Opportunistic Scheduling in Heterogeneous Networks: Distributed Algorithms and System Capacity

Transcription

1 Opportunistic Scheduling in Heterogeneous Networks: Distributed Algorithms and System Capacity arxiv:202.09v2 [cs.it] 5 Feb 202 Dor-Joseph ampeas, Asaf Cohen and Omer Gurewitz Department of Communication Systems Engineering Ben-Gurion University of the Negev Beer-Sheva, 8405, Israel {kampeas,coasaf,gurewitz}@bgu.ac.il February 26, 209 Abstract In this work, we design and analyze novel distributed scheduling algorithms for multi-user MIMO systems. In particular, we consider algorithms which do not require sending channel state information to a central processing unit, nor do they require communication between the users themselves, yet, we prove their performance closely approximates that of a centrally-controlled system, which is able to schedule the strongest user in each time-slot. Possible application include, but are not limited to, modern 4G networks such as 3GPP LTE, or random access protocols. The analysis is based on a novel application of the Point-Process approximation, enabling the examination of non-homogeneous cases, such as non-identically distributed users, or handling various QoS considerations, which to date had been open. Introduction Consider the problem of scheduling users in a multi-user MIMO system. For several decades, at the heart of such systems stood a basic division principle: either through TDMA, FDMA or more complex schemes, users did not use the medium jointly, but rather used some scheduling mechanism to ensure only a single user is active at any given time. Numerous medium access (MAC) schemes at the data link layer also, in a sense, fall under this category. Modern multi-user schemes, such as practical multiple access channel codes or dirty paper coding (DPC) for Gaussian broadcast channels [], do allow concurrent use of a shared medium, yet, to date, are complex to implement in their full generality. As a result, even modern 4G networks consider scheduling groups of users, each of which employing a complex multi-user code [2, 3].

2 Hence, scheduled designs, in which only a single user or a group of users utilize the medium at any given time, are favorable in numerous practical situations. In these cases, the goal is to design an efficient schedule protocol, and compute the resulting system capacity. In this work, we derive the capacity of multi-users MIMO systems under distributed scheduling algorithms and diverse user distributions.. Related Work Various suggested protocols in the current literature follow the pioneering work of [4]. In these systems, at the beginning of a time-slot, a user computes the key parameters relevant for that time-slot. For example, the channel matrix H (Figure (a)). It then sends these parameters to a central processing unit, which decides which user to schedule for that timeslot. This enables the central unit to optimize some criterion, e.g., the number of bits transmitted in each slot, by scheduling the user with the best channel matrix. This is the essence of multi-user diversity. In [5], the authors adopted a zero-forcing beamforming strategy, where users are selected to reduce the mutual interference. The scheme was shown to asymptotically achieve the performance of DPC, hence is asymptotically optimal. Another asymptotically optimal scheme is given in [6] for a large number of users and antennas. In [7], the authors proposed a scheduling algorithm which selects the user with the maximal λ, where λ denotes the minimum eigenvalue of the Wishart matrix HH. [8] proposed a scheduling scheme that transmit only to a small subset of heterogeneous users with favorable channel characteristics. This provided near-optimal performance when the total number of users to choose from was large. Scaling laws for the sum-rate capacity comparing maximal user scheduling, DPC and BF were given in [9]. Additional surveys can be found in [0, ]. Subsequently, [2] analyzed the scaling laws of maximal base station scheduling via Extreme Value Theory (EVT), and showed that by scheduling the station with the strongest channel among stations (Figure (b)), one can gain a factor of O( 2 log ) in the expected capacity compared to random or Round-Robin scheduling. Extreme value theory and order statistics are indeed the key methods in analyzing the capacity of such scheduled systems. In [3], the authors suggested a subcarrier assignment algorithm (in OFDM-based systems), and used order statistics to derive an expression for the resulting link outage probability. Order statistics is required, as one wishes to get a handle on the distribution of the selected users, rather than the a-priori distribution. In [4], the authors used EVT to derive throughput and scaling laws for scheduling systems using beamforming and various linear combining techniques. [5] discussed various user selection methods in several MIMO detection schemes. The paper further strengthened the fact that appropriate user selection is essential, and in several cases can even achieve optimality with sub-optimal detectors. Additional user-selection works can be found in [6, 7, 8, 9]. In [20, 2], the authors suggested a decentralized MAC protocol for OFDMA channels, where each user estimates his channels gain and compares it to a threshold. The optimal threshold is achieved when only one user exceeds the threshold on average. This distributed scheme achieves /e of the capacity which could be achieved by scheduling the strongest user. The loss is due to the channel contention inherent in the ALOHA protocol. [22] 2

3 extended the distributed threshold scheme for multi-channel setup, where each user competes on m channels. In [23] the authors used a similar approach for power allocation in the multi-channel setup, and suggested an algorithm that asymptotically achieves the optimal water filling solution. To reduce the channel contention, [20, 24] introduced a splitting algorithm which resolves collision by allocating several mini-slots devoted to find the best user. Assuming all users are equipped with collision detection (CD) mechanism, the authors also analyzed the suggested protocol for users that are not fully backlogged, where the packets randomly arrive at with total arrival rate λ and for channels with memory. [25] used a similar splitting approach to exploit idle channels in a multichannel setup, and showed improvement of 63% compared to the original scheme in [20]..2 Main Contribution In this work, we suggest a novel technique, based on the Point of Process approximation, to analyze the expected capacity of scheduled multi-user MIMO systems. We first briefly show how this approximation allows us to derive recent results described above. However, the strength of this approximation is in facilitating the asymptotic (in the number of users) analysis of the capacity of such systems in different non-uniform scenarios, where users are either inherently non-uniform or a forced to act this way due to Quality of Service constrains. We compute the asymptotic capacity for non-uniform users, when users have un-equal shares or when fairness considerations are added. To date, these scenarios did not yield to rigorous analysis. Furthermore, we suggest a novel distributed algorithm, which achieves the maximal multiuser diversity without centralized processing or communication among the users, and without any collision detection mechanism. The rest of this paper in organized as follows. In Section 2, we describe the system model and related results. In Section 3, we describe the Point of Process technique and briefly show how it is utilized. In Section 4 we analyse the non-uniform scenario. In Section 5 we describe the distributed algorithm and analyze its performance. Section 6 concludes the paper. 2 Preliminaries We will deal with the following model: y = Hx + n where y C r is the received vector and r is the number of receiving antennas. x C t is the transmitted vector constrained in its total power to P, i.e., E[x x] P, where t is the number of transmitting antennas. H C r t is a complex random Gaussian channel matrix such that all the entries are random i.i.d. complex Gaussian with independent imaginary and real parts, zero mean and variance /2 each. n C r is uncorrelated complex Gaussian noise with independent real and imaginary parts, zero mean and variance. In MIMO uplink model, we assume that the channel H is known at the transmitters, and they send their 3

4 User Base Station H rxt User User Base Station H rxt Base Station H rxt User Base Station H rxt User Base Station H rxt User (a) (b) Figure : (a) Multi-user MIMO. (b) MU-MIMO stations. channel statistics to the receiver. I.e., the channel output at the receiver consist of the pair (y, H). Then, the receiver lets to the transmitter with the strongest channel to transmit in the next slot. In MIMO downlink model, we assume that the channel H is known at the receivers, and they send their channel statistics to the transmitter, so he can choose the receiver that will benefit most from his transmission. Moreover, we assume that the channel is memoryless, such that for each channel use, an independent realization of H is drawn. Through this paper, we use bold face notation for random variables. 2. MIMO Capacity Distribution [26, 27] and [28] show that when the elements of the channel gain matrix, H, are i.i.d. zero mean with finite moments up to order 4 + δ, for some δ > 0 then the distribution of the capacity follows the Gaussian distribution by the CLT, with mean that grows linearly with min(r, t), and variance which mainly influenced by the power constraint P. With the observation that the channel capacity follows the Gaussian distribution, we would like to investigate the extreme distribution that the capacity distribution follows, and retrieve the capacity gain when letting a user that holds a channel with maximum capacity among all other users channels, utilize a time slot. 2.2 Extreme Value Analysis for the Maximal Value In this section we provide Extreme Value Theorem (EVT), shown in [29],[30] and [3], that later be used for asymptotic capacity gain analysis. In implementing this model for user capacities dataset, the choice of block size can be critical. The choice amounts to a trade-off between bias and variance. Blocks that are too small mean that the approximation by the 4

5 limit distribution is likely to be poor, leading to bias in estimation and extrapolation. On the other hand, large blocks generate few block maxima, leading to large estimation variance. Theorem ([32, 29, 3]). (i) Suppose that x,..x n is a sequence of i.i.d random variables with distribution function F (x), and let M n = max(x,..., x n ). If there exist a sequence of normalizing constants a n > 0 and b n such that as n, Pr(M n a n x + b n ) i.d. G(x) () for some non-degenerate distribution G, then G is of the generalized extreme value distribution type G(x) = exp { ( + ξx) /ξ} (2) and we say that F (x) is in the domain of attraction of G, where ξ is the shape parameter, determined by the ancestor distribution F (x) with the following relation. (ii) Let h be the following reciprocal hazard function h(x) = F (x) f(x) for x F x x F. (3) where x F = inf{x : F (x) > 0} and x F = sup{x : F (x) < } are the lower and upper endpoints of the ancestor distribution respectively. Then shape ξ is obtained as the following limit, d x xf h(x) ξ. (4) dx (iii) If {x n } is an i.i.d. standard normal sequence of random variables, then the asymptotic distribution of M n = max(x,...x n ) is of Gumbel distribution, Specifically, Pr(M n a n x + b n ) e e x where and a n = (2 log n) /2 (5) b n = (2 log n) /2 2 (2 log n) /2 (log log n + log 4π). (6) 5

6 f c MIMO Capacity Distribution C Figure 2: MIMO capacity distribution for m=32 transmitting antennas and n=28 receiving antennas vs Gaussian Distribution with µ = 2 and σ = 0.03 (red line). For completeness, a sketch of the proof is given in Appendix A. Similarly, if {x n } follows Gaussian distribution with mean µ and variance σ 2, then the above theorem results in and [ b n = σ (2 log n) 2 2 It follows that for a Gaussian distribution, a n = σ(2 log n) 2 (7) ] (2 log n) 2 [log log n + log(4π)] + µ. (8) a n = σ(2 log n) 2 0, which implies that M n b n σ(2 log n) 2 + µ. (9) 2.3 Multi-User Diversity Assuming MIMO uplink model, i.e., perfect CSI of users at the receiver, then the expected capacity that we achieve by choosing the maximal user in each time slot will follow the 6

7 50 f c Block Max Capacity Distribution C Figure 3: Maximal capacity distribution, when choosing the maximal capacity among 500 capacities that following the Gaussian distribution simulated in Figure 2, with µ = 2 and σ = The red line is the corresponding Gumbel distribution plotted in range [µ + 2σ, µ + 5σ]. expected value of Gumbel distribution with parameters a, b [2], i.e., E[M ] (a) = b + a γ (0) (b) = σ(2 log ) 2 (2 log ) 2 [log log log(4π)] + µ + γσ(2 log ) 2 where γ is Euler-Mascheroni constant, (a) follows from Gumbel distribution, (b) follows from (7) and (8). Hence, for large enough, ( ) E[M ] = σ(2 log ) 2 + µ + O log 3 Distributed Algorithm A major drawback of the previous method is that a base station must receive a perfect CSI from all users in order to decide which user is adequate to utilize the next time slot, which may not be feasible for a large number of users. Moreover, the delay caused by transmitting CSI to the base station would limit the performance. 7

8 In this section, we begin our discussion from a distributed algorithm, shown in [22], in which stations do not send their channel statistics to the base station, yet the performance is asymptotically equal to that of the previous section. We provide an alternative analysis to this algorithm, that will serve us later in this paper. The algorithm goes as follows. Given the number of users, we set high capacity threshold such that only a small fraction of the users will exceed it. In each slot, the users estimate their own capacity. If a user capacity is greater than the capacity threshold, he utilizes the slot. Otherwise, the user keeps silent in that slot. The base station can successfully receive the transmission if no collision occurs. Let C av denote the expected capacity. Thus, for sufficiently large we obtain the following. Proposition. The expected capacity when working in single user in each slot follows C av = [ ke k] ( u k/ + a + o(a /k ) ) () where u k/ is a threshold such that k out of users will exceed it on average, and a is normalizing constant following (5). Due to the distributed nature of the algorithm, we expecting collisions to occur. Thus, we express the expected capacity C av of this scheme as where and C av = ( Pr(unutilized slot)) E[C C > u k/ ] ( Pr(unutilized slot)) = ke k (2) E[C C > u k/ ] = u k/ + a + o(a /k ) (3) Hence, we need to analyze the expected capacity gained when letting a user with above threshold capacity to utilize a slot, and the probability that a single user utilizes the slot. We choose to prove the above through point of process method [3, 33]. With the point of process we can model and analyze the occurrence of large capacities, which can be represented as point process, when considering the users index along with the capacity value. Later in this paper, this method will allow us to analyze the non-uniform case as well. The following subsections are provided in order to prove Proposition. 3. Threshold Arrival Rate Assume that x,..., x n is a sequence of i.i.d random variables with distribution function F (x), such that F (x) is in the domain of attraction of some GEV distribution G, with normalizing constants a n and b n. We construct a sequence of point processes P, P 2,... on [0, ] R by P n = {( i n, x i b n a n 8 ), i =, 2,..., n }

9 P n t Point Process for Gaussian Distribution 0 B t Figure 4: Point process for Gaussian distribution with = 000 users, in which all samples are normalized with a n, b n constants, with floor threshold b l = 4. As we can see, only a small fraction of small users are above this threshold. In particular, we obtain the expected number of arrivals to set B by using (4). and examine the limit process, as n. Notice that large points of the process are retained in the limit process, whereas all points x i = o(b n ) can be normalized to same floor value b l. Theorem 2 ([34, 33, 3]). Consider P n on the set [0, ] (b l + ɛ, ), where ɛ > 0, then P n P as n where P is a non-homogeneous Poisson process with intensity λ(t, x) = ( + ξx) ξ + where x is the sample value, and t is the index of occurrence. For completeness, a proof in Appendix C. Let Λ(B) be the expected number of points in the set B, which can be obtained by integrating the intensity, Λ(B) = λ(b)db. (4) Since we are interested in sets of the form b B B v = [0, ] (v, ) 9

10 where v > b l, for which Λ(B v ) = Λ([0, ] (v, )) = = = t=0 t=0 t=0 [ x=v = ( + ξv) /ξ + λ(t, x)dxdt ( + ξx) /ξ + ( + ξv) /ξ + dt where a + denotes max{0, a}. Since the exceedance of large capacities can be modeled by Poisson process, that is, capacities exceeds high threshold u continuously and independently at a constant average rate Λ(B u ), we have the following. Corollary. The length of the inter-arrival time, which is the distance between the indexes of two successive arrivals, follows the exponential distribution with mean Λ(B u ). Corollary 2. Given k exceedances {(t j, x j )} k j= with x j > u, j, over time period (0, T ), then the approximating Poisson process function applied to the above exceedances of u is ] x=v Pr (N T = k) = r λ(t i, x i ) exp{ T Λ([0, ] (u, ))}. i= 3.2 Tail Distribution Focusing on points of the process P n that are above a threshold, we wish to examine their conditional distribution given that they exceeded high threshold. For any fixed v > b l let u n (v) = a n v + b n, and let x > 0, then ( xi b n Pr (x i > a n x + u n (v) x i > u n (v)) = Pr > x + v x i b n a n a n = Pr(P n (t) > x + v P n (t) > v) Pr(P (t) > x + v P (t) > v) ) > v where P n (t) and P (t) to be the corresponding excess value x i b n a n at index t, and the corresponding excess value at time t in the limit process respectively. The last step obtained from convergence in distribution shown in Theorem 2. 0

11 Hence, Pr(P (t) > x + v P (t) > v) = Λ(B x+v) Λ(B v ) [( ) x = + ξ + ξv ) = [( + ξ xσv where σ v = + ξv. Hence, the limiting distribution for large threshold Pr (x i > u n (v) + a n x x i > u n (v)) follows generalized Pareto distribution, GP D(a n σ v, ξ). + + ] /ξ ] /ξ Let y un(v) be a non-negative random variable which represents the excess over threshold u n (v), i.e., y un(v) = (x u n (v)) +. Since that for the Gaussian case ξ 0, (55) reduces to Pr(y un(v) y y un(v) > 0) = e y an (5) for all y 0. Hence, the tail of Gaussian distribution is well approximated by exponential distribution with rate parameter λ = a n, as shown in Figure 5. Hence, by taking expected value on the obtained exponential distribution, Proposition follows. Remark. Similarly to max-stability property, threshold stability ensures us that the limit distribution holds as long as we choose v > b l. Hence, observing conditional distribution with large enough threshold on the limit Poisson distribution we obtain a stability of the form of the Poisson distribution. 3.3 Threshold Estimation In order to achieve distributed model we suggest that given the total number of users, the end users estimates if their capacity belongs to top p users, hence, can decide if their capacity is adequate to utilize the next time slot. Hence, we would like to estimate a threshold u p such that only a fraction p of all users will exceed that threshold. Assuming that the capacity following Gaussian distribution Φ(x), with mean µ and variance σ 2, then from Proposition we derive the following. Proposition 2. The expected capacity obtained by setting p = k/, where k is the expected number of users such that their capacity is above threshold u p, out of total users is ( ) [ ( ( ) )] k E[C C > u p ] = µ + σ 2 log log 2π 2 log + log[2π] + σa + O ( 2). k (6)

12 f y Capacity Distribution given C threshold y Figure 5: Tail of Gaussian distribution, statistics of 722 observation out of that exceed threshold 3.5, which is Φ(3.5) of the observations. Dashed line is obtained by analyzing conditional distribution of Gaussian capacity given that capacity is above threshold. the solid line obtained from (5). In both the threshold was derived from (20), Proof. Let erfc ( ) denote the complementary inverse error function. The threshold u p such that Φ(u p ) = p = k is given by u p = µ + 2σ erfc (2p) (7) = µ + ( ) 2k 2σ erfc ( ) [ ( ( ) )] k = µ + σ 2 log log 2π 2 log + log[2π] + O ( 2). (8) k where the last equality used a Taylor series expansion. Substituting u p in (3), Proposition 2 follows. When we set large threshold, we can obtain the estimated threshold and the corresponding expected capacity by using extreme value theorem, which yield very similar results as we can see in Figure 7 and Figure 8. The user estimate a threshold u p that is near x F such that only a fraction p of the largest maximal capacities, among all maximal capacities, will exceed. Since we have only users, and we are interested only in maximal observations, then in order to gain sufficient amount of statistics, we suggest to divide logically the observations to blocks such that in each block there are observations, as we see in Figure 6(b). From the stability law of GEV, the maximum in each block is still well approximated by GEV distributions. Thus, 2

13 we can set a threshold u p, such that only a fraction p = k among maximal observations will exceed that threshold on average, assuming all x that satisfies a p x + b p > u p are in the tail corresponding to the upper tail of Gumbel distribution. Since 0 p and p = p, the threshold estimation via extreme value theorem is defined only for sufficiently small p when is large. Later in the paper we show that our interest is on p log, hence, threshold estimation using extreme value theorem holds. The return level u p is the p quantile of the ancestor distribution for 0 < p <, and has return period n = p observations. Hence, the user estimates the return level by setting quantile function Hence, for such u p we have G 0 (u p ) = p. G(u p ) = exp{ e (u p b / p )/a / p } = p (9) and we obtain that the maximum likelihood of the estimated return level u p is u p = b / p a / p log { log( p)} + o(a / p ). (20) The o(a / p ) error derived from Gumbel approximation error, shown in Appendix A. Substituting u p in (3) we have the following. Corollary 3. The expected capacity obtained by setting p = k/, where k is the expected number of maximal users to exceed u p, out of total maximal users, following E[C C > u p ] = (2 log k ) 2 (2) { (2 log k ) 2 log log( k } ) + +(2 log ) 2 + o (log. ) k Note that the first and second terms are converging to infinity at same rate. The expected capacity is mainly influenced by p. When p = k/ 0 the return level u p is high, but we may have idle time slots, since we may throw all extremes that weren t high enough. Similarly, if p = k/ the return level u p is low, hence, a small probability that no user will exceed u p, though, a non maximal user may decide to utilize next time slot, hence, a lower expected capacity is achieved. In the following section we derive a fare threshold trade-in, such that with high probability at least one user will exceed threshold, yet, only the maximal user utilize the next time slot, with high probability. 3

14 (a) (b) Figure 6: (a) k users exceeds threshold out of observations. (b) Partitioning to bins,such that in each bin there is approximately users, and among this maximal users we set a threshold such that on average only the largest k maximal users will exceed that threshold.. E C C u k Expected Capacity when k users exceeds on average k Figure 7: Threshold algorithm expected capacity gain for = 000 users, when setting threshold such that k users exceeds on average by (7)(solid line) and by (20) (dashed line), comparing to block maxima expected capacity gain (dot-dashed line). 4

15 Gaussian threshold estimation vs. Block maxima threshold estimation Threshold Figure 8: Threshold u p estimated by inverse error function derived from (7) (dot-dashed line), compering to block maxima threshold estimation u p derived from (20) (dashed line), when setting p = log / and p = log / respectively. 3.4 Throughput Analysis Since we are working with single user in each time slot, we define that a collision occurred if more than one user is trying to utilize a time slot. Similarly, we define that a time slot is idle if no user utilizing it. [20, Proposition 4] Shows that under the above constraint the optimal threshold that achieves maximum throughput obtained by demanding that only one user will exceed on average. This is also clear from Figure 7, for both threshold estimators. Proposition 3. For a threshold u p, p = k/ we have: Pr ( unutilized slot ) = ke k (22) Proof. The probability that more than two out of users will exceed u p, for p = k/, follows ( ) Pr( collision ) = ( Φ(u p )) j (Φ(u p )) j j j=2 [ ( = k ) ( ) ( k + k ) ] e k (k + ). 5

16 This also implies that the number of users exceeding threshold follows the Binomial distribution, hence, converges towards the Poisson distribution as goes to infinity. Similarly, under the same settings, the probability of an idle slot follows Pr( idle slot ) = e k. ( k ) Since ( Pr ( unutilized slot ) = Pr idle slot ) collision Proposition 3 follows. In particular, (23) implies that the system will be idle e of the time when setting the optimal threshold. 4 Heterogeneous Users In this section we assume that each user is located at different distance from the receiver, thus, his transmission traverses a different path, and experiences different attenuation, delay and phase shift. In our setting, the different gain loss each user experiences is reflected in different mean capacity and capacity variance each user sees. We analyze this model through the point of process approach under the distributed threshold scheme, where the i th user capacity follows Gaussian distribution with mean µ i and variance σi 2. Let Cav nu (u) denote the expected capacity in non-uniform environment. Thus, we obtain the following. Proposition 4. The expected capacity when working in single user in each slot in a nonuniform environment follows where and Cav nu Λ (u) = Λe ( ) Λ i B[0,/] [u, ) (u + σ i a + o(a )) Λ i= ( ) u (σ i b +µ i ) Λ i B[0,/] [u, ) = e σ i a, (23) Λ = Λ i (B [0,/] [u, ) ), (24) i= u is a threshold greater than zero that we set for all users, and a, b follows (5) and (6) respectively. 6

17 E C C u Expected capacity non uniform users u Figure 9: Solid line represents the expected capacity for = 000 users in non-uniform environment, where the channel capacity of each user follows Gaussian distribution with σ i U[0.03, 3] and µ i U[ 2, 2 + ], by the analysis in Proposition 4. Dashed line represents the expected capacity when all users have the same channel capacity as the capacity of the strongest user. Dot-dashed line represents the capacity when all users have the same channel capacity as the capacity of the mean user. Similar to the uniform setting, C nu av (u) = ( Pr(unutilized slot)) E[C C > u] (25) Thus, we analyze the expected capacity gain when letting a user with above threshold capacity to utilize a slot and the probability that a single user utilizes a slot in a nonuniform environment. Note that the computation of Cav nu is different from the uniform case, since users are not uniform, hence their probabilities to pass the threshold are different. For a Gaussian memoryless channel, every samples of the i th user can be represented as a point process for sufficiently large, as shown in Theorem 2, with ( Λ i (B [0,/] [up, )) = lim + ξ u (σ ) ib + µ i ) ξ ξ 0 σ i a = u (σ i b +µ i ) e σ i a arrivals to threshold u on average, where a and b follows (5) and (6) respectively. Assuming independent users, where each user exceeding threshold by Poisson process at average rate Λ i (B [0,/] [u, ) ), i =, 2,...,. When considering slot intervals, in which each user may exceed threshold once at most, then the total number of exceedances for users is equivalent to sum of independent Poisson-distributed random variables, hence, will follow 7

18 Poisson distribution with average rate Λ = i= e u (σ i b +µ i ) σ i a. Hence, for non-uniform MIMO channels, the number of exceedance can be represented as the following Poisson process ( ) { } Λk Pr N i = k = exp Λ k! i= where N i is the number of exceedances of the i th user in a very short interval, in which a single arrival is possible at most. The i.i.d. case can be obtained by placing σ i = σ and µ i = µ, i =, 2,..., in (26), achieving the expression in Proposition 3. In order to prove Proposition 4, we first prove the two claims below. Claim. Given that a single arrival to the threshold occurred, then the expected capacity for non-uniform users follow E[C C > u, N i = ] = i= i= ( ) Λ i B[0,/] [up, ) (u + σ i a + o(a )). Λ Proof. Since N, N 2,..., N are independent Poisson random variables with rate parameters ( ( ( ) Λ B[0,/] [up, )), Λ2 B[0,/] [up, )),..., Λ B[0,/] [up, ) respectively, then N i j= ( N j = k Binom k, Λ ) i. Λ Hence, the probability that the i th user exceeded threshold u given that a single exceedance occurred is ( ) Pr N i = N j = = Λ ( ) i B[0,/] [u, ). (26) Λ j= By Proposition, given that the i th user arrived to threshold, he contributes (u + σ i a + o(a )) to the expected capacity, hence, Claim follows. Claim 2. The probability of unutilized slot for non-uniform users follows { } Pr( unutilized slot ) = exp Λ + k =2 { } Λk exp Λ k!. 8

19 Proof. The probability of collision for non-uniform users follows ( ) Pr( k users exceeds u) = Pr N j = k k =2 = k =2 k =2 j= k Λ Λ e k! The probability of idle slot for non-uniform users follows ( ) Pr( idle slot ) = Pr N j = 0 = e Λ j= (27) (28) Since, Claim 2 follows. Pr( unutilized slot ) = Pr( idle slot more than one user exceeded ) 4. Weighted Users In this section, we derive the expected capacity when applying grade of service (GOS) to the users. In our setting, the grade of service is simply reflected in the exceedance probability applied to each user. Hence, given probability vector p R, each user sets a threshold corresponding to his exceedance probability by using (7) or by using (20). Hence, the arrival rate will correspond to the GOS applied on each user. Let Cav GOS denote the expected capacity in a non-uniform environment, when GOS applied on users. Thus, we obtain the following. Claim 3. The expected capacity with GOS in non-uniform environment follow C GOS av where and = Λ ( p) Λ ( p) e i= Λ (p ( ) i) i B[0,/] [upi, ) Λ ( p) (σ i [ b/pi a /pi log log ( p i ) + a ] + µi + o(a /pi ) ) Λ (p i) i (B [0,/] [upi, )) = b +b /pi e a ( log( p i )) a /p i (29) Λ ( p) = i= where p i is the exceedance probability of the i th user. Λ (p i) i (B [0,/] [upi, )) (30) 9

20 Proof. Since C GOS av = ( Pr(unutilized slot)) E {C C i > u i i=,2,..., } We analyze the following. In (23), we express the threshold arrival rate as a function of threshold u. Next, based on (20), we wish to set a unique threshold u pi for each user, such that the i th user will exceed his threshold with probability p i, and obtain the following. Λ (p ( ) i) up i σ i b µ i i B[0,/] [upi, ) = e σ i a = exp { (b/pi + b ) + a /pi (log log( p i )) a = e b +b /pi a ( log( p i )) a /p i. Since the arrival to threshold of each user can be modeled as Poisson process, assuming independent users, the total arrival to threshold rate is the sum of all users rate, thus, we have Λ ( p) i= e (b +b/p i ) a ( log( p i ) a /p i ). Since a slot is utilized only when a single arrival to threshold occurs in the point of process model, we have We notice that each user that exceeds threshold contributes a different capacity, corresponding to his threshold, hence, in order to obtain E {C C i > u i i=,2,..., } we average upon users the capacity that each user has. That is, } E {C C i > u i i=,2,..., } = Pr(the i th user exceeded)c ui i= where C ui is the capacity of the i th user, given that the i th user exceeded. 4.2 Equal Time Sharing of Non-Uniform Users We can achieve a proportional fairness by setting for each user a threshold that is relative to his own sample maxima probability, i,e. setting p i =, i =, 2,...,. Let Cav pf denote the expected capacity in a non-uniform environment, where and we set equal exceedance probability to all users. Hence, we obtain the following. Claim 4. The expected capacity with proportional fairness following C pf av = Λ () e Λ () i= ( (σ i [b + a log log ( ))] ) + µ i + o(a ). where ( )) a Λ () = e 2 b a ( log (3) 20

21 Since proportional fairness is a special case of GOS, it can be obtained by setting p i = / in (29) and substitute u pi for his estimated form showed in (20). Remark. For a channel with memory, the average arrival rate Λ i (B u, t) for each user is apparently time-dependent, but since it is a separable function of time and space, given the time dependency function f t, we can derive easily the expected capacity and the utilization probability under the same technique. Remark. Notice that in the non-uniform centralized downlink scheme there is no collision, and we are interested in the capacity gain that each user yields when he is maximal. The expression in (26) gives us the expected number of times that the i th user is maximal, and the capacity gained by letting him utilize the slot. Hence, we obtain the expected capacity in non-uniform environment for downlink model. 5 Collision Avoidance In this section we show an algorithm reaching the optimal capacity. [20, 24] shows a splitting algorithm that cope with collision when collision detection mechanism is available to users, by dividing each slot into mini-slots, such that a collision can be resolved in the next minislot. In our model, we assume that the users are only able to detect if the channel is being used in mini-slots resolution. If a collision occur we assume the whole slot is lost. First, we wish to minimize the idle slot probability, that without any enhancement will occur /e of the time. Next, we suggest an algorithm that copes with the resulted collision probability. From (23), it is easy to see that we the idle slot probability goes to zero when setting k = log as follows, Pr( idle time slot ) e log = / 0. However, when setting a threshold such that log users will exceed on average, we have to deal with log users on average, that find themselves adequate for utilizing next time slot. To overcome this problem, we suggest to rate users that exceeded the threshold by the distance they reached from the threshold. The set of values above the threshold is divided to l bins: [u p, u p + t ), [u p + t, u p + t 2 ),..., [u p + t l, ), numbered,..., l, respectively. A user which passed the threshold checks in which bin its expected capacity lies. If the bin index is i, it waits l i mini-slots and checks the channel. If the channel is clean, it transmits its data. In order to achieve uniform distribution over the bins, we set the bins boundaries by the exponential limit distribution that we found in (5) such that the i th bin boundaries follows t i = (2 log ) /2 log(i/l), i =, 2,..., l. From now on, we assume that the probability for a user who passed the threshold to fall in a specific bin is for all bins. l 2

22 P i Users Into Bins distribution i Figure 0: Distribution of users inside our bin when bin boundaries was set by (32) Proposition 5. In the suggested enhanced scheme, the probability of unutilized slot follows Pr(unutilized slot) = l j= m= ( ) ( k m ) m ( ) m k m where m is a realization of the number of uses who passed the threshold. ( l ) ( l j l ) m Proof. In order to analyze the probability of unutilized slot, we let E be the event that a single user occupies the maximal bin J, and denoting l for the total number of bins. Hence, for a fixed k users who exceeded threshold, we have Pr ( unutilized slot ) = Pr(E ) (32) l ( ) ( ) k i l j = k l l We notice that when k is not fixed, it should be represented as a random variable which follows the binomial distribution with parameters n = and p = k/, as follows from (23). Hence, by using complete probability formula, we have Pr(unutilized slot) = Pr(E ) (33) l ( ) ( ) m ( ) m ( ) ( ) m k k l j = m. m l l j= m= j= 22

23 This suggests that we can achieve small collision probability as we like, by increasing the number of bins, as the following claim asserts. Claim 5. In the enhanced algorithm the probability of unutilized slot converges to zero as l increases. Proof. If there are k users above threshold and l bins then the probability that all k fall into different bins is ( ) ( 2 ) (... k ) = l l l k j= ( j ) l Using that k/l e k/l is tight bound when k is small compared to l, we have k ( j ) l j= k e j/n j= { } k j = exp n j= = e k(k )/2l Hence, the probability of collision in any bin is e k(k )/2l, which is going to zero as l increases. Hence, Claim 5 follows. 5. Analyzing the Delay Regardless of collisions that may occur, we analyze the expected time that took the maximal user decide that he is the most adequate to utilize the time slot, which is equivalent to the expected index of the maximal occupied bin, out of l bins. In order to obtain this, we order the bins in descending order, such that bin corresponds to the maximal value. Since we choose k <<, on average only a small group of users will exceed threshold, thus, we can express the probability that bin j is maximal, without using extreme distributions. Let J denote the index of the maximal user bin, we obtain the following. Proposition 6. For a random number of users that exceeded threshold u p, the expected maximal bin index J follows E[J] = l j= m= ( ) ( k m ) m ( ) m ( ) m k l j. l 23

24 E C 4.5 Bin scheme algorithm performance vs. Block maxima performance Figure : Bottom line - Threshold scheme expected capacity for users, setting threshold that on average log users exceeds threshold, placing them into ( log ) 2 bins with the boundaries obtained in (32).Top line is the optimal centralized scheme performance. Proof. Given k users that exceeded threshold we obtain E[J k users exceeded] = = l Pr(J > j k user exceeded) (34) j= l ( ) k l j. l j= as we can see in Figure 2(a). To derive the expected maximal bin index, J, for random k we use the complete probability formula as follows. E[J] = = l j= m= l j= m= Pr(k = m) Pr(J > j k = m) (35) ( ) ( k m ) m ( ) m ( ) m k l j. (36) l Hence, Proposition 6 follows. Remark. The probability that bin J = j is maximal for a fixed k users who exceeded the 24

25 Probability Maximal index Probability for Deterministic Probability Maximal index Probability for Random k index (a) (b) index Figure 2: (a) maximal index simulation and analysis for deterministic k, where the blue line follows (38). (b) maximal index simulation and analysis for random k, where the red line follows (37). threshold follows Pr(J = j) = = Pr(k = m) Pr (J = j k = m) (37) m= m=0 ( ) ( ) m ( k k m ) m (( ) m ( l j + l j l l ) m ). Similarly, for random k = m users who exceeded threshold we have (( ) m ( ) m ) l j + l j Pr(J = j) =. (38) l l 6 Conclusion In this paper, we present a distributed scheduling scheme, for exploiting multiuser diversity in a non-uniform environment, where each user has a different location, therefor will experience different fading channel. We characterized the scaling law of the expected capacity and the system throughput by point of process technique, and present a simple analysis for the expected value and throughput when applying GOS upon users, and presented an enhancement for the distributed algorithm in which the expected capacity and throughput reaches the optimal capacity, for a small delay price. 7 Appendix A In this section we derive the constants a n and b n, for the Gaussian case. 25

26 Proof. We denote the standard normal distribution function and density function by Φ and φ respectively, and notice the relation of the tail of Φ, for positive values of x, from Taylor series: Φ(x) φ(x) (39) x with equality when x. First, we wish to find where ξ converges to. I.e., to what distribution type the maxima of Gaussian distribution converges. Thus, we use the relation in (39) to derive the shape parameter of Gaussian maxima, ξ d [ ] φ(x)/x dx φ(x) d dx x 0 we substitute ξ 0 in (2), and find the limit distribution from extreme value theory Pr(M n u) = [Φ(u)] n (40) [ = exp ( + ξ u b ] ξ n ) a n ξ 0 u bn exp[ e ( ) an ]. That is, the maxima of Gaussian random variables converges to Gumbel distribution, where u = a n x + b n. For retrieving the normalizing constants, a n and b n, as can be found rigorously at [29, Theorem.5.3.], we use a well known log approximation for large values of x, and apply it to (40), i.e., hence, apply ξ 0 to (42), thus, log[ ( x)] x log[φ(u)] n = n log( [ Φ(u)]) n ( Φ(u)) (4) So, in oreder to satisfy (43), we shell take Φ(u) = n e x. Using again the tail relation (39), we obtain, n ( Φ(u)) ( + ξx) ξ (42) n ( Φ(u)) ξ 0 e x. (43) n e x φ(u) u 26

27 or n e x applying log function on (44) will lead us to u φ(u) x (44) log n x + log u log φ(u) 0 (45) we substitute φ(u) for a Normal density function, 2π e 2 u2 in (45), hence, log n x + log u + u2 log 2π + 0 (46) 2 2 and by substitute x = u bn a n in (46) and rearrange it a little, we obtain, ( ) u bn + log u + a n 2 log 2π + u2 2 log n and since u 2 has the main influence on the left hand side, it implies that hence, by applying log to (47), we obtain u 2 2 log n 2 log u log 2 log log n 0 (47) or log u = (log 2 + log log n) + o(). (48) 2 We place (48) in (46),and rearrange it a little to obtain u 2 = 2 log n [ (log n) x + + (49) ] 2 (log n) (log 4π + log log n) + o( log n ) and hance, [ u = 2(log n) x (50) (2 log n) ] (log 4π + log log n) 2 + o( (2 log n) log n ) = (2 log n) 2 x + (2 log n) (2 log) 2 (log log n + log 4π) + o ( = a n x + b n + o(a n ) (log n) 2 ) 27

28 which means that (40) follows for a n = (2 log n) 2 and b n = (2 log n) 2 (2 log n) 2 [log log n + log(4π)]. 2 8 Appendix B two alternative proofs for Proposition 8.0. Direct Tail Analysis In this section we show an alternative approach for Gaussian tail estimation. We would like to examine the capacities that rests within the long tail, i.e., capacities that reached a high threshold u, that is, Pr(x δ u x + u x > u) From Bayes theorem F x x>u = F x,x>u(u + δux). F x (u) Since that under properly normalization, the capacity of one user is well approximated by a Normal distribution, then we can obtain the asymptotic behavior of the tail by using relation (39), hence, Pr(x δ u x + u x > u) = ( Φ(u)) [Φ(δ u x + u) Φ(u)] = ( Φ(u)) [( Φ(u)) ( Φ(δ u x + u))] = u [ φ(u) φ(u) u φ(δ ] ux + u) + o(u 3 ) δ u x + u u = δ u x + u φ(δ ux + u) + o(u 3 ) φ(u) u = δ u x + u e δuux e δux o(u 3 ) (5) Notice that if we pick a threshold u that grows linearly at rate θ(n), δ u that diminishes at rate θ(n ), and x grows more slowly that u, such that δ u x 0 and δ u u λ, where λ is arbitrary positive constant, then (5) reduces to a limit distribution, which has an exponential form for, i.e., Φ x x>u (δ u x + u) = e λ ux. (52) 28

29 Similarly, we can obtain the expected capacity given that it is above threshold u, E[C C > u] = xdφ x x>u (x) hence, E[C C > u] = Φ x (u) u xdφ x (x) = ( Φ(u)) 2π = ( Φ(u)) e u2 2 2π u u xe x2 2 where the last inequality follows from (39) Tail Analysis - Point of Threshold For tail approximation of F we can also use extreme value analysis when u is near the upper end point x F, as can be found rigorously in [3]. Let s assume that the limit representation {F (a n x + b n )} n i.d. G(x) = exp[ ( + ξx) ξ + ] (53) holds for large n and for all x such that a n x + b n is near x F. Thus, there is a threshold u near x F such that all x that satisfies a n x + b n > u are in the tail corresponding to the upper tail of a generalized extreme value (GEV) distribution, G(x). It follow that F (a n x + b n ) G n (x) or, for y = a n x + b n ( ) F (y) G y bn n. (54) a n Due to the max-stability property [30], which is a property satisfied by distributions for which the operation of taking sample maxima leads to an identical distribution, apart from a change of scale and location. (54) is still GEV distribution. Hence for some µ, σ > 0, ξ parameters F (y) i.d. exp { n ( + ξ ( )) } y µ ξ using a well-known approximation from Taylor series, e x > + x, we obtain, F (y) [ ( )] y µ ξ + ξ n σ 29 σ + +

30 which is the tail corresponding to the upper tail of a GEV distribution. Alternatively, we let y u be a non-negative random variable which represents the excess over threshold u, i.e., y u = (x u) +. Hence, Pr(y u > y y u > 0) = Pr(x > u + y x > u) (55) = F (u + y) F (u) [ + ξ(u + y b n n)/a n ] ξ + [ + ξ(u b n n)/a n ] ξ + = [ + ξy/σ u ] ξ + where σ u = a n + ξ(u b n ) + o(a n ). It follows that y u y u > 0 follows generalized Pareto distribution, GP D(σ u, ξ), with scale parameter σ u and shape parameter ξ. 9 Appendix C Proof. (Theorem 2) Let N n (B) and N(B) be the number of points of P n and P respectively in set B. Assuming that for any n disjoint sets B, B 2,..., B n, with B i C, i =, 2.., n, then N(B ), N(B 2 ),..., N(B n ) are independent random variables. we will show that as n and E(N n (B)) E(N(B)) Pr(N n (B) = 0) Pr(N(B) = 0). Thus, we take B v = (0, ] (v, ), such that the i th point of P n is in B v if x i b n a n i.e., if x i > a n v + b n. The probability of this is F (a n v + b n ). Hence, the expected number of such points is > v E[N n (B v )] = n[ F (a n v + b n )] log [F (a n v + b n )] n log G(v) = ( + ξv) ξ + = Λ(B v ) = E[N(B v )]. 30

31 Similarly, the event N n (B v ) = 0 can be expressed as { } xi b n {N n (B v )} = v, i =,..., n So a n = {x i a n v + b n i =,..., n} Pr(N n (B v ) = 0) = {F (a n v + b n )} n G(v) = exp[ ( + ξv) /ξ + ] = exp[ Λ(B v )] = Pr(N(B v ) = 0) References [] H. Weingarten, Y. Steinberg, and S. Shamai, The capacity region of the gaussian multiple-input multiple-output broadcast channel, Information Theory, IEEE Transactions on, vol. 52, no. 9, pp , [2] S. Sesia, I. Toufik, and M. Baker, Lte the umts long term evolution, From Theory to Practice, published in, vol. 66, [3] I. S. for Local and metropolitan area networks, Part 6: Air Interface for Broadband Wireless Access Systems Amendment 3: Advanced Air Interface. pub-ieee-std, May 20. [4] R. nopp and P. Humblet, Information capacity and power control in single-cell multiuser communications, in IEEE International Conference on Communications, Seattle, vol., 995, pp [5] T. Yoo and A. Goldsmith, On the optimality of multiantenna broadcast scheduling using zero-forcing beamforming, Selected Areas in Communications, IEEE Journal on, vol. 24, no. 3, pp , [6] R. Zakhour and S. Hanly, Min-max fair coordinated beamforming via large system analysis, IEEE International Symposium on Information Theory Proceedings, pp , 20. [7] C. Chen and L. Wang, Enhancing coverage and capacity for multiuser mimo systems by utilizing scheduling, Wireless Communications, IEEE Transactions on, vol. 5, no. 5, pp ,

32 [8]. Jagannathan, S. Borst, P. Whiting, and E. Modiano, Scheduling of multi-antenna broadcast systems with heterogeneous users, Selected Areas in Communications, IEEE Journal on, vol. 25, no. 7, pp , [9] M. Sharif and B. Hassibi, A comparison of time-sharing, dpc, and beamforming for mimo broadcast channels with many users, Communications, IEEE Transactions on, vol. 55, no., pp. 5, [0] G. Caire, Mimo downlink joint processing and scheduling: a survey of classical and recent results, in Proc. Workshop on Information Theory and Its Applications, [] B. Hassibi and M. Sharif, Fundamental limits in mimo broadcast channels, Selected Areas in Communications, IEEE Journal on, vol. 25, no. 7, pp , [2] W. Choi and J. Andrews, The capacity gain from intercell scheduling in multi-antenna systems, Wireless Communications, IEEE Transactions on, vol. 7, no. 2, pp , [3] L. Wang, C. Chiu, C. Yeh, and C. Li, Coverage enhancement for ofdm-based spatial multiplexing systems by scheduling, in IEEE Wireless Communications and Networking Conference, WCNC. IEEE, 2007, pp [4] M. Pun, V. oivunen, and H. Poor, Opportunistic scheduling and beamforming for mimo-sdma downlink systems with linear combining, in IEEE 8th International Symposium on Personal, Indoor and Mobile Radio Communications, 2007, pp. 6. [5] J. Choi and F. Adachi, User selection criteria for multiuser systems with optimal and suboptimal lr based detectors, Signal Processing, IEEE Transactions on, vol. 58, no. 0, pp , 200. [6] M. Airy, S. Shakkattai, and R. Heath Jr, Spatially greedy scheduling in multi-user mimo wireless systems, in the Thirty-Seventh Asilomar Conference on Signals, Systems and Computers, vol.. IEEE, 2003, pp [7] C. Swannack, E. Uysal-Biyikoglu, and G. Wornell, Low complexity multiuser scheduling for maximizing throughput in the mimo broadcast channel, in Proc. Allerton Conf. Communications, Control and Computing, [8] G. Primolevo, O. Simeone, and U. Spagnolini, Channel aware scheduling for broadcast mimo systems with orthogonal linear precoding and fairness constraints, in IEEE International Conference on Communications, vol. 4, 2005, pp [9] T. Yoo, N. Jindal, and A. Goldsmith, Finite-rate feedback mimo broadcast channels with a large number of users, in Information Theory, 2006 IEEE International Symposium on. IEEE, 2006, pp