this is a model of scheduling in 3G wireless technologies, such as CDMA2000 3GlxEV-DO downlink scheduling.) We intro- duce an algorithm which seeks to ...
Optimal Utility Based Multi-User Throughput Allocation subject to Throughput Constraints Matthew Andrews
Lijun Qian
Alexander Stolyar
Bell Laboratories. Lucent Technologies Murray Hill, NJ 07974. andrews @research.bell-labs .com
Prairie View A&M University Prairie View. TX 77446. lqian @pvamu.edu
Bell Laboratories. Lucent Technologies Murray Hill. NJ 07974. s tolyar~rt.srarch.bell-labs.com
Abstrucl- We consider the problem of scheduling multiple users sharing a time-varying wireless channel. (As an example, this is a model of scheduling in 3G wirelw technologies, such as CDMAZUOO 3GlxEV-DO downlink scheduling.) We introduce an algorithm which seeks to optimize a concave utility
function
E,H,(R) of
the
user throughputs R,, subject
to
certain lower and upper tlwoughpui bounds: Rrzn 5 R, 5 RYar, The algorithm, which we call the Gradient algorithm with MinimundMaximum Rate constraints (GMR) uses n rob# counter mechanism, which modifies an algorithm sohhg the corresponding uncanslrained problem, to produce the algorithm solving the problem with throughput constraints. Two important special cases of the utility function? are log R, and corresponding to the common Proportional Fairness and
E,
E,&,
Throughput Maximization objectives. We study the dynamics of user throughputs under GMR algorithm. and show that GMR is asymptotically optimal in the following sense. If, under an appropriate scaling, the throughput vector R ( t ) converges to a fixed vector R' ns time t m then R* is an optimnl solution to the optimization problem described above. We also present simulation results showing the algorithm performance. Kai M W ~ Sand p h s e s : Scheduling. wireless, CDMA, 3G, time varying channel, QoS, Gradient algorithm, Proportional Fair, Maximum Throughput, rate constraints. guaranteed rate
-
In the wireless context, the traffic flows correspond to the downlink data flows from ;t baseslation to multiple mobile users. (See Figure 1). If the signal-to-noise ratio (SNR) for user i is high, then if user i is picked for service in time slot t it can receive data at a high rate. Conversely. if the signal-tonoise ratio is low it can only receive data at a low rate. The scheduler resides at the basestation. In the EV-DO system the scheduler knows the feasible service rates because each user measures a pilot signal-to-noise ratio and reports back to the basestation in a Data Rate Control message (DRCI. We shall sometimes use DRCa(t)to denote the rate at which user i can be served at time t if ir is chosen for service. We emphasize that this rate is both iuer-depeadenr and time-vuping. In this paper we shall work in an infinifely backlogged model in which for each flow there is always data available for service. The set of all feasible long-term average service rate vectors R = ( R I , . . R I ) is called the system rule region, I/. Rate region is a convex poIyhedron in the positive orthant. Past work in lhe infinitely backlogged model has considered scheduling algorithms that optimize (over rate region V - ) a certain utility funciion of the average rates R. For example. the Proportional Fair algorithm [20J, [3] aims to optimize the function log 8.However, this optimization provides no guarantees to individual m m Each user may a1 times receive unacceptably bad service. In this paper we demonstrate how to rectify this problem by presenting scheduling algorithms that optimize a utility function of the rate vector RI subject to mininzun~ aad maxitnurn rate constrainls on the individual components Ri. More precisely, we are interested in solving the following optimization problem: I
Ci
1. INTRODUCTION
We consider a variable channel scheduling model that is motivated by the 3GI.xEV-DO system for high-speed wireless data. A channel serves 1 IraficJuws and operates in discrete (slotted) time. In each time slot a sclzeduler chooses one flow to serve. The channel slare is random, and it determines the service rates of each flow in the current time slot, if that flow is chosen for service.
mxx H ( R )
(1)
subject to
R
E V-
(2 1
3 ly": i E I:
(31
Ri 5 Rimax, i E I !
(4)
Fli
for utility functions of the form .
H ( R )= X H d R i ) !
Fig. 1. A wireless system.
a
2415
(5)
where each Hi(.) is an increasing concave continuausiy differentiable function defined for 3: 2 0. (We also allow the case liin,ln Hija) = Hi(0) = -w.) The rate constraint parameters RPi" and RY"" are fixed constants such that 0 5 Kyi" 5 Kinax and Rynx > 0. A. ONTResirlts
In Section IV we propose an algorithm, called the Gradient algorithm with MinimumMaximum Rate constraints (GMR) that aims to solve the problem (1)-(4).In time slot t. GMR always serves Row, i E arcmas eitiTijf)~JlljHi(t)jDRC,(t) , iEI
where & ( t ) is the current average service rate received by queue i , T,(t) is a rokerz counter for queue i. and ui > 0 is a parameter. The token counter Ti(t.),which we define preciscly in Section IV, is the key mechanism by which we enforce the rate constraints. In each time slot it is incremenred at rate either RFin or HT"" and it is decremented whenever flow 1; is served. if in some finite time interval, flow i receives service less than RFin then Ti(?) has a positive drift and so flow i is more likely to be served. Zf flow i receives service more than Rrax then Ti@)has a negative drift and so flow i is less likely to be served. (See Remark 2 in Section V on the form of GMR algorithm.) There are two special cases of lhe utility (objective) function that are of particular interest. 1) The Proportional Fair objective' which corresponds to H(R)= log Ri. In this special case, GMR always serves flow,
Ci
We refer to this special case of the algorithm as Proportional Fair with MinirnutdMaxitnn?n Rates (PFMR). We remark that if the rate constraints are trivial. i.e. = 0 and ,Fax = 00 for all i, then PFMR reduces to the standard Proportional Fair algorithm of 1201, 131 [i.e. it always serves flow i E argmaxiEf DRCi(t)/Ri(t). 2) The Throughput Maximization objective which corresponds to H ( R ) = C i R i . In this case, GMR always serves flow, i E argrnax e u i T i ( t ) ~ ~ ~ i ( t ) . iEI
and we refer to the algorithm as Maxifnnurn Thiouphput with MinimundMuximuin Rates (MTMR). In Section V we show asymptotic optimality of GMR in the sense that if: under an appropriate scaling, R [ t )converges to a 'The reason this objective can be desirable i s that multiplying u e r i's rate by sonis factor c has the same effect on the ohjective as niulriplying user j's rate by e. Alternatively, if Ri are the rates h a t maximize this objective, then for any other feasible set of rates Ri that satisfy the constraints we must havz,
fixed vector R' as t -, MI then R* is a solution to the problem (1)-(4). (See Remarks 2 and 3 in Section V on this notion of asymptotic optimality. and on tbe convergencc properties of GMR and related algorithms.) In Section VI we present simulation resulw to illustrate rhe behavior of PFWR and MTMR. B. Motiiniion for Minitiiutn/~QsimiirriR a e Constrainrs A guarantee on minimum bandwidth is arguably the simplest possible Quality-of-Service guarantee. Therefore we believe it is natural that subscribers to an expensive mobile highspeed data service would expect such an assurance. Other reasons why we fed it is important to provide minimum rare constraints are: 1) Some applications need a minimum rate in order to perform well. For example. streaming audio and video can become unusable if the bandwidth is too small. 2) Even for static TCP-based applications such as web browsing if the bandwidth is too small then we typically get a large queue buildup which can lead to TCP timeouts and poor performance. Such effects were discussed by Chakravony et al. in 171. 3 ) Providing a minimum rate guarantee can help to smooth out the effects of a variable wireless channel. 4) Providing a minimum rate can allow us to ensure that a slot-based service such as EV-DO is no worse than circuit-based data systems such as wireline dialup or 3G1X wireless service. , 5 ) By setting RYln differently for different users we can ensure that high-paying premium customers receive better service than regular customers. At first, ir might seem counterintuitive that a system operator
would want to provide maximum rate constraints, However, some possible reasons are: 1) If a user has only paid for a cheap data service. the operator might wish to cap their data rate in order to give them an incentive to upgrade to a more expensive premium service. 2 ) The first user that signs up for a data service will have the system J1 to themselves. Then, as more customers sign up, the service to the first user will typically deteriorate as it has to share bandwidth with all the new users. One solution to this deterioration is to cap all users' maximum rates to some reasonable level so that they do not experience perfromance degradation when more users subscribe. (Of course, doing this may involve wasting time slots which might not be so desirable.) We remark that if the system operator does not wish to have maximum rate constraints then this is easily accomplished by setting RTax to infinity (or some suitably large value).
C. Previous bbrk The most closely related work is hat of C161, U71, 151, where the Gradient algorithm was studied and proved optimal (under various models and orher assumptions) for ow problem
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wirhoi41 mimimim and maximum rate constraints. (See [SI for the results in our model setting.) This “unconstrained” Gradient algorithm is a special case of GMR. We therefore have a very natural question: ‘ Y h y not use the unconstrained Gradient algorithm for our problem as well. and deal with the rate constraints simply by modifying the utility function in a way that penalizes rate consuaini violations?’ As we demonstrate later in the paper, such an approach does not work well. KoughIy, the reason is that an algorithm with a modified utility function typically “overreacts” to temporary rate constraint “violations,” and this sipnificmtly degrades the achieved value of the utility function. The problem of utility based throughput allocation was also previously considered in [1SIt [19]. In particular, these papers addressed throughput allocation subject to certain constraints. There are two key differences between our algorithm and those in [IS]. [19]. First, GMR optimizes a concave utility function of the average throughputs. while schemes in [18], [19] are i n essence restricted to linear utility functions. Secondly. our token counter mechnism for average rate constraints enforcement is substantially different from the stochastic approximation based schemes of ri81, L191. A solution to the problem of maintaining service races in desired proportions to one another was presented in [121. A higher-level problem in which each user has a finite amount of data to serve and it leaves the system once this data is served is studied in [13]. Recall that we are concerned with the service rates provided to each user. Our model assumes that whenever a user is scheduled it always has data to serve. Now we would like to contrast our algorithm with algorithms for a different, much studied, model, where each user has a queue that is fed by an arrival process. In this setting, the most widely studied algorithms XK Mat-Weight type algorithms @I. [SI, [l]. In our setting, the “queue-length-based” version of such an algorithm always serves the user that maximizes DRC,(t)Qi(t)where Q i ( t )is the amount of data in the queue for user i . This algorithm is known to be stable’, which roughly means that it keeps the queues from running away to infinity whenever possible. Other stable algorithms include the “delayty-based” version of MaxWeight [SI, [11 and the EXP algorithm [lo], [ 11). The “delay-hasad” MaxWeight always serves the user that maximizes DRCi(t)tVi(t) where W i ( t ) is the Head-ofLine delay for user 2. EXP is a more complex algorithm that aims to have more control over the delay distributions. Despite the difference of the above problem from ours, we note that the above stable algorithms such as Max-Weight could be applied. for example, to provide nainimiirn rate conuaints in our problem, since they could operate on the token counters rather than the actual queue lengths. However. in this case all token counter stabilty means is that the minimum rate constraints are indeed enforced. Such a solution would nur oplimize an objective function subject to minimum rate ’We reinark that in this model Proportional Far is known to bz stable [141.
m
constraints. We finally remark that in our model the channel rate process is governed by a stationary stochastic process. The problem of* scheduling over a non-stationary wireless channel is addressed in [15). 11. VARIABLE CHANNEL SCHEDULING MODEL
We consider the following model introduced in [l]. There is a finite set I = { 1,2: , , I } of “traffic” flows served by a charrnel. (We will use the same symbol I for both the set and the number of its elements.) Each flow i consists of discrete crrsrotners (“bits of data”): which we sometimes call type i customers. In this paper we assume that there is always a sufficient “supply” of customers of each flow to serve. The system operates i n discrete time t = 0: 1; 3 ; .. .. By convention, we will identify an (integer) time t with the unit time interval [t.i. f I), which will sometimes be referred to as ~
the tiriie slot 1. The channel has a finite set of channel sfafes M . In each time slot, the channel is in one of the states ‘m E A4; and the sequence of states m ( t ) ; t = 0 , 1 , 2 , .. ., forms an (irreducible) finite state Markov chain with stationary distribution (ST,,, m E M } , niEM
If at time t the channel is in state 7n E M and it chnose.s queue i for servtce. then an integer number pT 2 0 of type i customers (e.g., bits of data} are served and depart the system at time t 1. We use pn1 = ( p y , . . , ,@) to denote
+
the corresponding vector of service rates, and we assume that for each flow i , h e r e is at least one state m such that pa > 0. (Sometimes, we will write D R C ~ ( = ~ )p;it’ for the service rate available to user i in slot t , if it were to be picked for service. This notation, which we already used in the Introduction. is standard in the CDMA2000 1xEVDO literature.) Note that the model in which channel state m(t) is in fact a combination of independently randomly varying (according to independent Markov chains) channel states mg(t) of individual users is essentially a special case of our model (up to very mild assuptions guaranteeing that Markov chain m ( t ) is irreducible). Suppose a stochastic matrix # = (r,hmiz ni. E M,i = 1: . . . I ) is fixed, which means that 4mi 2 0 for all nz and i , and dnLi= 1 for every m. Consider a Sfafic Service Split (SSS) scheduling rule, parameterized by the matrix Q. When the server is in state nz, the SSS rule chooses €or service queue i witb probability dmi.(Sometimes, we call the matrix q5 itself an SSS rule.) Clearly, the vector 21 = (VI,. . . : 211) = v(i), where
gives the long term average service rates allocated to different flows under an SSS rule 4. We define the system rate region to be the set V7 of all vectors ~ ( 4for ) all possible SSS rules 4. Thus V is the set of
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long-term service race vectors which the system is capable of providing. Rate region 17 is a convex closed bounded poIyhedron in the posilive orthanr. (See 141.) By I/* we denote the subset of maximal elements of V : namely. 11 E V’ if conditions L* 5 ,U (componentwise) and .IJ f I/ imply ‘U = U . Clearly, V* is a part of the outer (“north-east”) boundary of
If. The subset C - V of elements c E 5J satisfying conditions Bin’” 5 ui 5 ,Fax (i.e. conditions (3) and (4)) for all d. is also a convex closed bounded set. Since each function H i ( R i ) is continuous and increasing, we have the foIIowing simple fact. Proposition 1: If Vcortdis non-empty. b e n at least one solution K’ of probkm (1)-(4) exists. If Vcondcontains at least one point of the set V*, then any solution K* E IT*. If all functions H i ( & ) are strictly concave (for example. Reiiiark. The choice of the units for different variables H i ( & ) = log(R,)),a solution R* is unique. and parameters used by the GMR algorithm is a matter of implementation convenience. One choice of the units (which 111. BASICNOTATION A N D CONVENTIONS is the one we used in the similations. presented in Section VI) The sets of real numbers and non-negative real numbers are is as follows. Amounts of data are measured in bits and the denoted by R and R, respectively; R’ and R: denote their time is measured in slots. Consequently. all data rates Ri(t). I times products. = D R C ? ( ~pLi(f). ). f y i I 1 , ,Fax , RtokeTb art: measuted For vectors x , g E RI, in bitdslot (or bits. if a calculation involves amount of data served or arrived at the corresponding rate in one slot, as in 3: . y x i y i is scalar product, (7)-(9)),Finally. token counters T;(t)are measured in bits, and i parameters ui are in llbits. As we mentioned earlier, we refer to the specjal cases z x y A (zlyl,.~. ? r c p y ~ ) is component-wise product, of the GMR algorithm. corresponding to utility functions exp(z) = (exp(x1):.. . ,esp(zJ)) . H ( R ) = Cilog(Ri)and H ( R ) = as PFMR and MTMR algorithms, respectively. By specializing (6): we see The Euclidean norm lllc[l A 6 defines metric 1 1 -~ yII on that RI. the PFMR scheduling rule is The gradient of the function H is denoted by VH, i.e.
xi&,
For a function E = ( 0 is a paranielel: The valiies of average rate R, are updated as in the Proportional Fair a l g o r i t h [20], 131: R,(t
+ I ) = (1- P)R(t)-I- PPL(t.) ,
(11)
The token counter Ti provides the key mechanism trying to ensure that the user i received (long term) service rate stays between €?Finand Ryax.The dynamics of the token counter process Tiit)(see (7)) is roughly described and interpreted as follows. There is a virtual “token queue” (which may take negative values) corresponding to each flow i. The rokens “arrive in the (token) queue” (i.e. Ti is incremented) at the rate Ryin or RYax per slot, if c(t)is positive or negative, respectively. (For this reason. we sometimes refer to flyin and Rynaxas the token rates.) If user i is served in slat t , then p”‘) = DRCj(i;)tokens are “removed from the queue” (i.e. Ti is decremented). Thus, if in a certain time interval, the average service rate of flow i is less than R r i n , the token queue size T; has “positive drift”, and therefore the chances of flow i being served in each time slot gradimllv increase. If the average service rate of flow i is above Rrax, then Tihas negative drift, thus gradually decreasing the chances of user
2418
i being picked for service. If average service rate of flow i is between RFin and Ayax.then T, has positive drift when T, is negative and negarive drift when T, is positive; as a result, in this case Tiwill stay “around 0.” V. USER THROUGHPUT DYNAMICS UNDER G M R WITH SMALL PARAMETERS @ ASD ai
-
~
E
bf, 1 3
l { n l P ( n )= m } 4 T m ,
(12)
OinitlP
where I { . } denotes here the indicator function. For each p, let R p ( , )and Ti;’(.)be the realizations of the throughput and token counter vector-processes corresponding to the GMR algorithm. They are uniquely defined by the realization mfi and the fixed initial states Rp(0)and T$(O).Finally, we extend the domain of functions R p ( t )and TP(t) to all real t 2 0 by adopting the convention that they arc constant w t h i n each time slot [i,-t 1) for all integer t, and consider the following rescaled rate and token counter trajectories:
+
T’(f)
= 8’(f/@),
T’(t)
= p T p ( f / p ) ,t
r ( t ) + R’ as t
2 0.
of vector-€unctions ( T = ( r ( t ) ,t 2 0): T = ( ~ ( t )t ,2 0)) is called a j714id sample path (FSP). if the uniform on A pair
compact sets (u.o.c.) convergence (& 7 3 ) --$ ( T I T ) holds for at least one sequence (TO,d)defined as abovc. (In our case, the u.0.c. convergence means that for any fixed b 2 0, the convergence is uniform over t E [O, GI.) We can now formulate the main result of this section.
-+00
and ~ ( t remains ) uniformly bounded for all f. 2 0. Then, R’ is a solution to the problem (1)-(4) and, moreover, R’ E \/con.d
In this section we consider the dynamics of user throughputs and token counters under the GMR algorithm when parameters and ai are small. Namely, we consider the asymptotic regime such that p converges to 0. and each q = &i with some fixed ai > 0. We study the dynamics offlitid sample paths (FSP), which are (roughly speaking) possible trajectories (r.(t),~ ( t of) )a random process which is a limit of the process (R(t/P);pT(t/,fi)) as /? 0. (Thus, r ( t ) approximates the behavior of the vector of ihroughputs K ( t ) when p is small and we “speed-up” time by the factor 1/p; ~ ( tapproximates ) the vector T ( t ) scaled down by factor p. and with 1/p time speed-up.) The main result of this section is a “necessary condition for throughput convergence” (Theorem l), which roughly says that if FSP is such that the vector of throughputs r ( t ) converges to some vector R* as t m, then R’ is necessarily a solution to the problem ( 1 )-(4). We now define the asymptotic regime and an FSP more precisely. Consider a sequence of positive values of p1converging to 0, and assume that a, = ,&vi, a, > 0, for each ,fi. (We will denote a = ( a l j . . ar).) For each /3 we consider a realization (that is, a fixed sample Dah) of the channel state process m P = (m@(t>: i; = 0: I:?,. . . j . We assume that the sequence of (fixed realizations) mP satisfies the law of large numbers condition, namely, that for any t > 0 and any n1
Theorem I: Suppose FSP ( r ;T ) is such that
n I/* f fl,
Rernark 1. It is easy to show using FSP properties described below in Lemmas 1 and 2, that if Vconcln V’ = 8, then for any FSP the vector ~ ( tcannot ) remain hounded and in fact /l~(t)ll i x as t 130. Therefore, the uniform boundedness of ~ ( talone ) implies that vcond ~ ‘Vi * # 8. Remurk 2. It will be easy to see from our proofs that Theorem 1 still holds if factor ea,zT1(t) in the GMR rule (6) is replaced by a(utT+((t))? where a(z)is an arbitrary continuous, strictly increasing function, such that a(0) = 1, a(z).1 0 as z J. --os. and a (.) 1 00 as z 00. Moreover, the theorem still holds if rule (6) has the following “additive form:”
-
where V(T) IS an arbitrary contlnuous. strictly increasing function, such that ~ ( 0 = ) 0, V(X) 1 -cm as z 1 -m, and v ( z >t cc as z t OG. In this paper we choose to work with the specific form (6) of GMR algorithm. to simplify the exposttion to some degree. and also because this specific “multiplicative” form shows good convergence properties in our simulations and is in fact very convenient for practical implementation, Remark 3. We remind that Theorem 1 does nob assert that the convergence of throughputs 7.(t) to the se1 of optimal solutions of the problem (1)-(4) in fact holds. (Our simulation experiments show good convergence properties of the GMR in the form (6)) Subsequently to the present work. it has been proved recently in [61 that such convergence does hold for a quite general Greedy Pritnal-Dual (GPD) algorithm. which, for our model, specializes roughly to the scheduling rule
We note however, that the GPD algorithm convergence proof
in [6] does not apply to the GMR algorihm (6). To prove Theorem 1, we will first describe the basic FSP properties in Lemmas 1 and 2. Then we prove two specid (increasingly general) cases of Theorem 1 in Lemmas 3 and 4. and conclude this section with the proof of Theorem 1 itself. Lemrno I : For any fluid sample path. all its component functions are Lipschitz continuous in [O,m), with the Lipschitz constant upper hounded by C I l ~ ( O ) l l , where C > 0 is a fixed constant depending only on the system parameters. I Proof is analogous to that in [ 5 ] .
+
Since all component functions of an FSP are Lipschitz, they are absolurely continuous, and therefore almost all points t 2 0 (with respect to Lebesgue measure) are such that all component functions of an FSP have derivatives. Lemnza 2; The family of fluid sample paths satisfies the following additional properties.
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(i) For almost all t 2 0 (with respect to Lebesgue measure) we have: r'(tj = v ( t ) - r(t.); (15)
for the optimization problem
where
subject to consuaints
v ( t ) E argmex[exp(a x ~ ( t x) )V H ( - r ( t ) )''U ]
~
V E \r
L'
(16)
> n m i n and
-
'L'
5 R"""
,
(22)
Moreover, the complimentary slackness conditions are satis7-'(t)= o ( 7 - t o k e y f ) - u ( t ) ) (17) fied for the (Lagrange multipliers) AYL''' and A y n r . Indeed, if for some i we have IZ; > then T: 5 0 (.otherwise, by where the components r,fQke"(t):1 = 1: . . . ~I of vector (17'b(lW? rj(t) could not possibly be constant). and therefore ?*token(i) are such that qt 5 1. This means that Rf > Ryin implies Ayin = 0. Using an analogous argument, we see that RZ < ,Tux implies = K,yin if ~ ~ > ( 0t )) r,tokerr(t) E [K"'": Rl"] if ~ ~ = ( 0: t ) (18) ,%'ax = 0 , Thus, by the Kuhn-Tucker theorem (cf. [2])! 6' = R' solves = ,y, if T i ( t ) < 0. the problem (.21)-(22), which is equivalent to the problem (ii) "Shift property," If ( r :T ) is an FSP>then for any d 2 0, { B ~ T O. ~ T )is also an FSP. (iii) L'Compactness,i' If a sequence of FSPS ( , r ( j ) , -+ ( r : ~uniformly ) on compact sets as j m, then ( T : T ) is This in turn means that point R* is a maximal point of the also an FSP. set Irco'ld (i.e., it lies on its outer - "north-east" - boundary). The proof of properties (i)(15) and (iii) is completely and that vector V H ( R * )is normal to the [convex) set VcoTrd analogous to that of the corresponding FSP properties in [5]. at point R'. This implies that R* is a solution to (1)-(4), Property (i)(17) is easy to verify directly. using the definition Since R* solves the problem (19). R' is a point on the of an FSP - we omit the proof to save space. The shift outer boundary of the entire rate region V. i.e. R* E V'. property (ii) (as well as compactness (iii)) is an inherent This implies that R" belongs to the (non-empty) intersection property of fluid sample paths, valid for FSPs defined in of Vcond and V * . I many different settings (see for example [4] for a proof); and it is easily verified directly as well. I L e n " 4: Suppose FSP ( T ; T ) is such that and
7
i
Lernnia 3: Suppose
(T: T )
r ( t ) E R'
is a stationary FSP, namely
~ ( fG ) R* and ~ ( t )T* for all t 2 0
Then, R* is a solution to the problem (1)-(4) and R* Vcond
E
n I/* # 8.
Proof. Let vector f 72' be defined as 11 = exp(a x T * ) . In view of property (15). it follows from r ( t ) H* that we have ~ ( t )R' as well. By (16). for almost all t 2 0 we have V(t)
f argmax[Q x W E
If
V H ( T ( t ) ) .] U .
We see (since ~ ( t )R' and r ( t ) R*) that v = R' solves the problem mnxlq x V H ( R * ).]U (19) G V
or. equivalently, the problem max[VH(R*).t,+ x~~~ VEV
where the vectors Am*"?Amaz components:
yill
~
E
U -
R: have
>
(20)
the following
max{(vi - I ) I I ~ ( R ~ ) :2O0, )
Ayas= -miii{(71i
- 1)iYi(R:),O} 20
Adding the constant --Amin . Kmi9&+ Anzuz . Rmaz to the objective function in 120), we see that 't: = R' maximizes the Lagrangian . (2, - Hmin ) - A""" . (v - n m a x 1 OH(R*1 . +
for all i 2 0
and ~ ( tremains ) uniformly bounded for all t 2 0. Then, R" is a solution to the problem ( 1144) and R" E Vcond r l V * # 8. Proof. As shown in the proof of Lemma 3. v ( t ) = A'. Then, it follows from I17)-(18) that Rr E [Rpin:Ryax] for
each i - otherwise q ( t )could not remain bounded. Consider a function T~(.). If q(0)2 0 and Rr = RTin then (from (17)(18)) ~ i ( t . = 1 ~ i ( 0 for ) t 2 0. If ~ i ( 0 2 ) 0 and Rf > .Fin then q ( t ) will decrease linearly at the rate Byn- R: until it hits 0, and then will stay at 0. Similarly, if ~ ~ (5 00, )q ( t ) either stays at ~i(0) (in the case Rf = AyaX)or increases linearly until it hits 0 and then stays at 0 (in the case R,' < R,"). Thus, for some fixed d 2 0 and a fixed vector T*, we must have ~ ( tx)T* for t 2 d. The time shifted path ( B d r : O d 7 ) is also an FSP, and, as we have shown above. it is stationary. An application of Lemma 3 completes the proof. I
Proof of Theorem 1. For each integer d 2 0: consider the FSP ( T ( ~ ) , T ( ~ = ) ) (odT;b?dT)r which is a time shifted version of ( r :T ) . Since all component functions of all FSPs ( ~ ( ~ 1T ,( ~ ) )are uniformly Lipschitz continuous (because Ilr(t)ll is uniformly bounded) and the sequence of functions dd)( .) converges uniformly to the function identically equal to R*, we can choose a subsequence (&I, dj))converging (uniformly on compact sets) to a path ( r " , ~ " such ) that r"(t.) R' and T O (.) being uniformly bounded. But. the path (TO,?) is also an FSP. Application of Lemma 4 completes
2420
I
the proof. V I . SIMULATIONS
A . A chieiiing m in iiiirriri rates
In this section we report on simulation results for our algorithms. The DKC traces that we use art. determined by a DKC predictor. At each time slot t . the predictor gives the value o f DRCi(t) for each user i based on user position and a simulated channel fading process. The possible values for D R C i ( t ) are (in kbits per second), (0; YE.4,76.8, 15:3,(i! 307.2, 814.4, 921.6, 1228.811843.2: 2457.6). The average viiluc of DACi(i.) for each user is presented in Figure 2.
We show plots for Proporuonal Fair 1201, E31. PkMR and MTMR. For the latter two algorithms we take RFIn = 9.tikbps for all L and so the token rate for each user is 1.2 x 9.Gkbps = ll.52kbps. For these initial plots we do not impose a maximum r a k constraint (i.e. we set RYax = x for all i). The length of the time slot is 1.6Gims. The value of C I ~ is 6.25 x I P for d l users. I 09 0.8 -
0.7
06
~
-
0.5 0.4 -
0.1 03 01
Rare (bgl
Proportional Fair
za
15
I
30
User
Fig. 2.
m
1
40 users
3s
10 users
The avzrage DRC value for each user.
04
0.3
The 3GlxEV-DO system for high-speed data has two fea-
tures that for simplicity of exposition we did not consider in 0.1 the earlier theoretical sections of this paper. (In practice they 240Q 4fXXl 9600 have little effect on actual system performance.) First: if the basestation decides to transmit at a low rate (e.g. 38.4kbpsh we are forced to assign multiple time slots (e.g. 16 slots) to lhe same user. Otherwise. the amounl of data transmitted would be too small, which would lead to implementation 09 problems. Second. if the basestation decides to serve user i, the os actual data transmission rate might be slightly different from Dl D R C i ( t ) ,due to the error-correcting coding schemes that are 06 employed. In our simulations we do take these two features os into account. 04 . 03We also remark that PFMR and MTMR as described in 02 Section IV are designed to provide constraints on the long ferm 01.received service rates. In practice we wish to bound the service O L 2400 1800 9600 I9200 38400 76800 153600 rate received over shorter time intervals. We achieve this by Rate (bp) slightly increasing the token rate, when Ti(t) 2 0. MTMR (For the definition of R;Oke'' see Equation 7.) In particular, for our simulations we set Rtoken= 1.1 x Rpin when T z ( t )2 0. Fig. 3. (Top) Proportional Fajr. Middle) PFMR with RYl" = 9.6kbps. We run the traces for 90 seconds. At the end of each 10 (Bottom) MTMR with R y i n = 9.6kbps. second interval we calculate the average data rate that the user received during that interval. We then plot the cumulative The top plots show the cumulative distribution functions distribution function of all these rates for all users. We discard for Proportional Fair. We can see that as the number of users the measurements for the first 10 seconds in order that our increases, the minimum of the race distribution decreases. results are not skewed by transient effects. The middle plot shows the curves for PFMR. We see that In Figure 3 we show cumulative distribution functions of now. regardless of the number of users. the minimum of the rhese rates on a logarithmic scale for 10, 20, 30 and 40 users. distribution is clipped at around 9.6kbps. We note that for ~
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the case of 10 users, the curve for PFMR is almost identical to the curve for Proportional Fair. This is because for this small number of users. Proportional Fair already achieves R r i n = 9,fjkbps for all users, Hence for PFMR the token levels remain small and so the two algorithms are essentially the same, However, for the cases of 20, 30 and 40 users, Proportional Fair cannot provide Flyin= 9.6kbps'for all users. In this case. some of the token levels in PFMR rise to become nonzero so as to increase the rates of the low raie users. In the bottom plot we show the curves for MTMR. Here we also clip the minimum of the distribution. However. the curves are a different shape from the curves for PFMR. This i s because more of the service is given to a few users with the best DKC values. We recall that the aim of PFMR is to maximize log Ki sub-iecr to the RFln constraints. The aim o f MTMR is to maximize Ci Ri, subject to the .flyin constraints. In the following table we show h e values of these objective functions for the case of 30 users. We see that Proportional Fair has a slightly higher value of log R ithan PFMR since it is not trying to satisfy the RY'" constraints. We also note that PFMR has a higher value of log Ri than MTMR but MTMR has a higher value of Ci Ri.
0.9
0.8 0.7 0.6
~
05 0.4 -
0.3
-
1500
15w0
30"
6W3w Rate (bps)
l2WW
210000
Proportional Fair
xi
xi
MTMR
1
0.9 -
In Figure 4 we change the minimum rate constraint so that RTin = 3O.Okbps for all users. We show cumulative distribution functions for 6, 8, 10 users under Proportional Fair. PFMR and MTMR. Once again, both PFMR and MTMR achieve the minimum rate constraints whereas Proportional Fair does not. (Note that we consider smaller numbers ofusers than before since no algoridim could provide a 30.0kbps rate constraint to larger numbers of users). In Figure 5 we study how PFMR peforms when we have a maximum raie constraint. In particular we show simulation results for the PFMR algorithm with Rifi" = 9.Gkbps and Rpax = 5O.Okbps for all users. (Compare this figure to the middle of Figure 3, which corresponds to exactly same simulation scenario, but with RYax = cc.) We see that users' throughputs are indeed "capped" at 50 kbps, as desired. B. The mken processes
In Figure 6 we illustrate the behavior of the token processes for PFMR in the 30 user case with RFin = Y.6kbps and RYax = 30. User 7 has small DRC values and so it would not be able to achieve '?I = 9.6kbps under Proporlional Fair. Hence the token level for this user stabilizes in a range that is strictly above zero. This increases the likelihood that user 7 is served in each time slot. i n contrast, user 29 has larger DRC values which means that it woiild achieve Rg" = 9.6kbps under Proportional Fair. For this user the token level falls to zero since it does not need "extra help" from the tokens to obtain the minimum service rate.
0.8 0.7 0.6 0.5 0.4
0.3 02 01
~
~
~
~
~
~
I
We note that in any finite time interval. Lhe user can actually get a rate that is slightly less than the token rate (which is 11.52kbps). If i n some interval the token level rises then its service rate is slightly less than than the token rate. If the token level falls then the service rate is sIightly higher than the token rate. We illustrate this in the plot of user 7 in Figure 6. Service rates are given for two intervals of length 10 seconds. This phenomenon is also the reason why the minimum rates for MTMR in Figure 3 are slightly less than 9.6kbps.However, we emphasize that for longer time intervals the minimum service rates become closer to the token rate whenever the token levels are bounded. In particular, consider a time interval [ t l : t 2 ] . Let T(t1) and T(t2) be the token levels at time t i and f.2.
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C. Differentiating w e r s
0.7
0.6 0.5
Fig. 5 .
PFMR with R f ' " = 9.6khps and
.Fax
= 50.Dkbps.
48W
Fig. 7.
9600
19200 R a t (bpi
48wO
76803
1536"
Service rates for two classes of users.
In Figure 7 we show that hy using PFMR we can achieve different values of Byin for different users. In particular we assume that in the 30 user case. two users have paid extra so that they are assigned an I?;'" value of 48kbps. The remaining users have R,"" = 9.Gkbps. The token rate for the two "premium" users is 1.2 x 48kbps = 57.6khps. We show in the figure that the distribution of rates for these users is lower bounded by 4Pkbps. For the remaining users. the minimum of the rate distribution is clipped around 9.6kbps.
D. Coniparision to enforcing rate consrraints via ?nadiJed utilip function
Recall that in Section 1°C we discussed that a plausible metbod for achieving minimum rates would be to modify the utility function so that rate constraint violations are penalized. As an example, suppose that we modify Proportional Fair so that it aims to maximize the utility function,
H ( R )=
T U ~ C(6101~)
User 29 Fig. 6. The token prmesszs for users 7 and 29.
respectively. and assume for simplicity that T(t) never hits 0 in this interval, Then the service rate that the user receives in the interval [ f ~ , is l ~equal ] to.
(Here we use 1.2 x RYln, as opposed to just RYln. as a threshoId below which we impose penalty on the "low" values of R r i n +The reason for doing that is the same as the reason for using 1.2 x Fly'' as the token rate in FFMR.) The corresponding Gradient algorithm always tries to serve flow, i E argmaxDRCi(t)x iEI
(Token rate)
+
T(t1)- T(t2) t? -t.l
If T(t1)and T(t2)remain bounded as tz - tl becomes large, then the second term approaches zero and so the service rate In Figure 8 we compare this algorithm with PFMR for the case approaches the token rate. (If T ( t )sometimes hits 0 in tz - t ~ . o i 30 users and .pin = 9.Gkbps. We can see that it is less boundcdness of T ( t l )and T(t2)implies that the service rate effective than PFMR at providing minimum rates and achieves significantly less system throughput. becomes ut !east the token rate.) 2423
I
0.9 0.8
0.7
-
I
~
~
0.6 -
05 0.4
0.3
0.2
0.1 0
~
~
I
Fig. 1. Comparison of PFMR with a modified version of Proportional Fair that attempts to enforce minimum rates via a penalty in the utility function.
VII. DISCUSSION
Scheduling Poticiss. In Pmc. of tiw 40th Annual Alierlon Conference
We have proposed the GMR algorithm and proved an optimalily result showing that if the user throughputs converge, hen the corresponding stationary throughputs do in fact maximize the desired utility function, subject to minimurdmaximum rate constraints. The rate constraints are enforced via a very generic token counter mechanism. We note that token counters provide a very natural overload detection and control mechanism. A large positive value of a user’s
.
AJalali. KPadovani. and R.Pankaj. Data Throughput of CDMA-HDR. a High Efficiency - Hish Data Rate Personal Comniunicatiau Wireless System. In Proc. os the IEEE Seminiirrital Veiric/rlnr Technology Conference, 1TC2000-Spring. Tolqo. Japan. ~ M n y2000. A. L. Stolyar. MaxWz’right Scheduling in a Generalized Switch: State S p c e Collapse and Workload Minimization in Heavy Traffic. Annals of .ipplied Pmhbili@, vo1.14(1). 1004.pp. 1-53. A.L. Stolyar. On thz Asymptotic Optimality of the Gradient Scheduling Algorithm for Multi-User Throughput Allocation. Operu?inns Researck. vo1.53(1). 2005. to appear. A. L. Stolyar. Maxinuzing Queueing Network Utility subject IO Stability: Greedy Primal-Dual Alprithm. 2001. submitted. K. Chakavorty. S. Katti. I. Pratt and 1. Crowcroft. Flow aggregation for enhanced TCP over wide area wireless. In Proc:.IEEE INFOCOM. 2003. L. Tassiulas and A. Ephremides. “Stahdity properties of constrained queueing system and scheduling policies for maximum throughput in multihop radio nztworks:’ lEEE Tmnsocriom of) :lutomutir Control. vol. 37. no. 12. pp. 1936 - 1948. Deczmhzr 1992. M.Andrews. K. Kumaran. K. Ramanan, A. Stolyar. R. Vijayakumar. and P. Whuring, ”Providing quality of service over a shared wireless link” IEEE Commitnicafions Magazine. Fehruarv 200 1 [IO] S. Shakkottai and A. St&ar. “Scheduiing algorithms for a mixture of real-timz and non-real-time data in HDR,“ in Pmceedinys of 17th Infemrionnl Teletrafic Congress (ITC-17). Salvador da Bahia. Brazil, 2001. pp. 793 - 804. [ 1I ] S. Shakottai and A. L. Stolyar. Scheduling for Multiple Flows Sharing a Time-Varying Channel: The Exponential Rule. h i d y t i r Melhods in Applied Pmbabilily. h i M e m a n of Fridrih Kqpelevich. Yu. M. &dim, Editor: American Mathematical Society Translations. Series 2. Volume 207, pp. 185-202. American Mathematical Society Providznce. RL 2002. [I21 S. Barst and P. Whiting. “Dynamic rate control algorithms for CDMA throughput optimization,” in Proceedings uf IEEE INFOCOM ‘01, Anchorage, AK. April 2001. [ 131 S. Borst. “User-lev4 psrformance of channel-aware scheduling algorithms in wireless data networks.“ in Proceedings of IEEE INFOCOM ‘03,San Francisco. CA, April 2003. 1141 M. Andrews. “lnstahility of the proportional fair scheduling algorithm for HDR.” IEEE Transacliom on Wireless Communirurions 3(5), 2 W . [IS] M. Andrews and L. Zhang. “Schzduling over non-stationary wireless channels with finit2 rate sets.” In Pmceedirrgs of IEEE INFOCOM ’#: Hong Kong. 2004. j16] R. Agrawal, V. S u b r a w a n . Optimality of Certain Channel Aware
token counter indicates that this user “needs help” to reach the desired minimum Ihroughput. A large number of such users indicates that air interface resources are “stretched” in trying to provide minimum rate for all users - this can serve as a trigger of an overload control action. Our simulation results show good performance and robustness of the algorithm, which? along with its simplicity and “compatibility” with the widely employed Proportional Fair algorithm. make this algorithm very attractive for practical use.
on CoJnnmuwicution. Coirrrol. and Compiriing. Munficella. Illinois, US.4. Ocrober 2002. [17] H. Kushner. P. Whiting. Asymptotic Properties of Proportional Fair Shanng Algorithms. In Pmc. of the 40th .-lnm~l.4llerron Conference on Cammimicurion, Control. and Cornputittg. Monticello, Illinois. USA, OctuOer 2002. [ 181 X.Liu. E.K.P.Cnong, N.B.Shroff. Opportunistic Transmission Scheduling with Resource-Sharing Constrajnts in Wireless Networks. IEEE Journal on Selected Areas in Comrmrraications. 19(10), 2001. pp. 2053-
2063. A Framework for Qpporlunistic Scheduling in Wireless K‘etworks. Compirfer Netwurh. ~01.41,1002, pp.451-174. [20] P. Viswanath. D. Tsz and R. Laroia. Opportunistic Beamforming using Dumb Antennas. IEEE Trmsactiwis on Information Theory. 48(6), 2002. [ 191 X.Liu. E.K.P.Chong, N.B.Shroff.
REFERENCES M. Andrews. K. Kunnran. K. Ramanan. A. L. Stolyar. R. Vijayitkumar. P. Whitins. Scheduling in a Queueing System with Asynchronously Varying Service Rates. Plobubiliry in Engineering und bdormofioiid Sciences. ~01.14.2004. pp.191-217. P.E.Gil1 and W.Murray Numerical Mrtliods for ConsfruinedOptimizarion. Academic Press. London. 1974.
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