Fairness via priority scheduling

Transcription

1 Fairness via priority scheduling Veeraruna Kavitha, N Heachandra and Debayan Das IEOR, IIT Bobay, Mubai, , India vavitha,nh,debayan}@iitbacin Abstract In the context of ulti-agent resource allocation probles, fairness is a paradig shift in the recent past An efficient scheduler always allocates resources to the best agent Soe of the agents, who are ost often in bad conditions, are starved and fair schedulers are defined in this context In this paper, we are interested in the actual gains obtained by the otherwise starved agents, due to fair schedulers We propose a new notion of fairness, via a constrained optiization, which directly indicates the gains In general, this constrained optiization is an infinite diensional proble and the priary contribution of this paper is to reduce it to a tractable finite diensional zero finding proble We indicate iterative algorith(s) which achieves the notion of fairness defined in this paper We also copare it with soe of the existing notions of fairness Index Ters Fairness, Resource allocation, Infinite diensional convex optiization, Stochastic approxiation, Wireless counications I INTRODUCTION In resource allocation probles, agents ay have hoogeneous (siilar) or heterogeneous job requireents In heterogeneous cases, one of the iportant aspects of perforance is the distribution of the accuulated gains across various agents Fairness is an appropriate easure of evaluating the resource allocation schees in such scenarios In any such probles, the resource allocated ight be coon across the agents but the utility derived by an agent depends upon the agent Further there can be rando variations in the utility derived based on the rando state of the agent, at the tie the resource is allocated For exaple, consider a wireless counication syste with a nuber of agents (obiles) copeting for resources fro a single base station The coon resource to be allocated could be a tie slot, while the utility derived could be the aount of inforation conveyed during the tie slot, which in turn depends upon the state of channel between the agent allocated and the base station Opportunistic resource allocation taes advantage of the tie variations in the states of the copeting agents and allocates resources to the users with best state This approach is shown to be advantageous when the rando tie variations in states of the agents are independent across the various tie slots at which the allocation decisions are ade This approach results in an efficient solution (see for eg, [5], [11] and references therein), wherein the overall utility accuulated, because of such allocations, over all the tie slots and over all the agents would be the axiu This is a desirable feature fro the point of view of the central unit (CU) allocating the resources This however can regularly deprive the agents, whose states are bad with high probability, resulting in very sall or alost negligible utility accuulations for the This is not acceptable fro the point of view of the starved agents and that lead to the notion of fairness Fairness is a well studied concept (see [5], [4], [3], [11] and references therein) The notion of proportional fairness is introduced by Kelly etal [1] and a recent survey of various aspects related to fairness is available in [2] Fairness is studied either via Jain s index ([3]) or is achieved by the CU optiizing a certain concave function of the accuulated utilities called α-fair function ([5], [11], [4]) Well nown notions of fairness, naely proportional fairness, ax-in fairness, etc, are achieved in this anner A recent wor ([4]) develops a faily of fairness functions, paraetrized by two nubers, which unify all the previously developed fairness easures They show that this is the only faily of functions satisfying four siple axios of fairness etrics In this paper, we focus on a copletely different view point related to the sae proble The central question ased here is How uch exactly is the iproveent of the accuulated utilities of the starved users because of fair allocation? We obtain soe answers in a very siple scenario, when one uses α-fair schedulers ([5], [11]) Few answers Let there be M agents with 1,, M} being a typical agent Let u = (u 1,, u M ) represent the vector of (rando) instantaneous utilities of all the M agents and let β = (β 1,, β M ) represent the scheduler in the following sense: β (u) represents the probability with which agent is allocated given that the current vector of instantaneous utilities equals u An α-fair resource allocation β is obtained by axiizing the following function ([5], [11]): ax β Γ α (ū (β)) with ū (β) := E[u β (u)], Γ α (ū) := ū1 α 1 α 1} 1 α + log(ū)1 α=1} Here ū (β) is the accuulated utility of agent and by [11, Lea 1] the above α-fair scheduler satisfies the following: [ u ū ;α = E u Π j 1 (ū ;α ) α u }] j (ū j;α ) α We first consider an extreely siplified scenario and obtain the following result about the accuulated utilities ū ;α } (proof in Appendix A)

2 Theore 1 Consider a scenario with instantaneous rates given by u = a X, where X } M are IID rando variables and a > 0 is a constant for every That is, the instantaneous utilities of the agents are identically distributed except for constant ultipliers i) When a proportional fair scheduler (when α = 1) is used the accuulated utilities are proportional to their ean value ie, to a : ū ;1 = a ū c with constant ū c := E[X Π j 1 X X j}] ii) With two agents (ie, M = 2), the one with larger a (wlg let a 2 > a 1 ) will have higher accuulated utility when an efficient scheduler (α = 0) is used and this accuulated utility decreases while that of the lower utility agent (ie, agent 1) increases as the fairness factor increases (ie, as α increases): a) ū 2;0 > ū 1;0 ; b) As α, ū 2;α and ū 1;α In a rather siplified scenario of Theore 1, we could predict soe properties of the accuulated utilities as a function of fairness factor, α We could probably extend soe of these results little ore, ay be when restricted to the setting of Theore 1 or probably under slightly weaer assuptions But it not clear whether these answers can be obtained under fairly general assuptions Alternatively, these answers can directly be obtained, if fairness is achieved by deanding a lower bound on the accuulated utilities of the starved users So, we propose a constrained optiization to achieve a fair scheduler (see Section III) In general, this is an infinite diensional linear optiization proble with linear constraints We reduce this to a finite diensional optiization proble and following are the intuitive ideas behind the sae The instantaneous efficient decisions (without considering fairness) always chose the one with axiu instantaneous utility It is clear that, to achieve fairness, one needs to deviate fro the efficient decisions for soe values of instantaneous utilities It is also intuitive that we obtain the best, even under the fairness constraints, if these deviations fro the efficient decisions are ade when the deviations result in inial losses In [9], [10], Kleinroc et al introduced a class of priority based scheduling algoriths in the context of queuing systes with ultiple classes of custoers, wherein a scheduling decision is ade after ultiplying the waiting tie of the longest waiting custoer of each class with a priority factor assigned to the class We observe that a siilar paraetrized class of schedulers can achieve the required onotone shift of efficient decisions and prove that achieving fairness is equivalent to choosing an appropriate priority factor We ae precise these concepts in this paper and show further that the fairness constraint optiization can be solved using a very siple iterative zero finding algorith, whose coplexity is siilar to the coplexity of the original α fair scheduling algoriths of [5] In Section II the proble is introduced In Section III one faily of fair schedulers are introduced while the other extensions and coparisons are discussed in Section V Conclusions and future directions are also discussed in the sae section Nuerical exaples are provided in Section IV and the proofs and related details are provided in Appendices II SYSTEM AND BACKGROUND A coon resource is shared by M agents Upon allocation of the resource, agent gains a utility u, where u is copletely deterined by its state h The state of the resource h is a rando variable and is independent across the agents The tie is divided into slots and resource is allocated once in a slot We are interested in the accuulated utilities, or equivalently the tie averages of the utilities and we study the sae via their expected values The state h for any agent is an identical and independent process across the tie slots We assue that the instantaneous utilities u } are deterinistic functions of the states h } and to siplify the notations we wor directly with the rando utilities u } We assue u U, where U is a subset of R The agents are interested in the average utility obtained by theselves: ū (β) = E[u β (u)] with u := (u 1, u 2,, u M ) The ai of a central unit (CU) is to obtain a scheduler β = (β 1,, β M ) which achieves a given goal with respect to these average utilities This resource allocation proble is a typical exaple of a ulti agent gae theoretic proble in which the central unit (CU) also participates The utility of an agent copeting for resources is obvious: it is the utility accuulated by the agent due to allocations (ū for agent ) On the other hand the utility of central unit (CU) in the gae theoretic context will depend upon its objective, lie for exaple, fairnessefficiency trade-off, level of fairness, etc, (see [11] for ore details) We do not pursue the gae theoretic approach in this paper, rather transfer the goals of the individual agents to the CU itself It is the CU which ensures that the fairness goals of all the agents are satisfied, whenever possible as would be evident in the sections below Efficient scheduler As already entioned the CU can have varying objectives depending upon the design If its gain is based solely upon the total utility transferred, irrespective of the agent that received the utility, it is would naturally schedule in every tie slot the agent with axiu instantaneous utility This is a situation in which the CU axiizes the su of the average utilities over an appropriate doain: ax Υ(β) β Ω with Υ(β) := [ ] ū (β) = E u β (u) (1) We next describe the doain of optiization Ω Let 2 represent the L 2 -nor, ie, u 2 = E[u 2 ] We extend this naturally to any β = (β 1,, β M ) L 2 (we still denote

3 this space by L 2 ) by β 2 = β 2 2 = E[β 2 ] for any β L 2 sae However, we show that a finite diensional scheduler optiizes the above proble and towards this, we consider the following sub faily of schedulers inspired by delay priority schedulers ([9]): Now the doain of optiization Ω L 2, is defined as below: Ω := β : Π U [0, 1] M ; β (u) = 1 for all ost all u} (2) The scheduler axiizing (1) is called an efficient scheduler and it is easy to see in this case that the efficient scheduler is given by Let ū eff β eff (u) = 1 arg ax u =} (3) denote the corresponding accuulated utilities, ie, ū eff := ū (β eff ) = E[u 1 arg ax u =}] (4) III FAIRNESS VIA CONSTRAINED OPTIMIZATION Consider a siple exaple scenario with two agents The states of one agent is better than the other with probability one, ie, assue u 1 > u 2 with probability one In this case, if CU uses efficient scheduler (3), then agent 2 is never allocated a resource and its average utility is zero Siilarly, if an agent has its utility lesser than the other agent with high probability, he would obtain non zero utility, but nevertheless would be a very sall quantity So, such agents are discouraged to use the facility offered by the CU This calls for a different scheduler which ensures that all the agents obtain satisfactory utility That is, the scheduler should ensure every agent obtains a fair share for hiself The fair shares could depend upon 1) the context (eg, in wireless counications, user with data transfer requests can wait longer than the user with a counication call request), 2) the price paid for service (eg, one class of users ight be willing to pay ore to obtain better service), 3) type/size of requests (eg, soe users ight require short jobs while others ight require service for longer duration) etc When one deviates fro the efficient scheduler (1) to obtain a required level of fairness, it is clear that efficiency is reduced So, the goal of this wor is to introduce a class of schedulers which axiize the efficiency under the constraint that fairness is aintained to a required level We begin by introducing the following notion of Θ- fairness (with a given Θ := (0, θ 2, θ 3,, θ M )), given by the following constrained optiization βθ := arg ax Υ(β) with Υ(β) := ū (β) β Ω subject to ū (β) θ for all 2 (5) Here without loss of generality, we assue ū eff 1 ū eff for all 2 In general, proble (5) is an infinite diensional constrained optiization proble and it is difficult to solve the Ω := β(b) = (β 1 (b), β 2 (b),, β M (b)) with β (b)(u) := 1 arg ax u b =}} (6) The intuitive reasons for this choice is already discussed in Introduction As seen fro (6), the faily of priority schedulers are paraetrized by the finite diensional vector b = (b 1, b 2,, b M ) Throughout we assue b 1 = 1 while b 1 for all, which obviously iproves the utilities of lower utility agents Assuptions Without loss of generality (wlg), we assue that the agents are arranged in the increasing order of their efficient utilities, ie, ū eff 1 ū eff 2 ū eff M We assue a wor conserving principle, ie, the CU always schedules the resource whenever there is a utility to be transferred (ie, β = 1 for all states with non zero utilities, ie, whenever u 0) We wor under the following assuptions A1 Assue P rob(u 0) = 0 and E[u 2 ] < for all 1 M A2 When θ > 0, we assue θ > ū eff for all > 1 A3 The set of constraints in proble (5) are achievable, in fact there exists at least one β Ω such that: ū (β) > θ for all with θ > 0 A4 The instantaneous utilities u are continuous rando variables for each, ie, they have density These processes are also independent across the agents We iediately have the following two results, whose proofs are in Appendix A Theore 2 There exists a solution for proble (5) in Ω, under the assuption A1 Theore 3 The optiizer of the constrained optiization (5) satisfies all the constraints under A1 and A2 We now consider two different cases: one when the utilities are continuous rando variables (which satisfies A4) and one when they are discrete We will see that Ω itself is finite diensional in the discrete case, while this is not true in the continuous case, but however a reduction is possible A Continuous utilities We begin with reducing the infinite diensional constrained optiization proble (with doain Ω) to that of a finite diensional one (Ω ) We also establish the uniqueness of the optiizer in Ω Theore 4 Assue A1-4 Then, 1) There exists a β Ω which is an optiizer of the constrained optiization proble (5), ie, there exists a b R M 1 and: β = β(b ), with β (b )(u) := 1 arg ax u b =}

4 By Theore 3, b satisfies the fairness constraints with equality, ie, ū (β ) = θ for all 2 2) Further for any 2, b = 1 if θ = 0, ie, for those agents without constraint 3) There exists a unique eleent in Ω which satisfies all the constraints This unique eleent is the unique optiizer in Ω for the constrained optiization proble (5) Proof: is in Appendix A A new fair scheduler : We propose the following two tie scale stochastic approxiation based algorith to obtain the zeros of fairness constraints equation Here represents the tie slot ( ) ū,+1 = ū, + µ u, 1 β =} ū, β = arg ax (b,u, ), b,+1 = b, + ɛ (ū, θ ) for all 1, for all with θ > 0, and with b, = 1 for all with θ = 0 The first iteration estiates the average utilities of agent, while the second iterate updates the scheduler representative b to converge towards a point that satisfies the constraints and hence is an optial scheduler The second iterate is updated at a slower rate: ɛ = Cµ 1 N,2N,3N,4N, }} for soe large N and suitable C 1 In Section IV we establish that the above algorith indeed obtains the required zeros asyptotically via soe nuerical exaples One can alternatively consider a single tie scale version, wherein the second iterate is updated fast and is updated directly with the help of the 1-estiate u, 1 =β, in place of ū, (for all with θ > 0): (7) b,+1 = b, + µ (u, 1 β =} θ ) (8) This algorith is also considered in Section IV The theoretical evidence to establish the convergence of these algoriths and choosing the best aong the possible variants, would be taen up as a future wor B Discrete Utilities Consider a case in which all agents have discrete utilities Agent can tae any one of N different utilities and let N := Π M N define the total possible nuber of instantaneous utility vectors u By indexing each one of u equivalently with a nuber 1 j N, one can rewrite the doain of optiization as a subset of [0, 1] NM as below: Ω = β = (β 1 (j),, β M (j))} j N with β (j) [0, 1],, j and such that β (j) = 1 for all j N} Interesting point to notice for discrete case is that, Ω by itself is a finite diensional subset However in this case, any conclusions of Theore 4 are not true and the technical reasons for the sae is discussed in Appendix A after the proof of Theore 4 We ainly do not have soe uniqueness properties and in Appendix B, via a siple exaple we discuss the issues and advantages related to discrete case However Theores 2, 3 are true and by virtue of these one can again obtain the optiizer as a (finite diensional) zero of fairness constraints equation One can odify the algorith (7), in an obvious way, to update directly the scheduler coponents β (j)} ( ) ū,+1 = ū, + µ u, 1 β a =} ū, for all 1, β a = a rando allocation generated with probability vector (β 1, (u ),, β M, (u )) β,+1 (u ) = β, (u ) + ɛ (ū, θ ) β,+1(u ) = 1 :θ >0 = arg ax for all with θ > 0 and β,+1 (u ); with u,, the efficient decision Of course the required projection has to be tae care in updates of β, } However we do have an issue: not all the zeros are optiizers (see Appendix B) Further the diension of Ω increases with the nuber of states and or the nuber of agents and this increases the coputational coplexity In such cases, one can approxiate the optial scheduling policy with that of an appropriate continuous counterpart and this procedure is explained in the technical report ([12]) Since there exists a unique zero in Ω in continuous case which is also the axiizer, we expect this to represent approxiately the optiizer of the discrete case also, whenever the diension is large IV ILLUSTRATIVE EXAMPLES A Continuous utilities: Exaple 1 By Theore 4, Θ-fairness is achieved by an unique zero of the equation representing fairness constraints in the finite diensional Ω We begin with a very siple exaple of uniforly distributed utilities and obtain the required unique zero, and thereby achieve the required Θ-fair scheduling, using the single tie scale algorith (8) We consider a three agent exaple, with agents having utilities u } that are uniforly distributed between 0 and u ax = [4, 18, 1] The efficient utilities of the three agents, for this exaple, can easily be estiated and they equal [1946, 0245, 0027] We can see that the agents 2 and 3 are starved and we would lie to obtain an Θ = [0, 06, 01] scheduler in Figure 1 We notice that the average utilities ū, as well as the scheduler representatives b 2,, b 3, converge within 4000 iterations to satisfy the fairness constraints For larger systes (as considered in the coing exaples) the convergence ight be slower, but would still (9)

5 2 3 bar u, Iterations, b 2,, b 3, b 2 run1 b 3 run1 b 2 run2 b 3 run Iterations, Fig 1 Exaple 1 (Unifor utilities): Two siulation runs with different initial conditions Convergence of iterates ū, in the left figure (satisfying the constraints) and Convergence of iterates [b 2,, b 3, ] in right figure With single tie scale algorith, ū, is used only for illustrative purposes Fig 2 Exaple 2 (Rayleigh states): Two siulation runs with different initial conditions Convergence of iterates ū, in the left figure and Convergence of iterates b in right figure Initial b b converges to Final utilities TABLE I CONTINUOUS UTILITIES EXAMPLE 2: DIFFERENT INITIAL CONDITIONS, BUT CONVERGENCE TO THE SAME LIMIT

6 be coparable to that of the existing fair schedulers ([5]) With two tie scale version (7), the convergence is uch slower B Continuous utilities: Exaple 2 We consider the context of wireless counications and loo at the following proble: M cellphones are copeting for resource fro a single central unit or base station In this context, h represents the channel state between the base station and user For any agent, h is a continuous rando variable and h } are independent across the agents and independent and identical across the tie slots We assue that h is a Rayleigh rando variable The axiu rate (capacity) at which data can be transferred over the counication channel is the utility u, derived by agent when allocated the resources This (capacity) utility u is a deterinistic function of h and is given by, u = log(1 + h 2 ) We consider the case with 4 agents (M = 4) and the Rayleigh paraeters of the 4 agents are given by [20, 15, 12, 10] The goal here is to see 1) The convergence of expected utilities for different agents 2) Existance of unique zero of the fairness constraints equation, ie, convergence of vector b irrespective of the initial condition The efficient scheduler results in the following accuulated allocations: ū eff = [3618, 1784, 0867, 0426] The fairness constraint vector Θ is set as, Θ = [0, 21, 13, 75] The other paraeters of the two-tie scale algorith (7) are µ = 1/ + 001/ 3, N = 100 and C = 50 That is, fairness paraeters b, } are updated once in 100 steps For the exaple at hand, we found these to be a good set of paraeters Rayleigh distributed rando variable h with paraeter σ is generated fro uniforly distributed rando variable U via the following transforation: Observations h = σ 2logU Figure 2 illustrates the convergence behavior of the iterates of algorith (7) while Table I studies the liits of the sae algorith In the left figure one saple in 250 saples is plotted while in the right figure one in 200 saples is plotted This is done for the clarity of the figures Fro table, it can be seen that the expected utilities converge and all the fairness constraints are satisfied within a negligible argin of error, irrespective of the initial conditions, as established by Theore 4 Thus this algorith can be used to obtain the required zero and thereby to ipleent the Θ-fairness for any given Θ C Discrete case We now consider an exaple with discrete utilities ie the case when A4 does not hold Here Ω itself is a finite diensional set but the coputational coplexity increases as the size N (see Section III-B) increases We present here a ethod, wherein the Θ-fair scheduler of the discrete scenario can be well-approxiated by the scheduler of an appropriate continuous case and this approxiation is good when the cardinality N is large We continue with the exaple of wireless counication proble fro the continuous case study, but with a difference Here the syste supports only a finite nuber of transfer rates, say cellphone can be supported by one of the rates given U = 0, log(1+δ 2 ), log(1+(2δ ) 2 ),, log(1+(n δ ) 2 )}, where δ > 0 The channel state h is Rayleigh as in previous exaple, but now the utility of the obile is u = axu U : u log(1 + h 2 )} Clearly as (δ, N ) (0, ), u log(1 + h 2 ) for alost all h and we approxiate such cases with the continuous case of the previous exaplewe consider one such case in which all the agents are having equal nuber of transfer rate choices, ie, N = N 1 for all In this experient we calculated δ } such that the Rayleigh density of every obile at δ N 1 (every channel state after this value is supported by the coon highest rate) equals 90 percentile value All the other paraeters are sae as in the previous exaple, including Θ We again use the algorith (7), even when it is a discrete case The results are tabulated in Table II Interestingly even up-to the case with N 1 = 20, all the constraints are (well) satisfied approxiately, by an eleent fro β(b) Ω, to which the algorith converges asyptotically Fro Theore 3 (which is true even for the case with discrete utilities) the optiizer satisfies the constraints More interestingly, the scheduler representative b is close to the corresponding one in the continuous case And we now in the continuous case that the (unique) zero of the constraints equation is the axiizer Thus we expect the (approxiate) zeros so obtained to be the Θ-fair schedulers even for the discrete case We do notice fro the sae table that for the case with N 1 = 5, the constraints are not satisfied Thus this ethod is not accurate, when the cardinality of the utility space is sall However in such cases the diension of Ω itself is sall and once can directly use the algorith (9) indicated for the discrete case V EXTENSIONS, COMPARISONS AND DISCUSSIONS One can generalize and define f-fairness for M-agents as in the following Let f = (f 1,, f M ) : R M + R M + be any given easurable function and define, f-fair scheduler as: β f := arg ax β Ω ū i (β) i subject to f (ū 1 (β),, ū M (β)) 0 for all (10)

7 N 1 Initial b b converges to Final utilities continuous 1, 14, 14, , 14, 14, , 14, 14, , 14, 14, TABLE II DISCRETE UTILITIES EXAMPLE: THE ALGORITHM (7) PERFORMS WELL FOR N 1 = 20 OR MORE Note that f (u) = u θ gives the notion of Θ-fairness introduced earlier One can explore other options via this general definition For exaple, we are interested in the following function: f (u 1, u ) = ū γ ū 1 for all > 1 Essentially we are interested in aintaining the ratio: ū ū 1 γ for all > 1 The functions of this type can equivalently be represented using the vector Γ := (1, γ 2,, γ M ) and the corresponding fairness can be tered as Γ- fairness All the proofs go through for this notion of fairness also We just ention few differences Theore 3 will go through in a siilar way if we assue as in Assuption A2 that we ai to proise better returns for starved agents, which are better than what they could obtain using efficient scheduler, that is: γ > ūeff ū eff for all > 1 1 In step 17 of Theore 4 we will have b = 1 + ϱ γ ϱ 1 for all > 1 To prove Theore 5 version for Γ-fairness, if possible consider two vectors b, c both of which satisfy the ratio constraints and such that b c Then if wlg ū 1 (b) > ū 1 (c), and then to aintain the ratios we will need recursively that ū (b) > ū (c) for all Which is not possible because of wor conserving principle When ū 1 (b) = ū 1 (c), then again recursively to aintain the ratios, we will have ū (b) = ū (c) for all In all, this eans we ight have b c but with ū (b) = ū (c) for all to satisfy the ratio constraints But this is iediately contradicted by the existing Theore 5 Coparison with existing notions of fairness The unconstrained optiization (with Θ = (0, 0,, 0)) gives efficient scheduler When Γ = (1,, 1) we obtain the ax in fairness as this ensures all agents have equal utility There exists a Γ which defines the proportional fair scheduler For exaple, for the case considered in Theore 1 given in the Introduction and using the sae theore, Γ defined by ( Γ = 1, a 2, a 3,, a ) M 1, a 1 a 2 a M achieves proportional fairness This is possible because for any given Γ there exists an unique vector of accuulated utilities This is true because of the wor conserving principle as explained in the previous paragraph while indicating the proof of Theore 5 for the case of Γ-fairness We can also achieve this via Θ-fairness as given below As entioned in the introduction, every α-fair scheduler satisfies the following equation ([11, Lea 1]) β,α (u) = 1 arg ax (ū ;α ) α u =} with ū ;α = E[u β,α (u)] This iplies α-fair scheduler is obtained by the priority schedulers of Section III with priority factors ( ) α ū1;α b = for all 2 ū ;α For every α there exists a Θ α such that the Θ α scheduler of this paper obtains the corresponding α fair scheduler and in fact Θ α = [0, ū 2;α, ū M;α ] Conclusions and Future directions In this paper, we introduced a notion of fairness which provides direct inforation about the iproveent of the accuulated utilities of the otherwise starved agents Via a series of theores we proved that this type of fairness can be achieved by algoriths whose coplexity is siilar to those already proposed in literature for the other notions The notion introduced here has soe connections to the research done before For exaple, in wireless counications which has to support both data calls (lengthy connections which can wait) and voice calls (ipatient and short calls, which dropoff if all the servers are busy), it is proposed to optiize the average waiting tie of a data call while ensuring that the drop probability of a voice call is below certain acceptable liit We also introduced Γ-fairness which ensures that the ratios of utilities of any pair of agents is better than that achievable using efficient scheduler Using this we can achieve the existing notion of ax-in fairness In an extreely siplified scenario (Theore 1), we showed that with a Proportional fair scheduler the accuulated utilities are proportional to their eans Once the ean of the instantaneous utilities (E[u 1 ],, E[u M ]) of all the agents are nown (one can easily estiate the ean), one can actually ipleent a fair scheduler which ensures that the accuulated utilities of various agents are proportional to their own ean values

8 For exaple, one can achieve this using Γ-fairness defined in this paper for ( Γ := 1, E[u 2] E[u 1 ],, E[u ) M 1] E[u M ] We would lie to coplete this analysis and investigate ore on these aspects in future In this paper we indicated few iterative algoriths to ipleent the new notion of fairness We would lie to study their convergence properties In view of Theore 4 alternatively, one ay obtain this as the optiizer in Ω and design an iterative algorith to achieve the sae The case with discrete utilities needs to be explored ore REFERENCES [1] F Kelly, A Maulloo, D Tan, Rate control for counication networs: shadow prices, proportional fairness, and stability, Journal of the Operations Research Society 49 (1998) [2] Wieran, Ada Fairness and scheduling in single server queues Surveys in Operations Research and Manageent Science 161 (2011): [3] R Jain, D M Chiu, and W R Hawe, A quantitative easure of fairness and discriination for resource allocation in shared coputer syste, Eastern Research Laboratory, Digital Equipent Corp, 1984 [4] T Lan, D Kao, M Chiang, and A Sabharwal, An axioatic theory of fairness in networ resource allocation, in Proceedings of IEEE INFOCOM IEEE, 2010, pp 19 [5] HJ Kushner, PA Whiting, Convergence of Proportional-Fair Sharing algoriths under general conditions, IEEE Trans Wireless Coun vol 3, no 4, pp , Jul 2004 [6] V K N Lau, Proportional Fair Space Tie Scheduling for Wireless Counications, IEEE Trans counications, vol 53, no 8, Aug 2005 [7] DG Luenberger, Convex Prograing and Duality in Nored Space IEEE Trans on Syste Science and Cybernetics, July 1968 [8] DG Luenberger, Optiization by vector space ethods, John Wiley and Sons, Inc, 1969 [9] Leonard Kleinroc, and Roy P Finelstein Tie dependent priority queues Operations Research 151 (1967): [10] Leonard Kleinroc, A delay dependent queue discipline, Naval Research Logistics Quarterly, volue 11, pages , 1964 [11] Veeraruna Kavitha, Eitan Altan, Rachid Elazouzi, Rajesh Sundareshan, Fair scheduling in cellular systes in the presence of noncooperative obiles accepted at IEEE Trans on Networing [12] Veeraruna Kavitha, N Heachandra, Debayan Das, Fairness via priority scheduling, technical report downloadable using APPENDIX A This appendix contains all the proofs of the theore given in the ain body of the paper It also contains the theores and leas used for establishing those theores Proof of Theore 1: The first part is easily evident by substituting a ū c } into the fixed point equation given above the theore For the next two parts, we have the case d with M = 2 When α = 0, because X 1 = X2 : ū 2;0 = a 2 E[X 2 1 a2x 2>a 1X 1}] = a 2 E[X 1 1 a2x 1>a 1X 2}] > a 2 E[X 1 1 a1x 1>a 2X 2}], because a 2 > a 1, (11) > a 1 E[X 1 1 a1x 1>a 2X 2}], because a 2 > a 1, = ū 1;0 Define ξ α := a 2 (ū 1;α ) α /a 1 (ū 2;α ) α and define ν α 1 = E[X 1 1 X1>ξ α X 2}] and ν α 2 = E[X 1 1 X1 ξ α X 2}] (12) Clearly either ν1 α and ν2 α with α or the vice-versa (note ν1 α + ν2 α is a constant for all α) Note (as in previous step) by fixed point equation that, ū α = a ν α for all and hence that ξ α = (a 1 /a 2 ) α 1 (ν1 α /ν2 α ) α When α = 0, while proving the first part of theore we see that ν2 0 ν1 0 (see inequality 2 in equation (11)) Now if ν2 α and ν1 α with α, then ν2 α /ν1 α and then ν2 α /ν1 α > 1 for all α 0 and hence we also have ξ α But then fro (12) ν1 α which is a contradiction Thus, ν1 α and ν2 α with α Proof of Theore 2: Clearly Ω is a convex subset of L 2 and L 2 is a Hilbert space Consider a saller subset of Ω which defines the constraints of (5) as below: Ω θ = β Ω : ū θ for all 2} By linearity of the ap β ū (β), Ω θ is also a convex subset Further it is a closed subset To see that, if there exists a subsequence β n = (β1 n,, βm n )} Ω θ with β n β in L 2 nor Then, β n β in probability and this iplies the existence of a subsequence along which β n β alost surely The alost sure convergence gives us that: 1 β(u) 1 and β (u) = 1 for alost all u Further by doinated convergence theore ū (β n ) ū (β) Thus ū (β) θ for all, and so β Ω θ and hence Ω θ is closed The function β Υ(β) = ū(β) is clearly linear and is bounded because by Cauchy-Schwartz inequality: Υ = sup β 2 1 u 2 < ū (β) sup β 2 1 β 2 u 2 Thus, Υ is a continuous linear functional and and hence is Gateaux differentiable function Thus 1, there exists a β Ω θ which axiizes Υ Proof of Theore 3: If say there exists a β = (β1,, βm ) which axiizes the optiization proble (5) without satisfying the constraint with equality That is to say for soe with θ > 0, u β(u)dp (u) θ = δ > 0 1 We have the following theore fro Functional analysis Theore Let K be a nonepty closed convex set of a Hilbert space V, and let J : V R be a convex Gateaux differentiable function If K is bounded, or if J is infinite at infinity, there exists at least one iniu of J over K, ie, J(u) = inf v K J(v) for soe u K

9 By A2, θ > ū eff given by (4) and this iplies, agent obtains ore because of non efficient scheduling decisions (ones with β(u) > 0 when arg ax u ), ie, P (A) > 0 where A := } u : β(u) > 0 when u 1 = ax u The reason for the above is: the non efficient decisions are obtained by shifting the efficient decisions of only the un constrained agents (agents with θ = 0) in favor of the agents with constraints on non zero easure set and there should be at least one such agent, assue without loss of generality, this to be the agent 1 This iplies there exists a δ 1 > 0, δ2 > 0 and a set A such that 2 P (B δ1, δ 2 ) > 0 where B δ1, δ 2 B δ1, δ 2 } := β(u) > δ 1 with ax u = u 1 and u 1 u > δ 2 Then a new scheduler policy is defined as the following: β on B β c δ1, δ = 2 (β1 + δ, β2,, β δ,, βm ) on B δ1, δ 2 } with 0 < δ δ < in δ1, udp It is easy to chec that, (u) ū ( β) = ū (β ) δ B δ1, δ2 u dp (u) > θ That is, the new policy still satisfies the constraint for agent Also, it does not alter the decisions of any other constrained user which iplies the constraints of the other users are also intact Further, ū ( β) = ū (β ) + δ (u 1 u )dp (u) B δ1, δ2 contradicting the optiality of β ū (β ) + δ δ2 P (B δ1, δ 2 ) > ū (β ) Proof of Theore 4 : Any scheduler β = (β 1, β 2,, β M ) satisfies that β 0 and β = 1 for alost all values of u = (u 1, u 2,, u M ) And so, Υ(β) = (u u 1 ) β (u)dp (u) + u 1 dp (u) (13) 2 M And the constraints are given by: u β (u)dp (u) θ for 2 2 Clearly A = n,l B 1 n, 1 and so, if such a δ 1, δ 2 does not exist then l P (A) = P ( n, β (u) > 1 with ax n u = u 1 and (u 1 u ) > 1 }) = 0 l The last ter in equation (13) is independent of the scheduler β and can be oitted for optiization and further considering the dual variables ϱ := (0, ϱ 2, ϱ 3,, ϱ M ) (note there is no constraint on the first agent and hence ϱ 1 = 0) we are interested in optiizing the following wrt β: L(β; ϱ) = [(u u 1 ) β (u)] dp (u) + 2 M 2 M ϱ ( u β (u)dp (u) θ ) (14) Note here that for agents with θ = 0, ie, for ones without constraint, we set ϱ = 0 for ease of notations For any fixed ϱ, we are equivalently interested in: ax β Ω 2 M [u (1 + ϱ ) u 1 ] β (u)dp (u) (15) Clearly the β;ϱ = (β1;ϱ,, βm;ϱ ) defined as below: β 1;ϱ(u) = Π 2 1 [(1+ϱ)u u 1] < 0} and β ;ϱ(u) = 1 arg ax 2 [(1+ϱ )u u 1]=}(1 β 1;ϱ(u)), for any 2 optiizes (15) It is easy see to that the above is sae as the following (1 := (1,, 1)): β ;ϱ(u) = β (1 + ϱ)(u) (16) with β (b)(u) defined as in (6), where b = 1 + ϱ By Lea 1, under A4 this is the unique axiizer for any given ϱ The doain Ω is a convex closed subset of the Banach space, L 1 The objective function and the constraints are both linear The constraints are functions fro Ω to Euclidean space R nc, where n c represents the total nuber of constraints We consider nored space R nc with usual partial order By Theore 2, which establishes the existence of optiizer and A3, the Duality theore on nored linear spaces, [7, Theore 5] is applicable (see also [8]) Further, the optiizer given by (16) is unique for each tuple ϱ = ϱ } and these two give us the following: By the first stateent of [7, Theore 5] ax β Ω; ū (β) θ, Υ(β) = in ax ϱ 0 β L(β, ϱ) and there exists a ϱ which satisfies the constraints and which achieves the iniu on the right hand side By Theore 2, there exists an optiizer β of (5) Then, by the second stateent of [7, Theore 5] β also axiizes L(β, ϱ ) By Lea 1 there exists an unique axiizer β ;ϱ for L(β, ϱ ) and hence (see (16)): β = β ;ϱ = β(1 + ϱ ) (17) Note here ϱ = 0 for θ = 0 By Theore 3 any axiizer satisfies the constraints and by Theores 5 and 3 there is a unique axiizer in Ω

10 Rears about discrete utility case: Equation (17) in the proof of Theore 4 is the crucial step to show that an eleent fro finite diensional Ω axiizes the constrained optiization (5) This step is true by Lea 1, which establishes the existence of an unique optiizer for the Lagrangian objective function L(β, ϱ) given by (14), with any arbitrary Lagrange ultiplier ϱ The lea is true only under A4, ie, when the instantaneous utilities are continuous rando variables One can easily construct exaples with discrete utilities when uniqueness fails and we present one such a siple exaple in Appendix B But the rest of the arguents would be sae for discrete rates and in this case the optiizer of (5) still axiizes L(β, ϱ ), however it can also be a convex cobination of all the axiizers of L(β, ϱ ) and hence ay not be an eleent fro Ω Lea 1 Under A4, for any given ϱ the scheduler given by (16) is the unique axiizer of (14) Proof: By adding the constant ter u 1 dp (u) to (14) and noticing that β 1 = 1 1 β we are equivalently interested in optiizing (with ϱ 1 = 0): L(β) := u (1 + ϱ )β (u)dp (u) The axiizer β ;ϱ given by (16) is clearly the point-wise axiizer of the ter inside the integration Define the following rando variables: = arg ax ũ = ax u (1 + ϱ ) u (1 + ϱ ), u = ax u (1 + ϱ ) and Under A4, P (u = u ) = 0 for any and hence P (u = ũ ) = = P (ax u = ũ ; = ) P (u = ax u ; = ) P (u = u ; = ) P (u = u ) = 0 (18) If there exists any other axiizer β of (14) and say P (A) > 0 with A := u : β ;ϱ(u) β(u)} Because u is a point-wise axiu value and ũ is the second axiu, A = B u : β;ϱ(u) β(u), u = ũ } with B := u : β;ϱ(u) β(u); u ũ > 1 } In view of (18) and because P (A) > 0, there exists at least one such that: P (B ) > 0 Because β = 1 for all u, L(β :ϱ) L( β)) = u dp (u) (u (1 + ϱ ) β (u))dp (u) = = = u β (u)dp (u) (u (1 + ϱ ) β (u))dp (u) β (u)(u u (1 + ϱ ))dp (u) A A β (u)(u u (1 + ϱ ))dp (u) β (u ũ )dp (u) = (u ũ )dp (u) B (u ũ )dp (u) B which is a contradiction to the optiality of β A 1 dp (u) = 1 P (B ) > 0 Theore 5 There exists at axiu one eleent in Ω which satisfies all the constraints of (5) Proof: Consider the case with M 1 constraints The general case goes through in a siilar way If possible, consider b c, such that both of the satisfy the constraint equations Recall by Theore 4 that c 1 = b 1 = 1 Define the following sets: A j, (b) := u j b j < u b } for all 1 j, n with j Define, A (b) := j A j, (b) Clearly, ū (b) = E[u β (b)(u)] = E[u 1 A(b)] Wlg say c M > b M Then clearly, A 1,M (b) = u 1 < u M b M } u 1 < u M c M } = A 1,M (c) Since agent M s constraint is satisfied at both b and c (ie, ū M (b) = θ M = ū M (c)), the above relation iplies there exists at least one 1 with 1 < 1 < M and such that A 1,M (b) A 1,M (c) b M b 1 Wlg let 1 = M 1 As, > c M c 1 b M > c M and c M > b M we have c M 1 > b M 1 b M 1 c M 1 These in turn iply (for agent M 1) that: A 1,M 1 (b) A 1,M 1 (c) and A M,M 1 (b) A M,M 1 (c) Again because ū M 1 (b) = θ M 1 = ū M 1 (c), there should be at least one 2 with 1 < 2 < M 1 and such that A 2,M 1(b) A 2,M 1(c) which iplies b M 1 b 2 > c M 1 c 2

11 Wlg let 2 = M 2 and this iplies c M 2 > b M 2, b M 1 > c M 1 and b M 2 c M 2 b M = b M 2 b M b M 1 b M 1 b M 2 > Thus we have (now for agent M 2) c M c M 1 c M 1 c M 2 = c M c M 2 A 1,M 2 (b) A 1,M 2 (c), A M,M 2 (b) A M,M 2 (c) and A M 1,M 2 (b) A M 1,M 2 (c) Thus and because ū M 2 (b) = ū M 2 (c) there exists an 3 = M 3 (wlg) such that A M 3,M 2 (b) A M 3,M 2 (c) b M 3 b M 2 > b M 3 c M 2 Continue in the sae way up to 3 and for ū 3 (b) = ū 3 (c) we will need that c 2 > b 2 so that b 3 /b 2 > c 3 /c 2 This eans (calculating as before) for agent 2 that: A 1,2 (b) A 1,2 (c), and A,2 (b) A,2 (c) for all This iplies that ū 2 (b) < ū 2 (c), which contradicts the fact that both of the satisfy all the constraints APPENDIX B: A DISCRETE UTILITY EXAMPLE We consider a siple case with two agents The first agent has u 1 20, 10} and P rob(u 1 = 10) = P rob(u 1 = 20) while for the second agent P rob(u 2 = 5) = 1 In this case the scheduler β = (β 1, β 2 ), with β 1 representing the probability with which agent is scheduled when its utility is 20 while β 2 represents the sae when agent 1 s utility is 10 In this case, ū 1 (β) = β β 2 and ū 2 (β) = 5 (05 (1 β 1 ) + 05 (1 β 2 )) For exaple, if the fairness constraint is ū 2 θ, it can be satisfied by the following two β s: (1, 1 θ/25) or (1 θ/25, 1) Thus uniqueness of zero of fairness constraints equation is not true in discrete case In view of Theore 3, it is easy to see that aong the above two, the optiizer of 5 is (1, 1 θ/25) This scheduler obtained by changing the β eff = (1, 1) (efficient scheduler) only in the second coponent This coponent corresponds to u 1 = 10 and hence is the one with lesser u 1 u 2 Thus, while choosing the optial zero of fairness constraint, one first needs to change the beta s (fro beta s corresponding to efficient scheduler) corresponding to lowest difference utilities and then to the next lower utilities and so on That is exactly the intuitive reason why priority type scheduler wors with continuous utilities We plan to explore further these ideas in future