Laxity-Based Opportunistic Scheduling with Flow-Level Dynamics and Deadlines
aa r X i v : . [ c s . I T ] O c t Laxity-Based Opportunistic Scheduling withFlow-Level Dynamics and Deadlines
Huasen Wu and Youguang Zhang
School of Electronic and Information EngineeringBeihang University, Beijing 100191, ChinaEmail: [email protected]
Xin Liu
Microsoft Research Asia, Beijing 100080, ChinaUniversity of California, Davis, CA 95616, USA
Abstract —Many data applications in the next generation cel-lular networks, such as content precaching and video progressivedownloading, require flow-level quality of service (QoS) guaran-tees. One such requirement is deadline, where the transmissiontask needs to be completed before the application-specific time.To minimize the number of uncompleted transmission tasks,we study laxity-based scheduling policies in this paper. Wepropose a Less-Laxity-Higher-Possible-Rate (L HPR) policy andprove its asymptotic optimality in underloaded identical-deadlinesystems. The asymptotic optimality of L HPR can be appliedto estimate the schedulability of a system and provide insightson the design of scheduling policies for general systems. Basedon it, we propose a framework and three heuristic policies forpractical systems. Simulation results demonstrate the asymptoticoptimality of L HPR and performance improvement of proposedpolicies over greedy policies.
I. I
NTRODUCTION
Opportunistic scheduling plays an important role in im-proving network resource efficiency and user experience. Alarge number of scheduling policies, such as Proportional Fair(PF) scheduler [1] and MaxWeight [2], have been proposedto balance between the system throughput and the level ofsatisfaction among different users. Most existing work focuseson the packet-level scheduling, where the number of users isassumed to be fixed and the performance is defined on thepacket-level, e.g., number of packets received in a unit timeor average delay of all received packets.On the other hand, file download and multimedia streamingbecome increasing popular in cellular networks [3]. The trafficgenerated by these applications is characterized by flow-leveldynamics and deadlines. This is because scheduling suchtraffic needs to be carried out across a greater temporalscale, during which the population of users may change. Inaddition, the transmission tasks should be completed beforetheir application-specific deadlines to maintain the requiredquality of experience (QoE). For example, in progressivedownloading, to achieve quasi-live streaming, a segment ofa video should be downloaded before the buffer depletes,which imposes a deadline of several seconds [3]. Therefore,we study opportunistic scheduling policies to minimize thedelay violation probability in wireless networks with flow-leveldynamics and deadlines.Flow-level scheduling has been considered in the literature.Similar to the packet-level scheduling, a critical issue is toguarantee stability if possible. Recent results show that themaximum stable region can be easily achieved by applyingsome simple rules such as Best-Rate (BR) rule [4]. Otherpapers investigate policies for minimizing the average trans-mission delay. In [5, 6], with the assumption of fast varyingchannel conditions, it is shown that combining opportunis-tic scheduling and the Shortest-Remaining-Processing-Time (SRPT) discipline in machine-job scheduling can minimize theaverage delay. However, the transmission delay may exceedthe user’s tolerant delay and become useless. In [7], the authorsstudy flow-level scheduling policies for maximizing delay-dependent utility functions. This model can be viewed asscheduling with soft deadline constraints. However, it requiresthe knowledge of channel states in the future, which may bedifficult in practice.Scheduling with deadlines has been investigated in machine-job scheduling literature. Policies such as Earliest-Deadline-First (EDF) and Least-Laxity-First (LLF) have been proposedand shown to be optimal for underloaded systems [8]. Namely,a feasible schedule can be obtained by EDF and LLF if thereare some off-line policies can do so. Other policies, e.g.,D over [9], have been proposed for overloaded systems andare shown to achieve the optimal competitive ratio. However,the temporal variation of data rate makes the design andanalysis of scheduling policies for wireless networks withflow-level deadlines challenging. In [10], the authors showthe Max C/I policy, which greedily serves the user withthe highest data rate, achieves the optimal competitive ratio,assuming a partial value model. In this model, one user doesnot require the completion of the entire transmission task andthe value is in proportion to the amount of data received. Inmany applications, however, it is required that at least certainpercentage of data should be received or it will be useless.In this paper, to minimize the number of uncompleted tasks,we study scheduling policies that balance between servingurgent users and maintaining multi-user diversity. We quantifythe urgency of transmission tasks with laxity and proposelaxity-based policies for scheduling file download traffic. Un-der the assumption of polymatroid capacity region [5], wepropose a Less-Laxity-Higher-Possible-Rate (L HPR) policyand show its asymptotic optimality in underloaded identical-deadline systems. To the best of our knowledge, this is the firsttheoretical result on wireless scheduling with deadlines usingthe entire value model. The insights obtained from this policycan serve as a guideline to design policies for general systems.Based on it, we propose a laxity-based policy framework andthree heuristic policies for practical systems. Through numer-ical simulations, we demonstrate the asymptotic optimalityof L HPR and the performance improvement of channel-and-urgency-aware policies.II. S
YSTEM M ODEL
We consider the flow-level scheduling with deadlines in thedownlink of a single cell. A sequence of users enter the systemand request to download files with deadlines. They departupon task completion or delay violation. The objective of thease station (BS) is to minimize the number of uncompletedrequests.
A. Traffic and Channel Model
Let I be the index set of all users entering the system.For each user i ∈ I , the download request is representedby a triple ( A i , F i, , D i ) , where A i , F i, , and D i denote thearrival time, the initial file size (in bits), and the deadline,respectively. All A i s, F i, s, and D i s are random variables.The difference between the deadline and the arrival time, i.e., D i − A i , reflects the delay tolerance of user i . We focus onfile download applications such as content precaching. Hence,we assume that the file size F i, is available as soon as user i arrives.All data are transmitted over a wireless channel from theBS to each user using Time-Division Multiplexing (TDM).The channel condition for each user is time-varying and ismodeled as a stationary stochastic process R i ( t ) ( t ∈ R ),where R i ( t ) ≥ denotes the instantaneous rate at which theBS can transmit to user i at time t . We assume a wirelesssystem with homogeneous channels, where R i ( t ) s ( i ∈ I ) arestatistically identical with ¯ R i = E [ R i ( t )] = ¯ R for all i ∈ I .For a more practical system with heterogenous channels, wecan transform the original system into an equivalent systemwith homogeneous channels using the scaling technique in[5]. Similar to [5], we normalize the data rate and the file sizewith respect to the average rate ¯ R . B. Scheduling Process
The BS schedules the transmission in a slotted manner. Timeis divided into time slots of length ∆ t and each slot is indexedby an integer n . In slot n , we let Q [ n ] ⊆ I be the index set ofusers present in the system, and Q [ n ] = |Q [ n ] | be the numberof users. For each user i ∈ Q [ n ] , we denote its residual filesize by F i [ n ] .At the beginning of the n -th slot, i.e., t n = n ∆ t , the BSallocates user i with data rate c i [ n ] . The rate vector, whichconsists of all c i [ n ] s ( i ∈ Q [ n ] ), stays in the capacity regioncorresponding to Q [ n ] [5]. Thus, the residual file size of user i evolves as follows: F i [ n + 1] = max { , F i [ n ] − c i [ n ]∆ t } , (1)where the initial value of residual file size is F i, .Note that the time scale separation [5] is applied here. Inother words, we assume that the channel conditions R i ( t ) s( i ∈ I ) vary infinitely fast. Then, a time slot can be dividedinto mini-slots, each of which is in the order of the channelcoherence time. If the BS schedules in each mini-slot, thenthe data rates allocated to the present users in each slot can beaveraged out and the rate vector stays in the capacity region.This assumption is critical because it captures the multi-userdiversity effect in a tractable manner. However, we note thatthe time scale separation is highly ideal, especially when theslot length ∆ t → . We will only use it for asymptotic analysisin Section III, and will relax this assumption when designingheuristic policies for practical systems in Section IV.The objective of the BS is to minimize the number ofusers violating their deadlines. However, designing optimalpolicies for such a scheduling problem is challenging evenwith the time scale separation argument. Therefore, in this paper, we first focus on the optimal policies for underloadedsystems under certain additional assumptions. We call a policy optimal in underloaded systems , if a feasible schedule can beobtained by it whenever there are some off-line policies can doso, following the convention in machine-job scheduling [8, 9].Then, we propose heuristic policies for more general systemsand evaluate their performance through simulations.III. A SYMPTOTICALLY O PTIMAL P OLICY IN I DENTICAL -D EADLINE S YSTEMS
In this section, we study the opportunistic scheduling prob-lem in an identical-deadline system with M users, all of whichrequest to download files before a same deadline, i.e., D i = D for i ∈ I = { , , . . . , M } .Due to the flow-level dynamics, designing optimal policiesis challenging even for the identical-deadline system. Mo-tivated by the idea of polymatroid capacity region [5], wepropose a laxity-based policy, referred to as L HPR, and proveits asymptotic optimality.
A. Polymatroid Capacity Region
To be self-contained, we briefly summarize the definitionas follows. Polymatroid capacity region [5] approximates theoriginal capacity region with its polymatriod outer bound.Consider the scenarios where all the channel conditions arei.i.d. processes across all users. Let g k be the achievable multi-user diversity gain when there are k active users, i.e., theratio between the maximum throughput achieved by the k -user system and the single user system. Assume that g k isconcavely increasing in k and let g ≡ . The polymatroidcapacity region is defined as follows: ¯ C k ≡ (cid:26) r ∈ [0 , k : ∀K ⊆ { , . . . , k } , X i ∈K r i ≤ g |K| (cid:27) . The polymatroid capacity region is the tightest polymatroidouter-bound region containing the original capacity region,which is shown in Fig. 1 for the 2-user case. Thus, the mini-mum delay violation probability obtained with the polymatroidcapacity region is a lower-bound of the practical system. O g g ( - , ) g g g ( , - ) g g g ,2 2 !" g g [ ] c n [ ] c n Fig. 1. Polymatroid capacity region for 2-user case [5].
B. Design of L HPR Policy
In order to minimize the number of uncompleted tasks, theBS should tradeoff between maintaining multi-user diversity(i.e., maximizing system throughput) and serving the moreurgent users. To quantify the urgency of a given user, weintroduce expected laxity . The expected laxity is similar to thelaxity defined in traditional job scheduling with the constantservice rate replaced by the expected rate. efinition 1 (Expected Laxity) In slot n , for each i ∈ Q [ n ] ,the expected laxity is defined as L i [ n ] = D i − n ∆ t − F i [ n ] g . (2)In the above definition, the term F i [ n ] /g is the expected timerequired to finish the task with the entire channel allocated touser i . Hence, the expected laxity represents the time that canbe allocated to other users without effecting the transmissionof user i . Users with smaller expected laxity are more urgent.Motivated by the intuition that the BS should allocate moreresource to more urgent users, we propose a Less-Laxity-Higher-Possible-Rate (L HPR) policy.
Definition 2 (L HPR policy) In slot n , when Q [ n ] = ∅ , sortall users in Q [ n ] in the ascending order of their expected laxityand let j ( i ) be the rank of user i . The L HPR policy serveseach user i ∈ Q [ n ] with data rate c i [ n ] = g j ( i ) − g j ( i ) − . (3)Note that with the assumption of concavely increasinggain, we have g j − g j − < g j +1 − g j , indicating thatwith L HPR policy, the user with less expected laxity isallocated with higher data rate. Moreover, the total data rateis P i ∈Q [ n ] c i [ n ] = g Q [ n ] , and thus the L HPR policy reachesthe maximum system throughput that can be obtained whenthe number of users is Q [ n ] . C. Asymptotic Optimality of L HPR Policy
In this subsection, we show that the L HPR policy isasymptotically (as the slot length ∆ t tends to 0) optimal forunderloaded systems with identical deadlines.It is worth noting that in constant-rate machine-job schedul-ing, the optimality of the LLF (or EDF) policy is shownwith the exchange argument, i.e., transforming any feasibleschedule into the one found by the LLF (or EDF) policy [8].However, this approach does not work for L HPR since thefeasible service rates depend on the number of users presentin the system. Rather than using the exchanging argument,we study the asymptotic optimality by examining the leastexpected laxity obtained by L HPR in every time slot.We examine the state of all users having entered the system,including both present users and completed users. Let Q ′ [ n ] denote the index set of users arriving before time t n , i.e., Q ′ [ n ] = { i : 1 ≤ i ≤ M, A i ≤ t n } . We notice that by setting the residual size F i [ n ] = 0 ,the expected laxity defined in (2) can also be applied forcompleted users. We further notice that the term n ∆ t in (2) iscommon for all users. For the sake of notation simplicity, weintroduce the virtual expected laxity for each user i ∈ Q ′ [ n ] ,which is defined as follows L ′ i [ n ] = L i [ n ] + n ∆ t = D − F i [ n ] g . Note that by this definition, the virtual expected laxity is L ′ i [ n ] = D for completed users since F i [ n ] = 0 .In order to show the asymptotic optimality of L HPR, weneed to examine the least expected laxity, or equivalently, the least virtual expected laxity, obtained by L HPR in every slot.The least virtual expected laxity is given by L ′ L HPR [ n ] = min i ∈Q ′ [ n ] L ′ i [ n ] , and the least-laxity-user is defined as the user having the leastvirtual expected laxity, i.e., i ∗ [ n ] = arg min i ∈Q ′ [ n ] L ′ i [ n ] . Note that there may be more than one user having the leastvirtual expected laxity and we let i ∗ [ n ] be the one with thesmallest index. This will not affect the analysis result since theperformance of L HPR is reflected by the value of the leastvirtual expected laxity.We expect that L HPR achieves the maximum least virtualexpected laxity in every time slot to ensure the optimality ofL HPR. Unfortunately, this is not always true because L HPRserves users with rate values from a discrete set { g , g − g , . . . , g M − g M − } , but other policies can reach larger leastvirtual expected laxity by using finer allocation. For example,for two users with L ′ [0] = L ′ [0] = D , the L HPR policyallocates data rates g and g − g to these two users and theleast virtual expected laxity of L HPR is L ′ L HPR [1] = D + g − g g ∆ t . However, one can allocate equal data rate, i.e., g ,to the two users and achieve the least virtual expected laxity D + g g ∆ t . This is larger than L ′ L HPR [1] due to the concavelyincreasing property of g k . Nevertheless, we can show that thedifference vanishes as the slot length ∆ t tends to 0 and thusL HPR is asymptotically optimal.First, we focus on the case where all users arrive at the sametime. Without loss of generality, we assume A i = 0 for all i = 1 , , . . . , M . Hence, for all n ≥ , Q ′ [ n ] = { , , . . . , M } .Recall that the key idea of the L HPR policy is allocatinglarger data rate to the user with less virtual expected laxity.Thus, we first define two relationships, “Used-to-be-Less-Than (ULT)” and “Indirectly-Used-to-be-Less-Than (I-ULT)”,which will play an important role in analyzing the performanceof L HPR.
Definition 3 (ULT and I-ULT)a) In slot n , for two users i and i , we say that i used-to-be-less-than (ULT) i before slot n , denoted as i ≤ n i ,if there exists an n ′ ( ≤ n ′ ≤ n ) such that L ′ i [ n ′ ] ≤ L ′ i [ n ′ ] .b) In slot n , for two users i and i , we say that i indirectly-used-to-be-less-than (I-ULT) i before slot n , denoted as i (cid:22) n i , if there exists a user sequence i ′ , i ′ , . . . , i ′ m , suchthat i ≤ n i ′ , i ′ ≤ n i ′ , . . . , and i ′ m ≤ n i . The lemma below shows that under the L HPR policy, ifa user ULT another user, its virtual expected laxity will notmuch exceed that of the other user.
Lemma 1
For an identical-deadline system under the L HPRpolicy, if users i and i satisfy i ≤ n i , then L ′ i [ n ] − L ′ i [ n ] ≤ ∆ t. (4)Lemma 1 can be proved by using the definition of ULT andtracing the virtual expected laxity of users i and i . It isomitted here due to the space limitation. Interested users arereferred to Appendix A.n order to provide a lower bound on the least virtualexpected laxity obtained by L HPR, we define a least-laxity-set , which contains the least-laxity-user i ∗ [ n ] and all otherusers I-ULT i ∗ [ n ] . Definition 4 (Least-Laxity-Set) The least-laxity-set in slot n is an index set Q [ n ] satisfying the following conditions:a) i ∗ [ n ] ∈ Q [ n ] ;b) For any i ∈ Q [ n ] , i ∈ Q [ n ] if and only if i (cid:22) n i ∗ [ n ] . Let Q [ n ] = |Q [ n ] | be the number of elements in Q [ n ] .By the definition of Q [ n ] , we know that from time slot totime slot n − , all the Q [ n ] largest data rates, i.e., { g , g − g , . . . , g Q [ n ] − g Q [ n ] − } , are allocated to users in Q [ n ] . Withthis property of Q [ n ] , we present the following lemma statinga lower bound on the least virtual expected laxity of L HPR.
Lemma 2
The least virtual expected laxity obtained byL HPR in time slot n is bounded as L ′ L HPR [ n ] ≥ min (cid:26) D − ( M − t, Q [ n ] (cid:20) X i ∈Q [ n ] L ′ i [0] + ng Q [ n ] ∆ tg (cid:21) − ( M − t (cid:27) . (5) Proof:
By the definition of I-ULT, for any i ∈ Q [ n ] , i = i ∗ [ n ] , we can find a sequence of different users, { i ′ , i ′ , . . . , i ′ m } , such that i ≤ n i ′ , i ′ ≤ n i ′ , . . . , and i ′ m ≤ n i ∗ [ n ] . Note that the sequence does not contain i or i ∗ [ n ] and thus m < M − . According Lemma 1, we have L ′ i [ n ] ≤ L ′ L HPR [ n ] + ( M − t. (6)Consequently, when there are some completed users in Q [ n ] , the least virtual expected laxity is bounded by L ′ L HPR [ n ] ≥ D − ( M − t. (7)On the other hand, when there are no completed users in Q [ n ] , as we have pointed out before, from time slot to n − ,all the Q [ n ] largest data rates are allocated to the users in Q [ n ] .Thus, the sum of virtual expected laxity is X i ∈Q [ n ] L ′ i [ n ] = X i ∈Q [ n ] L ′ i [0] + ng Q [ n ] ∆ tg . (8)Note that from (6), we have L ′ L HPR [ n ] ≥ Q [ n ] X i ∈Q [ n ] L ′ i [ n ] − ( M − t. (9)Finally, combining (7), (8), and (9) , we know that (5) istrue.Next, using Lemma 2, we show the asymptotic (as the slotlength ∆ t → ) optimality of L HPR in underloaded identical-deadline systems.
Theorem 1
Assume that for i = 1 , , . . . , M , A i = 0 , D i = D > . When the slot length ∆ t → , the L HPRpolicy achieves the maximum least-laxity at any time t and isasymptotically optimal in underloaded systems. Proof: For given t , we divide the time into N slots andthe slot length ∆ t = t/N . According to Lemma 2, as N → ∞ and ∆ t → , the least virtual expected laxity at time t satisfies L ′ L HPR [ N ] ≥ min (cid:26) D, Q [ N ] (cid:20) X i ∈Q [ N ] L ′ i [0] + g Q [ N ] tg (cid:21)(cid:27) . (10)Because g Q [ N ] t is the maximum throughput that Q [ N ] userscan obtain in a duration t , there are no other feasible schedulescan obtain larger least virtual expected laxity than L HPR.Consequently, when the arrival sequence is schedulable, theL HPR policy will generate a feasible schedule as ∆ t → and is asymptotically optimal in underloaded systems.This conclusion can be extended to the case with identicaldeadlines but different arrival times, which is stated by thefollowing theorem. Theorem 2
Assume that for i = 1 , , . . . , M , D i = D , A i The asymptotically optimal policy L HPR is based onthe idealized assumption of polymatroid capacity region, andcannot be implemented in practical systems. In this section,we propose practical heuristic laxity-based policies. A. Policy Structure First, for a TDM system, typically at most one user can beserved in each time slot. We assume that the slot length ∆ t is sufficiently small and the channel state is constant withinone time slot. Let R i [ n ] = R i ( t n ) be the data rate supportedby user i in slot n . Then, in slot n , using policy Π , the BSchooses user Π( S [ n ]) to serve based on the network status S [ n ] , which is given by S [ n ] = {Q [ n ]; ( A i , F i [ n ] , D i ) , i ∈ Q [ n ]; R i [ n ] , i ∈ Q [ n ] } . Furthermore, another issue is that a practical system maybe overloaded, i.e., not all download tasks can be completedbefore their deadlines. Serving users which are likely to expiremay waste the chance to finish other download tasks. Thus,according to the expected laxity, we divide the present users Q [ n ] into two groups, G (+) δ [ n ] and G ( − ) δ [ n ] , which are givenby G (+) δ [ n ] = { i : i ∈ Q [ n ] , L i [ n ] ≥ δ } , and G ( − ) δ [ n ] = { i : i ∈ Q [ n ] , L i [ n ] < δ } . The users in G (+) δ [ n ] will be served by trading off betweenthe data rate and the urgency. For the users in G ( − ) δ [ n ] , theyill be served only if G (+) δ [ n ] = ∅ , so that we do not waste ontasks that are unlikely to be finished. Moreover, for G ( − ) δ [ n ] ,we will simply serve the user with the highest data rate tomaximize the system throughput. Specifically, we propose thefollowing policy framework: Π( S [ n ]) = arg max i ∈G (+) δ [ n ] κ i R i [ n ] U ( L i [ n ]) , if G (+) δ [ n ] = ∅ , arg max i ∈G ( − ) δ [ n ] κ i R i [ n ] , if G (+) δ [ n ] = ∅ , (11)where κ i > is a constant for distinguishing priorities ofdifferent applications, and U ( · ) is a decreasing function thatquantifies the urgency based on the expected laxity L i [ n ] de-fined by (2). With the structure proposed above, we can obtaindifferent policies by designing different urgency functions. B. Laxity-based Heuristic Policies We construct the urgency function in (11) to obtain differentpolicies. First, note that when δ < , the expected laxity L i [ n ] for i ∈ G (+) δ [ n ] may be negative. We deal with this issue byusing an approximation L ǫi [ n ] , which is given by L ǫi [ n ] = max { L i [ n ] , ǫ } , where ǫ > is a small constant.We propose three heuristic polices based on polynomial,exponential, and logarithm urgency function, and refer to themas L-MaxWeight, L-Exp, and L-Log, respectively. a) L-MaxWeight U L-MaxWeight ( L i [ n ]) = ( L ǫi [ n ]) − α , where α > . b) L-Exp U L-Exp ( L i [ n ]) = exp (cid:26) − β i L ǫi [ n ] ζ + ( ¯ L [ n ]) η (cid:27) , where β i > , ζ > , and ¯ L [ n ] = |G (+) δ [ n ] | P i ∈G (+) δ [ n ] β i L ǫi [ n ] . c) L-Log U L-Log ( L i [ n ]) = 1log( ζ + β i L ǫi [ n ]) , where β i > and ζ > .Rigorous analysis for the performance of the above heuristicpolicies is challenging and we will evaluate their performancethrough simulations in the next section.V. S IMULATION R ESULTS In this section, we evaluate the performance of the pro-posed laxity-based policies through simulations. We presentsimulation results on the schedulability and delay violationprobability. A. Simulation Settings1) Traffic Model: We consider both identical-deadline sys-tems with finite users and stationary-arrival systems. In theidentical-deadline system with deadline D , we assume thatthere are M users with arrival times uniformly distributed inthe interval [0 , aD ] . a ∈ [0 , is a constant for controlling thedistribution of the arrival time. In the stationary-arrival case, we assume that the users arriveaccording to a Poisson process with rate λ . Moreover, for each i ∈ I , we set the deadline D i = A i + ξF i, /g , where ξ > is the maximum acceptable stretch factor. Note that the stretchfactor is defined as the ratio between the practical delay and theexpected delay in an ideal situation where the entire channelis occupied by a given user [11]. Hence, the metric ξ providesan indication of how much delay the application can tolerate.For the file size, we apply the model proposed in [12] forFTP traffic, where the file size follows truncated lognormaldistribution with mean 2 Mbytes, standard deviation 0.722Mbytes, and maximum size 5 Mbytes. The size is normalizedwith respect to the average rate. 2) Channel Model: We use a continuous transmission ratemodel [7, 13] and assume that the data rate supported by eachuser is given by R i [ n ] = B log (cid:0) γ i [ n ] (cid:1) , where B is thebandwidth, and γ i [ n ] is the received SINR of user i in slot n . Rayleigh fading channel is assumed for each user and thusthe received SINR γ i [ n ] follows exponential distribution. Weset B = 800 KHz and the expectation of SINR E γ i [ n ] = 0 dB. Note that these values only have a marginal effect sincethe data rate is normalized with respective to the average rate.To obtain multi-user diversity gains used in L HPR, we set g = 1 due to the normalization of the data rate, and obtain g k ( k > ) by evaluating the throughput of the Max C/I schedulerwith k users. 3) Policy Parameters: The parameters used by the proposedlaxity-based policies are summarized in Table I. We set δ = − < since the users with negative (but not toolarge absolute value) expected laxity may still be able tobe completed if they experience good channel conditions inthe following slots. Other parameters are similar to those inconventional packet level scheduling policies [13]. TABLE IP ARAMETERS USED BY SCHEDULING POLICIES Common L-MaxWeight L-Exp L-Log κ i = R i α = 1 β i = 0 . β i = 10 δ = − ζ = 1 ζ = 10 ǫ = 0 . η = 0 . B. Schedulability under Different Policies Fig. 2 shows the number of schedulable realizations underdifferent scheduling policies. From Figs. 2(a) and 2(b), wecan see that the proposed L HPR achieves the largest numberof schedulable realizations among all policies. By tracingthe scheduling results of each realization, we find that inthe identical deadline system, a realization is schedulableunder L HPR as long as it is schedulable under some otherpolicies. In contrast, a schedulable realization under L HPRis not necessarily schedulable under other policies. Theseresults demonstrate the asymptotic optimality of L HPR inunderloaded identical-deadline system. Similar phenomenonoccurs in the stationary-arrival system, though we cannot provethe asymptotic optimality of L HPR in such a general system.The proposed laxity-based policies, L-MaxWeight, L-Exp,and L-LLF, outperform the greedy policy Max C/I in max-imizing the number of schedulable realizations. Comparingthe performance of L-MaxWeight, L-Exp, and L-LLF in theidentical-deadline system and the stationary-arrival system,we can see that L-MaxWeight and L-Exp outperform L-LLFin the identical-deadline system, but the situation is reversedn the stationary-arrival system. This is because the varianceof the expected laxity in the identical-deadline system ismuch smaller than that in the stationary-arrival system. Theurgencies quantified by the logarithm function are similaramong different users and L-LLF behaves as Max C/I in theidentical-deadline system. But the L-LLF policy provides abetter tradeoff when the variance of the expected laxity islarge and performs better in the stationary-arrival system.We emphasize that in the presented range, no realizationis schedulable under EDF and LLF, which are unaware ofthe channel conditions (EDF is not evaluated in the identical-deadline system since all users have the same deadline). EvenMax C/I, which makes scheduling decisions based only onchannel conditions, can perform much better than EDF andLLF. This shows the value of channel condition knowledge inimproving the scheduling performance. 120 140 160 180 200 220 2400500100015002000250030003500400045005000 Deadline D (seconds) N o . o f s c h e d . r ea li za ti on s L HPRL−MaxWeightL−ExpL−LogMax C/I (a) Identical-deadline system Stretch factor ξ N o . o f s c h e d . r ea li za ti on s L HPRL−MaxWeightL−ExpL−LogMax C/I (b) Stationay arrival systemFig. 2. Number of schedulable realizations ( M = 15 , a = 0 . , and λ =0 . ). C. Delay Violation Probability Fig. 3 depicts the delay violation probability under differentpolicies. It is surprising at the first sight that from Fig. 3(a),the delay violation probability of L HPR is larger than that ofMax C/I when D < . This is because the L HPR policytries to maximize the least expected laxity of the system byprioritizing the most urgent user. Thus, when a realization isunschedulable, resource is wasted and many users will violatetheir deadline constraints. Similar problems occur in otherlaxity-based policies.In the stationary-arrival system, the proposed laxity-basedpolicies outperform the greedy Max C/I policy. For example,the delay violation probability of L-LLF is only 25% of that ofMax C/I. In addition, the delay violation probability turns tobe about − under L HPR and L-LLF when ξ = 5 . Whilesimilar probability is obtained by Max C/I until ξ = 7 , whichrequires additional 40% delay.Again, we point out that the channel-oblivious policies, EDFand LLF, perform rather badly compared with the channel-aware policies. Particularly, in the identical-deadline systemunder LLF, many users achieve very close expected laxity.Hence, most of users miss their deadlines and the delayviolation probability only slightly decreases as D increases.VI. C ONCLUSION AND F UTURE W ORK In this paper, we study laxity-based policies for schedulingfile downloading traffic which is characterized with flow-leveldynamics and deadlines. Under the idealized assumption ofpolymatroid capacity region, we propose an asymptotically 160 180 200 220 240 26010 −5 −4 −3 −2 −1 Deadline D (seconds) p v i o l L HPRL−MaxWeightL−ExpL−LogMax C/ILLF 180 190 200 210 22010 −2 −1 (a) Identical-deadline system −5 −4 −3 −2 −1 Stretch factor ξ p v i o l L HPRL−MaxWeightL−ExpL−LogMax C/IEDFLLF (b) Stationary-arrival systemFig. 3. Delay violation probability ( M = 15 , a = 0 . , and λ = 0 . ). optimal policy, referred to as L HPR. We also propose heuris-tic policies, L-MaxWeight, L-Exp, and L-LLF, for practicalsystems. Comparative study between the proposed laxity-basedpolicies and traditional greedy policies such as Max C/I, EDF,and LLF, demonstrates that the performance can be improvedby intelligently balancing the multi-user diversity and theurgent users.We mainly focus on designing optimal policies for under-loaded systems and assume that all completed tasks have samevalue. However, in practice, it is possible that not all tasks canbe finished before their deadlines and different tasks may havedifferent values. In the future, we will study algorithms thatestimate the schedulability of the coming sequence, drop usersto avoid resource wasting, and maximize obtained utility whenthe system is possibly overloaded.A PPENDIX AP ROOF OF L EMMA HPR, in any time slot n ′ , if L ′ i [ n ′ ] − L ′ i [ n ′ ] ≤ ∆ t , then L ′ i [ n ′ + 1] − L ′ i [ n ′ + 1] ≤ ∆ t. (12)This is because if L ′ i [ n ′ ] − L ′ i [ n ′ ] ≤ , since L ′ i [ n ′ + 1] ≤ L ′ i [ n ′ ] + ∆ t and L ′ i [ n ′ + 1] ≥ L ′ i [ n ′ ] , then (12) follows.Otherwise, < L ′ i [ n ′ ] − L ′ i [ n ′ ] ≤ ∆ t , and the BS willallocate more resource to user i than i . Hence, L ′ i [ n ′ + 1] − L ′ i [ n ′ ] ≤ L ′ i [ n ′ + 1] − L ′ i [ n ′ ] and L ′ i [ n ′ + 1] − L ′ i [ n ′ + 1] ≤ L ′ i [ n ′ ] − L ′ i [ n ′ ] ≤ ∆ t .By the definition of ULT, we know that i ≤ n i implies theexistence of an n ′ ( ≤ n ′ ≤ n ), such that L ′ i [ n ′ ] − L ′ i [ n ′ ] ≤ < ∆ t ) . Thus, the desired results follows.A PPENDIX BP ROOF OF T HEOREM A = 0 and D > . For given time t > , let i last ( t ) be the last user entersthe system before time t , i.e., i last ( t ) = max { i : 1 ≤ i ≤ M, A i < t } . We divide the interval [0 , t ] into N time slots, each of whichwith length ∆ t = t/N . Assume that the i -th ( ≤ i ≤ i last ( t ) )user arrives in the n i -th slot, i.e., in the interval ( n i ∆ t, ( n i +1)∆ t ] , and can be served from time slot n i + 1 . The initialvirtual expected laxity of user i is L ′ i [ n i + 1] = L ′ i ( A i ) = D − F i, g . We study the lower bound on the least virtual expectedlaxity of L HPR by examining the least-laxity-set Q [ N ] .imilar to the proof of Lemma 2, we know that for all i ∈ Q [ N ] , L ′ i [ N ] ≤ L L HPR [ N ] + ( M − t. (13)Assume that the users in the least-laxity-set Q [ N ] = { i , i , . . . , i Q [ N ] } are sorted in the ascending order of theirarrival times. If some of the users in Q [ N ] is completed, thenthe sum of virtual expected laxity in time slot N is boundedby L L HPR [ N ] ≥ D − ( M − t. (14)Otherwise, by the definition of Q [ N ] , we know that fromtime slot n i j + 1 to n i j +1 ( j = 1 , , . . . , Q [ N ] − ), all the j largest data rates { g , g − g , . . . , g j − g j − } are allocated tothe users in Q [ N ] , and from time slot n i Q [ N ] + 1 to N − , allthe Q [ N ] largest data rates { g , g − g , . . . , g Q [ N ] − g Q [ N ] − } are allocated to the users in Q [ N ] . Thus, the sum of virtualexpected laxity is X i ∈Q [ N ] L ′ i [ N ] = X i ∈Q [ N ] L ′ i [ n i + 1] + ∆ t Q [ N ] − X j =1 g j ( n i j +1 − n i j )+∆ tg Q [ N ] ( N − n i Q [ N ] − . (15)Then, as N tends to infinity, the slot length ∆ t = t/N tendsto 0. From (13), we know that the virtual expected laxity of allusers in Q ( t ) tends to the least virtual expected laxity at time t , i.e., L L HPR ( t ) . 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