Evaluating the Average-case Performance Penalty of Bandwidth-like Interfaces

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1 Evaluaing he Average-case Performance Penaly of Bandwidh-like Inerfaces Björn Andersson Carnegie Mellon Universiy ABSTRACT Many soluions for composabiliy and composiionaliy rely on specifying he inerface for a componen using bandwidh. Some previous works specify period (P and budge (Q as an inerface for a componen. Q/P provides us wih a bandwidh (he share of a processor ha his componen may reques; P specifies he ime-granulariy of he allocaion of his processing capaciy. Oher works add anoher parameer deadline which can help o provide igher bounds on how his processing capaciy is disribued. Ye oher works use he parameers α and where α is he bandwidh and specifies how smoohly his bandwidh is disribued. I is known [4] ha such bandwidh-like inerfaces carry a cos: here are askses ha could be guaraneed o be schedulable if asks were scheduled direcly on he processor, bu wih bandwidh-like inerfaces, i is impossible o guaranee he askse o be schedulable. And i is also known ha his penaly can be infinie, i.e., he use of bandwidh-like inerfaces may require he use of a processor ha has a speed ha is k imes faser, and one can show his for any k. This brings he quesion: Wha is he average-case performance penaly of bandwidh-like inerfaces? This paper addresses his quesion. We answer he quesion by randomly generaing askses and hen for each of hese askses, compue a lower bound on how much faser a processor needs o be when a bandwidh-like scheme is used. We do no consider any specific bandwidh-like scheme; insead, we derive an expression ha saes a lower bound on how much faser a processor needs o be when a bandwidh-like scheme is used. For he disribuions considered in his paper, we find ha (i he experimenal resuls depend on he experimenal seup, (ii his lower bound on he penaly was never larger han 4.0, (iii for one experimenal seup, for each askse, i was greaer han 2.4, (iv he hisogram of his penaly appears o be unimodal, and (v for implici-deadline sporadic asks, his lower bound on he penaly was exacly 1. Caegories and Subjec Descripors D.4.7 [Operaing Sysems]: Organizaion and Design The auhors reain copyrigh. Real-ime sysems and embedded sysems; G.4 [Mahemaical Sofware]: Algorihm design and analysis General Terms Algorihms, Performance, Theory Keywords Real-ime, Composabiliy, Composiionaliy 1. INTRODUCTION Consider a askse τ scheduled on a single processor wih Earlies-Deadline-Firs (EDF. Assume ha asks are arbirarydeadline sporadic asks, i.e., a ask τ i is characerized by, D i, and, wih he inerpreaion ha τ i generaes a sequence of jobs wih a leas ime unis beween wo consecuive arrivals of jobs of τ i and each job of τ i has execuion ime a mos and each job has a deadline D i ime unis relaive o is arrival. I is known [5] ha if for all posiive i holds ha τ i τ max( D i + 1, 0 (1 hen he askse is schedulable. (A askse is schedulable, if for each jobse ha i can generae, for each schedule ha EDF can generae for his jobse, i holds ha all jobs mee deadlines; noe ha because we assume EDF wih arbirary ie-breaking, here may be more han one valid schedule for a jobse scheduled by EDF. The resul above (in Eq. 1 is well-known and allows sofware praciioners o efficienly verify, before run-ime, ha all iming requiremens will be me a run-ime. This resul works assuming ha he enire askse is known o a single person (or schedulabiliy analysis ool and ha he sysem does no undergo design changes and ha asks do no use more resources han saed by heir parameers. The real-ime sysems research communiy undersood hese limiaions of using Eq. 1 as a schedulabiliy es and using EDF a run-ime. The research communiy undersood ha i was necessary o monior he execuion of a ask o see if i execued more han i was expeced o, and also o monior a ask o see if i generaed jobs more frequenly han i was expeced o. In addiion, he research communiy undersood ha in sysem inegraion, i is ofen advanageous o describe a se of asks wih relaed funcionaliy wih a simpler descripion. Therefore, he research communiy creaed a large se of soluions o achieve his. Typically, hese soluions work as follows. A componen (someimes called a

2 server ask and someimes called a subsysem is characerized by a bandwidh parameer and some oher parameers. One or more asks are assigned o a componen. And each ask is assigned o exacly one componen. Then, a each insan a run-ime, a roo scheduler (someimes called global scheduler selecs a componen and a local scheduler in his componen selecs a ask in his componen. This seleced ask execues on he processor. The bandwidh of a componen is characerized as a share of he processor (for example, Componen 1 should use a mos 20% of he processor. One could use a run-ime mechanism ha guaranees ha his bandwidh is allocaed o each componen in each ime inerval (even infiniely small ime inervals. Bu his would require infiniely many conex swiches which would make such a soluion impracical. Therefore, he soluions presened in he research lieraure use anoher parameer as well: server-period. Some soluions ensure ha in a ime inerval of duraion a leas as large as he server period, a componen is allocaed processing ime being a leas as large as is bandwidh (i.e., bandwidh muliplied by server period. Many schemes in he lieraure suffer from a socalled blackou period; for such a scheme, he guaraneed allocaion is slighly less. Some schemes have a parameer, server deadline, which can be used o conrol he duraion of his blackou period. Common o hese schemes, however, is ha before run-ime, a schedulabiliy es is performed on he roo scheduler; i akes he bandwidh and poenially oher parameers of each componen and deermines if he roo scheduler will be able o allocae enough bandwidh o each componen. These bandwidh-like schemes have several advanages. Firs, hey achieve isolaion. This is imporan because i is ofen very difficul o find he wors-case execuion ime of a ask. Wih hese schemes, one can be sure ha if he execuion ime of a job would exceed is esimaed wors-case execuion ime, hen i does no jeopardize iming guaranees of jobs in oher componens. Second, hey allow hard and sof real-ime asks o be execued on a single processor. Third, hey provides a simple inerface for sysem inegraors. In paricular, he concep of bandwidh is easy o undersand for laypersons. (For example: Componen 1 is assigned 10% of he processor; Componen 2 is assigned 70% of he processor; Componen 3 is assigned 15% of he processor. Fourh, hey allow differen schedulers o be used in differen componens (e.g., he local scheduler in Componen 1 may be EDF and he local scheduler in Componen 2 may be RM and he local scheduler in Componen 3 may be FIFO. Fifh, some real-ime operaing sysems suppor heir run-ime mechanisms. Sixh, he run-ime policing has low ime and space complexiy. Given all hese advanages, i may seem ha bandwidh-like schemes offer a good foundaion for real-ime sysems. Unforunaely, bandwidh-like schemes can wase an infinie amoun of resources. I can be seen as follows: Consider a askse τ wih n asks, τ 1, τ 2,..., τ n and hese asks have he parameers =,D i = i, = 1. If hese asks are scheduled direcly on he processor (i.e., wihou componens, i can be seen ha he askse is EDF-schedulable according o Eq. 1. Le us now discuss heir behavior in a sysem wih componens and wih a bandwidh-like scheme. Suppose ha here are n componens and one ask in each componen; specifically, ask τ i is assigned o componen i. In order for he roo scheduler o be schedulable, i is required ha he sum of bandwidh of he componens is a mos 1. The bandwidh required by a componen depends on he acual scheme used (he blackou period maers and he predicabiliy of supply of he roo scheduler maers; bu for all bandwidh-like schemes, i holds ha he required bandwidh of a componen is a leas as large as he sum of he densiy of he asks in he componen. This yields ha componen i requires a leas he bandwidh /D i. Hence, componen i requires a leas he bandwidh 1/i. Consequenly, in order for he roo scheduler o be schedulable, i mus hold ha i {1..n} 1/i 1. I is easy o see ha (for n 2 his condiion is false and hence he sysem is no schedulable wih a bandwidh-like inerface. Le he processor be k imes faser. Then, in order for he roo scheduler o be schedulable, i mus hold ha i {1..n} 1/i k. Leing n approach infiniy yields ha i {1..n} 1/i approaches ln n and hence i approaches infiniy. Thus, even using a processor ha is k imes faser canno guaranee ha he sysem is schedulable wih a bandwidh-like inerface. We can do his reasoning for any k and hence we obain ha bandwidh-like inerfaces can generae an infinie wase of resources. This observaion (ha bandwidh-like inerfaces can wase an infinie amoun of resources is known in he lieraure [3, 4]. Wha is no known, however, is how well bandwidhlike schemes perform in he average-case as compared o a scheme ha performs fla scheduling. Exploring his quesion is he goal of his paper. 2. FORMULATING THE PROBLEM From Eq. 1 i can be seen ha for a askse scheduled wih EDF, a processor speed max >0( τ i τ max( D i + 1, 0 (2 is sufficien o mee deadlines. From he discussion in he previous secion, i can be seen ha if each ask is in is own componen, hen wih a bandwidhlike scheme, a processor speed τ i τ is necessary o mee deadlines. For a askse τ, le spdf(τ be defined as: min(d i, (3 τ i τ min(d i, max >0( τ i τ max( D i +1,0 Here spdf(τ should be read as speed-up facor. Inuiively, spdf(τ indicaes a lower bound on how much faser he processor needs o be in order for a bandwidh-like scheme o make he askse τ schedulable. We have already seen ha here is a askse such ha spdf is infinie. This paper explores he quesion: Wha is spdf(τ for ypical askses? The nex four secions consider differen assumpions and derive expressions for spdf(τ. For he firs hree nex secions, we generae asks randomly and obain hisograms of spdf(τ. For he fourh nex secion, we give a simple analyic expression of spdf(τ and hence do (4

3 no run experimens. Afer hese four secions, we commen on he experimenal resuls. 3. EVALUATION FOR THE SPECIAL CASE OF TASKSETS WITH INFINITE MINI- MUM INTER-ARRIVAL TIMES In his secion, we consider he special case where for each ask τ i in τ, i holds ha =. For his case, spdf(τ can be compued as: max >0( τ i τ θ( D i where θ is he sep funcion (i reurns 1 if is inpu is nonnegaive and i reurns 0 if is inpu is negaive. Look a he denominaor. Noe ha his sep funcion only changes for hose ha are equal o a D parameer. I can be seen ha we only need o check hose ha are equal o a D parameer. Using his observaion yields: max τj τ ( τ i τ θ(d j D i D j Look a he denominaor. Observe ha we only need o include he erms where D j D i 0; he oher ones are zero. Wih his observaion, addiional rewriing yields: max τj τ ( τ i τ s.. D i D j D j We will explore spdf(τ for randomly-generaed askses. We do i wih wo ypes of askse generaion: similar asks and very differen asks. Wih he askse generaion similar asks, we generae asks as follows: D i = random(1, 10, = random(1, 10, where random(a, b generaes a random number, a real number, in he range [a,b] and i does so wih a uniform disribuion. If > D i hen we swap he values so ha D i. Afer ha, we normalize he densiy of he askse so ha i is 0.999; i.e., we muliple he C-parameers of all asks so ha he densiy becomes I can be seen ha his does no change spdf(τ; we do i simply so ha we ge askse parameers ha are easier o inerpre. Then we sor he asks in ascending order of he D parameer. Wih he askse generaion very differen asks, we generae asks as follows: = 10 random(0,4 random(1, 10, D i = 10 random(0,4 random(1, 10, where random(a, b generaes a random number, a real number, in he range [a,b] and i does so wih a uniform disribuion. If > D i hen we swap he values so ha D i. Afer ha, we normalize he densiy of he askse so ha i is 0.999; i.e., we muliple he C-parameers of all asks so ha he densiy becomes Then we sor he asks in ascending order of he D parameer. The resuls for boh similar asks and very differen asks are shown in Figure 1. Each subplo shows a hisogram of 3000 randomly generaed askses. (5 (6 (7 4. EVALUATION FOR THE GENERAL CASE, ARBITRARY-DEADLINE SPORADIC TASKS In his secion, we consider he general case, i.e., i is no required ha T parameers are infinie. I is also assumed ha he askse has arbirary deadlines; i.e., D i is allowed o be less han, or equal o, or greaer han. Recall from Eq. 4 ha τ i τ min(d i, max >0( τ i τ max( D i +1,0 Noe ha he denominaor has an expression which considers all posiive. One can see, however, ha if only hose for which here exiss a ask τ j and a non-negaive ineger k such ha = k T j + D j are considered, hen he calculaion yields he same resul. One can also see ha if only values of ha are a mos 10 max τi τ ( + D i are considered, hen he calculaed value is an approximaion of spdf(τ ha has an error of a mos 10%. We will use his o obain an approximaion of spdf(τ. We will explore spdf(τ for randomly-generaed askses. We do i wih wo ypes of askse generaion: similar asks and very differen asks. Wih he askse generaion similar asks, we generae asks as follows: D i = random(1, 10, = random(1, 10, = random(1, 10, where random(a, b generaes a random number, a real number, in he range [a,b] and i does so wih a uniform disribuion. If > D i hen we swap he values so ha D i. Afer ha, we normalize he uilizaion of he askse so ha i is 0.999; i.e., we muliple he C-parameers of all asks so ha he uilizaion becomes Then we sor he asks in ascending order of he D parameer. Wih he askse generaion very differen asks, we generae asks as follows: = 10 random(0,4 random(1, 10, D i = 10 random(0,4 random(1, 10, and = 10 random(0,4 random(1, 10, where random(a, b generaes a random number, a real number, in he range [a,b] and i does so wih a uniform disribuion. If > D i hen we swap he values so ha D i. Afer ha, we normalize uilizaion as menioned above. Then we sor he asks in ascending order of he D parameer. The resuls for boh similar asks and very differen asks are shown in Figure 2. Each subplo shows a hisogram of 3000 randomly generaed askses. 5. EVALUATION FOR THE GENERAL CASE, CONSTRAINED-DEADLINE SPORADIC TASKS In his secion, we consider he general case, i.e., i is no required ha T parameers are infinie. I is also assumed ha he askse has consrained deadlines; i.e., D i is allowed o be less han or equal o. The resuls for boh similar asks and very differen asks are shown in Figure 3. Each subplo shows a hisogram of 3000 randomly generaed askses. (8

4 (a Tasks similar and τ =2. (b Tasks similar and τ =10. (c Tasks similar and τ =100. (d Tasks similar and τ =1000. (e Tasks no similar and τ =2. (f Tasks no similar and τ =10. (g Tasks no similar and τ =100. (h Tasks no similar and τ =1000. Figure 1: Hisogram for spdf for differen ways of generaing askses randomly for he case ha T =.

5 (a Tasks similar and τ =2. (b Tasks similar and τ =10. (c Tasks similar and τ =100. (d Tasks similar and τ =1000. (e Tasks no similar and τ =2. (f Tasks no similar and τ =10. (g Tasks no similar and τ =100. (h Tasks no similar and τ =1000. Figure 2: Hisogram for spdf for differen ways of generaing askses randomly for he case ha s finie. Arbirary-deadline sporadic asks.

6 (a Tasks similar and τ =2. (b Tasks similar and τ =10. (c Tasks similar and τ =100. (d Tasks similar and τ =1000. (e Tasks no similar and τ =2. (f Tasks no similar and τ =10. (g Tasks no similar and τ =100. (h Tasks no similar and τ =1000. Figure 3: Hisogram for spdf for differen ways of generaing askses randomly for he case ha s finie. Consraineddeadline sporadic asks.

7 6. EVALUATION FOR THE GENERAL CASE, IMPLICIT-DEADLINE SPORADIC TASKS In his secion, we consider he special case where for each ask τ i in τ, i holds ha =. Recall from Eq. 4 ha τ i τ min(d i, max >0( τ i τ max( D i +1,0 Using = on his expression yields ha: Rewriing yields: τ i τ max >0( τ i τ max( +1,0 τ i τ max >0( τ i τ T i (9 (10 (11 I can be seen ha he denominaor is maximized for. This yields: Simplifying yields: τ i τ τ i τ (12 1 (13 For his reason, here is no need o run experimens for implici-deadline sporadic asks. 7. COMMENTING ON THE EXPERIMEN- TAL RESULTS Recall ha spdf is a lower bound on how much faser a processor needs o be in order for a bandwidh-like scheme o mee deadlines. And recall ha Figure 1, Figure 2, and Figure 3 provide hisograms for spdf. From hese figures, i can be seen ha (i he experimenal resuls depend on he experimenal seup, (ii his lower bound on he penaly was never larger han 4.0, (iii for one experimenal seup, for each askse, i was greaer han 2.4, (iv he hisogram of his penaly appears o be unimodal, and (v for implicideadline sporadic asks, his lower bound on he penaly was exacly 1. Le us now discuss wo subfigures in Figure 1 in order o undersand he resuls. Consider Figure 1a. I shows an experimenal seup wih jus wo asks so i is easy o see deails. Consider he askse wih he larges observed spdf. By inspecing his askse, we see ha is spdf is We will now discuss how his resul was obained. The askse conains 2 asks and i urns ou he denominaor (of Eq. 7 is maximized for j = 1. For he askse, i holds ha D 1 = The askse is as follows: C 1 = 0.58, D 1 = 1.08, C 2 = 4.58, D 2 = Consider Eq. 7 again. Recall ha here is a denominaor wih a max-expression. I urns ou ha for his askse, his expression is maximized for j = 1. We also know ha D 1 = Applying his knowledge on Eq. 7 yields: ( (14 τ i τ s.. D i / We can compue he sum in he denominaor; i is he sum of C for asks wih D Using values yields ha i is 0.58 and hence he denominaor is 0.58/ This yields ha he denominaor is and we have already seen ha he numeraor is Hence, This yields: spdf = (15 Consider Figure 1h. I shows he experimenal seup where he larges observed spdf occurred. By inspecing his askse, we see ha is spdf is We will now discuss how his resul was obained. The askse conains 1000 asks and i urns ou he denominaor (of Eq. 7 is maximized for j = 148. For he askse, i holds ha D 148 = In order o give an idea of wha he askse looks like while no consuming oo much space, we lis below only he firs ask and also lis asks wih D and wih C > The lising is as follows: C 1 = 0.24, D 1 = 2.06,..., C 13 = 0.43, D 13 = 4.12,..., C 30 = 0.59, D 30 = 6.28,..., C 58 = 0.24, D 58 = 8.70,... C 88 = 0.25, D 88 = 12.25,... C 102 = 0.45, D 102 = 14.28,... C 106 = 0.23, D 106 = 15.03,... C 119 = 0.27, D 119 = 16.71,... C 134 = 0.22, D 134 = 19.56,... C 145 = 0.58, D 145 = 21.05,... C 147 = 0.20, D 147 = Consider Eq. 7 again. Recall ha here is a denominaor wih a max-expression. I urns ou ha for his askse, his expression is maximized for j = 148. We also know ha D 148 = Applying his knowledge on Eq. 7 yields: ( (16 τ i τ s.. D i /21.94 We can compue he sum in he denominaor; i is he sum of C for asks wih D Using values yields ha i is and hence he denominaor is 5.621/ This yields ha he denominaor is and we have already seen ha he numeraor is Hence, This yields: spdf = (17 8. RELATED WORK The lieraure on hierarchical scheduling, composabiliy, and composiionaliy is vas; here we only survey some of he previous work. When rae-monoonic scheduling was developed as a comprehensive framework, i was recognized ha many sysems have sofware whose resource consumpion is hard o characerize; e.g., i is hard o find is wors-case execuion ime or minimum iner-arrival ime. For his reason, reservaion-based frameworks were developed (see for example [13]. The run-ime behavior of such a framework is as follows: A ask may be associaed wih a server ask and his server ask is scheduled as a normal ask (for example wih an execuion ime, ofen called budge, and a period

8 and if a ask is associaed wih a server ask hen i is only allowed o execue when he server ask execues. In his way, if a ask τ i is in a server ask and if τ i experiences an execuion overrun or arrives more ofen han expeced hen is impac on oher asks is bounded and i is bounded by he parameers of he server ask. Laer works creaed such reservaion for EDF; one example of ha is he consanbandwidh server (CBS [1]. Researchers realized ha reservaion-based frameworks can be used o form hierarchical scheduling; some work ha did so wih EDF include [14, 9]. Wih he focus on hierarchical scheduling, Feng and Mok developed [11] a resource model which saes ha a roo scheduler supplies, in a ime inerval of duraion, a leas ( k α k unis of execuion o componen k. Shin and Lee developed [15] anoher model where a componen k is characerized by is period and execuion ime and wih his, presened a so-called supply-bound funcion; his work could be used for componens ha use fixedprioriy as a local scheduler or componens ha use EDF as a local scheduler. Laer works have focused on resource sharing, specifically he quesion, if a ask τ lock execues wihin a criical secion and is curren budge has reached zero, wha should he scheduler do? There may be oher asks ha will reques his criical secion and hese requess can be graned only if he ask ha currenly holds ha criical secion (τ lock has released i. In order for he ask (τ lock o release he criical secion, i mus be able o execue and in order for his ask (τ lock o execue, i mus have a curren budge greaer han zero. Differen soluions for his have been developed; see for example [8, 7, 6]. Clearly, server parameers mus be seleced in order o ensure schedulabiliy; his has been he focus of [2]. Recall ha early in his paper, we poined ou ha bandwidhlike schemes can suffer from poor performance and we have suggesed ha using he demand-bound funcion is an inerface ha does no suffer from his drawback. A similar poin has been made in [12, 10]; hey also discuss approximae policing of such an inerface. 9. CONCLUSIONS Bandwidh-like schemes can cause a performance penaliy. I was known ha for cerain askses, his performance penaly is infinie bu i was no known how large his penaly is for average askses. Therefore, his paper generaed askses randomly and compued spdf which is a lower bound on how much faser a processor needs o be in order o mee deadlines if a bandwidh-like scheme is used. Through he experimenaion in his paper, i can be seen ha (i he experimenal resuls depend on he experimenal seup, (ii his lower bound on he penaly was never larger han 4.0, (iii for one experimenal seup, for each askse, i was greaer han 2.4, (iv he hisogram of his penaly appears o be unimodal, and (v for implici-deadline sporadic asks, his lower bound on he penaly was exacly 1. Acknowledgmen Copyrigh 2016 ACM This maerial is based upon work funded and suppored by he Deparmen of Defense under Conrac No. FA C-0003 wih Carnegie Mellon Universiy for he operaion of he Sofware Engineering Insiue, a federally funded research and developmen cener. [Disribuion Saemen A] This maerial has been approved for public release and unlimied disribuion. Please see Copyrigh noice for non-us Governmen use and disribuion. DM REFERENCES [1] L. Abeni and G. Buazzo. Inegraing mulimedia applicaions in hard real-ime sysems. In RTSS, [2] L. Almeida and P. Pedreiras. 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