can be achieved by a set of extreme power allocation from P E, that is F F E. In other words that every feasible flow rate allocation can be achieved only with an appropriate scheduling, and without power control. Theorem 1 If the rate is a linear function of the SINR, then for arbitrary values of parameters of constraint set (1), we have that F E = F. (Proof in appendix) The theorem says that every feasible rate allocation, thus including the Pareto efficient ones, can be achieved with 0 P MAX power control, and with an appropriate scheduling. Hence 0 P MAX is at least as good as any other power control, and power adaptation is not needed. To interpret this finding, consider a UWB MAC protocol presented in [5] where both power adaptation and scheduling is used. Any rate achieved by this MAC protocol could be achieved with another protocol that would not adapt power and would use an appropriate scheduling. Comment. One could be tempted to interpret Theorem 1 by saying that in the case of a linear rate function, adapting power can be directly mapped into scheduling time shares. This is not correct. Indeed, in the linear regime, the rate of a link is a linear function of the SINR, but it is not a linear function of the transmission powers of interfering nodes. Therefore, it is not obvious that the gain from power adaptation can be completely achieved by making linear combinations through scheduling. The intuitive explanation lies in the fact that, since the SINR is non-linear in interference, when increasing the transmitted power of a node, this has a more important effect on increasing the signal than on increasing interference (which yields a strictly positive second derivative, as shown in the proof). We conjecture that there are non-linear rate functions that yield the same conclusion. We next consider a scenario where there are no constraints on average consumed power (or equivalently P MAX P MAX ), and we prove that power adaptation is strictly suboptimal. In other words, if any node at any time uses less than maximum power for a transmission, then there exists an alternative schedule with 0 P MAX power assignment which yield higher rate for at least one flow, and higher or equal rates for other flows. The finding is more precisely formulated in the following theorem: Theorem 2 Consider an arbitrary network where the rate is a linear function of the SINR, and an arbitrary schedule α and a set of power allocations p n for that network. If for some n, α n > 0 and power allocation p n P E then the resulting average rate allocation f is not Pareto efficient on F. (Proof in appendix) The theorem says that a Pareto efficient allocation cannot be achieved if in any time slot a power allocation different from 0 P MAX is used. Applying the finding on the framework of [5] we can conclude that there exists a different schedule that does not use power control and that improves the performance of a network. The main use of Theorem 1 and Theorem 2 is the following corollary: 7

9 S1 D1 R 1 MAX (0,P ) MAX MAX (P,P ) P 1 d D2 S2 MAX (P,0) l R 2 P 2 Figure 1. A simple example of a network with 2 links. The topology of the network is given on the left. Node S 1 sends to node D 1 while node S 2 sends to D 2. The feasible rate set for this network is given in the middle. The lighter region in dashed lines represents the set of feasible rates that can be achieved without scheduling, only with power adaptation. The lighter region in full lines represents an increase that is achieved by scheduling and without power adaptation (0 P MAX power control). The darker region in dashed lines is the same example without scheduling and with power control, but this time with additional average power constraints. Again the darker region in full lines represents an improvement introduced by scheduling. We see that the second protocol cannot achieve Pareto efficient rates of the feasible rate set, except for the three rate allocations. But these three rate allocations are achieved with power allocations (0, P MAX ), (P MAX, 0) and (P MAX, P MAX ) which belong to 0 P MAX power strategy. In the figure on the left, the feasible set of average consumed power under minimum rate constraints is depicted in gray. The region in full lines represents average power consumption achievable with scheduling and without power adaptation, and the region in dashed lines represents average power consumptions achievable without scheduling and with power adaptation. All average powers belonging to this set can be achieved without power adaptation. It remains as a future work to further investigate the trade-off between scheduling and power adaptation by incorporating costs of different power control and scheduling protocols. 6 Appendix 6.1 Proof of Theorem 1 The outline of the proofs is as follows: we first take an arbitrary linear objective function of the form U( x, p) = i µ i x i i λ i p i, we maximize it on T, and we show that the maximization point has to be generated with powers from P E. This is showed in Lemma 1. Next, we use the property of convex sets saying that for every Paretto efficien point there exists a plane touching the set in that point. Since this Pareto efficient point is also the maximizer of a linear objective function that corresponds to the plane, it follows that it has to be generated with 0 P MAX. We start by proving some properties of the optimal point with respect to a linear objective function: Lemma 1 Consider a function U( x, p) = i µ i x i i λ i p i for some arbitrary vectors µ, λ. Then: 1. There is a unique maximum U = U( x, p ) on set T, 2. The maximum (f, x, p ) T E, 3. If some i, µ i > 0 and for all j, λ j = 0, then for arbitrary α and {p n (s)} 1,,N, such that for some n, α n > 0 and 0 < p n i < Pi MAX, and the resulting ( x, p) we have U( x, p) < U( x, p ). Proof of 1): Both function U( x, p) and set T are convex, hence the maximum is attained in some (f, x, p ) T. We also know there exist α, {p n (s)} 1,,N that satisfy (1). Proof of 2): Let us first assume there is a single system state S = {s} hence there is no randomness in the system. We use an approach similar to [4, 5]. Without loss of generality, we fix all α(s), {p n (s)} 1,,N to arbitrary values, except p 1 1 (s), and we consider a function p 1 1 (s) V i µ i x i i λ i p i as a function of a single free variable p 1 1 (s). We then have the following derivatives: V p 1 1 (s) = 2 V = ( p 1 1 (s))2 µ 1 α 1 (s) h 11 (s) η 1 (s) + β k 1 pn k (s)h k1(s) λ 1α 1 (s) (2) µ i α i (s) p 1 i (s)h ii(s)h 1i (s) ( i=2 I η i (s) + β ) 2 (3), k i p1 k (s)h ki(s) i=2 I µ i α i (s) 2p 1 i (s)h ii(s)h 2 1i ( (s) η i (s) + β ) 3 (4). k i p1 k (s)h ki(s) We first suppose that for all i, µ i > 0. It is easy to see from (4) that regardless of the values of other variables, the second derivative is always positive, V (p 1 1 (s)) is always convex, hence the maximum is attained for p 1 1(s) {0, P MAX }. Next we suppose, without loss of generality, that for some m we have µ 1 0,, µ m 0. Then clearly the optimal is to have x 1 = 0,, x m = 0. Then by setting µ 1 = 0,, µ m = 0, the new optimization problem has the same maximum as the old one, and we again have that the optimal values belong to {p n (s)} 1,,N P E, and x X E. At this point we proved the second claim under assumption that there is no randomness in the system. We next relax this assumption. From the above we know that for every state s S there is a power allocation from P E that maximizes the utility. Since averaging over S is a linear op- 9

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