Comparison of Loss ratios of different scheduling algorithms
aa r X i v : . [ c s . O S ] J a n Comparison of Loss ratios of different schedulingalgorithms
Sudipta Das, Lawrence Jenkins and Debasis Sengupta.
Abstract —It is well known that in a firm real time systemwith a renewal arrival process, exponential service times andindependent and identically distributed deadlines till the end ofservice of a job, the earliest deadline first (EDF) schedulingpolicy has smaller loss ratio (expected fraction of jobs, notcompleted) than any other service time independent schedulingpolicy, including the first come first served (FCFS). Variousmodifications to the EDF and FCFS policies have been proposedin the literature, with a view to improving performance. Inthis article, we compare the loss ratios of these two policiesalong with some of the said modifications, as well as theircounterparts with deterministic deadlines. The results includesome formal inequalities and some counter-examples to establishnon-existence of an order. A few relations involving loss ratiosare posed as conjectures, and simulation results in support ofthese are reported. These results lead to a complete pictureof dominance and non-dominance relations between pairs ofscheduling policies, in terms of loss ratios.
Index Terms —Firm real time system, real time queue, EarliestDeadline First, First Come First Served, service time dependentscheduling, admission control, loss ratio comparison.
I. I
NTRODUCTION I N real time systems consisting of aperiodic jobs, such asweb server, network router or real time database; it istypically not known when the job will arrive or what willits service time and its deadline be. If too many jobs arrivesimultaneously, the system becomes overloaded and the jobsbegin to miss their deadlines. The service requirements for thejobs are often not known beforehand, and hence are specifiedin probabilistic terms. So a fundamental problem in suchsystems is to schedule a set of jobs such as to allow themaximum possible number of jobs to meet their respectivedeadlines.In this article, we consider various scheduling algorithms forfirm real time systems (i.e., systems where a job must leavethe queue after its deadline [5]) with a single processor andan aperiodic workload, under their commonly used model asa single server queue with an infinite buffer [7]. The infinitebuffer ensures that there is no upper limit on the maximumnumber of jobs that can remain in the system. We adopt theusual assumption that every job is ready as soon as it isreleased, it can be preempted at any time and it never suspendsitself. Moreover, we assume that the deadline of a job is tillthe end of its service, that the context switch overhead isnegligibly small when compared with the service times of thetasks, and that the number of priority levels is unlimited. We
S. Das and L. Jenkins are with the Department of Electrical Engineering,Indian Institute of Science, Bengaluru, India.D. Sengupta is with the Indian Statistical Institute, Kolkata. also assume that job arrivals follow a renewal process with rate λ (i.e., the mean inter-renewal time is /λ ), the service timesare independent and identically distributed random variableswith mean /µ , and the relative deadlines are also independentand identically distributed random variables with mean /δ .The service time and absolute deadline of a job are assumedto be known at its arrival epoch. For example, in a Web server,the name of the requested URL can be used to look up thelength of the requested page and, hence, estimate the servicetime of the request.In the above set up, the simplest scheduling policy is theFirst Come First Served ( FCFS ) policy, which stipulates thatjobs be serviced in the order of their arrival. A more complexscheduling policy that has some attractive optimality propertiesis the Earliest Deadline First (
EDF ) policy [8]. According tothis policy, jobs that have arrived and await execution are keptin a ready queue, sorted in ascending order by their absolutedeadlines. When the processor finishes a job, the first jobin the queue is selected for execution. When a job arrives,it is inserted in the proper position of the queue (breakingties arbitrarily). A variant of the
EDF policy provides forpreemption of the currently running job by a newly arrived job,if the absolute deadline of this job is earlier than that of thecurrently running job. If it is assumed that a job can always bepreempted, and that there is no cost of preemption, then it canbe shown that preemptive
EDF is the optimal policy withinthe class of non-idling service-time independent preemptivepolicies [6], i.e.,
EDF can produce a feasible schedule of aset of jobs J with arbitrary release times and deadlines ona processor iff J has feasible schedule. Also, it has beenshown that
EDF stochastically minimizes the loss ratio (i.e.,the fraction of jobs not completed) in both preemptive andnon-preemptive models [10], [11].There have been attempts to reduce the loss ratio by con-trolling admission of newly arriving jobs in the queue, througha scheduling test. Prominent examples of this innovation areutilization based admission controller [1] and the exact admis-sion controller [8]. The Utilization based admission-controllerfor aperiodic jobs is pessimistic in the sense that it sometimesdenies admission to a job even if that job can be scheduled atthat instant. It can be shown that a utilization based admission-controller also passes some jobs that would not be completedbefore their respective deadlines. The exact admission con-troller (
EAC ) seeks to remove these shortcomings at the costof increased computational complexity ( O (log n ) for EAC asopposed to O (1) for the utilization based admission controller)[2].While an admission controller takes into account the history of jobs already in the queue, a particular decision regardingadmission may appear to be unduly conservative in the light ofevents that follow that decision. If the decision to serve a jobis deferred till the epoch of it being served, then that decisioncan be made on the basis of additional information. Here, weconsider a simple modification to scheduling protocols, calledthe early job discarding ( EDT ) technique. The
EDT does notcheck the scheduling feasibility of a job on its arrival, butrather admits each incoming job into the system, inserts the jobin an appropriate place of the queue according to the protocolbeing used and lets the system evolve. It discards a job at theepoch of its getting the server from the head of the queue,irrespective of it being a fresh job or a previously preemptedjob requesting the server again, if it is clear at that momentthat the job cannot be completed before the deadline. Thename early job discarding technique reflects the fact that itdiscards a job before its deadline epoch. It should be notedthat
EDT may not be feasible in applications that demandguaranteed completion of jobs once they are admitted to thequeue. On the other hand, even where it is feasible, the valueof this common sense belt-tightening step in improving theperformance of a scheduling policy has never been formallystudied. We show that this step can be more effective thanadmission controllers in cutting down the loss ratio.In this article, we compare the performances of differentscheduling strategies in terms of the job loss ratio. We showthat, under a purely random environment, the inclusion of EDTor EAC in the FCFS and EDF scheduling policy reduces theloss ratio. We also prove that
EDF along with
EDT has smallerloss ratio than all other scheduling algorithms considered here.This article is organized as follows. In Section II, possibledominance relations of the scheduling strategies in terms ofloss ratio are discussed. Special attention to systems withdeterministic job deadlines is given in Section III. Someconcluding remarks are provided in Section IV, while proofsof all the result are presented in the appendix.II. S
OME DOMINANCES AND NON - DOMINANCES
Let α Gsp denote the loss ratio of the system under schedulingpolicy sp and relative deadline distribution G . The followingproposition follows from Theorem 1 of Towsley and Panwar[10]. Proposition
1. In an
G/M/ /G queue, the loss ratio underthe EDF scheduling policy is smaller than that under any otherservice time independent scheduling policy with deadline tillthe end of service.In particular,
EDF produces smaller loss ratio than
FCFS ,i.e., α GEDF ≤ α GF CF S .In this section, we look for similar dominance relationsbetween pairs of the scheduling policies
FCFS , EDF , FCFS-EDT (FCFS along with EDT),
EDF-EDT (EDF along withEDT),
FCFS-EAC (FCFS along with EAC) and
EDF-EAC (EDF along with EAC). The only relevant work that we could access in this connection is asimulation study in [1], where it was found that EDT works marginally betterthan the utilization based admission controller in a particular situation.
Proposition
2. In a
G/G/ /G queue, the loss ratio under the EDF scheduling policy can only reduce when Early DiscardingTechnique is used, i.e., α GEDF - EDT ≤ α GEDF . Proposition
3. In a
G/G/ /G queue, the loss ratio underthe FCFS scheduling policy can only reduce when EarlyDiscarding Technique is used, i.e., α GF CF S - EDT ≤ α GF CF S . Proposition
4. In a
G/G/ /G queue, the loss ratio underthe EDF scheduling policy can only reduce when ExactAdmission Control is used, i.e., α GEDF - EAC ≤ α GEDF . Proposition
5. In a
G/G/ /G queue, the loss ratio underthe FCFS scheduling policy can only reduce when ExactAdmission Control is used, i.e., α GF CF S - EAC ≤ α GF CF S . Proposition
6. In a
G/M/ /G queue, the loss ratio underthe EDF-EDT scheduling policy is less than that of
EDF-EAC scheduling policy, i.e., α GEDF - EDT ≤ α GEDF - EAC . Proposition
7. In a
G/M/ /G queue, the loss ratio underthe EDF-EDT scheduling policy is less than that of
FCFS-EDT scheduling policy, i.e., α GEDF - EDT ≤ α GF CF S - EDT . Proposition
8. In a
G/G/ /G queue, the loss ratios underthe FCFS-EDT and
FCFS-EAC scheduling policies are iden-tical, i.e., α GF CF S - EAC = α GF CF S - EDT .The following example shows that there is no dominancerelation between the loss ratios under the
EDF and
FCFS-EAC (or
FCFS-EDT ) scheduling policies, i.e., neither of theinequalities α GF CF S - EAC ≤ α GEDF and α GEDF ≤ α GF CF S - EAC hold in general.
Counter-example 1.
Consider the
M/M/ queue with dead-line till the end of the service, where the relative deadline hasthe exponential distribution with mean equal to 16 times themean service time ( /δ = 16 /µ ). The loss ratios, plotted inFigure 1 as a function of the normalized arrival rate ( λ/µ ),show that the inequality α ExpEDF ≤ α ExpF CF S - EAC holds forsmall arrival rates, while the inequality α ExpF CF S - EAC ≤ α ExpEDF holds for large arrival rates. The values of the loss ratios arecomputed on the basis of simulations of about one millionarrivals. Thus, neither of α GEDF and α GF CF S - EAC uniformlydominates the other.The results of propositions 1 and 7 give rise to the questionof possible superiority of EDF over FCFS in terms of loss ratioeven under exact admission control. We could not prove thisdominance relation. However, simulation results summarizedin Figure 2 appear to support this. We ran the simulations forthe Poisson arrival process with a wide range of normalizedarrival rates (with λ/µ varying from 0 to 4) and four types ofrelative deadline distributions, namely exponential, uniform,log-normal and two-point. The mean ( /δ ) of the deadlinedistribution was varied from /µ to /µ . In the case of thelog-normal distribution, the coefficient of variation was chosenas 1 for all values of δ , while in the case of the two-pointdistribution, the probabilities 0.9 and 0.1 were assigned to λ / µ ) Lo ss R a t i o FCFS−EAC−ExpEDF−Exp
Fig. 1. Loss ratios of the
FCFS-EAC and
EDF scheduling algorithms forexponential relative deadline with /δ = 16 /µ and various normalized arrivalrates ( λ/µ ). λ / µ Exponential Distribution µ / δ N o r m a li z ed Lo ss R a t i o λ / µ Uniform Distribution µ / δ N o r m a li z ed Lo ss R a t i o λ / µ Lognormal Distribution µ / δ N o r m a li z ed Lo ss R a t i o λ / µ Two−point Distribution µ / δ N o r m a li z ed Lo ss R a t i o Fig. 2. Loss ratios for various deadline distributions under the
EDF-EAC scheduling policy normalized by loss ratio under the
FCFS-EAC schedulingpolicy, for various values of normalized arrival rate ( λ/µ ) and normalizedmean relative deadline ( µ/δ ). the points / (9 δ ) and /δ , respectively, for all values of δ .The values of the loss ratios were computed on the basis ofsimulations of about one million arrivals. On the basis of thesefindings, we make the following conjecture. Conjecture
1. In an
G/M/ /G queue, the loss ratio underthe EDF-EAC scheduling policy is less than that of
FCFS-EAC scheduling policy, i.e., α GEDF - EAC ≤ α GF CF S - EAC .The findings of this section are summarized in Figure 3. Inthis figure, an arrow extending from the scheduling policy sp to the policy sp indicates that α Gsp ≤ α Gsp , a double headedarrow indicates equality of the loss ratios, while a pair ofarrows facing each other indicates that there is no dominancerelation. The dashed arrow represents a conjectured relation.A key finding that emerges from Figure 3 is the superiorityof EDF-EDT over the other scheduling policies, in terms ofloss ratio. III. D
ETERMINISTIC DEADLINE
We now assume that the deadline distribution is degenerate,i.e., the deadline is completely deterministic. It is easy to seethe following facts.1) The
FCFS and
EDF scheduling policies are equivalent.
EDF-EDT EDF-EAC EDF FCFS
Proposition 2 Proposition 6 Proposition 7 Conjecture 1 Counter Example 1
FCFS-EAC FCFS-EDT
Proposition 4 Proposition 8 Proposition 5 Proposition 1
Fig. 3. Relationship between various scheduling algorithms in terms of orderof loss ratios, for stochastic relative deadlines.Fig. 4. Relationship between various scheduling algorithms in terms of orderof loss ratios, for deterministic relative deadlines.
2) The
FCFS-EDT and
EDF-EDT scheduling policies areequivalent.3) The
FCFS-EDT and
FCFS-EAC scheduling policies areequivalent.4) The
EDF-EDT and
EDF-EAC scheduling policies areequivalent.In view of the above facts, the relations depicted in Figure 3simplify to those given in Figure 4.Let α Detsp denote the loss ratio of the system under schedul-ing policy sp and deterministic relative deadline. Movaghar[9] showed that the loss ratio for the FCFS scheduling policyis bounded from below by the corresponding ratio for the casewhere the deadline is deterministic. In particular, the followingproposition follows from Lemma 5.1.3 of Movaghar [9]. Proposition
9. In an
M/M/ /G queue with a specifiedmean deadline till the end of service, the loss ratio under the FCFS scheduling policy happens to be the minimum when thedeadline distribution is degenerate, i.e., α DetF CF S ≤ α GF CF S .The above result provides a connection between two lossratios shown in Figures 3 and 4. Questions about similar otherconnections arise naturally. An optimality result of the type ofProposition 9 for the EDF scheduling policy was conjecturedin [7], and we state it below. λ / µ Exponential Distribution µ / δ N o r m a li z ed Lo ss R a t i o λ / µ Uniform Distribution µ / δ N o r m a li z ed Lo ss R a t i o λ / µ Lognormal Distribution µ / δ N o r m a li z ed Lo ss R a t i o λ / µ Two−point Distribution µ / δ N o r m a li z ed Lo ss R a t i o Fig. 5. Loss ratio for deterministic deadline normalized by loss ratios forvarious deadline distributions under the
EDF scheduling policy, for variousvalues of normalized arrival rate ( λ/µ ) and normalized mean relative deadline( µ/δ ). Conjecture
2. In an
M/M/ /G queue with a specifiedmean deadline till the end of service, the loss ratio under the EDF scheduling policy happens to be the minimum when thedeadline distribution is degenerate, i.e., α DetEDF ≤ α GEDF .We were unable to find either a proof of the above conjec-ture or a counter-example to disprove it. However, we con-ducted extensive simulations for a number of relative deadlinedistributions. We considered Poisson arrival process with awide range of normalized arrival rates (with λ/µ varyingfrom 0 to 4), and four types of relative deadline distributions,namely exponential, uniform, log-normal and two-point. Themean ( /δ ) of the deadline distribution was varied from /µ to /µ . As in the case of simulations run for Conjecture 1 ,the coefficient of variation of the log-normal distribution wasfixed as 1 for all values of δ , and the points / (9 δ ) and /δ ofthe two point distribution were assigned probabilities 0.9 and0.1, respectively, for all values of δ . The values of the lossratio were computed on the basis of simulations of about onemillion arrivals. The results, summarized in Figure 5, supportthe above conjecture.We looked for a similar result for the EDF-EDT schedulingpolicy, but were unable to find either a proof or a counter-example. We state it in the form of a conjecture, which issupported by the simulation results summarized in Figure 6.The model of this experiment is the same as before. Conjecture
3. In an
M/M/ /G queue with a specified meandeadline till the end of service, the loss ratio under the EDF-EDT scheduling policy happens to be the minimum whenthe deadline distribution is degenerate, i.e., α DetEDF − EDT ≤ α GEDF − EDT .The following three counter-examples complete the set ofconnections between the loss ratios depicted in Figures 3and 4.
Counter-example 2.
Consider the
M/M/ queue with dead-line till the end of the service, where the relative deadline iseither deterministic with value /µ or exponentially distributedwith mean /µ . Loss ratios plotted in Figure 7 as a function λ / µ Exponential Distribution µ / δ N o r m a li z ed Lo ss R a t i o λ / µ Uniform Distribution µ / δ N o r m a li z ed Lo ss R a t i o λ / µ Lognormal Distribution µ / δ N o r m a li z ed Lo ss R a t i o λ / µ Two−point Distribution µ / δ N o r m a li z ed Lo ss R a t i o Fig. 6. Loss ratio for deterministic deadline normalized by loss ratiosfor various deadline distributions under the
EDF-EDT scheduling policy, forvarious values of normalized arrival rate ( λ/µ ) and normalized mean relativedeadline ( µ/δ ). λ / µ ) Lo ss R a t i o FCFS−EDT−ExpFCFS−Det
Fig. 7. Loss ratios of the
FCFS-EDT,Exp and
FCFS,Det schedulingalgorithms for mean relative deadline /δ = 2 /µ and various normalizedarrival rates ( λ/µ ). of the normalized arrival rate ( λ/µ ), show that the inequality α DetF CF S ≤ α ExpF CF S - EDT holds for small arrival rates, whilethe inequality α ExpF CF S - EDT ≤ α DetF CF S holds for large arrivalrates. The values of the loss ratio were computed on the basisof simulations of about one million arrivals. Thus, neither of α DetF CF S and α GF CF S - EDT uniformly dominates the other.
Counter-example 3.
Consider the
M/M/ queue with dead-line till the end of the service, where the relative deadlineis either deterministic with value /µ or exponentially dis-tributed with mean /µ . The loss ratios, plotted in Figure 8as a function of the normalized arrival rate ( λ/µ ), show thatthe inequality α DetF CF S ≤ α ExpEDF - EDT holds for small arrivalrates, while the inequality α ExpEDF - EDT ≤ α DetF CF S holds forlarge arrival rates. The values of the loss ratio were computedon the basis of simulations of about one million arrivals. Thus,neither of α DetF CF S and α GEDF - EDT uniformly dominates theother.
Counter-example 4.
Consider the
M/M/ queue with dead-line till the end of the service, where the relative deadlineis either deterministic with value /µ or exponentially dis-tributed with mean /µ . The loss ratios, plotted in Figure 9as a function of the normalized arrival rate ( λ/µ ), show thatthe inequality α DetF CF S ≤ α ExpEDF - EAC holds for small arrival λ / µ ) Lo ss R a t i o EDF−EDT−ExpFCFS−Det
Fig. 8. Loss ratios of the
EDF-EDT,Exp and
FCFS,Det scheduling algorithmsfor mean relative deadline /δ = 16 /µ and various normalized arrival rates( λ/µ ). λ / µ ) Lo ss R a t i o EDF−EAC−ExpFCFS−Det
Fig. 9. Loss ratios of the
EDF-EAC,Exp and
FCFS,Det scheduling algorithmsfor mean relative deadline /δ = 16 /µ and various normalized arrival rates( λ/µ ). rates, while the inequality α ExpEDF - EAC ≤ α DetF CF S holds forlarge arrival rates. The values of the loss ratio were computedon the basis of simulations of about one million arrivals. Thus,neither of α DetF CF S and α GEDF - EAC uniformly dominates theother. IV. C
ONCLUDING R EMARKS
In this paper, we have proved some dominance relationsbetween various scheduling algorithms in terms of their re-spective loss ratios. We have also proved, through counter-examples, the non-existence of a dominance relation betweensome pairs of scheduling algorithms. A few possible domi-nance relations are left as conjectures, supported by extensivesimulations. These relations help one construct a comprehen-sive dominance structure of scheduling algorithms in terms ofloss ratios, parts of which were given in Figures 3 and 4. Thiscombined structure is shown in Figure 10.An intuitive explanation of the smaller loss ratio of EDF-EDT in comparison to EDF-EAC is the fact that, whiletaking the decision regarding admission of a job, an admissioncontroller takes into account the history of jobs already inthe queue . Some of these decisions may appear to be undulyconservative in the light of events that follow. By deferring thedecision of discarding a job till the epoch of its being served,EDT is able to take into account additional information.
EDF-EDT-D, FCFS-EDT-D, EDF-EAC-D, FCFS-EAC-D EDF-EDT EDF-EAC EDF FCFS-EDT, FCFS-EAC
FCFS
Proposition 2 Proposition 6 Proposition 4 Counter example 2 Proposition 5 Counter example 3 Conjecture 1 Proposition 1 Conjecture 3 Counter example 1 Proposition 9 Proposition 2 Counter example 4 Conjecture 2 Proposition 7
EDF-D, FCFS-D
Fig. 10. Relationship between various scheduling algorithms in terms oforder of loss ratios, for stochastic and deterministic relative deadlines.
The result of Proposition 6 indicates that EDF-EDT maybe preferred where an early guarantee of completion of jobsis not essential. On the other hand, if an admission controllermust be used, then Propositions 4 and 6 specify limits to itsperformance from both sides.The result of Towsley and Panwar [10] on the optimalityof the EDF scheduling policy among the class of all service-time independent policies does not hold in the presence ofEDT, which makes the scheduling policy dependent on servicetime. This fact gives rise to the question of possible optimalityof EDF among the modified class of scheduling policies thataccommodate EDT. The result stated in Proposition 7 partiallyanswers that question, and keeps open the possibility of overalloptimality. This issue may be taken up for research in future.A
PPENDIX AP ROOFS OF PROPOSITIONS
Proof of Proposition 2.
Consider a finite number of jobarrivals, and arrange all the jobs in order of their departureepochs under
EDF . Observe that the departure order of
EDF-EDT is the same as that under
EDF . A job that is discardedunder EDT would have missed the deadline in any case. Onthe other hand, the act of discarding a particular job can onlyreduce the waiting times of the subsequent jobs (arrangedas above). Consequently, the act of discarding that job canonly reduce the number of subsequent jobs missing theirrespective deadlines. This argument holds for every singleevent of discarding of jobs under EDT. Thus, for any givenconfiguration of a finite number of jobs, the proportion of jobsmissing deadline under
EDF-EDT is less than or equal to thatunder
EDF . It follows that the expected proportion of jobs (outof the first n arrivals for any fixed n ) is less for EDF-EDT than for
EDF . The stated result is obtained by taking the limitof the expected proportions as n goes to infinity. Proof of Proposition 3.
The result can be proved along thelines of the proof of Proposition 2.
Proof of Proposition 4. [3] Consider a variation of the
EDF-EAC scheduling policy, where a job that does not satisfy theadmission criterion of
EAC is not discarded at the time of admission, rather it is merely tagged for eventual rejection atthe epoch of its getting the server. Note that this modificationdoes not change the completion status of any job, but the orderof the jobs (tagged or untagged) getting the server becomesthe same as that under
EDF . We shall show that the modified
EDF-EAC procedure produces a smaller loss ratio than
EDF .Now, consider the first n arrivals. Permit the first taggedjob of the list to be served, allowing for possible preemptionunder EDF . The fact that the tagged job would have beendenied admission under
EAC indicates that either this job or atleast one job that arrived earlier but is located down the queuewould miss deadline. On the other hand, providing serviceto the tagged job can only increase the waiting times of all the subsequent jobs in the ordered list (including those whicharrived after the first tagged job), and this increase may triggerfurther cases of missed deadline. Thus, the act of providingservice to the tagged job can only increase the number ofjobs missing deadline. This argument holds for every singleevent of providing service to the successively tagged jobs.Thus, for any given configuration of a finite number of jobs,the proportion of jobs missing deadline under
EDF-EAC isless than or equal to that under
EDF . Hence, the expectedproportion of jobs (out of the first n arrivals) is less for EDF-EDT than for
EDF . The stated result is obtained by taking thelimit of the expected proportion as n goes to infinity. Proof of Proposition 5.
The result can be proved along thelines of the proof of Proposition 4.
Proof of Proposition 6. [3] Consider the task of schedulingthe first N jobs in a G/M/ /G queue. For n = 0 , , , . . . , N ,let P n denote the scheduling policy, where the jobs arescheduled according to the EDF-EAC policy for the first n arrivals, and there is a switch to the EDF-EDT policy beforethe arrival of the next job. Note that P corresponds to the EDF-EDT policy, while P N corresponds to the EDF-EAC policy. If we can show that the expected count of completedjobs is a decreasing function of n , then the stated result willfollow by allowing N to go to infinity.In order to compare the policies P n − and P n , considerthree cases depending on the admission status of the job J that corresponds to the n th arrival.C ASE
1. Let J be admissible according to EAC . In thiscase, the pattern of service provided under the policies P n − and P n would be identical.C ASE
2. Let J be inadmissible according to EAC owing tothe fact that its own deadline is too short for its completion.In this case also, the pattern of service provided under thepolicies P n − and P n would be identical. The only differenceis that P n would not admit J , while P n − would admit it butdiscard it at the epoch of its getting server.C ASE
3. Let J be inadmissible according to EAC becauseof the fact that its admission would result in non-completionof service to another job. Consider the entire history of arrivalsand server utilization subsequent to the arrival of J . Let J ′ bethe label of the first job (arrived before or after J ) that fails tocomplete because of the admission of J under P n − . Let usdenote by J ∗ the job immediately preceding J ′ in the queue according to P n . Regarding the arrival epoch of J as time 0,let τ and τ + τ ∗ be the remaining aggregated service times ofthe jobs having absolute deadlines earlier than those of J and J ′ , respectively. Let X and X ′ be the remaining service timesof J and J ′ , respectively, at time 0. Let d , d + d ∗ and d + d ∗ + d ′ be the absolute deadlines of J , J ∗ and J ′ , respectively. In case J ′ immediately follows J in the queue of P n − (i.e., there isno job with the label J ∗ ), we set τ ∗ and d ∗ as zero.We shall show that, given these circumstances, X is stochas-tically smaller than X ′ .For any set of fixed and positive values of τ , τ ∗ , d , d ∗ and d ′ satisfying τ < d and τ + τ ∗ < d + d ∗ , consider the event E = { τ + X ≤ d ; τ + X + τ ∗ ≤ d + d ∗ ; τ + τ ∗ + X ′ ≤ d + d ∗ + d ′ < τ + X + τ ∗ + X ′ } . Note that this event represents the conditions that J and J ∗ can be completed before their respective deadlines accordingto P n − , while J ′ can be completed under P n but not under P n − . If the service rate is µ , it is easy to see that the jointdensity of X and X ′ given E is f X,X ′ | E ( x, x ′ ) = µ e − µ ( x + x ′ ) P ( E ) , x ≤ a, x ′ ≤ b < x + x ′ , where a = min { d − τ, d + d ∗ − τ − τ ∗ } , b = d + d ∗ + d ′ − τ − τ ∗ (with a ≤ b ) and P ( E ) is the unconditional probability Z Z x ≤ a, x ′ ≤ b 1. Let J be successfully serviceable as of the timeof its arrival, if it is scheduled according to FCFS-EDT . Itfollows that it is successfully serviceable as of the time of itsarrival, if it is scheduled according to EDF-EDT also. In thiscase, the completion status of all the jobs under the policies P n − and P n would be identical, even though the two policiesmay place J in different positions of the queue.C ASE 2. Let J not be successfully serviceable as of thetime of its arrival, if it is queued according to EDF-EDT . Itfollows that it is not successfully serviceable as of the time of its arrival, if it is scheduled according to FCFS-EDT also. Inthis case also, the completion status of all the jobs under thepolicies P n − and P n would be identical.C ASE 3. Let J be successfully serviceable as of the timeof its arrival, if it is queued according to EDF-EDT but notso under FCFS-EDT . Let J ′ be the label of the first job(arrived before or after J ) that fails to complete after beingsuperseded by J under P n . Let X and X ′ be the remainingservice times of J and J ′ , respectively, at time 0. It can beproved along the lines of the proof of Proposition 2.6 that X is stochastically smaller than X ′ , and that the expected totalnumber of completed jobs is smaller under P n − than under P n .The proof is completed by combining the findings of thethree cases for fixed N (which establishes that P has smallerloss ratio than P N ), and then allowing N to go to infinity. Proof of Proposition 8. Consider a variation of the FCFS-EAC scheduling policy, where a job that does not satisfy theadmission criterion of EAC is not discarded at the time ofadmission, rather it is merely tagged for eventual rejection atthe epoch of its getting the server. Note that this modificationdoes not change the completion status of any job, but the orderof the jobs (tagged or untagged) getting the server becomes thesame as that under FCFS . The fact that the tagged job wouldhave been denied admission under the FCFS-EAC procedureindicates that this job, if served, would have missed its owndeadline. Thus, this job would also be discarded under FCFS-EDT . It can be seen that, out of the first n arrivals, the setof jobs that would be discarded under FCFS-EDT is preciselythe set of jobs tagged as above. It follows that, for any givenconfiguration of n job arrivals, the proportion of jobs missingdeadline is the same under FCFS-EAC and FCFS-EDT . Theresult follows by taking expectation of this proportion andallowing n to go to infinity.R EFERENCES[1] Tarek Abdelzaher and C. Lu. Schedulability analysis and utilizationbounds for highly scalable real-time services. 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