Constant Bandwidth Servers with Constrained Deadlines

Transcription

1 Consan Bandwidh Servers wih Consrained Deadlines Daniel Casini Scuola Superiore S. Anna isa, Ialy Luca Abeni Scuola Superiore S. Anna isa, Ialy Alessandro Biondi Scuola Superiore S. Anna isa, Ialy ABSTRACT Tommaso Cucinoa Scuola Superiore S. Anna isa, Ialy The Hard Consan Bandwidh Server (H-CBS) is a reservaion-based scheduling algorihm ofen used o mix hard and sof real-ime asks on he same sysem. A number of varians of he H-CBS algorihm have been proposed in he las years, bu all of hem have been conceived for implici server deadlines (i.e., equal o he server period). However, recen promising resuls on semi-pariioned scheduling ogeher wih he demand for new funcionaliy claimed by he Linux communiy, urge he need for a reservaion algorihm ha is able o work wih consrained deadlines. This paper presens hree novel H-CBS algorihms ha suppor consrained deadlines. The hree algorihms are formally analyzed, and heir performance are compared hrough an exensive se of simulaions. CCS CONCETS Compuer sysems organizaion Real-ime operaing sysems; Embedded sofware; KEYWORDS Real-ime scheduling, resource reservaions, emporal isolaion ACM Reference forma: Daniel Casini, Luca Abeni, Alessandro Biondi, Tommaso Cucinoa, and Giorgio Buazzo Consan Bandwidh Servers wih Consrained Deadlines. In roceedings of Inernaional Conference on Real-Time Neworks and Sysems, Grenoble, France, Ocober 2017 (RTNS 17), 10 pages. hps://doi.org/10.475/123_4 1 INTRODUCTION Several real-ime sysems are characerized by muliple compuaional aciviies (asks) ha require o be emporally isolaed among hemselves, meaning ha he emporal behavior of a ask is no affeced by he misbehavior of some oher ask, for insance due o execuion overruns. This is he case of open environmens [24], where independenly developed sofware componens need o be This work has been parially suppored by he RETINA Eurosars rojec E ermission o make digial or hard copies of par or all of his work for personal or classroom use is graned wihou fee provided ha copies are no made or disribued for profi or commercial advanage and ha copies bear his noice and he full ciaion on he firs page. Copyrighs for hird-pary componens of his work mus be honored. For all oher uses, conac he owner/auhor(s). RTNS 17, Ocober 2017, Grenoble, France 2017 Copyrigh held by he owner/auhor(s). ACM ISBN /08/06... $15.00 hps://doi.org/10.475/123_4 Giorgio Buazzo Scuola Superiore S. Anna isa, Ialy giorgio.buazzo@sanannapisa.i execued in isolaion. An effecive approach for achieving emporal isolaion is he resource reservaion mechanism [21], according o which each ask is assigned a fracion of he oal processor capaciy (called reservaion bandwidh). In his way, each ask running in a reservaion wih bandwidh α 1 behaves (from a emporal poin of view) as if i were execuing alone on a virual processor wih speed α imes he one of he physical processor, independenly of he behavior of he asks running in oher reservaions. Resource reservaion is especially useful o execue hard realime asks ogeher wih sof real-ime asks characerized by execuion imes subjec o high variaions (and for which i is difficul o find an upper bound) [9]. Resource reservaion is also adoped o implemen hierarchical scheduling [6, 24], where each reservaion can handle one or more asks and schedule hem according o a local scheduling policy. A reservaion is ypically implemened hrough a reservaion server, which is a kernel mechanism ha manages he reservaion bandwidh by allocaing a ime budge Q every period for he execuion of he served asks. Several reservaion servers have been proposed in he real-ime lieraure using differen policies for managing he budge, boh under fixed-prioriy schedulers and he Earlies Deadline Firs (EDF) algorihm [11]. Exising deadlinebased servers have no been designed o manage consrained deadlines. However, recen resuls on semi-pariioned scheduling [8, 13] highlighed he need for ad-hoc soluions considering reservaion servers having deadlines lower han heir reservaion periods. In fac, in hese works, whenever a reservaion canno be allocaed in any processor, i is spli ino muliple chunks using he C=D spliing scheme [10]. According o his scheme he resuling chunks have always a consrained deadline, hus implicily requiring reservaion servers wih a relaive deadline shorer han heir period. The need for a heoreically sound framework for deadline-consrained reservaions has also recenly come ou 1 in he conex of on-going developmens of he SCHED_DEADLINE scheduling class of he Linux kernel [19]. Conribuions. This paper proposes hree novel exensions o he popular Hard Consan Bandwidh Server (H-CBS) [7], in order o cope wih consrained deadlines: he firs one allows for ensuring he same wors-case guaranees as a sporadic real-ime ask ha is configured wih he same parameers of he server; he second and hird ones build upon a sufficien schedulabiliy es o improve he server performance in erms of average-case and 1 hps://lkml.org/lkml/2017/2/10/611

2 Figure 1: T sbf i () RTNS 17, Ocober 2017, Grenoble, France probabilisic merics, hus being more suiable for sof real-ime sysems. The presened approaches are compared boh in erms of analyical properies and empirical performance by simulaion. aper Srucure. The paper is organized as follows. Secion 2 inroduces he sysem model, he adoped noaion, and reviews he needed background. Secion 3 presens a general formulaion o describe differen varians of he H-CBS algorihm. Secion 4 highlighs shorcomings of exising sae-of-he-ar reservaion algorihms when dealing wih consrained deadlines. Secion 5 presens he novel reservaion algorihms exending he H-CBS o work wih consrained deadlines. Secion 6 presens some simulaion resuls for validaing he proposed approach. Finally, Secion 7 draws he conclusions and illusraes possible fuure work on he opic. 2 SYSTEM MODEL AND BACKGROUND This paper considers an arbirary number of virual processors running ino a pariioned muliprocessor sysem. Each virual processor is implemened by a reservaion server r i = (Q i, D i, i ) characerized by a budge Q i, a relaive deadline D i, and a period i. Each reservaion server has a bandwidh α i = Q i i and a consrained deadline less han or equal o is period (D i i ). If R k is he se of reservaions running on he k-h physical processor, is oal uilizaion is U k = r i R k α i. The workload running ino a reservaion r i consiss of a se Γ i of sporadic asks scheduled according o a local scheduling policy, where each ask τj i is characerized by a wors-case execuion ime C i j, a relaive deadline Di j, and a periodti j. For he sake of simpliciy, he superscrip in ask parameers denoing he reservaion will be omied when referring o a generic reservaion or i is clear from he conex. The server index is also removed whenever a single reservaion server is considered. Reservaion servers are assumed o be scheduled according o preempive EDF and asks are assumed o exchange daa hrough asynchronous (i.e., non blocking) communicaion mechanisms. In his paper, he proposed formulaions for handling servers wih consrained deadline are based on he H-CBS algorihm. 2.1 Background A reservaion server, alhough guaraneeing a desired bandwidh α i, may inroduce variable execuion delays on he served asks due o he fac ha, once he budge is exhaused, he server s asks will no be scheduled unil he nex replenishmen ime. The wors-case service delay i depends on he server parameers and he used budge managemen policy. Having a bounded wors-case service delay is a fundamenal propery for guaraneeing he schedulabiliy of he workload execuing inside he server [22]. In his work, a server is said o implemen a hard reservaion if i is able o guaranee boh a bandwidh α i and a wors-case (bounded) service delay i Server Schedulabiliy. The schedulabiliy of a consraineddeadline reservaion server can be verified hrough he rocessor Demand Crierion (DC) [3]. The DC builds on he concep of 2 Noe ha his definiion differs o he one proposed by Rajkumar e al. [23]. 2Q i Q i dbf i () C i 0 i 0 D i D. Casini e al. dbfi e (, i, E i ) Figure 1: Approximae C i + demand c i bound funcion of a sporadic ask C i demand bound funcion (dbf), 0 which α i in E α i + D i ihis case represens he wors-case compuaional demand of a reservaion (b) in any inerval [0, ]. When he dbf demand e i (, i, of E a i ) server does no exceed he one of a sporadic ask wih he same parameers, is dbf becomes: 2C i + i D i db f i () C= i Q i, (1) i The processor demand crierion 0 isdrecalled i α i in E α i + D i itheorem 2.1. (d) Theorem 2.1 (rocessor Demand Crierion (From [3])). A se R of consrained-deadline reservaions, whose demand does no exceed he one of a sporadic ask wih he same parameers, is EDFschedulable iff D, db f i () τ σ i R Q Q + σ 2Q wih D = r i R { = D i + f T i : L f N 0 }, where L is he maximum analysis inerval (see [3, 25] for more deails abou L )). The DC has a pseudo-polynomial compuaional complexiy if U < 1 (exponenial if U = 1). Sufficien polynomial-ime schedulabiliy ess for consrained-deadline reservaions can be derived by exploiing approximae demand bound funcions as defined by Fisher e al. [18]. Such approximae demand bound funcions wih a single disconinuiy (see Figure 1) are defined as follows: { Qi + α db f i () = i ( D i ) if D i, dbf T (2) 0 i () oherwise. A sufficien es exploiing he definiion of db f i () is repored below. Theorem 2.2. A reservaion 4C i se R k of periodic consrained-deadline reservaions wih U 1 is EDF-schedulable 3C i if r 2C i R k, αi + α j 1 (3) C i r j R:j i D j D i wih α 0 D i = Q i D i i, and T i + D i 2T i + D i 3T i + D i Qi = Q i + α j ( j D j ) r j R k :j i D j D i roof. From Theorem 2.1, if 0 r i R k db f i () holds, hen he sysem is schedulable. By definiion, db f i () db f i (). Hence, if 0 τ i R k db f i () holds, hen he sysem is schedulable. The funcion db f i () is characerized by a disconinuiy a = D i and by a linear par wih slope α i for > D i. Then, he funcion db f () = r i R k db f i () is consiued by a sequence of n = R disconinuiies. Le δ 1,..., δ n be he ordered sequence of disconinuiies. Since in any inerval [δ i, δ i+1 ] he funcion db f () is eiher consan or linear (wih a slope less or equal

3 τ 4 τ 4 arrives Consan Bandwidh Servers wih Consrained Deadlines Figure 1: Transien Example RTNS 17, Ocober 2017, Grenoble, France sbf i () 2Q i Q i 0 i dbf i () Figure 2: Supply bound funcion of a generic periodic reservaion server 2C i C i han U, wih U 1 by0assumpion) D i T i + ind boh i casesi is sufficien o check he condiion db f (δ i ) δ i. Hence, (c) he following es can be considered: dbfi e (, i, E i ) r i CR i k +, c i db f j (D i ) D i. r j R k C i Then, subsiuing he firs branch of Equaion 2 he es becomes: 0 α i E α i + D i i r i R k, Q j + (D i (b) D j ) Q j D i. dbf e r j R k :D i (, i, E j D j i i ) Finally, applying some algebraic ransformaions he condiion can be rewrien as: 2C i C r i i R k, α i + α j 1 r0 j R k D:j i D j i α D i i E α i + D i i The heorem follows. (d) Local schedulabiliy. Guaraneeing he schedulabiliy of he workload running inside a server requires he definiion of he supply bound funcion (sbf). The sbf (shown in Figure 2) models he minimum amoun of ime available in a reservaion r i in every ime σ Q Q + σ 2QOherwise, + σ he server ransis o he IDLE sae. inerval of lengh and can be derived by idenifying he minimum ime allocaed o r i in he wors-case scenario. The ineresed reader can refer o [24] for furher deails. 3 HARD CONSTANT BANDWIDTH SERVERS Across he las wo decades of lieraure on reservaion algorihms, various incarnaions of he H-CBS algorihm have been proposed by differen auhors. To shed he ligh on he exising proposals and provide a common basis for presening he resuls of his work, his secion presens a generalized formulaion of he H-CBS. The formulaion is based on a se of saic rules and anoher se of mea dbf T i () some even. The server is assumed o keep rack of he sae of he workload (pending or ready). As long as i has ready workload o execue, he server is agnosic wih respec o he execuion requess of he workload, which are managed according o an inernal (server-independen) policy, such as firs-in-firs-ou (FIFO) or EDF. A generalized formulaion for he H-CBS. A H-CBS server is described by hree parameers (Q, D, ), where Q is he maximum server budge, D is he server relaive deadline, and is he server period (also named as he reservaion period). The server works by racking wo dynamic sae variables: he curren budge q (also known as runime) and he scheduling deadline d (also known as server absolue deadline). A any poin in ime, he server can be in one of he following saes: IDLE, READY, and THROTTLED. The H-CBS algorihm relies on EDF scheduling and is subjec o he following rules: R1 Iniially he server is in he IDLE sae, where he budge and he scheduling deadline are iniialized o (q,d) = (0, 0). R2 When he server sars having ready workload o execue a ime : If he server is in he IDLE sae, a new budge and scheduling deadline are generaed as (q,d) = generae(q,d, ). Then, he server will ransi o he READY sae a ime w = wakeup(q,d, ). R3 A any poin in ime, he server in he READY sae ha has he earlies scheduling deadline d is seleced for execuion. R4 Whenever he server execues he workload for δ ime unis, is budge is decreased as q = q δ. Furhermore, he server noifies he consumed budge o he sysem by means of he funcion accoun(δ,d). R5 When he budge exhauss (i.e., q = 0), he server ransis o he THROTTLED sae. R6 While a server is in he THROTTLED sae, a ime p = d + D (end of he curren reservaion period) he server budge is recharged o q = Q and a new scheduling deadline is assigned as d = d +. Sill a ime p, if he server has ready workload o execue, hen i ransis o he READY sae. R7 Whenever he server is in he READY sae and sops having ready workload o execue, i ransis o he IDLE sae. 3.1 The classic H-CBS Among all he proposals, he mos common varian of he H-CBS was firs proposed by Marzario e al. [20] in 2004, and laer reformulaed by Abeni e al. [2]. Such an algorihm was designed for implici deadlines only and can be expressed by defining generae(q,d, ) as follows: { q (q,d) if < d generae(q, d, ) = rules. Such mea rules, which allow differeniaing beween he various versions, are hen discussed o insaniae he exising varians Also, wakeup(q,d, ) = and accoun(δ,d) has no effec. (Q, + ) oherwise. of he H-CBS, as well as he novel ones proposed in Secion 5 o In 2014, Biondi e al. [7] showed ha his formulaion is affeced cope wih consrained deadlines. by a schedulabiliy issue whenever he H-CBS is adoped in conjuncion o oher scheduling mechanisms ha can inroduce blocking A served workload is said 4C i o be pending from he ime i is released (i.e., enering an empy 3C i service queue) o he ime a which imes (e.g., non-preempive secions, locking proocols, ec.), hus i complees is execuion 2C (i.e., i leaving he service queue empy). perurbing he sandard EDF scheduling of he servers. The auhors also proposed a soluion o he idenified issue, which can be Furhermore, he workloadc is i said o be ready when i is pending and can make progress, ha is, i is no suspended waiing for 0 expressed by defining D i T i + D i 2T i + D i 3T i + D i wakeup(q, d, ) = min(, d q/α) and { q (Q,d generae(q, d, ) = α + ) if < d q α, (5) (Q, + ) oherwise. As i can be noed by looking a he differences beween (4) and (5), he laer formulaion always recharges he budge a he maximum value Q. The difference in he wakeup(q,d, ) funcion was provided o guaranee a bounded service delay of 2( Q). α, (4) 1

4 (b) D RTNS 17, Ocober 2017, Grenoble, France 3.2 The revised H-CBS While he radiional H-CBS algorihm ries o re-use he curren budge (and generaes a new scheduling deadline when he curren budge is no usable), he revised H-CBS [1] (again, defined for D = ), insead, ries o re-use he curren scheduling deadline as much as possible (a he cos of decreasing he budge) and is based on he following definiion of he generae() funcion: generae(q,d, ) = (max{q, Q (d )},d) (6) Moreover, wakeup(q,d, ) = and accoun(δ,d) has no effec. 3.3 The H-CBS-SO algorihm The self-suspending ask model [14] has been inroduced o capure possible self-suspensions of he execuion (waiing for some even) ha are explicily caused by he ask behavior, i.e., no imposed by he scheduler or oher scheduling mechanisms. Represenaive applicaions of such a model are in he conex of semaphore-based locking proocols or in he use of hardware acceleraors. Noably, self-suspending asks received a lo of aenion in he las years, being he corresponding schedulabiliy analysis challenging and affeced by several flaws in he lieraure [15]. A simple approach o analyze self-suspending asks is he suspension-oblivious analysis, where suspension imes are pessimisically accouned as execuion imes for he purpose of checking he sysem schedulabiliy. In 2015, Biondi e al. [4] showed ha he suspension-oblivious analysis is no compaible wih he sandard H-CBS algorihm. To reconcile he H-CBS algorihm wih a simple analysis for selfsuspending asks, he same auhors proposed a varian of he H-CBS denoed as H-CBS-SO (H-CBS for suspension-oblivious analysis). The H-CBS-SO mainains a logical queue, denoed as SS-QUEUE, ha keeps rack of he servers ha have pending bu no ready workload. The SS-QUEUE follows he EDF ordering. The key difference wih respec o he sandard H-CBS (boh he formulaions presened in Secion 3.1 are compaible) lies in he accoun(δ, d) funcion, which in he H-CBS-SO is defined as: { qss = q accoun(δ, d) = SS δ if d SS d, (7) no effec oherwise; where q SS and d SS denoe he budge and he deadline of he server a he head of he SS-QUEUE, respecively. The servers ino he SS-QUEUE are subjec o rules R5 and R6 described in Secion 3, wih he only difference ha in rule R6 he server does no ransis o he READY sae (noe ha, according o rule R7, a server is in he IDLE sae as long as i is ino he SS-QUEUE). Furhermore, whenever a server leaves he SS-QUEUE, rule R1 is no riggered. 4 ROBLEM DEFINITION The main objecive of his work is he developmen of a reservaion server wih he following feaures: (i) providing he budge o he server load wihin a relaive deadline less han or equal o he server period; (ii) guaraneeing a leas a budge Q every period ; (iii) guaraneeing a bounded wors-case service delay ; and (iv) guaraneeing he schedulabiliy of a se of such servers. Noe ha such feaures canno be achieved by using he radiional formulaions of he Hard Consan Bandwidh Server, because heir mea rules D < D = D < D < Figure 2: H-CBS: = Wors + D Case 2Q Delay Q Q D. Casini e al. Q = 2( Q) Q Figure 1: H-CBS: Wors Case Delay (a) = ( ɛ) + (D Q) + D Q Figure ɛ 3: H-CBS: Wors Case Delay Q D = Figure 3: Wors-case Figure 1: delays H-CBS: considering: 1Wors Case Delay (a) he original H- CBS (D=); (b) a soluion which discards he budge whenever he server becomes IDLE. Inse (c) shows he desirable = ( ɛ) + (D Q) (a) + D Q ɛ Q wors-case delay Figure wih 3: consrained H-CBS: Wors deadlines. Case Delay D < (c) (b) = + D 2Q Figure Q 2: H-CBS: Wors Case Delay Q Q = 2( Q) D Q D (e.g., see Equaion 4) leverage uilizaion-based (b) EDF schedulabiliy heory, which is valid only for implici deadlines. The insaniaion Figureof 2: such H-CBS: mea-rules Wors Case is obained Delay manipulaing he condiion U 1 which is necessary and sufficien o guaranee schedulabiliy when dealing wih implici deadlines. For his reason, if any server simply execues according o is bandwidh Q = 2( Q) Q he sysem is implicily schedulable. A naive soluion ha avoids leveraging D = any specific schedulabiliy es could consider o deplee he whole budge whenever a reservaion server remains wihou ready workload. According o his 1 approach, wakeup(q, d, ) = and accoun(δ,d) has no effec. Also, (a) generae(q,d, ) is defined as: (c) Figure 3: H-CBS: { (0,d) Wors Case if Delay < d D +, generae(q, d, ) = (Q, + D) oherwise. In doing so, however, he wors-case service delay would be much higher han he one ha would be experienced wih a radiional H-CBS (shown in Figure 3, inse (a)). Inse (b) of Figure 3 illusraes he wors-case service delay of such a naive soluion. The desirable wors-case service delay of a consrained-deadline server is shown in Figure 3, inse (c). A second naive soluion could consis in seing he period of a classical H-CBS equal o he relaive deadline D hus leveraging he so called densiy-based schedulabiliy es. For insance, Equaion 4 could be reused considering D in place of and he raio Q D in place of 1 α. Unforunaely, he condiion r i R Q i D i 1 is only sufficien o guaranee servers schedulabiliy and highly pessimisic. In paricular, when using a consraineddeadline server in he conex of C=D Semi-ariioned scheduling, a single zero-laxiy reservaion would occupy a whole processor! The nex secion proposes new soluions for reservaion servers ha can be used in his conex. D D (8)

5 Consan Bandwidh Servers wih Consrained Deadlines 5 ROOSED SOLUTIONS This secion presens hree soluions for realizing a H-CBS server wih consrained deadlines. The firs one is based on he H-CBS-SO algorihm presened in Secion 3.3, and ensures he server schedulabiliy wih any safe schedulabiliy es for EDF wih consrained deadlines. The second one explicily builds upon he sufficien EDF schedulabiliy es of Theorem 2.2 and is objecive is o improve he server performance wih respec o he firs soluion, in erms of probabilisic performance merics (such as deadline-miss raio of sof real-ime workload, response ime perceniles, ec.) and average-case performance (such as average hroughpu). Finally, an improved varian of he second soluion is also presened, which adops he same raionale of he revised H-CBS presened in Secion 3.2. Finally, differences among he proposed soluions are discussed. 5.1 A soluion based on H-CBS-SO As illusraed in Secion 4, he key issue wih he exising H-CBS formulaions concerns he case in which a server is suspended due o he lack of ready workload o execue. A simple insigh can be leveraged o overcome such an issue: since he H-CBS-SO was conceived o deal wih self-suspending asks running upon he server, a safe behavior can be achieved by reaing any server suspension in he same manner. The resuling algorihm is named as H-CBS D -W and is defined as follows. Firs, a logical EDF-ordered queue, named S-QUEUE, is provided. Whenever a server ransis o he IDLE sae, hen i is insered ino he S-QUEUE. The same definiion of he accoun(δ, d) funcion repored in Equaion (7) is adoped, where in his case q SS and d SS are respecively he budge and he deadline of he server a he head of he S-QUEUE. Analogously o he H-CBS-SO, he servers ino he S-QUEUE are subjec o rule R5. Also, when rule R5 applies o a server, he laer is removed from he S-QUEUE. Finally, he following definiions are adoped o insaniae he mea-rules inroduced in Secion 3 for a server r i : wakeup(q, d, ) = (9) and { (q,d) if ri is ino he S-QUEUE, generae(q, d, ) = (Q i, + D i ) oherwise. (10) The following lemma shows ha his approach makes he H- CBS D -W compaible wih any EDF schedulabiliy es for consraineddeadline sporadic asks. Lemma 5.1. If he H-CBS D -W is adoped, he processor demand of a reservaion r i R k never exceeds he one of a consraineddeadline sporadic ask wih wors-case execuion ime Q i, minimum iner-arrival ime i and relaive deadline D i. roof. As long as a reservaion has ready workload, rules R5 and R6 guaranee ha a server r i (i) does no execue for more han Q i ime unis every period i, and (ii) will always have a relaive deadline equal o D i. Whenever a server sops having ready workload, according o he definiion of he H-CBS D -W, he server is insered ino he S-QUEUE, where i is sill subjec o rule R5, so prevening r i o consume more han Q i budge unis wihin is RTNS 17, Ocober 2017, Grenoble, France curren period. Such rules ake effec because he budge of r i is decremened by he accoun(δ, d) funcion whenever a sporadic ask wih he same parameers of r i would have execued according o EDF (i.e., when r i is a he head o he S-QUEUE and is deadline is shorer han all he ones of he servers ha are in he READY sae). By he definiion of he generae(q,d, ) funcion, he server parameers do no change if r i resars having ready workload when i is ino he S-QUEUE. Finally, consider he case in which r i is no ino he S-QUEUE and resars having ready workload a ime. Le p be he end of he r i s reservaion period in which r i sopped having ready workload (enering he S-QUEUE). If p, hen according o Equaion (10) he server resars execuing wih budge Q i and deadline + D i : noe ha his behavior is compaible wih a sporadic arrival of a ask configured wih he same parameers of r i. Oherwise, if < p, hen r i mus be in he THROTTLED sae and hence canno execue unil ime p, as a corresponding sporadic ask would do. The following lemma bounds he maximum service delay of a H-CBS D -W server. Lemma 5.2. Consider a se R k of H-CBS D -W reservaions ha are EDF-schedulable. The service delay of each server r i R k is no larger han i = i + D i 2Q i. roof. Consider workload released a ime o be execued upon a server r i. The service delay is maximized in he scenarios where (i) r i has no available budge a ime ; and (ii) he following periodic insance of he server, beginning a ime p, sars providing service o he workload as lae as possible wihou violaing he server schedulabiliy, i.e., a ime p +D i Q i. Among such scenarios, he wors-case service delay occurs when he disance beween imes and p + D i Q i is maximal, which happens when (i) he firs insance (beginning a ime p i ) deplees all is budge as soon as possible and (ii) ime coincides wih he earlies ime in which such a budge is depleed. Following he definiion of he H-CBS D -W, he budge is consumed only when he server provides service or by means of he accoun(δ, d) funcion (see Equaion (7)). In boh he cases, he even discussed in (ii) canno happen before ime = p i + Q i. The corresponding delay resuls equal o (p + D i Q i ) (p i + Q i ) = D i + i 2Q i and is illusraed in Figure 3-c. The above wo lemmas imply ha he H-CBS D -W algorihm (i) guaranees he highes wors-case schedulabiliy performance, as he server behaves no worse han a sporadic ask, and (ii) provides he shores possible wors-case service delay, as idenified in Secion 4. However, hese feaures come a he cos of consuming he server budge even when he server is no acually execuing (noe ha rule R4 may perform a double budge accouning by means of he accoun(δ,d) funcion). This drawback leaves room for improvemen in erms of average-case performance and oher merics ha are no relaed o he wors-case server behavior. A pragmaic improvemen. A possible way o ge rid of he double budge accouning inroduced by he H-CBS D -W algorihm may consis in adoping a mechanism similar o he one proposed in he CASH [12] algorihm. CASH was designed for reclaiming unused budge a run-ime and relies on an EDF-ordered queue of

6 RTNS 17, Ocober 2017, Grenoble, France D. Casini e al. servers wih spare budge. For insance, he CASH mechanisms can be inegraed wih he H-CBS D -W by considering he servers ino he S-QUEUE as servers wih spare budge, and conexually (and carefully) revising he budge accouning rules of he server wihou affecing he wors-case performance of he server. Due o lack of space, his alernaive is lef as a conjecure o be invesigaed in fuure work. 5.2 An analysis-based soluion The key idea of he algorihm proposed in his secion is o leverage a schedulabiliy es for consrained-deadlines servers o solve he problem idenified in Secion 4. This new algorihm is denoed as H-CBS D. As discussed in Secion 2.1.1, he exac schedulabiliy es under EDF wih consrained deadlines (Theorem 2.1) has a pseudo-polynomial complexiy. Therefore, if such a es is used as a heoreical foundaion o develop some algorihmic rules of he H- CBS D, hen he resuling algorihm would in urn require a pseudopolynomial complexiy, hus poenially leading o a high run-ime overhead. To keep he algorihm overhead low, hence favoring is pracical applicabiliy, i is possible o leverage he (sufficien) linear-ime schedulabiliy es presened in Theorem 2.2. Noe ha his choice impacs on he admission es o be adoped for each reservaion r i R k, which mus be based on Theorem 2.2: clearly, his reduces he schedulabiliy performance of he sysem wih respec o he case considered in he previous secion. The H-CBS D explois Theorem 2.2 o design a new generae funcion for he generalized H-CBS formulaion in Secion 3. The wakeup(q, d, ) = and accoun(δ, d) funcions have no effec, meaning ha (i) he budge is always immediaely available as soon as i is recharged and ha, (ii) he budge is no decreased when he server is IDLE. Deriving he generae funcion. Given a se of reservaion servers R k, a any poin in ime each reservaion r i R k is characerized by a pair of dynamic sae-variables: he absolue deadline d i and he curren budge q i. When such parameers are used for a differen ime insan, say, hey are referred o as q i and d i. The goal of he following paragraphs is o derive a proper definiion for he generae funcion by exploiing such dynamic parameers. Firs, Lemma 5.3 is provided o bound he demand generaed by a H- CBS D server wihin a ime inerval in which he generae funcion is no invoked. Then, such a bound is used o derive a schedulabiliy es (Lemma 5.4) for he sysem wihin he same inerval of ineres. Finally, Lemma 5.4 is used o derive a definiion for he generae funcion ha guaranees o always assign a schedulable pair (q,d). Lemma 5.3 (Run-ime demand funcion). Le be an arbirary poin in ime afer a call o generae(). Le > he firs ime afer in which rule R2 riggers a call o he generae funcion for reservaion r j. Then, <, he processor demand of r j is bounded by is run-ime demand funcion, defined as: { rd f j (,q j q,d j ) = j + α j ( d j ) if d j 0 oherwise (11) where q i and d i are he curren budge and he curren absolue deadline a ime, respecively. roof. Similarly as argued in he proof of Lemma 5.1, as long as he server has ready workload, rules R5 and R6 guaranee ha he server behaves as a sporadic ask wih he same parameers of he server. As a consequence, for [,d j ), he generaed demand is zero, as he firs insance of he server wihin [, ] has deadline a ime d j. Since he inerval of ineres sared a ime, where he budge of r j is q j, he demand generaed in [,d j ] is q j Q i. By exploiing he linear demand approximaion inroduced in Equaion (2), from ime d j on, he demand of r j can be bounded as q j + α i ( d j ). A any poin in ime, if he server sops having pending workload, i eiher ransis o he IDLE sae or i is in he THROTTLED sae. Consider he firs case. To coninue o generae workload, he server mus ransi back o he READY sae, which however involves a call o generae() (see rule R2). By hypohesis, his canno happen in [, ). Similarly, if he server is in he THROTTLED sae, by rule R6 i will ransi o he IDLE sae as soon as is budge is replenished: herefore, he same argumen used above applies. The lemma follows. To reduce cluer when presening he following resuls, he funcion sae(r i, ) is defined o reurn he sae of server r i a ime. Furhermore, he following reservaion sub-ses are defined: R r k ( ) = {r i R k : sae(r i, ) = READY}, and Rk w ( ) = {r i R k : sae(r i, ) = THROTTLED r i has ready workload} Saring from Lemma 5.3, Lemma 5.4 is derived wih he aim of formulaing a novel definiion for he generae funcion of H- CBS D. In paricular, given a reservaion r i which is applying rule R2, Lemma 5.4 allows verifying if he use of a given budge and a given absolue deadline guaranees he sysem schedulabiliy. Lemma 5.4. Consider a reservaion se R k ha is schedulable according o Theorem 2.2. Le be an arbirary poin in ime afer a call o generae. Le > he firs ime afer in which any of he servers in R k riggers a call o he generae funcion. The reservaion se does no experience deadline misses < if: D, rd f j (,q j,d j ) + rd f i (,q i,d i )+ r i R w k ( ) db f i ( w i )+ r i {R r k ( )\{r j }} r i {R k \{R r k ( ) R w k ( )}} db f i ( ) (12) where > is he nex ime in which a reservaion r i R k riggers a call o generae(), i w = d i D i + i, D = {d i } {d i + i }, r i R r k ( ) r i R w k ( ) and r i R k, q i andd i are heir curren budge and absolue deadline a ime. roof. The lemma follows from he processor demand crierion recalled in Theorem 2.1 provided ha all he erms in Equaion (12) are valid demand bounds. Under he hypohesis on imes and, by Lemma 5.3 he firs wo erms are valid demand bounds for he reservaion servers in he READY sae (se R k r ( )). A reservaion server r i in he THROTTLED sae wih absolue deadline d i will experience a full budge replenishmen a i w = d i D i + i, and canno generae demand unil he laer ime. From ime i w, since

7 Consan Bandwidh Servers wih Consrained Deadlines RTNS 17, Ocober 2017, Grenoble, France he generae funcion is no called wihin [, ], is maximum demand is he same of he one generaed by a sporadic ask wih he same parameers of he server, hence is bounded by db f i ( i w ). If a reservaion server r i is in he IDLE sae, i mus call he generae funcion before saring generaing workload. Hence, is demand in he inerval of ineres is always zero, and consequenly he fourh erm is a valid bound. The se D includes all he disconinuiies of he adoped demand bounds. The lemma follows. Lemma 5.4 guaranees he sysem schedulabiliy unil he nex call o generae funcion occurs. The idea of he following resuls is o o use Lemma 5.4 wihin he definiion of he generae funcion o ensure he same schedulabiliy condiion also when generae is called. Le q and d o be he vecor of budges and deadlines for reservaions r i {R k r \ {r j }} a ime, respecively. Based on Lemma 5.4, i is possible o define a funcion o deermine wheher a pair (q j,d j ) can be used by a reservaion server r j wihou affecing he sysem schedulabiliy: { check(q,d rue if Lemma 5.4 is saisfied,q j,d j ) = false oherwise Then, he generae funcion is defined as follows (for a sake of simpliciy and homogeneiy wih respec o previous formulaions we omi he dependency from some parameers): generae(q j,d j, ) = (q j,d j ) if check(q,d,q j,d j ) < d j (Q j, + D j ) if check(q,d,q j,d j ) check(q,d, Q j, j D, ) < d j (max(0,q j ),d j ) oherwise where D j = + D j, and q j = Q j ( (d j D j )). The raionale behind his funcion is ha: (13) (1) The check funcion can be used o verify wheher he curren budge q j can be safely re-used mainaining he curren absolue deadline d j. (2) Oherwise, check is used o verify wheher is safe o perform a full budge replenishmen q j = Q j a he curren ime, wih deadline + D j. (3) Finally, if he firs wo iems fail, he deadline remains unchanged and he curren budge is se o he value ha i would have reached if i always execued in [d j D j,min(d j D j + Q j, )] Wih he algorihm definiion in place, Lemma 5.5 guaranees he schedulabiliy of H-CBS D. Lemma 5.5 (Safey in he wors case behavior). The budge and absolue deadline pairs assigned by generae as defined for he H-CBS D does no violae schedulabiliy. roof. Since Lemma 5.4 guaranees schedulabiliy unil he nex call o generae occurs and he generae funcion uses Lemma 5.4, he safey of he firs wo cases of Equaion (13) is auomaically guaraneed. Considering he hird branch of Equaion (13), he reservaion server r i which is calling he generae funcion mainains he same deadline d i and a budge q i obained considering ha i execued while i was acually IDLE. I is worh noing ha r i is guaraneed wih a budge q i qi (by means of a previous call o generae which exploied Lemma 5.4, if any, or by he admission es), regardless of he sae of r i. Then, he sysem remains schedulable if r i requires less demand. The lemma follows. Lemma 5.6 shows ha he H-CBS D algorihm guaranees o each reservaion r i a full budge replenishmen and an absolue deadline + D i o each server r i which wakes-up afer a leas i afer he las replenishmen. Lemma 5.6. Whenever H-CBS D is adoped, a any poin in ime such ha d i D i + i, he algorihm guaranees a pair (q i,d i ) = (Q i, + D i ) for each reservaion r i. roof. A he sysem sarup, a pair (qi,d i ) is implicily guaraneed by he admission es (Lemma 2.2). When he firs wo branches of he generae funcion are considered, he reservaion-se is guaraneed exploiing Lemma 5.4. I can be observed ha he proof of Lemma 5.4 would follow also wihou adding he erm: db f i ( ), r i {R r k Rw k } because each reservaion r i {R k r Rw } does no consume budge k a ime. Then, in his case he presence of ha erm is sufficien o reserve a pair (qi,d i ) o each server passing from IDLE o READY a leas afer i since he las replenishmen. Conversely, when considering he hird branch of generae he demand required by a reservaion r i is always lower or equal han he one ha i would required if i would be READY (he budge is consumed even if he reservaion is no acually execuing). Hence, a pair (qi,d i ) is implicily guaraneed by he previous call(if any) o generae which did no use he hird branch, or by he admission es. The lemma follows. As a consequence of Lemma 5.6, he hird branch of he generae funcion is never aken whenever i is called afer a leas i afer he las replenishmen (a leas he condiion for choosing second branch always succeeds). Lemma 5.7 bounds he maximum servicedelay of H-CBS D. Lemma 5.7 (Bounded-delay). Consider a reservaion se R k of H-CBS D reservaions ha are EDF-schedulable. The service delay of each server r i R k is no larger han i = i + D i 2Q i. roof. According o he general H-CBS formulaion presened in Secion 3, whenever new workload arrives a ime while he server is IDLE, generae is called. Le us analyze separaely he wors-case delay of he algorihm when he hree differen branches of he generae funcion are aken. Firs branch. The budge q i is available o be used wihin he absolue deadline d i. The wors-case scenario occurs when he period insance of he server sars execuing as lae as possible, i.e., a ime = d i q i. Hence, he wors-case delay is equal o. Since d i D i, hen = D i. Second branch. A budge Q i is available o be used wihin + D i. In he wors-case he server sars execuing a = + D i Q i, leading o a delay = = D i Q i. Third branch. The budge q is available o be execued wihin he curren absolue deadline d i. By definiion, q is he budge ha he

8 RTNS 17, Ocober 2017, Grenoble, France D. Casini e al. server would has if i always execued from is las replenishmen o ime. The scenario which maximizes he delay occurs when new workload arrives in he firs ime insan in which q = 0 and he following insance of he server execues as lae as possible. This siuaion occurs when he workload arrives a a = d i D i +Q i and he server sars execuing a = d i + i Q i. The wors-case delay in his case is = = d i + i Q i (d i D i +Q i ) = i +D i 2Q i. Finally, he global wors-case delay allowed by he algorihm is = max(,, ) = i + D i 2Q i. The lemma follows. Improving he available budge. Alernaively, he firs branch of generae could be defined o follow a principle similar o he one adoped in he revised CBS [1]: mainain he curren scheduling deadline and use he maximum budge q j allowed by Lemma 5.4. Such a budge can be derived as described in Lemma 5.8. Lemma 5.8. Consider a reservaion se R k ha is schedulable according o Theorem 2.2. Given an absolue ime in which a reservaion r j ransied from IDLE o READY wih an absolue deadline d j, he maximum safe budge q j which preserves he algorihm properies is: q s j = min D d j q j (), where D is defined in Lemma 5.4, and q j () = α j ( d j ) K + (14) wih K = r i {R r k \{r j }} rd f i (,q i,d i ) + r i R w k r i {R r k Rw k } db f i ( ). db f i ( w i )+ roof. The lemma follows direcly as a consequence of Lemma 5.4, wriing rd f j (,q j,d j ) in is exended form (by means of Equaion 12, subsiuing he firs branch of is definiion, and hen considering only absolue-deadlines greaer or equal han d j ) and solving wih respec o q j. In his case he generae funcion is defined as follows: min(q j,max(0,q s j )) if < d j q s j > 0 (Q j, + D j ) if < d j q s j 0 generae(q j,d j, ) = check(q,d, Q j, j D, ) (max(0,q j ),d j ) oherwise where D j = +D j. The algorihm obained by leveraging Lemma 5.8 is named H-CBS D -R. I is worh noing ha using H-CBS D -R each call o generae may cause he calling reservaion o reques a demand higher han he one of a sporadic ask wih parameers (Q j, D j, j ) (bu sill guaraneeing schedulabiliy), hus implemening a sor of budge-reclaiming mechanism. 5.3 Discussion The proposed soluions provide he same wors-case guaranees, namely boh of hem are able o provide he same wors-case delay and o mainain schedulabiliy among servers. However, hey differ in many poins: Applicabiliy. The H-CBS D -W is independen of he schedulabiliy es used for admiing reservaions. This implies ha i can be used boh under global, pariioned and semi-pariioned scheduling. Conversely, he H-CBS D and H-CBS D -R algorihms leverage he approximaed schedulabiliy analysis presened in Lemma 5.4, which is a processor-demand based analysis suiable only for pariioned or semi-pariioned EDF-based scheduling. Moreover, he schedulabiliy es used for admiing reservaion is consrained o be he same as he one presened in Theorem 2.2. Average-case performance. H-CBS D and H-CBS D -R leverage he schedulabiliy analysis o improve he average-case performance and probabilisic merics (ypically used for sof real-ime workload) wih respec o H-CBS D -W. In paricular H-CBS D -R poenially reclaims some spare budge whenever he generae funcion is called. Implemenaion and Complexiy. Algorihm H-CBS D -W can implemen all he mea-rules in consan-ime. On he oher hand, i may require he implemenaion of an addiional server queue, he S-QUEUE. Differenly, H-CBS D and H-CBS D -R do no require addiional daa srucures. Also, wakeup(q,d,) can sill be implemened in consan-ime and accoun(q,d,) does no require any implemenaion. As a drawback, he generae funcion has linear complexiy. 6 SIMULATION RESULTS As previously shown (see Lemmas 5.2 and 5.7), all he algorihms presened in his paper have he same wors-case performance. Whenever each reservaion server is used o schedule a single ask wih wors-case execuion ime smaller han or equal o han he maximum budge, minimum iner-arrival ime larger han or equal o he reservaion period, and he same relaive deadline of he server, hen all is deadlines are guaraneed o be respeced (his is someimes refferred as hard schedulabiliy propery of he CBS). Oherwise, some deadlines can be missed: in his case, he probabiliy o respec or miss a deadline depends on he used algorihm. To es he performance of he hree scheduling algorihms when serving his kind of real-ime asks, an exensive simulaion sudy has been carried ou and is presened in his secion. The algorihms have been esed in a widely variable se of siuaions, by randomly generaing a workload composed by realime asks wih random execuion and iner-arrival imes. The workload running ino each reservaion server consiss of a single sporadic ask τ i. The workload (i.e., he compuaion ime and he iner-arrival ime of each job) is conrolled by he following parameers: (1) The raio C r beween he average compuaion ime c i of a ask τ i and he maximum runime Q i of he server used o schedule i: c i = C r Q i. (2) The raio a beween he average uilizaion of a ask τ i and he uilizaion of he server used o schedule i: c i = a Q i p i i, where p i is he average iner-arrival ime of τ i. (3) The variance of he execuion and iner-arrival imes. The ses of servers and asks used in he simulaions have been generaed as follows. Given n = 5 reservaions 3 and a arge oal uilizaion U = i Q i i, budges and periods have been generaed using he Randfixedsum algorihm [17]. The periods i of he reservaions are disribued beween 5000 and Then, he relaive 3 Simulaions wih a differen number of reservaion servers have been performed, showing ha he number of reservaions does no affec he resuls. Hence, only resuls for n = 5 are repored.

9 Consan Bandwidh Servers wih Consrained Deadlines RTNS 17, Ocober 2017, Grenoble, France deadlines of each reservaion server have been generaed wih uniform disribuion in he inerval [Q i + β( i Q i ), i ], wih β = 0.4. Reservaion ses ha are no schedulable according o Theorem 2.2 have been discarded. All he ses generaed using his approach wih U 0.7 resuled o be schedulable according o Theorem 2.2. For U = 0.75, abou 2% of he generaed reservaions ses are no schedulable according o Theorem 2.2, whereas hey are all schedulable performing an exac schedulabiliy es; he siuaion becomes more ineresing for U = 0.9, when 49% of he generaed server ses are no schedulable according o Theorem 2.2 bu only 16% of he server ses are really no schedulable using an exac es. This fac has an impac on he usabiliy of he proposed algorihms, since (as previously discussed) H-CBS D -W can be used wih any schedulabiliy es while he oher wo algorihms require o use Theorem 2.2 o check he servers schedulabiliy 4. Afer generaing he reservaion-ses, he ask τ i served by each server has been generaed considering variables execuion imes, disribued according o a uniform random variable wih average c i = C r Q i. Similarly, he iner-arrival imes have been considered o be variable and disribued according o a uniform random variable c wih average value p i = i i aq i. The execuion imes are uniformly disribued in [c i sc i 2,c i + sc i 2 ] and he iner-arrival imes are uniformly disribued in [p i sp i 2,p i + sp i 2 ], where cs i = Q i sz, ps i = i sz, and sz is a parameer in he ask se generaion. The relaive deadline of each ask τ i is equal o he relaive deadline of he server used o schedule i. The generaed ses have been simulaed by using even-based simulaor. For each se of parameers, (U, a, C r, sz), 100 differen reservaion-ses have been generaed, and he number of missed deadlines for each ask has been couned. The raio of oal missed deadlines has been compued for each run, averaging resuls over he 100 repeiions (also compuing he 90% confidence inerval). The simulaor has been validaed by esing ses of asks and servers generaed as described above ogeher wih servers ha have been correcly dimensioned o respec he hard schedulabiliy propery of he CBS 5. For hose, he simulaions correcly repored 0 missed deadlines. Impac of he Toal Uilizaion U. In a firs se of simulaions, he impac of he oal servers uilizaion U = i Q i i has been evaluaed by varying U in he inerval [0.5, 1] wih seps of 0.1 while keeping C r, a and sz consan. The simulaions showed ha U had no significan impac on he percenage of missed deadlines. This is consisen wih he emporal isolaion propery provided by he CBS algorihm: he real-ime performance of a ask depends on he parameers of he ask and on he server used o schedule i, bu does no depend on he oher asks and servers running in he sysem. Due o his resul, in he following simulaions he oal uilizaion has been fixed o U = 0.7. Impac of he Server Load. More ineresing is he case in which he varied parameer is he parameer a (which is he raio beween 4 These resuls have been obained by generaing 1000 random ask ses according o he explained algorihm, and checking heir schedulabiliy according o Theorems 2.2 and Q i larger or equal han he wors-case execuion ime of he served ask and i smaller or equal han he minimum iner-arrival ime, wih he same relaive deadline. deadline hi raio deadline hi raio sz= cr=0.6 U= H-CBS D -W 0.2 H-CBS D H-CBS D -R Figure 4: Deadline hi raio vs a. a sz= a=0.6 U= Figure 5: Deadline hi raio vs C r. C r H-CBS D -W H-CBS D H-CBS D -R he average uilizaion of he served ask and he bandwidh of he reservaion: his value can be considered an esimaion of he server load ). When a approaches 1, in he average case he served ask ends o use all he execuion ime provided by he server, wih a significan number of insances exceeding he provided budge and compleing in subsequen periods, hus missing heir deadlines. Decreasing a, he number of respeced deadlines increases. Figure 4 shows he fracion of deadlines ha have been respeced as a funcion of a, wih C r = 0.6 and sz =. For small values of a all he algorihms are able o respec almos all of he deadlines. When a increases, H-CBS D -R coninues o respec a very high percenage of deadlines, because i provides a sor of budgereclaiming, as discussed in Secion 5.3, while he performance of he oher wo algorihms degrade. Since H-CBS R -W decreases he budge of a he server even if he ask does no execue, from Figure 4 i resuls o perform worse. Impac of he Average Execuion Time. Figure 5 shows he fracion of respeced deadlines as a funcion of C r, wih a = 0.6 and sz =. In his siuaion, he performance of H-CBS D and H- CBS D -R are similar, while H-CBS R -W performs slighly worse. All he algorihms are srongly penalized by large C r values. Impac of he Execuion and Iner-Arrival Times Variance. Figure 6 shows he fracion of respeced deadlines as a funcion of sz, wih a = 0.6 and C r = 0.6. Again, H-CBS R -W misses more deadlines han he oher wo algorihms. The plo also shows ha he number of misses deadlines increases wih sz, confirming ha large variances in he execuion and iner-arrival imes have a negaive impac on performance. The above observaions are also confirmed in Figure 7, showing he deadline hi raio vs boh C r and sz, in a summarizing 3D plo spanning across a differen region of he parameers space, wih C r from 0.7 o 1.1, sz from 0.6 o and a = 0.6.

10 RTNS 17, Ocober 2017, Grenoble, France D. Casini e al. deadline hi raio cr=0.6, a=0.60, U= SZ H-CBS D -W H-CBS D H-CBS D -R Figure 6: Deadline hi raio as a funcion of sz. deadline hi raio (a=0.60, U=0.7) H-CBS D -W H-CBS D H-CBS D -R SZ CR Figure 7: Deadline hi raio as a funcion of sz and C r. 7 CONCLUSIONS AND FUTURE WORK This paper proposed hree differen algorihms which exend he Hard Consan Bandwidh Server o work wih consrained deadlines. All he hree algorihms showed having he same wors-case properies. However, he firs one is a generic soluion independen of he adoped schedulabiliy es, whereas he ohers build upon a specific sufficien schedulabiliy es and showed having beer performance in erms of probabilisic merics (i.e., deadline hi raio). In he lieraure, many CBS-based reservaion algorihms were presened, bu none of hem is able o deal wih consrained deadlines. 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