Scheduling variable rate links via a spectrum server

Scheduling Variable Rate Links via a Spectrum Server Chandrasekharan Raman, Roy D. Yates and Narayan B. Mandayam WINLAB, Dept of ECE Rutgers University Piscataway, NJ {chandru, ryates, narayan}@winlab.rutgers.edu Abstract— We consider a centralized Spectrum Server that coordinates the transmissions of a group of links sharing a common spectrum. Links employ on-off modulation with fixed transmit power when active. In the on state, a link obtains a data rate determined by the signal-to-interference ratio on the link. By knowing the link gains in the network, the spectrum server finds an optimal schedule that maximizes the average sum rate subject to a minimum average rate constraint for each link. Using a graph theoretic model for the network and a linear programming formulation, the resulting schedules are a collection of transmission modes (sets of active links) that are time shared in a fashion that is reminiscent of spatial reuse patterns in cellular networks. In the special case when there is no minimum rate constraint, the optimal schedule results in a fixed dominant mode with highest sum rate being operated all the time. In order to offset the inherent unfairness in the above solution, we introduce a minimum rate constraint and characterize the resulting loss in sum rate when compared to the case when there is no minimum rate constraint. We also investigate alternate fairness criteria by designing scheduling algorithms that achieve max-min fairness and proportional fairness. It is shown that the max-min fair rate allocation maximizes the minimum common rate among the links. Simulation results are presented and future work is described. I. I NTRODUCTION Since the earliest days of radio regulation, spectrum management has been driven by improvements in technology, from improved filters and frequency stability that allowed more channels to be created, to sophisticated logic and radio techniques that created the worldwide phenomenon of cellular. More recently, however, a new paradigm has emerged in which regulation has driven technology. A relatively small regulatory experiment in “open spectrum” that began in the ISM (Industrial Scientific, Medical) bands has spawned an impressive variety of important technologies and innovative uses, from cordless phones and wireless LANs to toll takers, meter readers and home entertainment products. This obvious success has further energized an already intense debate about regulatory strategy by introducing a new set of issues and beliefs, and while this debate displays intensely held regulatory and economic viewpoints, it inevitably turns on the old-fashioned fulcrum of technological capability as well. Ultimately, the capacity of the open access bands, and the quality of service they can offer, will depend on the degree to which radios can be designed to adapt to a wide variety of conditions. As a consequence, radios in future wireless systems are envisaged to be ‘smart’ and ‘interference aware.’ Such radios, often referred to as cognitive radios, are expected to have the ability to cooperate and dynamically share spectrum among several interfering radios. In addition to the degree of flexibility and adaptability of these radios, the need for global information regarding signals in space, time and frequency plays a prominent role in successful cooperation and coexistence. In this paper, we introduce the notion of a Spectrum Server which can serve as an information aid to enable coexistence of radios in a shared environment. Specifically, these radios could be made to cooperate by the centralized spectrum server which can determine neighborhood and interference information from measurements from the radios and enable efficient coordination. The spectrum server could then ‘advise’ a set of links, so that spectrum can be used efficiently. There are many different ways in which the spectrum server can coordinate a set of radios in a wireless network [1], [2]. In this work, we consider the problem of scheduling transmissions for a group of links which have a fixed transmission power, under the objective of maximizing the sum rate obtained by the links. We also address issues of fairness by deriving scheduling algorithms that result in max-min fair and proportional fair rate allocations. Max-min fair scheduling of rates have been studied extensively in the context of flow control of sources in a network [3]. Proportional fair scheduling has been studied in the context of multiuser diversity [4] and downlink scheduling for HDR [5]. But to the best of our knowledge, it has not been studied in the context of our framework. Scheduling transmissions in a wireless network has been studied in various contexts. In [6], a joint scheduling and power control strategy is proposed to maximize network throughput and energy efficiency of the system. Their algorithm selects candidate subsets of concurrently active links, and applies the distributed power control algorithm [7] to find the minimal power vector. Another direction in this problem is addressed in [8], [9], where the authors look at the crosslayer issues of routing, scheduling and power control. In [10], a centralized MAC protocol is proposed but the objective is to maximize a utility function. The authors in [11] introduce the concept of transmission modes and develop a framework for integrated link scheduling and power control policies to Fig. 1. Graph of network showing the nodes and directed links maximize the average network throughput, when each link is subject to an average power constraint and each node is subject to a peak power constraint. The authors assume a model in which the data rate of a link is a linear function of the signalto-interference ratio at the receiver. In contrast, we consider transmitters with a fixed power onoff modulation and devise schedules that maximize the system throughput. We assume that we obtain a non-zero rate in the links for any non-zero signal-to-interference ratio (SIR). The optimization problem, subject to minimum rate constraints in the individual links, is posed as a linear program. If the link gains are known to the spectrum server, it can schedule the transmissions among the links to maximize the system throughput. It is shown that when there is no minimum rate constraint, a fixed set of links (called the dominant mode) which maximizes the sum rate is operated all the time. In order to offset the inherent unfairness in the above solution, we introduce a minimum rate constraint and characterize the resulting loss in sum rate when compared to the case when there is no minimum rate constraint. We also investigate alternate fairness criteria by designing scheduling algorithms that achieve maxmin fairness and proportional fairness. We show that the maxmin fair rate allocation can be obtained in one step by solving a linear program which maximizes the minimum common rate among the links. The proportional fair schedule is obtained by solving a non-linear convex optimization program. The paper is organized as follows. In section II, we describe the system model. The problem formulation and analytical results are described in section III. We present the max-min fair schedule in section IV and the proportional fair schedule in section V. The simulation results are presented in section VI. We conclude in section VII with pointers to future work. Before we explain the system model, we comment on the notation of this paper. We use boldface lowercase characters Fig. 2. Graph of network showing transmission mode corresponding to (1 0 1 0) for vectors and boldface uppercase for matrices. If a is a P vector, aT denotes its transpose and aT b = i ai bi represents the inner product of the vectors a and b. The vector of all zeros and all ones are represented by 0 and 1 respectively. II. S YSTEM M ODEL Consider a wireless network with N nodes forming L logical links sharing a common spectrum. The network can be represented as a directed graph G(V, E), where the nodes in the network are represented by the set of vertices V of the graph and the links are represented by a set of directed edges E. Therefore the cardinalities |V| = N and |E| = L. A directed edge from a node m to node n implies that n wishes to communicate data to node m. We consider the scenario where the spectrum server coordinates the activity of the set of L links to share the spectrum efficiently. Define the set of transmission modes T = {0, 1, . . . , M − 1}, where M = 2L denotes the number of possible transmission modes. Then the mode activity vector ti of mode i is a binary vector, indicating the on-off activity of the links. If ti = (t1i , t2i , . . . , tLi ) is a mode activity vector, then  1, link l is active under transmission mode i, tli = 0, otherwise. (1) Note that there are M possible transmission modes including the mode in which all links are off. Figure 1 shows a representative network and Figure 2 shows particular transmission mode for the set of links. Let the transmitter power on a link l be Pl . If Glk is the link gain from the transmitter of link k to the receiver of link l and σl2 is the noise power at the receiver of link l, the SIR γli at the receiver of link l in transmission mode i is given by tli Gll Pl (2) γli = P 2. k∈E,k6=l tki Glk Pk + σl The link gain between a transmitter and receiver takes into account the path loss and attenuation due to shadow fading. We assume that the link gains between each transmitter and receiver are known to the spectrum server. The data rate in each link depends on the SIR in that link. We assume that the transmitter can vary its data rate, possibly through a combination of adaptive modulation and coding. In particular, for a given mode, the transmitter and receiver on a link employ the highest rate that permits reliable communication given the link SIR in that mode. For purposes of this study, we assume that the transmission of other links are treated as Gaussian noise and that a transmission on link l is reliable in a given mode i with a data rate cli = log(1 + γli ). (3) We emphasize here that we do not consider any minimum SIR threshold required at each receiver, i.e., associated with each transmission mode i, a non-zero γli defines some rate on the link l. Let xi be the fraction of time that transmission mode i is active and rl be the average data rate of link l. Each link has a minimum average data rate requirement rlmin . The average data rate in link l is the time average of the data rates of all the transmission modes that include link l. Thus, X cli xi , (4) rl = i Since C is a matrix with non-negative entries, the constraint 1T x = 1 can be replaced by the constraint 1T x ≤ 1 since the optimum x, say xopt , will satisfy 1T xopt = 1. Otherwise, we could scale xopt up so that the objective function is increased. We denote the optimal value 1T Cxopt as copt (0). A. No minimum rate constraint We now consider the special case when rmin = 0, i.e., when there is no minimum rate requirement for any of the links. Theorem 1: When rmin = 0, the solution to problem (7) is xopt = [0 0 . . . 1 . . . 0 0]T , where the position of 1 corresponds to the transmission mode with the maximum sum rate. The optimal objective value is the maximum column sum of the rate matrix C. Hence, the optimal strategy is to always operate the transmission mode with the maximum sum rate. Proof: The proof of the theorem is straightforward. Since rmin = 0, any x satisfying (7b) and (7c) is feasible, as (7a) is trivially satisfied. Since 1T C represents the row-vector of column sums of C, the objective function 1T Cx is some convex combination of column sums of the matrix C. Thus, 1T Cx = = or in vector form, r = Cx, (5) where C = [cli ] is an L×M matrix with non-negative entries, such that column i indicates the rate obtained by each link in mode i. ≤ L X M X cli xi l=1 i=1 L M X X xi i=1 M X III. M AXIMUM S UM R ATE S CHEDULING We are interested in maximizing the sum of the average data rates over all links l = 1, 2, . . . , L, subject to constraints on the minimum rate for each link. The optimization problem can be posed as the linear program (LP): max subject to 1T r (6) r = Cx, r ≥ rmin , (6a) (6b) 1T x = 1, (6c) x ≥ 0. (6d) P The objective function 1T r = i ri is the sum of average rates of the individual links. The constraint (6b) represents the minimum rate constraint and (6c) is the normalization for the schedule. The variables in the LP (6) are r and x. Rewriting the LP in terms of the variable x only, we get copt (rmin ) = max subject to 1T Cx (7) Cx ≥ rmin , 1T x ≤ 1, (7a) (7b) x ≥ 0. (7c) cli (9) l=1 xi max i i=1 = max i (8) L X cli L X cli (10) l=1 (11) l=1 P where the equality in (11) is true since i xi = 1. Equality holds in (10) when x = xopt = [0 0 . . . 1 . . . 0 0]T where PL the position of 1 in xopt is î = arg maxi l=1 cli . Hence the proof. Depending on the geometry of the links, the dominant transmission mode can be a single active link or a collection of geographically separated links. However, one implication of the above theorem is that the links that are not a part of the dominant transmission mode are starved. So, the system is not fair in terms of providing non-zero data rates to all the links. B. Non-zero minimum rate constraint In the case when rmin is non-zero, any x satisfying (7b) and (7c) may not be feasible. There is an additional constraint in (7a) which has to be met. Hence the optimal objective value cannot exceed copt (0). We now characterize the loss in sum rate due to the minimum rate constraint. We begin by writing the dual problem for the LP. The Lagrangian for the LP (7) is L(x, u, v) = 1T Cx + uT (Cx − rmin ) + v(1 − 1T x), (12) where u ∈ RL and v ∈ R are the dual variables. The Lagrange dual is g(u, v) = = = sup L(x, u, v) For all modes j ∈ / T̂ , the nonzero interference gains Glk and the monotonicity of the fraction P/(cP + σ 2 ) imply that (13) x≥0 T −u rmin + v + sup (1T C + uT C − v1T )x (14) x≥0  −uT rmin + v, 1T C + uT C − v1T ≤ 0 (15) ∞, otherwise γlj < γ̄lj = P −rTmin u + v subject to CT (1 + u) ≤ v1, u ≥ 0, v ≥ 0. (16a) (16b) = −rTmin u∗ + v ∗ ≥ −rTmin u∗ + copt (0). Since copt (0) − copt (rmin ) ≤ rTmin u∗ , the loss in sum rate is at most rTmin u∗ . An interpretation of the dual variable u∗ is that it can be viewed as the amount of rate loss for a unit increase in rTmin . This is analogous to the dual prices interpretation, in which the dual variables are interpreted as the price paid for using the limited resources (primal variables), the constraints of which are specified in the primal problem. C. Maximum sum rate schedule with high SNR links We can examine the special case of high SNR links when each link transmits with a large power P . Let us define a set of modes T̂ = {il : tlil = 1, tkil = 0 for all k 6= l} . In mode il , link l transmits in isolation and thus we call T̂ = {i1 , i2 , . . . , iL } the set of isolation modes. When the transmit power P is high, all links have high SNR and a link l achieves a high rate when transmitting in the isolation mode il . However, in a shared (non-isolation) mode j∈ / T̂ , links will have interference-limited SIRs and relatively low data rates. These observations lead to the following theorem. Theorem 2: If the interference gains Glk are all non-zero, then for sufficiently large transmit power P , the solution to (7) is time sharing among the transmission modes in T̂ . Proof: If P is the transmit power in all links l ∈ E, from (2) the SIR γlj of link l in transmission mode j is given by tlj Gll P 2. j∈E,k6=l tkj Glk P + σl γlj = P (17) . γlj < γ̄ = max max γ̄lj . j∈ / T̂ (18) l (19) It follows from (3) that clj ≤ c̄ = log(1 + γ̄), (16) By strong duality [12, Chapter 5], the optimal value of the dual problem in (16) is equal to copt (rmin ). Let (u∗ , v ∗ ) be the solution of (16). Since by Theorem 1, copt (0) is the maximum column sum of C and u ≥ 0, we have according to (16a), v ∗ ≥ copt (0). Therefore, the optimal value of (16) copt (rmin ) j∈E,k6=l tkj Glk We can thus upper bound the SIR γlj of any link l in any transmission mode j ∈ / T̂ as Thus the dual problem for the LP (7) is minimize Gll j∈ / T̂ . (20) Note that c̄ serves as an upper bound for the rate that can be obtained by any link l in a shared mode j ∈ / T̂ . However, in a mode il ∈ T̂ in which only link l is active, γlil = Gll P = γl (P ), σl2 (21) a monotone increasing function of P . Let us define cl (P ) = log(1 + γl (P )). (22) as the data rate obtained when link l transmits with power P in the isolation mode il . Since cl (P ) is a monotone increasing function of P , there exists a transmit power P ∗ , such that P > P ∗ implies cl (P ) > Lc̄ for all links l. Now, let us suppose that P > P ∗ , but x is an optimal schedule for problem (7) with xj > 0 for a shared mode j∈ / T̂ . Consider a new schedule x′ given by   i=j 0 x′i = xi + xj /L i ∈ T̂ (23)   xi otherwise The schedule x′ reallocates the time xj in mode j equally to the isolation modes il in T̂ . In particular, an isolation mode il ∈ T̂ will now be active for time xj (24) x′il = xil + . L We now show that every link l receives a positive rate increase by switching to schedule x′ . Under schedule x, a link l obtains rate X X cli xi . (25) cli xi = clj xj + clil xil + rl = i i∈{j,i / l} Under schedule x′ , link l obtains rate X X cli xi . cli x′i = clil x′il + rl′ = i (26) i∈{j,i / l} For link l, the difference in rates is rl′ − rl = clil (x′il − xil ) − clj xj (27)  c lil (28) − clj xj . = L However, P > P ∗ implies that in the isolation mode il , link l obtains rate clil = cl (P ) > Lc̄. (29) It follows that rl′ − rl > 0 for all links l. This contradicts the optimality of schedule x in that every link achieves a strictly higher rate under schedule x′ . IV. M AX - MIN FAIR RATE SCHEDULING The maximum sum rate scheduling is biased towards links that have the best quality (i.e., least interference) and is unfair to the other links that are not a part of the dominant transmission mode. To address this, we will consider two other fairness criteria in deriving scheduling strategies - max-min fair and proportional fair. In this section, we present the maxmin fair [3] schedule. Definition 1: A vector of rates r is said to be max-min fair if it is feasible and for each l ∈ E, rl cannot be increased while maintaining feasibility without decreasing rl′ for some link l′ for which rl′ ≤ rl . Formally, for any other feasible allocation r̃, with r̃l > rl , there must exist some l′ such that r̃l′ < rl′ ≤ rl . In the context of flow control of sources in a communication network, iterative algorithms for computing max-min fair rate vectors exist [3]. Such iterative algorithms use a ‘progressive filling’ technique that starts with all rates equal to zero and increases the rates until one or several link capacity limits are reached. In order to obtain the max-min fair schedule in our setting, we begin by formulating the LP to maximize the minimum common rate in all the links. We will then show that the solution to this LP results in the max-min fair solution. The LP which maximizes the minimum common rate among the links is r∗ = max subject to rmin r = Cx, (30) (30a) r ≥ rmin 1, 1T x = 1, (30b) (30c) x ≥ 0. (30d) Before proving that the above LP results in the max-min fair schedule, we state the following theorem: Theorem 3: If the link gains Glj are all non-zero, then the LP (30) which maximizes the minimum common rate among the links results in all links getting the same rate r∗ , i.e., r∗ = r∗ 1. The proof of Theorem 3 appears in the appendix. We now show that the schedule obtained by solving (30) is max-min fair. Theorem 4: The solution x∗ obtained by solving the LP (30) results in the max-min fair solution for the maximum sum rate problem (7). Proof: Our objective is to seek a max-min fair solution in the set (7a)-(7c). Denote the set by S. Let us consider the solution x∗ of (30) which results in the rate allocation r∗ = r∗ 1. Now, for any feasible x ∈ S, there can be only three different possibilities: 1) r such that the rates in all links rl ≤ r∗ , l ∈ E. 2) r such that for some links rl < r∗ and some links rl′ ≥ r∗ for l, l′ ∈ E. 3) r such that the rates in all links rl > r∗ , l ∈ E. The third possibility can be ruled out since it contradicts the optimality of (30). From Definition 1 of max-min fairness, it follows that r∗ 1 is the max-min fair rate vector when the first two possibilities hold. V. P ROPORTIONAL FAIR SCHEDULING The max-min fair schedule derived in the previous section leads to global fairness. In this section, we discuss a fairness criteria which leads to fairness of individual links. Definition 2: A vector of rates r is proportional fair if it is feasible, i.e., Cx = r for x such that 1T x = 1 and x ≥ 0, and if for any other feasible vector r′ , the aggregate of proportional change is negative. X r′ − ri i ≤ 0. (31) ri i In [13], Kelly proposed proportional fairness in the context of rate control for elastic traffic. It can be shown that the proportionally fair vector is the one that maximizes the sum of logarithms of the utility functions. Hence, to obtain the proportional fair rates, we solve the following non-linear optimization problem with linear constraints X max log rl (32) l subject to r = Cx, 1T x = 1, (32a) (32b) x ≥ 0. (32c) The objective function of the above non-linear optimization problem is increasing and strictly concave. The constraint set is linear and hence the problem is a convex optimization problem [12]. This implies that the problem has a unique global maximum over the constraint set. The solution for such problems can be found out by gradient search algorithms [12]. VI. S IMULATION RESULTS We present some simulation results in this section. Though the analytical results are true for more general cases, we present simulation results for some specific cases to illustrate our findings. The simulation set-up is a 50×50 grid. The links are of fixed lengths and placed at random locations in the grid. The interference gain Glj between the transmitter of link j and the receiver of a link l is given by Glj = d−4 lj , where dlj is the separation distance between the transmitter and receiver. The transmit powers are fixed for all transmissions and the link geometries are characterized through the signal-to-noise ratio (SNR) at the receiver for that link (in the absence of interference). In the case of maximum sum rate scheduling with no minimum rate constraint, the transmission mode with the highest sum rate is chosen. The links which are not a part of this transmission mode are not operated at all. In the special case when the noise at the receiver is high, the denominator in the SIR expression is dominated by the receiver noise. This approximates the case when there is no interference from the 50 50 Transmitter Receiver 45 Transmitter Receiver 45 4 40 40 35 35 3 30 30 5 25 2 3 25 20 1 20 2 15 4 15 1 10 10 5 5 0 0 5 10 15 20 25 30 35 40 45 50 0 5 0 5 10 Fig. 3. Set of five links each of length d = 10. The dominant transmission mode at SNR=10 dB is shown in solid lines. 15 Fig. 4. 20 25 30 35 40 50 A set of source-destination pairs Sum rate Link 2 Link 5 Link 1, 3, 4 12 10 Average Rates (bits/sec/Hz) neighboring links. Hence the best policy would be to turn on all the links in order to maximize the sum rate in all the links. As the SNR in each link increases, the interference from neighbors also increases. Then the best transmission mode is that which has the highest sum rate among all the other transmission modes. The set of links chosen follows spatial reuse patterns that are reminiscent of those used in cellular networks. Figure 3 shows a set of links and the dominant transmission mode at SNR = 10 dB. The links in the dominant mode are shown in solid lines. In the case of maximum sum rate scheduling with nonzero minimum rate constraint, we see that more than one transmission mode is operated since there is a minimum rate requirement for each link. The best mode is selected for most of the time and the mode which includes the poorer quality links are turned on for a fraction of time just enough to satisfy their minimum rate requirement. Most of the transmission modes are thus not used at all since it is best to use the dominant mode during all other times except when a minimum rate should be guaranteed to some poor quality links. We now discuss an illustrative example of this case. Figure 4 shows a set of source-destination pairs. When rmin is zero at 20 dB received SNR, the mode consisting of links {2, 5} is always operated. But as the common minimum rate rmin increases from zero, an additional set of modes is operated to satisfy the minimum rate requirement for each link. The schedules of the individual transmission modes as shown in Figure 6 varies with the minimum rate so that the minimum rate constraint in each of links is maintained. Notice that only five different modes are active. When rmin is increased in steps, we observe that the same set of modes is operated. After a certain rmin value, say r′ , a different set of modes has to be operated in order to obtain a feasible schedule. Until then the rate of all the modes falls linearly with increase in rmin . The break point in the sum rate curve occurs at r′ . We zoom in to the graph for rmin values ranging from 1.4 to 1.58 45 8 6 Max min rate 4 2 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 r min Fig. 5. Variation of sum rates and individual rate as a function of rmin in Figure 7. We observe that mode {1} and mode {3} are active for equal amounts of time since link 1 and 3 transmit in isolation in this mode, and they require the same minimum rate to transmit. The fraction of time mode {4} is active increases as the fraction of time mode {2, 4} transmits decreases to compensate for the increase in rate in link 4. Finally, the rates in all the links are same at r∗ ≈ 1.58. The rates corresponding to maximum of the minimum common rate r∗ is shown in the Figure 5. All the links end up getting the same rates in this case. The schedule and the rate allocation vector the same as those obtained when we solve (7) with rmin = r∗ 1. The comparison of scheduling schemes under different optimization settings is shown in Figure 8. From the Figure, we notice that only in the case of maximum sum rate with no minimum rate constraint, there exist links with zero obtained rate. In the case of the max-min fair solution, all the links end up getting the same rate. Table I shows a numerical comparison of the sum rate and individual 6 dominant mode {2,5} mode {2,4} mode {1} mode {3} mode {4} 0.8 5 max sum rate, r =0 min max sum rate, r =1 min Rates in bits/sec/Hz Fraction of time the mode is active 1 0.6 0.4 proportional fair rate max min rate 4 3 2 0.2 1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 r 0 0 Fig. 6. Schedule of different transmission modes at various rmin values when maximizing sum rate of links dominant mode {2,5} mode {2,4} mode {1} mode {3} mode {4} Fraction of time the mode is active 1 0.8 0.6 0.4 0.2 0 1.4 1.42 1.44 1.46 1.48 1.5 1.52 1.54 1.56 1.58 1.6 r min Fig. 7. Schedule of different modes - zoomed in for higher rmin values link rates among the various links in the network. VII. D ISCUSSION AND 1 2 3 4 5 6 Link indices min C ONCLUSION We introduced the notion of a Spectrum Server, which allocates a schedule for a set of links in a wireless network, which is modelled as a directed graph. We observe that this problem formulation can also be applied to the case where the links operate in non-overlapping frequency ranges. In this case, some of the link gains Glj may be zero. The model can be easily extended to the case when there are bidirectional links between two nodes (if we assume there are separate transmitters and receivers), in which case the number of transmission modes will be 22L . In this case, interference at the receiver which is colocated with the transmitter in another link is very high. This may result in schedules in which one of the bidirectional links are active at any given time. If there is a restriction that only one of the bidirectional links can be active Fig. 8. Comparison between rates of the links under different settings at any given time (or when each node has a single transceiver). In this case, the number of transmission modes will be only 3L . In our work, the problem of maximizing the sum rate in all the links subject to minimum rate constraint is posed as a linear program. The solution to the linear program gives the optimum schedule for the transmission modes in the network. But for a network involving L links, there is an exponential number of transmission modes and thus the LP we solve involves an exponential number of variables. However, we conjecture that almost always there are only very few active transmission modes as corroborated by our simulation results. In the present work, the whole process is centralized since the spectrum server just solves the linear program to compute the schedules for each of the links. A heuristic algorithm to find the best schedule with less measurement overhead would be an interesting future work to address. This may involve links reporting the interference seen by them from all the other links. This would also be a first step to finding a completely distributed scheduling algorithm. The centralized approach proposed in this paper would then serve as an upper bound to the performance of such distributed algorithms. Throughout this paper, we have assumed that the spectrum server has knowledge of the link gains. This involves measurement of link gains by the spectrum server. An interesting issue is how coarse the measurement can be and how it affects the scheduling algorithm. If the link gains are modelled by a time-varying fading process, then finer measurements would be very expensive. The problem formulation in this work yields itself to many optimization problems. One such example is to minimize the sum fraction of times the links are on so that the total transmit power in all the links is minimized. Another example is to maximize the sum rate in all the links subject to the condition that all the links are active for equal amount of time. In this work, we assume that there is a single hop com- TABLE I C OMPARISON OF SUM RATE AND INDIVIDUAL RATES Sum rate Link 1 Link 2 Link 3 Link 4 Link 5 rmin = 0 rmin = 0.5 rmin = 1 10.8987 0 5.3814 0 0 5.5173 10.2035 0.5 4.5640 0.5 0.5 4.1394 9.5082 1.0 3.7466 1.0 1.0 1.0 munication between the source destination pairs. Yet another interesting issue would be to find the solution for the maximum sum rate problem if we have a schedule with multiple hops between the source and destination. While we have primarily considered links of equal length for the purposes of numerical illustration, it is of interest to study the performance of the various scheduling algorithms for the case of links of unequal lengths as well. VIII. ACKNOWLEDGEMENT This work is supported in part by the NSF under grant number NeTS-0434854 and by the Defense Spectrum Office (DSO) of the Defense Information Systems Agency. The first author thanks Jasvinder Singh for useful discussions on the material in this paper. A PPENDIX Proof of Theorem 3: The LP which maximizes the minimum common rate is max subject to rmin (33) rmin = 1.58 (Max-min fair solution) 7.9 1.58 1.58 1.58 1.58 1.58 Proportional fair solution 7.0632 0.8324 1.9581 1.3314 1.9516 0.9896 sets T1∗ , T2∗ and T3∗ such that T1∗ T2∗ T3∗ = = = {i ∈ T ∗ : til = 0, for all l ∈ L2 }, ∗ {i ∈ T : til = 0, for all l ∈ L1 }, T ∗ \{T1∗ ∪ T2∗ }. (34) (35) T1∗ and T2∗ contain active transmission modes which consist of links only from L1 and L2 respectively, and T3∗ contains transmission modes which consist of links in both L1 and L2 . We consider two cases below. A. Case (i): T1∗ is non-empty There exists an active transmission mode i ∈ T1∗ consisting of links only from L1 . Consider the mode i′ with activity vector ti′ given by  1, for all l ∈ L2 , tli′ = (36) 0, otherwise. Mode i′ consists of all links from L2 . Therefore, cli′ > 0 for l ∈ L2 . In the optimal schedule x∗ , we know that x∗i > 0 ∗ but x∗iP is ′ may be zero. The rate in link l under schedule x ∗ rl = k clk xk . Define for some fixed ǫ1 > 0, the feasible schedule r = Cx, (33a) r ≥ rmin 1, 1T x = 1, (33b) (33c) x̂ = [x∗1 . . . x∗i − ǫ1 . . . x∗i′ + ǫ1 . . . x∗M−1 ]T . x ≥ 0. (33d) For sufficiently small ǫ1 , the schedule x̂ will be feasible. Now, for l ∈ L2 , the rate r̂l due to schedule x̂ is X r̂l = clk x̂k = r∗ − cli ǫ1 + cli′ ǫ1 (37) Let r∗ be the optimal value of (33), corresponding to a schedule x∗ and a set of active transmission modes T ∗ = {i ∈ T : x∗i > 0}. Note that the idle transmission mode with the all zero activity vector would never be a part of T ∗ because, if it were, we can improve the rates of links in L2 and this contradicts that r∗ is the optimal solution of (33). It is required to prove that at optima, the rate vector Cx = r∗ 1. We assume the contrary that the solution to (33) leads to unequal rates over the set of L links. We can then partition the sets of links E into two disjoint non-empty sets L1 = {l ∈ E : rl > r∗ } and L2 = {l ∈ E : rl = r∗ }. This in turn induces a partition on the set T ∗ of all active transmission modes for the optimal solution into three disjoint k Since cli′ > 0 for l ∈ L2 , r̂l = r∗ + cli′ ǫ1 . (38) Thus, we conclude that r̂l > r∗ , l ∈ L2 . Note that ǫ1 needs to be chosen such that for all l ∈ L1 , r̂l > r∗ . The choice of ǫ1 such that cli ǫ1 < minl∈L1 rl − r∗ ensures that rˆl > r∗ for l ∈ L1 . We can thus improve the rates in all links in L2 . This contradicts the optimality of r∗ . We denote this step as Increase(1). B. Case (ii): T1∗ is empty In this case, if T3∗ is empty, then T ∗ = T2∗ and hence all rates are equal, and the proof is complete. Thus we consider only the case of T3∗ being non-empty. For an active mode j ∈ T3∗ , there exist non-empty subsets of L1 and L2 , namely M1 and M2 such that the activity vector tj is given by   1, l ∈ M1 ⊆ L1 , 1, l ∈ M2 ⊆ L2 , tlj = (39)  0, otherwise. Consider the mode j ′ for which the activity vector tj ′ is given by  1, if l ∈ M2 , tlj ′ = (40) 0, otherwise. We have assumed that all link gains Glj are non-zero, that is there is lesser interference for links in M2 in mode j ′ than in mode j due to a lesser number of active links in mode j ′ . Thus for links l ∈ M2 , clj = < Gll Pl 2 k∈M1 ∪M2 ,k6=l tkj Glk Pk + σl Gll Pl P ′ 2 = clj . k∈M2 ,k6=l tkj ′ Glk Pk + σl P (41) Since j ∈ T3∗ , x∗j > 0. For some ǫ2 > 0, we define a feasible schedule x̂ = [x∗1 . . . x∗j − ǫ2 . . . x∗j ′ + ǫ2 . . . x∗M−1 ]T . Under schedule x∗ and x̂, link l obtains rate X rl = clk x∗k k and r̂l = X clk x̂k k respectively. Thus, the difference r̂l − rl = (x̂j − x∗j ′ )clj ′ + (x̂j − x∗j )clj = ǫ2 (clj ′ − clj ). 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