On the Aloha throughput-fairness tradeoff

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1 On the Aloha throughput-fairness traeoff 1 Nan Xie, Member, IEEE, an Steven Weber, Senior Member, IEEE Abstract arxiv: v1 [cs.it] 5 May 2016 A well-known inner boun of the stability region of the slotte Aloha protocol on the collision channel with n users assumes worst-case service rates all user queues non-empty). Using this inner boun as a feasible set of achievable rates, a characterization of the throughput fairness traeoff over this set is obtaine, where throughput is efine as the sum of the iniviual user rates, an two efinitions of fairness are consiere: the Jain-Chiu-Hawe function an the sum-user α-fair isoelastic) utility function. This characterization is obtaine using both an equality constraint an an inequality constraint on the throughput, an properties of the optimal controls, the optimal rates, an the fairness as a function of the target throughput are establishe. A key fact use in all theorems is the observation that all contention probability vectors that extremize the fairness functions take at most two non-zero values. Inex Terms multiple access; ranom access; Aloha; stability; throughput-fairness traeoff; Jain fairness; α-fair; proportional fair. I. INTRODUCTION We investigate the throughput fairness traeoff for the slotte Aloha meium access control MAC) protocol [1], [2] serving n users contening on a share collision channel. Throughput fairness traeoffs naturally arise in settings of share access to a constraine resource, where maximum use of the resource is at os with fair access to the resource, on account of the inefficiency incurre in resource contention. In the setting of Aloha, this incurre inefficiency takes the form of waste slots in which either no user contens ile) or multiple users conten collision). Trivially, maximum throughput of one successful packet per time slot is achieve by the unfair allocation granting one user access an shutting out all other users, while the maximally fair allocation granting each user equal access achieves a throughput that ecays to zero in the number of users. Our focus is on characterizing the traeoff connecting these two extreme points. Although moern MAC protocols in use toay are far more complex an more sophisticate than Aloha, many of them nonetheless retain at their core the notion of ranom access, which is the efining characteristic of Aloha. It is therefore natural, in our opinion, to first analyze the throughput fairness traeoff in ranom access in the canonical setting of slotte Aloha before seeking to characterize such traeoffs uner more complicate protocols. The authors are with the Department of Electrical an Computer Engineering, Drexel University, Philaelphia, PA 19104, USA. N. Xie nx23@rexel.eu, S. Weber sweber@coe.rexel.eu contact author).

2 2 One ifficulty precluing this goal from being achieve is that the stability region for slotte Aloha on the collision channel remains unknown, in spite of 40+ years of effort. Because of this, we employ a well-known inner boun on the stability region, obtaine by assuming each of the user s queues is nonempty, thereby yieling a worst-case effective service rate seen by each user. This inner boun is known to be tight for all special cases for which the stability region of slotte Aloha is known. Even with this simplifying assumption, however, the throughput fairness problem is still nontrivial on account of the fact that the inner boun cannot be escribe explicitly. Rather, the inner boun is given as the image of the function mapping contention probability vectors controls) to worst-case) packet transmission rates, over the set of all possible controls. A. Relate work The throughput fairness traeoff literature is quite large an iverse, stemming from its relevance to a wie variety of isciplines, incluing queueing theory, communication networks, optimization, an economics. As such, we restrict our iscussion to only the most pertinent prior work. Specifically, we summarize prior work on each of the two fairness metrics use in this paper, namely, the Jain-Chiu-Hawe function an the α-fair utility function. The Jain-Chiu-Hawe fairness measure [3], hereafter simply Jain s fairness, measures the fairness of an n-vector x = x 1,..., x n ), representing in our context the vector of user rates, as the normalize istance from x to the all-rates-equal ray passing from the origin through the point 1. This metric has been wiely aopte, e.g., [4], [5]. The α-fair parameterize family of utility functions was introuce to the networking community in [6], but is nearly ientical to the classic isoelastic utility function in economics [7]. The α-fair family of utility functions has foun profitable use in characterizing throughput fairness traeoffs an resource allocation policies in wire an wireless networks, an in that sense may be viewe as part of the larger boy of work terme network utility maximization NUM), e.g., [8], [9], [10], [11]. The basic concept in NUM is to associate with each user a utility often assume to be concave increasing) that epens upon the resources allocate to the user, an seek a feasible resource allocation that maximizes the sum-user utility. In essence, the concavity of the utility function captures the law of iminishing returns for each user, an thus optimizing sum utility over all feasible allocations yiels a solution that is fair in the sense that all users enjoy a common marginal utility. Returning to α-fair utility functions, the parameter α 0 controls the concavity of the utility function, where α = 0 correspons to a linear utility function no iminishing returns), α = 1 is a logarithmic utility function so-calle proportional fair utility), an as α the utility-optimal resource allocation is the so-calle max-min fair allocation. Given this, it is natural to think that increasing α woul trae sum-user throughput for fairness, although recent work [12], [13], [14], [15] has ientifie counter-examples. Recent work has aresse throughput fairness traeoffs using both these fairness measures in the context of ownlink scheuling [15], [5]. In contrast, our focus is on uplink, an this funamental ifference limits the applicability of many of the results in [15], [5] to our setting. An axiomatic approach to fairness is given in [16], with an insightful iscussion contrasting Jain s fairness an α-fairness.

3 3 TABLE I SUMMARY OF RESULTS #/Result Title/Description II Moel an problem statement Lem. 1 All-rates equal ray s geometric an algebraic properties III Properties of optimal controls Prop. 1 Schur-concavity of fairness measures in rate space Prop. 2 Majorization properties uner throughput constraint Cor. 1 Sufficiency to optimize over Λ or S in control space) Prop. 3 Properties of controls in S 2 uner throughput constraint Prop. 4 Sufficiency to optimize over the restricte set in Def. 1 IV Jain-Chiu-Hawe fairness traeoff Prop. 5 T-F traeoff uner Jain s fairness when n = 2 Prop. 6 Monotonicity properties of the Jain s objective over S 2 Thm. 1 T-F traeoff uner Jain s fairness for general n 2 Thm. 2 No change uner throughput inequality constraint Alg. 1 Incremental plotting of T-F traeoff for a sequence of n s Thm. 3 Properties of the Jain T-F traeoff V α-fair network utility maximization α 1) Prop. 7 T-F traeoff uner α-fairness when n = 2 Prop. 8 Monotonicity property of the α-fair objective over S 2 Thm. 4 T-F traeoff uner α-fairness for general n 2 Thm. 5 Change uner throughput inequality constraint Thm. 6 Properties of the α-fair T-F traeoff B. Outline an contributions The primary contribution of this paper is a characterization of the throughput fairness T-F) traeoff for n users employing slotte Aloha on a collision channel. This is one through six theorems: Theorem 1 2) gives the T-F traeoff uner Jain s fairness with a throughput equality inequality) constraint an Theorem 3 gives properties of the optimal controls, optimal rates, an the T-F traeoff itself. Theorem 4 5) gives the T-F traeoff uner α-fairness with a throughput equality inequality) constraint, an Theorem 6 gives properties of the optimal controls, optimal rates, an the T-F traeoff itself. This rest of the paper is organize as follows. The moel an problem statement are introuce in II, while III contains results common to both fairness measures. Builing upon III, the next two sections IV, V) aress the Aloha throughput-fairness traeoff uner Jain s an α-fairness respectively. Finally VI offers a brief conclusion. Three appenices follow the references, holing long proofs from III, IV, an V respectively. Table I lists all the results in the paper, an Table II provies general notation.

4 4 TABLE II GENERAL NOTATION Symbol n [n] x p Meaning number of users; efault vector length positive integers up to n vector of user arrival rates vector of user contention probabilities xp) worst case service rates uner control p 2) u = 1 n 1 m e i x, y) uniform contention probability vector rate vector for p = u II-D) unit vector with 1 in position i [n] Eucliean istance between x an y Λ Aloha stability region inner boun 1) Λ the bounary of the set Λ 3) S close stanar unit simplex II-C) S probability vectors 4); efficient controls, c.f., 3) T x) sum-user throughput of x 5) F x) fairness measure of x: F J 7) or F α 8) {θ t} n t=1 critical throughputs 6) Vp) the set of non-zero values in p Def. 1) pp s, k, n ) restricte control vectors in Def. 1 S 1 efficient controls with Vp) = 1 Def. 1) S 2 efficient controls with Vp) = 2 Def. 1) S 1,2 efficient controls with Vp) {1, 2} Def. 1) α parameter in α-fair utility functions 9) θ F θ) target throughput optimize fairness given target throughput θ II. MODEL AND PROBLEM STATEMENT This section is ivie into the following subsections: an introuction of some general notation in II-A, a iscussion of the Aloha protocol an the collision channel in II-B, efinition of the Aloha stability region Λ A an its inner boun Λ in II-C, an the efinitions of throughput an fairness in II-D. A. General notation All vectors are lowercase an bol an are by efault of length n. Inequalities between two vectors are unerstoo to hol component-wise. We write [n] to enote {1,..., n} for n N. The unit vector with a one in position i is enote e i, for i [n]. The all-one vector is enote by 1, the uniform istribution 1 n1 is enote u, an the all-zero vector is enote by 0. Eucliean istance is enote x, y). Carinality of a set V is enote V. We sometimes write z to enote 1 z. Table II lists frequently use notation; aitional notation will be explaine at first use.

5 5 B. The Aloha protocol an the collision channel Recall a MAC protocol specifies a mechanism to coorinate competing users access to the share channel; we consier the finite-user slotte Aloha MAC protocol operating on a collision channel. The protocol parameters are n, x, p), where i) n N is the number of users, ii) x R n + is an n-vector enoting the inepenent arrival rates of users ata packets, which we henceforth call the rate vector, an iii) p [0, 1] n is an n-vector inicating the user contention or channel access) probabilities, which we henceforth call the control vector. Each user has an associate packet queue that can hol an infinite number of packets, store in orer of arrival. Each packet will be remove from the queue if an only if it has just been successfully transmitte. The channels are error-free. Time is slotte an synchronize. At the beginning of each time slot, every user with a non-empty queue, say user i [n], contens for channel access to the common base station by transmitting its hea-of-line packet with a fixe probability p i, inepenent of anything else. The collision channel assumption means the state of the channel in each time slot may be classifie as i) ile no one attempts to transmit, either because of having an empty queue or electing not to transmit), ii) collision more than one user transmits, an all attempte transmissions fail), or iii) success precisely one user transmits, an this attempte transmission succees). This ternary feeback is error-free an instantaneous at the en of each time slot. C. The stability region Λ A an its inner boun Λ An important yet still open problem is the queueing-theoretic stability region also calle the network layer capacity region [17, pp. 28]) of this moel, enote Λ A A for Aloha), which contains all arrival rate vectors x that can be stabilize by the protocol, i.e., for each x Λ A there exists a control vector p that stabilizes each of the n queues. The stability region is open even for the case of inepenent arrival process an n > 2 users. A summary of the history of this problem is provie in [18], with compelling recent work incluing [19], [20] among others. As Λ A is unknown, we employ a suitable inner boun on Λ A as a proxy for the stability region of slotte Aloha. This inner boun, enote Λ below, has been prove to coincie with the exact stability region for all special cases for which the stability region is known [21], [22]), an has been conjecture [23, V], [18, V Thm. 2]) to in fact be the stability region, Λ A. The set Λ is efine as: Λ x Rn + : p [0, 1] n : x i p i 1 p j ), i {1,..., n}. 1) The expression p i j i 1 p j) is the worst-case service rate for user i s queue, namely the service rate assuming all other users have non-empty queues an thus all users are eligible for channel contention. In particular, user i s transmission is successful in such a time slot if user i elects to conten with probability p i ) an each other user j i oes not conten each with inepenent probability 1 p j ). Clearly, Λ is an inner boun, since an arrival rate that is stabilizable uner the worst-case service rate is certainly stabilizable uner a better service rate. It may be shown [24, II, Prop. 2] that an equivalent efinition of Λ is to change all the inequalities to equality, i.e., x Λ j i

6 6 if an only if there exists a p [0, 1] n for which x = xp), where x i p) p i 1 p j ), i [n]. 2) j i We refer to such a p as a critical compatible) control for x. 1 Base on the above efinition of Λ, testing whether or not a caniate x is or is not in Λ is equivalent to the solvability of x = xp) over p [0, 1] n. The efinition of Λ is therefore implicit, in the sense that testing membership x Λ requires establishing the existence or not) of a suitable control p. When aressing throughput fairness traeoffs we will be optimizing an objective function over Λ, which thus becomes the feasible set for the optimization. The implicit characterization of Λ is what makes the corresponing throughput fairness traeoff optimization problem non-trivial. The natural solution, which we employ, is to make p [0, 1] n the optimization variable, thereby requiring the corresponing nonlinear compositions on both the throughput an fairness functions, i.e., T xp)) an F xp)), efine below. To emphasize this istinction, we refer to x as a rate vector in rate space, an p as a control vector in control space. The bounary of Λ in R n + is enote Λ an is characterize [25] as Λ = x Rn + : p S : x i = p i 1 p j ), i {1,..., n}, 3) where S {z R n + : n i=1 z i 1} enotes the stanar unit simplex, an its face, enote n S {z R n + : z i = 1}, 4) is the set of probability vectors on [n]. Thus, Pareto efficient throughputs, i.e., x Λ, are achieve by an only by controls that are probability vectors, i.e., p S. For this reason, we call S the set of efficient controls. It may be helpful to visualize Λ an its bounary Λ using Fig. 2 IV-A) for the n = 2 case, where they are shown as the light blue shae area an the brown curve respectively. In aition, the following lemma the proof of which is straightforwar an is omitte), use in some proofs, is relevant to Λ in that it implies: a) geometrically, the ray from the origin through 1 the all-rates equal ray) resies insie Λ until it hits the bounary Λ at x = θn n 1 see 6) an the iscussion below), shown in Fig. 2 as the black ot, an b) there only exists) two one) controls) p for any rate vector x on this ray segment that lies insie on the bounary of) Λ, in the sense of 2). Lemma 1: Let an integer n 2 be given. The function p1 p) n 1 for p [0, 1] is increasing when p [0, 1/n] an ecreasing when p [1/n, 1], with the maximum 1 n 1 1 n 1 n) attaine at p = 1/n. j i i=1 D. Throughput an two fairness measures The sum-user throughput of any rate vector x Λ is efine as: n T x) x i. 5) i=1 1 More generally, we efine a compatible control for x as a control vector p for which x xp). In this paper we only employ critical compatible controls, an as such we often refer to p satisfying x = xp) simply as a control for x.

7 7 Note T x) [0, 1] since, by the efinition of the collision channel, there is at most one successful transmission on the channel in each time slot. We efine the vector θ = θ 1,..., θ n ) with θ 1 = 1 an θ t 1 1/t) t 1, t {2,..., n} 6) as the vector of critical throughputs. Observe 1 = θ 1 > > θ n > 1/e. Define the rate vector m θn n 1 = xu) associate with θ n, i.e., m is the rate vector for the uniform control u, with corresponing throughput T m) = θ n. Geometrically, m is the unique intersection of the ray from the origin through 1 the all-rates equal ray) with Λ. The fairness of x is enote F x); we will employ the following two fairness efinitions in this paper. The first, Jain-Chiu-Hawe fairness [3], henceforth referre to simply as Jain s fairness an enote F J x), is a now classic means of quantifying the fairness of a resource allocation x: F J x) = T x)2 n x 2. 7) The Jain s fairness function has the following properties: i) scale invariance, i.e., F J βx) = F J x) for any β R ++ ; an ii) bouneness, i.e., F J [1/n, 1], with F J βe i ) = 1/n for any i [n] an F J β1) = 1 for any β R ++. The secon fairness measure, the α-fair sum-user utility function, efine as n F α x) U α x i ), 8) i=1 for α 0, is the sum-user utility of the allocation x, where the common) per-user utility functions are efine, for α R, as: 2 U α x) = logx), α = α x1 α, α 1. 9) Maximization of sum-user utility over a set of feasible allocations, for any concave increasing utility function U α x), often implicitly enforces a throughput fairness traeoff. For example, the cases α = 0, 1, have corresponing optimal solutions that maximize throughput, proportional fairness log-utility), an max-min fairness, respectively. It is for this reason that we refer to F α x) as a fairness function. Observe that uner the throughput equality constraint T x) = θ, the objective F J x) is inversely proportional to F 1 x), i.e., F α x) in 8) with α = 1, an as such maximizing F J x) uner T x) = θ is equivalent, in the sense of having the same extremizers, to minimizing F 1 x). Even though F α only possesses the esirable properties of a utility function for α 0, this equivalence allows us to stuy extremizers of F J an F α α 0) uner a unifie framework, as in Prop. 4 in III. The general throughput-fairness traeoff for slotte Aloha, using the proxy stability region Λ as the feasible set of arrival rate vectors, is the Pareto frontier of the parametric plot T x), F x)) over x Λ. An equivalent alternate 2 Note that lim α 1 U αx) = ±1/0, i.e., is unefine, an not equal to U 1 x) = log x. One way to rectify this iscrepancy is to moify the efinition to inclue a constant shift, e.g., Ũ αx) 1 1 α x 1 α 1 ), which is known as the isoelastic utility function in economics. As is conventional in the networking literature, we omit this constant as it has no effect on the extremizers.

8 8 formulation of the throughput fairness traeoff is to seek to maximize F x) over x Λ such that T x) = θ, for θ 0, 1) a target throughput constraint. We omit θ = 0 an θ = 1 as target throughputs as both correspon to trivial ege cases. In fact, we will aress two types of throughput constraints in this paper: i) a throughput equality constraint T x) = θ, an ii) a throughput inequality constraint T x) θ. The equality constraint is use, as mentione above, to characterize the throughput fairness traeoff, while the inequality constraint amits a natural operational interpretation: allocate resources as fairly as possible subject to the sum throughput exceeing a minimum requirement. As we will show, there are parameter regimes wherein these two problems are the same, an regimes where they are ifferent. Finally, observe that Λ, F x), an T x) are each permutation invariant, an as such any extremizer x that maximizes fairness uner a throughput constraint is permutation invariant, meaning any permutation of x is likewise an extremizer. Further notes about notation. Auxiliary functions typically name as f 1, f 2, etc.) use in proofs are unerstoo to be internal meaning a ifferent function with the same name might be use in a ifferent proof. The following inequality about the natural logarithm function is frequently use in the paper: log1 + z) z, for all z > 1, 10) which is strict unless z = 0. Finally, we use F θ) to represent the maximum fairness for a given target throughput θ, which is not to be confuse with F x) efine in 7) an 8). III. PROPERTIES OF OPTIMAL CONTROLS We use the framework of majorization in III-A to establish that it suffices to restrict the control space from [0, 1] n to the set of efficient controls, namely S 4), an then use Karush-Kuhn-Tucker KKT) conitions in III-B to establish structural properties of those controls that extremize F α x) for α, 1] [1, ) uner a throughput constraint. A. A majorization approach We aress the Aloha T-F traeoff problem through the lens of majorization [26], the origins of which are roote in questions of fairness. Majorization efines a partial orer on the set of vectors with the same length an sum of components. More precisely, a is majorize by b, enote a b, if k i=1 a [k] k i=1 b [k] for all k [n], where a [k] is the k th component of a sorte in nonincreasing orer. For example, the quasi uniform probability vectors in S) below are majorize as [26, pp. 9]: 1 n,..., 1 ) ) 1 n n 1,..., 1 n 1, 0 1 2, 1 ) 2, 0,..., 0 1, 0,..., 0). 11) As the above example suggests, in many contexts the statement x y may be interprete as x is more fair than y, in the sense that the components of vector x are more nearly equal than those of y. It is therefore natural to try to stuy our T-F traeoff within the framework of majorization. The class of Schur concave or convex) functions are

9 9 symmetric functions that preserve majorization, i.e., F is Schur concave convex) if F x) F y) F x) F y)) for all x, y) such that x y. The following result, taken from [16] c.f. Thm. A. 4 in Ch. 3 of [26]), inicates the relevance of Schur concavity to our problem note Schur concavity is preserve uner summation, c.f. 8)). Proposition 1: The Jain s fairness function 7) an α-fair utility function 9) for α 0 are Schur concave in x. Remark 1: An immeiate consequence of this result is that it allows us to restrict the set of feasible controls from [0, 1] n to [0, 1) n. First, observe that if there are multiple users contening with probability one, then the corresponing rate vector is x = 0, an as such T x) = 0, meaning such points cannot achieve any target throughput θ 0, 1). Secon, if there is a unique user, say i, with p i = 1 i.e., p j [0, 1) for all j i), then x = π i e i, where π i = j i 1 p j). But, such an x majorizes every other feasible point in rate space, an thus will not maximize either of our fairness objectives. The following result establishes two key facts. First, it suffices to consier only efficient controls, p S, for maximizing fairness uner a throughput equality) constraint. Secon, there is no majorization relationship among any two efficient controls that both satisfy the throughput constraint. Thus, majorization oes not by itself solve the T-F traeoff optimization problem. Proposition 2: Fix the number of users n an the target throughput θ θ n, 1). Define the hyperplane H θ = {x R n + : T x) = θ} of rate vectors with throughput θ. Define Λ θ = Λ H θ, Λ θ = Λ H θ, an Λ int θ as the set of stable, stable efficient, an stable inefficient rate vectors with throughput θ, respectively. Then 1) for any x Λ int θ, there exists some x Λ θ such that x x; 2) for any istinct x, x both in Λ θ, it hols that x x an x x. The proof is foun in Appenix I-A. One consequence is the following. = Λ θ \ Λ θ Corollary 1: When maximizing either Jain s fairness 7) or the α-fair objective 8) over Λ subject to a throughput equality constraint T x) = θ for θ [θ n, 1), it suffices to restrict the feasible set the set of points on the bounary of Λ that satisfy the throughput constraint, i.e., to Λ θ efine in Prop. 2). This then implies an optimal control, p, efine in IV-B, is in S. This corollary follows almost immeiately from Prop. 1 an Prop. 2 item 1)) taking into account the fact that p S iff xp) Λ [25]. An inepenent proof is given in Appenix I-A for the case of Jain s fairness, highlighting the geometric intuition behin the result. B. Optimal controls uner a throughput constraint In this subsection we present two results that apply to both the Jain s fairness analysis in IV an the α-fair analysis in V. First, we efine some useful restrictions of the feasible set of controls in Def. 1; this restriction is an essential component in most of our subsequent proofs. Secon, in Prop. 3 we present some properties associate with the throughput constraint T xp)) = θ over this restricte set. Finally, Prop. 4 establishes that the optimal controls for both fairness objectives will lie in the restricte set in Def. 1. Definition 1: Let p [0, 1) n be a control, an efine the following:

10 10 1) Vp) = i [n] {p i} \ {0}. Thus Vp) Vp) ) enotes the set number) of istinct nonzero values 3 in p. 2) S 1 = {p S : Vp) = 1} enotes the set of efficient controls with exactly one istinct nonzero value. Note S 1 consists of all vectors p an their permutations) of the form p i = 1/n for i [n ] an p i = 0 for i {n + 1,..., n}, for n [n]. 3) S 2 = {p S : Vp) = 2} enotes the set of efficient controls with exactly two istinct nonzero values. These two values are enote p s, p l for small an large, respectively) with 0 < p s < p l < 1. Moreover, any such p has a total of n nonzero components, of which k take value p s an n k take value p l, for some k [n 1] an some n {2,..., n}, an p s 0, 1/n ). Since p S, it follows that kp s +n k)p l = 1, or equivalently, p l = p l p s, k, n ) 1 kp s n k. 12) We call p s, k, n ) the three free parameters which together characterize a p S 2, an write pp s, k, n ) to enote a p with those parameters. The rates associate with controls p s, p l are enote x s, x l, respectively, with x s = x s p s, k, n ) p s 1 p s ) k 1 1 p l ) n k x l = x l p s, k, n ) p l 1 p s ) k 1 p l ) n k 1 13) an it is easily shown that x s < x l. 4) S 1,2 = {p S : Vp) 2} enotes the set of efficient controls with at most two istinct nonzero values. Because p S it follows that Vp) 0, an thus S 1,2 = S 1 S 2. Observe S 1 may be viewe as the limiting case of S 2 as p s 1/n. Therefore S 1,2 may equivalently be efine as the closure of S 2 an thus p S 1,2 may also be parameterize by p s, k, n ) with the moification that p s 0, 1/n ]. In fact, we will use S 1,2 an S 2 interchangeably with the former highlighting Vp) {1, 2} an the latter emphasizing p s can take the bounary value 1/n. Following the pp s, k, n ) parameterization in Def. 1, we further efine the following shorthans to be use: r x = r x p s, k, n ) x l = p l1 p s ) x s p s 1 p l ) r p = r p p s, k, n ) 1 p s 1 p l. 14) The following proposition gives properties of the solution of the throughput equality constraint T xp)) = θ over p S 2. Leveraging the p s, k, n ) parameterization in Def. 1, we efine for fixe n {2,..., n}): T p s, k, n ) T xpp s, k, n ))) 15) Rk, n ) {T p s, k, n ) : p s 0, 1/n ]} 16) 3 Vp) is the number of istinct nonzero values, not the number of inices taking nonzero values.

11 11 for p s 0, 1/n ] an k [n 1]. Note Rk, n ) is the set of achievable throughputs over p S 2 with fixe k, n ), i.e., the image of T p s, k, n ) over p s 0, 1/n ]. This image is a subinterval of [0, 1] on account of the continuity of T p s, k, n ) in p s. Proposition 3: Assume p S 2 is parameterize using p s, k, n ) as in Def. 1. 1) Fix k, n. The throughput T p s, k, n ) is monotone ecreasing in p s 0, 1/n ], an as such at most one p s 0, 1/n ] will solve T p s, k, n ) = θ. This unique p s, when it exists, is enote by p s k, n, θ), an is the solution to T p s k, n, θ), k, n ) = θ, 17) which can be expresse as an orer-n polynomial in p s ) equation. 2) Now only fix n. The range of achievable throughputs for a given k is Rk, n ) = [θ n, θ n k), which is an increasing in k) neste sequence of intervals: R1, n ) Rn 1, n ). 3) For θ [θ t, θ t 1 ), for some t {2,..., n}, the set of k, n ) pairs for which there exists p s 0, 1/n ] such that T p s, k, n ) = θ is D t,n n {t,...,n} {k, n ) N 2 : k {n t + 1,..., n 1}}, 18) an is illustrate in Fig. 1 left). k n n 1 D t,n k = n 0 1 k = n 0 t +1 1 =1 2 =1/2. t 1 n 0 1 t. t t t 1 t n 0 n n 0 t +1 n 0 n 0 k. n 0 2 n 0 1 n 0 1/e ) 1 2 ) ) ) k n 0 t ) n 0 t +1 ) n 0 2 ) n 0 1 k Fig. 1. Left: Illustration of the region D t,n 18) to scale, the figure shows the case t = 4 an n = 12, with the value n = 8 selecte on the n axis). Right: Illustration that k {n t + 1,..., n 1} is necessary an sufficient when n t) for θ [θ t, θ t 1 ) to intersect Rk, n ) = [θ n, θ n k) shown as soli vertical intervals) in 16). The proof is in Appenix I-B. The following proposition shows that optimal controls for both the Jain s fairness an α-fair objectives will lie in the restricte set of Def. 1.

12 12 Proposition 4: Consier the following two extremization maximization or minimization) problems, each parameterize by α, 1] [1, ) an θ 0, 1): extremize F α xp)) : T xp)) θ, extremize F α xp)) : T xp)) = θ. 19) p [0,1) n p [0,1) n i) For both the inequality an equality constraine problems above, a necessary conition for p to extremize 19) is Vp) 2. ii) For the inequality constraine problem: if an optimizer p of 19) left) has the property that Vp ) = 2, then the throughput constraint hols with equality, i.e., T xp )) = θ. The proof is in Appenix I-B. IV. JAIN-CHIU-HAWE FAIRNESS TRADEOFF Recall from II-D that maximizing F J x) 7) uner a throughput equality constraint T x) = θ is equivalent, in the sense of having the same extremizers, to minimizing F 1 x) 8), i.e., α = 1, uner the same constraint. As mentione in II-C, any x Λ may be expresse as xp) 2) for some p [0, 1] n. Thus, an equivalent formulation of the Jain throughput fairness optimization problem for n users with target throughput θ 0, 1) is: min p [0,1) n F 1xp)) = 1 2 n x i p) 2 s.t. T xp)) = θ. 20) i=1 This section is comprise of three subsections. We give: i) preliminary results in IV-A, ii) the main results in IV-B, an iii) some aitional properties of the Jain throughput-fairness traeoff in IV-C. A. Preliminary results We start with the special case n = 2. Proposition 5: The throughput fairness traeoff uner Jain s fairness metric, for n = 2 users, is FJ 1, θ 0, 1 2 θ) = ]. 21) θ 2 θ 2 +2θ 1, θ 1/2, 1) Proof: For the n = 2 case we may use a irect approach instea of solving 20)), since the set Λ may be written explicitly i.e., parameter-free) as Λ = {x R 2 + : x 1 + x 2 1} [21], illustrate in Fig As evient from the figure, the constraine feasible set is the intersection of the throughput constraint line for general n, a hyperplane) H θ = {x : x 1 + x 2 = θ} with Λ. Define the maximum fairness line {x : x 1 = x 2 } for general n, the ray emanating from the origin 0 passing through 1), on which F J x) = 1. In the case of θ 0, 1/2], we see Λ H θ intersects this ray, i.e., F J x) = 1 is feasible. In the case of θ 1/2, 1), F J x) = 1 is not feasible, but the fairness is easily shown to be monotone increasing on H θ as x moves towars x 1 = x 2 c.f., Fig. 8 in the proof of Cor. 1 in III-A for general n), an as such, the optimal fairness is achieve at the two points for which H θ intersects Λ = {x R 2 + : x 1 + x 2 = 1}. These two equations together yiel the solutions x 1, x ) 2) = θ± 2θ 1 2, θ 2θ 1 2, from which the maximum fairness may be compute to be the secon expression in 21). 4 As an asie, the stability inner boun Λ is known to be exact, i.e., Λ A = Λ, for the case n = 2 [21].

13 13 x = {x 1,x 2 ): p x 1 + p x 2 =1} H = {x 1,x 2 ):x 1 + x 2 = {x 1,x 2 ):x 1 = x 2 } 0.2 H 1/3 H 3/5 0.0 x Fig. 2. Illustration of the proof of Prop. 5, the Jain throughput fairness traeoff for n = 2 users. Shown are the set Λ, its bounary Λ, two throughput constraint hyperplanes H θ for θ {1/3, 3/5}, an the maximum fairness line {x 1, x 2 ) : x 1 = x 2 }. The constraine feasible set Λ H θ bol line segments) intersects the maximum fairness line on which F J x) = 1) for θ 1/2, but not for θ > 1/2. The basic iea in establishing the Jain throughput-fairness traeoff Thm. 1) is to first apply Cor. 1 in III-A to restrict the feasible set from p [0, 1) n to S, then apply Prop. 4 in III-B to further restrict it to S 1,2, an finally Thm. 1 is prove by employing Prop. 3 in III-B an Prop. 6 below, the proof of which is foun in Appenix II-A. Leveraging the p s, k, n ) parameterization of p in Def. 1, recall the efinition of T p s, k, n ) in 15) in III an observe the Jain objective F 1 xp)) in 20) may be written as F 1 p s, k, n ) F 1 xpp s, k, n ))). 22) Prop. 6 establishes two key monotonicity properties of the objective 22) uner the throughput equality constraint over the restricte set p S 2. Proposition 6: Uner the constraints p S 2 with p = pp s, k, n )) an T p s, k, n ) = θ, the objective F 1 p s, k, n ) 22) obeys the following two monotonicity properties for all k, n ) D t,n efine in 18): 1) F 1 p s, k, n ) < F 1 p s, k + 1, n ) 2) F 1 p s, k, n ) < F 1 p s, k + 1, n + 1). In Fig. 1 left), the two monotonicity results show F 1 is ecreasing in k along any vertical line fixe n ), an along any iagonal line with unit slope fixe n l = n k). B. Main results For general n, θ), where n > 2 an θ 0, 1), we are not able to obtain an explicit expression for the throughput fairness traeoff, primarily because there is no known explicit characterization of Λ for n > 2. If x is an optimal

14 14 rate vector, i.e., a minimizer of 20), then we refer to any p satisfying xp ) = x as a corresponing optimal control. The main theorem of this subsection is an implicit characterization of this traeoff, meaning we characterize p for each θ as the solution of a polynomial equation), from which we can compute F 1 xp )). We reiterate the permutation invariance of both x an p. Theorem 1 Throughput fairness traeoff uner Jain s fairness): The throughput fairness traeoff for n 2 users uner Jain s fairness metric, with a throughput equality constraint T x) = θ, for θ 0, 1), inclues three regimes, illustrate in Fig. 3, parameterize by θ: 1) if θ < θ n, then the maximum fairness is F J = 1, achieve when every user receives equal rate: x ip ) = θ/n. 2) if θ = θ t for some t [n], then p = 1/t) t i=1 e i, with the corresponing maximum fairness F J = t/n. The function T F ) = 1 1 ) nf 1 23) nf is a monotone, ifferentiable, an convex interpolation between the points {θ t, t n )} t [n]. 3) if θ θ t, θ t 1 ) for some t {2,..., n}, then p = p se 1 + p l t i=2 e i where p l = p lp s, k, n ) accoring to 12) with k = 1, n = t, an p s the unique real root on 0, 1/t) of the following orer-t) polynomial in p s ) equation: p s 1 p l ) t p s ) 2 1 p l ) t 2 = θ. 24) The proof is foun in Appenix II-B. The T-F traeoff plots for n = {1,..., 4} users are illustrate in Fig. 4 right) where regime 1) is omitte. Remark 2: It can be verifie that in the statement of Thm. 1, regime 2) can be merge into 3) by allowing 24) to be solve for p s on 0, 1/t]. They are state separately for conceptual clarity an better consistency with the proof of Thm. 2. In aition, regime 2) is where we have a close-form expression for both the extremizer an the optimize objective. Regime /e 1/2 n n 1 2 1,T 0 Regime 1 ) ) ) Regime 3 Fig. 3. Illustration of the three regimes, parameterize by θ, in Thm. 1: regime 1 is θ 0, θ n), regime 2 is θ = θ 1,..., θ n), an regime 3 is n t=2 θt, θ t 1). As motivate in II-D, the throughput inequality constraint is natural from the operational perspective of wishing to maximize fairness subject to a minimum throughput requirement. As may be intuitive, this moification to the

15 15 constraint feasible set) has no effect on the solution, as shown in the following theorem. Theorem 2: The solution in Thm. 1 of the Jain s throughput fairness traeoff 20) is unaffecte by changing the throughput equality constraint to an inequality constraint T xp)) θ. The proof is foun in Appenix II-B. C. Properties of the Jain T-F traeoff As can be seen from Thm. 1, the extremizer p = pp s, k, n ), with p s solving T p s, k, n ) = θ in 24), has the property that n, the total number of active users i.e., users with nonzero contention probabilities), equals t, where θ [θ t, θ t 1 ), for t {2,..., n}. In fact, because 24) oes not epen on n, the total number of users in the system, one can easily verify that, if θ θ n l for some integer l {1,..., n 2}, then the extremizer p is as if the total number of users in the system were n l, except that l zeros nee to be pae in orer to make p an n-imensional vector. It follows that the maximum Jain s fairness satisfies FJ θ; n) = 1 l ) FJ θ; n l), θ θ n l, 25) n where our notation highlights F J is a function of θ an is parameterize by n. One use of the recursive relationship 25) is that it enables incremental plotting of the T-F traeoff for a sequence of values of n {2,..., n max }. From Thm. 1 if θ [θ n, θ n 1 ) then n = n, meaning, at the optimum, every user in the system is active. We therefore call the interval [θ n, θ n 1 ), for each n N, the active throughput interval, meaning all n users are actively contening uner the optimal control for any target throughput θ in this interval. This observation is the root iea in the Jain T-F plotting algorithm Alg. 1), which returns a plot of the Jain T-F traeoff over θ 0, 1) for all n {2,..., n max }. Naturally, the interval [θ n, θ n 1 ) must be iscretize for each n. Fig. 4 left) illustrates Alg. 1 for n max = 4 users. First, the plot of F J θ; 2) over θ [θ 2, θ 1 ) i.e., the active interval for n = 2, thick blue) is scale using 25) to obtain FJ θ; 3) an F J θ; 4) over the same interval thin blue for both). Then, the plot of F J θ; 3) over θ [θ 3, θ 2 ) i.e., the active interval for n = 3, thick purple) is scale to obtain FJ θ; 4) over the same interval thin purple), an so on. Note first that, for each n, at θ = 1 the maximum Jain s fairness is the minimum possible, i.e., F J = 1/n, corresponing to the fairness when only one user say i) contens for access i.e., x = p = e i ), as x = e i is the unique up to permutation) rate vector in Λ achieving θ = 1. Secon, for each n, for any θ θ n the maximum Jain s fairness is the maximum possible, i.e., F J = 1, corresponing to all n users contening with equal probability, uniquely achievable by the rate vector x = θu. The Jain T-F traeoff for each n up to 4 users is shown in Fig. 4 right). The following theorem gives some properties of the optimal controls, optimal rates, an the Jain T-F traeoff. Theorem 3: The Jain T-F traeoff for n 2 users, over θ [θ n, 1), has the following properties: 1) For fixe n, the small an large contention probabilities of the optimal control, p sθ), p l θ), an the corresponing optimal rates, x sθ), x l θ), are piecewise ecreasing an increasing, respectively, in θ. More precisely, fix t {2,..., n} an θ [θ t, θ t 1 ). Then:

16 16 Algorithm 1 Jain T-F traeoff for all n {2,..., n max } 1: for n = 2,..., n max o 2: Plot F J θ; n) = 1 for θ [0, θ n) 3: for θ [θ n, θ n 1 ) o 4: Compute p sθ) solving T p s, 1, n) = θ i.e., 24) in Thm. 1 with t = n) 5: Compute F J θ; n) = F Jxpp sθ), 1, n))) using 2), 7), an Def. 1 6: en for 7: Plot F J θ; m) = n m F J θ; n) for m {n,..., n max} 8: en for a) Both p s an x s are continuous an ecreasing over each interval [θ t, θ t 1 ), but are not monotone over [θ n, 1). In particular, i) p s θ) θ < 0, x s θ) θ < 0, ii) at θ = θ t they take values p sθ t ) = 1/t, x sθ t ) = θ t /t, an iii) at θ = θ t 1 they take value p sθ t 1 ) = x sθ t 1 ) = 0. b) Both p l an x l are continuous an increasing over [θ n, 1), but neither is ifferentiable at each θ t for t {2,..., n 1}. In particular, i) p l θ) θ > 0, x l θ) θ > 0, ii) at θ = θ t they take values p l θ t) = 1/t, x l θ t) = θ t /t, an iii) at θ = θ t 1 they take value p l θ t 1) = 1/t 1) an x l θ t 1) = θ t 1 /t 1). 2) For fixe n, the T-F traeoff curve is ecreasing in θ, i.e., θ F J θ; n) < 0. 3) For fixe θ, the T-F traeoff curve is ecreasing in n, i.e., FJ θ; n) > F J θ; n + 1). 4) For fixe n, the T-F traeoff curve is continuous but nonifferentiable at {θ t } n 1 t=2, i.e., F J θ; n) θ θ t = F J θ; n) θ θ t, but θ F J θ; n) θ θt θ F J θ; n) θ θt for each t {2,..., n 1}. 5) For fixe n, the T-F traeoff curve is piecewise convex in θ, i.e., 2 θ F 2 J θ; n) > 0, for θ [θ t, θ t 1 ) with t {2,..., n}. The proof is foun in Appenix II-C. Fig. 5 shows p sθ), p l θ) left) an x sθ), x l θ) right), illustrating property 1) in Thm. 3. Properties 2) through 5) in Thm. 3 can be seen from Fig. 4 right). Finally, we mention that a plot of the interpolate function T F ) 23) in Thm. 1 not shown) on the actual T-F traeoff in Fig. 4 woul show the interpolation lies above the true traeoff, an is tight only at the critical throughputs θ. V. α-fair NETWORK UTILITY MAXIMIZATION In this section we investigate the throughput-fairness traeoff within the framework of α-fair utility functions [6], [7]. Recall the objective F α for α 0), the α-fair utility function U α, the throughput function T, an the mapping between a control p an a rate vector xp) given in 8), 9), 5), an 2) respectively. The optimization uner a throughput equality constraint is: max p [0,1) n F αxp)) = n U α x i p)) s.t. T xp)) = θ. 26) i=1 We solve this problem for α 1. In this following we give i) preliminary results in V-A, ii) the main results in V-B, an iii) some aitional properties of the α-fair throughput-fairness traeoff in V-C.

17 17 FJ ; n) FJ ; n) /4 2/ /2 1/ /3 1/3 1/ /e / /e Fig. 4. Left: Illustration of using Alg. 1, leveraging the Jain fairness recursion 25), to incrementally plot the Jain T-F traeoff for n max = 4 users. Vertical grilines inicate the θ t s for t {2, 3, 4}. Horizontal grilines inicate the maximum fairness at the θ t s for each t [n] an each n n max. The T-F traeoff for the active throughput intervals thick curves) nee to be compute first, after which the rest parts thin curves) can be obtaine by scaling. Right: Thm. 1 regimes 2) an 3)): T-F traeoff uner Jain s fairness for n = 1 blue), 2 orange), 3 green), an 4 re). p ) 1.0 n 0 = x ) 1.0 n 0 = / /3 1/ / /3 4/ Fig. 5. Illustration of property 1) in Thm. 3: Optimal controls p sθ) left, lower/thinner branches), p l θ) left, upper/thicker branches) an optimal rates x s θ) right, lower/thinner branches), x l θ) right, upper/thicker branches) versus target throughput θ, for n = 4 users. Vertical grilines inicate the θ t s: θ 2, θ 3, θ 4 ) = 1 2, 4 9, 27 ) 0.5, , ). Horizontal grilines inicate the corresponing optimal controls 64 left) an optimal rates right) when θ = θ t. Shown also are the optimal number of active users n for ifferent ranges of θ. A. Preliminary results We start with the special case n = 2. Proposition 7: The throughput fairness traeoff uner α-fairness α 1), for n = 2 users, is 2 2 ) α 1 α 1 θ, θ 0, 1 2 ] α > 1 1 ) 1 α ) ) 1 α θ+ 2θ 1 Fαθ) α θ 2θ 1 2, θ 1 2 =, 1) 2 log 2 θ, θ 0, 1 2 ] α = 1 2 log 2 1 θ, θ 1 2, 1) 27)

18 18 Proof: The proof resembles that of Prop. 5 in IV-A. The all-rates equal ray {x : x 1 = x 2 } can still be viewe as the maximum fairness line as the maximum α-fair objective is attaine by points either on this line or closest to this line, subject to the throughput constraint x 1 + x 2 = θ. This follows from the Schur-concavity of the objective Prop. 1 in III-A) an the proof of) Cor. 1 in III-A. Therefore, when θ 1/2, the maximizer is on the ray {x : x 1 = x 2 } an hence x 1, x 2) = θ 2, θ 2 ); when θ > 1/2, the maximizer is obtaine by fining the points on the bounary of Λ that satisfy the throughput constraint as they are the closest to the all-rates equal ray, see Fig. 8), which gives x 1, x ) 2) = θ± 2θ 1 2, θ 2θ 1 2. Substitution of the expressions of the maximizers into the objective yiels 27). The basic iea in solving the throughput-fairness traeoff uner α-fairness Thm. 4) is to first apply Cor. 1 in III-A to restrict the feasible set from p [0, 1) n to S, an then apply Prop. 4 in III-B to further restrict it to S 1,2. The optimization problem is solve with the ai of Prop. 8 shown below, which establishes a key monotonicity property of the objective in 26) uner the throughput equality constraint over the restricte set p S 2. It plays a similar role to that of Prop. 6 in proving Thm. 1 IV-B). Leveraging the p s, k, n ) parameterization of p in Def. 1 an the efinition of T p s, k, n ) in 15) in III-B we efine Proposition 8: Uner the constraints p S 2 F α p s, k, n ) F α xpp s, k, n ))). 28) with p = pp s, k, n )) an T p s, k, n ) = θ, the objective F α p s, k, n ) 28) for α 1 is increasing in k for k {1,..., n 1} when n is hel fixe. Thus the maximum of F α p s, k, n ) is attaine when k = n 1. The proof is foun in Appenix III-A. B. Main results For general n, θ), where n > 2 an θ 0, 1), we will again give an implicit characterization of the T-F traeoff uner α-fairness when α 1. The main theorem of this subsection is a characterization of the optimal control p for each θ as the solution of a polynomial equation) from which we can compute F α xp )). Theorem 4 Throughput-fairness traeoff uner α-fair when α 1): The throughput fairness traeoff for n 2 users uner α-fairness when α 1, with a throughput equality constraint T x) = θ, for θ 0, 1), inclues two regimes, parameterize by θ: 1) if θ θ n, then the maximum fairness is Fαθ) n log ) n θ, α = 1 = n ) α 1, 29) θ, α > 1 n α 1 achieve when every user receives equal rate: x i p ) = θ/n. 2) if θ θ n, 1), then p = p se 1 + p l n i=2 e i where p l = p lp s, k, n ) accoring to 12) with k = n 1, n = n, an p s the unique real root on 0, 1/n) of the following polynomial equation n 1)p s ) 2 1 p s ) n n 1)p s )1 p s ) n 1 = θ. 30)

19 19 The proof is foun in Appenix III-B. The T-F traeoff plots for n = {1,..., 4} users are illustrate in Fig. 6. Observe the ifference between regime 1) in Thm. 4 for α-fairness when α 1 an regime 1 in Thm. 1 for Jain s fairness: although the maximizers are the same, the objective is increasing in θ in the former, whereas it is constant in the latter. Observe also the asymmetry between regime 2) in Thm. 4 an regimes 2) an 3) in Thm. 1: k = n 1 an n = n for all θ θ n, 1) in the former, while k = 1 an n = t for θ [θ t, θ t 1 ) in the latter. Thus, the optimal control vector p for α-fairness has n 1 users with small contention probability p s an one user with large contention probability p l for n always equal to n, while the optimal control vector p for Jain s fairness has one user with p s an n 1 users with p l, for n etermine by the active throughput interval containing θ. Similar to IV-B, we now aress the case where the throughput constraint in 26) is an inequality T xp)) θ. Theorem 5: If the throughput equality constraint is change to an inequality constraint T xp)) θ then the solution in Thm. 4 of the α-fair utility maximization problem 26) when α 1 is only affecte in the first regime, namely when θ θ n. More precisely, if θ θ n, then the maximum fairness is inepenent of θ an is given by ) n n log Fαθ) θ n, α = 1 = ) α 1, 31) n n α 1 θ n, α > 1 where the maximizer in the control space is a uniform vector p = u. The proof is foun in Appenix III-B. C. Properties of the α-fair T-F traeoff The follow theorem gives some properties of the T-F traeoff for the α-fair objective. Theorem 6: The T-F traeoff for n 2 users uner α-fairness for α 1, with target throughput θ θ n, 1), has the following properties: 1) For fixe α an n, the smaller p s) an larger p l ) components of the optimal control are ecreasing an increasing in θ respectively, i.e., p s θ) θ < 0, p l θ) θ > 0. The smaller x s) an larger x l ) components of the corresponing optimal rate vectors are likewise ecreasing an increasing in θ, i.e., x s θ) θ < 0, x l θ) θ > 0. 2) For fixe α an n, the maximum α-fair objective F α) is ecreasing in θ i.e., θ F αθ; n) < 0, an is continuous an ifferentiable. For n = 2, Fαθ; 2) is concave i.e., 2 θ Fαθ; 2) < 0). For n > 2, there exists 2 a throughput threshol θ α n) such that F αθ; n) is convex concave) in θ for θ < θ α n) θ > θ α n)). 3) For fixe α an θ θ n, 1), the maximum α-fair objective is ecreasing in n, i.e., F αθ; n) > F αθ; n + 1). The proof is foun in Appenix III-C. Fig. 7 shows p sθ), p l θ) left) an x sθ), x l θ) right), illustrating property 1) in Thm. 6. Fig. 6 illustrates properties 2) an 3) for the cases of α = 1 left) an α = 2 right). VI. CONCLUSION We have presente six theorems that characterize the throughput fairness traeoff uner slotte Aloha, using both Jain s fairness measure Theorems 1-3), an the α-fair measure Theorems 4-6). The key property enabling

20 20 F1 ; n) F1 1; 1) F1 2; 2) -5 F1 3; 3) F2 ; n) F2 1; 1) F2 2; 2) F2 3; 3) F1 4; 4) F2 4; 4) Fig. 6. Illustration of Thm. 4 an properties 2) an 3) in Thm. 6: T-F traeoff uner α-fairness when n = 1 blue), 2 orange), 3 green), an 4 re) users, for α = 1 left) an α = 2 right). Vertical grilines inicate the θ t s an horizontal grilines inicate the corresponing optimal α-fair objective for each n at θ = θ n i.e., Fαθ n; n). Shown as cyan ots are the inflection points upon which the T-F curves transitions from convex ecreasing to concave ecreasing, for n > 2. The thresholing θ αn) is compute using 143). p ) 1.0 x ) / /3 1/ / /3 4/ Fig. 7. Illustration of property 1) in Thm. 6: Optimal controls p sθ) left, lower/thinner branches), p l θ) left, upper/thicker branches) an optimal rates x sθ) right, lower/thinner branches), x l θ) right, upper/thicker branches) versus target throughput θ, for n = 2 blue), 3 purple), an 4 yellow) users. Vertical grilines inicate the θ t s: θ 2, θ 3, θ 4 ) = 1 2, 4 9, 27 ) 0.5, , ). Horizontal grilines inicate the 64 corresponing optimal controls left) an optimal rates right) when θ = θ t. Different from the case of Jain s fairness see Fig. 5 in IV-C where only the plots for n = 4 users are shown), here n = n hols irrespective of the value of θ. the analysis is Prop. 4, which reuces the set of potential extremizers of the fairness functions from [0, 1) n to S 1,2, i.e., those controls taking at most two nonzero values. Theorems 1 an 3 aress the case of a throughput equality constraint, T x) = θ, an Theorems 2 an 4 aress the case of a throughput inequality constraint T x) θ. The main point is that the throughput fairness traeoff is the same for both types of constraints for θ θ n ). The key ifference between the Jain an α-fair traeoff uner a throughput constraint θ [θ t, θ t 1 ) is in the nature of the optimal controls: to maximize the Jain fairness objective requires n = t active users, of which k = 1 use a small contention probability an rate an t 1 use a large contention probability an rate, while to maximize the α-fair