Stochastic Analysis of Periodic Real-Time Systems *

Transcription

1 Stochastic Analysis of Peiodic Real-Time Systems * José Luis Díaz Daniel F Gacía Kanghee Kim Chang-Gun Lee Lucia Lo Bello José Maía López Sang Lyul Min Oazio Miabella Abstact This pape descibes a stochastic analysis method fo geneal peiodic eal-time systems The poposed method accuately computes the esponse time distibution of each tas in the system, thus maing it possible to detemine the deadline miss pobability of individual tass, even fo systems with maximum utilization facto geate than one The method unifomly coves both fixed-pioity scheduling (such as Rate Monotonic) as well as dynamic-pioity scheduling (such as Ealiest Deadline Fist) and can handle abitay elative deadlines and execution time distibutions The accuacy of the method is poven by compaing the esults fom the analysis with those obtained fom simulations, as well as othe methodologies in the liteatue 1 Intoduction Taditional scheduling algoithms and analysis methods, such as pocesso utilization analysis [16, 11] and esponse time analysis [4, 19], focus on stict had deadlines, by which a system is deemed schedulable only if evey instance (called a job) of evey tas is guaanteed to meet its * The autho names ae listed in alphabetical ode since this pape is a collaboative wo based on two diffeent papes submitted to the confeence sepaately One pape was witten by José Luis Díaz, José Maía López and Daniel F Gacía, and the othe was witten by Kanghee Kim, Lucia Lo Bello, Chang-Gun Lee, Sang Lyul Min, and Oazio Miabella José Luis Díaz, José Maía López and Daniel F Gacía ({jdiaz,chechu,daniel}@atcuniovies), Depatamento de Infomática, Univesidad de Oviedo (33204, Gijón, Spain) Kanghee Kim (him@achisnuac) and Sang Lyul Min (symin@dandelionsnuac), School of Compute Science and Engineeing, Seoul National Univesity (Shillim-dong San 56-1, Kwana- Gu, Seoul, Koea), suppoted in pat by the Ministy of Science and Technology unde the National Reseach Laboatoy pogam and by the Ministy of Education unde the BK21 pogam Lucia Lo Bello and Oazio Miabella ({llobello,omiabel}@ diitunictit), Dipatimento di Ingegneia Infomatica e delle Telecomunicazioni, Facoltà di Ingegneia, Univesità di Catania (Viale A Doia 6, Catania, Italy) Chang-Gun Lee (cglee@eeengohio-stateedu), Depatment of Electical Engineeing, Ohio State Univesity (2015 Neil Avenue, Columbus, OH 43210, USA) deadline Although this deteministic timing guaantee is needed in had eal-time systems, it is too stingent fo socalled soft eal-time applications that equie only a pobabilistic guaantee that the deadline miss atio of a tas is below a given theshold Fo such soft eal-time applications, we need to elax the assumption that evey instance of a tas equies the wost-case execution time and analyze system behavio fom a statistical point of view Pogess has ecently been made in the analysis of ealtime systems unde the stochastic assumption that jobs fom a tas equie vaiable execution times Reseach in this aea can be categoized into two goups depending on the appoach it taes to facilitate the analysis The methods in the fist goup intoduce a wost-case assumption to simplify the analysis (eg, the citical instant assumption in Pobabilistic Time Demand Analysis [18] and Stochastic Time Demand Analysis [7, 6]) o a estictive assumption (eg, the heavy taffic condition in the Real-Time Queueing Theoy [12, 13]) Those in the second goup, on the othe hand, assume a special scheduling model that povides isolation between tass so that each tas can be analyzed independently of othe tass in the system (eg, the esevationbased system addessed in [1] and Statistical Rate Monotonic Scheduling [2]) In this pape, we popose a stochastic analysis method that does not intoduce any wost-case o estictive assumptions into the analysis, and is applicable to geneal pioity-diven eal-time systems The method is geneal in the sense that it coves geneal pioity-diven systems including both fixed-pioity systems such as RM [16] and DM [15] and dynamic-pioity systems such as EDF [16] (Fist In Fist Out is also coveed since it is consideed a special case of EDF whee all the jobs have a constant elative deadline) The analysis method can handle any peiodic tas set consisting of tass with abitay elative deadlines (including elative deadlines geate than the peiods) and abitay execution time distibutions The analysis method is based on Maov pocess modeling, which enables us to eason pobabilistically about the steady-state behavio of the system even in the case of possible oveload It povides both analytical and numeical solutions fo the deadline miss pobabilities of tass by

2 computing the complete pobability function (PF) of the esponse time of each tas The est of the pape is oganized as follows In Section 2, the elated wo is descibed in detail In Section 3, the system model assumed and the notations used thoughout the pape ae given Section 4 descibes the stochastic analysis method we popose and also explains vaious appoximation techniques to educe its computational oveheads In Section 5, we compae expeimental esults fom the poposed analysis method with those obtained fom simulations and othe methodologies Finally, in Section 6, we conclude the pape with diections fo futue eseach 2 Related wo Seveal studies have addessed the vaiability of tas execution times even in deteministic schedulability analysis Fo example, a multifame model was poposed by Mo and Chen [17] in which the execution time of a tas may vay geatly fom one instance to anothe assuming that this vaiation follows a nown patten The patten is given as a finite list of numbes, and the execution times of successive instances ae geneated fom the list Fom this model, new utilization bounds which impove those of Liu and Layland [16] ae deived fo fixed-pioity peemptive scheduling Howeve, since this model is aimed at poviding a deteministic timing guaantee, it is still pessimistic In the attempt to povide pobabilistic guaantees to ealtime tass with vaiable execution times, some esevationbased models that povide isolation between tass have been studied These models include the esevation-based system addessed by Abeni and Buttazzo [1] and a modification of Rate Monotonic Scheduling poposed by Atlas and Bestavos [2], which is called Statistical Rate Monotonic Scheduling (SRMS) Both assume esevation-based scheduling algoithms so that the analysis can be pefomed as if each tas had a dedicated (vitual) pocesso That is, fo each tas, a guaanteed budget of pocesso time is povided in evey peiod [1] o supe-peiod (the peiod of the next low-pioity tas which is assumed to be an intege multiple of the peiod of the tas in SRMS) [2] Theefoe, the deadline miss pobability of a tas can be analyzed independently of othe tass assuming the guaanteed budget Howeve, the stochastic analysis methods developed fo these systems ae not applicable to geneal pioitydiven systems due to the adoption of non-pioity-diven scheduling algoithms o the modification of the oiginal pioity-diven scheduling ules A moe geneal appoach is to model the system as a single seve queue, and ty to apply the esults of classical queueing theoy The eal-time system we ae tying to analyze is made up of peiodic independent tass scheduled by a peemptive pioity-diven schedule on a unipocesso Appaently, this system is a multiclass queue in which each class has deteministic inte-aival times and abitay sevice times (D/G/1) Howeve, this model cannot account fo the elative phases of the tass, which influence the statistical distibution of the esponse times Futhemoe, the few esults available fo multiclass queues assume specific distibutions (Poisson, nomal, etc) fo the inte-aival times The case of geneal distibutions of inte-aival times is usually addessed unde heavy-taffic conditions (ie, the aveage utilization is close to one) and, even in this case, the esults ae not valid fo deteministic inte-aival times, because this is a special case of a geneal distibution One extension of the classical queueing theoy to deal with eal-time issues was poposed in the Real-Time Queueing Theoy [12] This analysis method is flexible in that it is not limited to a paticula scheduling algoithm and can be extended to eal-time queueing netwos Howeve, the analysis method assumes that the system is unde heavytaffic conditions In addition, it only consides one class of clients, ie, the inteaival times and execution times ae identically distibuted fo all the tass This model does not fit well with the peiodic tas model Anothe appoach to the statistical chaacteization of eal-time systems is to extend an existing esponse time analysis, substituting the fixed execution times with andom vaiables Following this appoach, Tia et al [18] poposed the Pobabilistic Time Demand Analysis (PTDA), which substitutes the sums of fixed execution times in the Time Demand Analysis [14] with convolutions of pobability functions (PFs) In this way they can obtain the PF of the esponse time of a tas assuming the wost-case scenaio on the tas elease times, ie, the citical instant The analysis is esticted to the fist activation of the tas, since the deadlines cannot be geate than the peiods This assumption is also made in the Stochastic Time Demand Analysis (STDA) by Gadne [6], which extends the PTDA to cove systems whee the deadlines may be geate than the peiods In this analysis, the pobability of deadline misses is computed fo each job eleased in the fist in-phase busy inteval, and the maximum of these pobabilities is chosen as an uppe bound on the pobability of deadline misses fo the coesponding tas Both analyses ae based on the sum of andom vaiables, and thus the use of convolutions to detemine its PF Howeve, the analyses cannot addess systems whee the maximum system utilization is geate than one In this case, the assumption that the busy inteval stating at the citical instant will contain the wost-case esponse time is no longe valid, so the deadline miss pobabilties computed by the analyses ae not actual bounds Recently, Kim et al [9] analyzed the stochastic behavio of a dynamic-pioity system combined with an oveun handling mechanism called andomized dopping Although this stochastic analysis method was developed to

3 compute the deadline miss pobabilities of tass fo such a system, it can still be used fo a pue dynamic-pioity system By modeling the system as a Maov pocess, it computes the stationay esponse time distibutions of all the jobs in a hypepeiod (which is defined as a peiod whose length is equal to the least common multiple of the peiods of all the tass) and thus the esponse time distibutions of all the tass Howeve, it deals with dynamicpioity systems only and the deivation of the Maov matix is complicated due to the abstaction of job goups and aggegated esponse times In this pape, we pesent a boade appoach, which stats fom the ideas of [18], [6] and [9] We povide a simple deivation method fo the Maov matix and both analytical and numeical solutions fo the esponse time distibutions of all the tass We also descibe a genealized famewo to deal with both fixed-pioity and dynamic-pioity systems extending the concept pesented in [9] This model can be moe easily undestood stating fom a simple one, in which the system is not seen as a set of peiodic tass, but as a set of jobs eleased in a given sequence Based on the simple model, a systematic method to compute the esponse time PF of any job is developed and pecisely descibed in [5, 8] In Section 41, we will summaize this method in the moe geneal context of a peiodic tas system, and will povide an oveview of the whole stochastic analysis Finally, note that ou stochastic analysis is useful not only fo soft eal-time systems, but also fo so-called pobabilistic had eal-time systems [3], whee a pobabilistic guaantee close to 100% suffices 3 System model and notation The system is modeled as a set of N independent peiodic tass S = {τ 1,τ 2,,τ N }, each tas τ i being modeled by (T i,φ i,c i,d i ), whee T i is the peiod of the tas, Φ i its initial phase, C i its execution time, and D i its elative deadline The execution time is a discete andom vaiable * with a nown pobability function (PF), denoted by f Ci ( ), whee f Ci (c) = P{C i =c} Since the value of C i is bounded, its PF can be stoed as a finite vecto of values [ f Ci (Ci min ),, f Ci (Ci max )] Each tas gives ise to an infinite sequence of jobs, and we will denote the j-th job of tas τ i by Γ The elease time of job Γ will be denoted by λ This time is deteministic and equal to Φ i +( j 1) T i Each job equies an execution time which is a andom vaiable whose distibution is given by f Ci ( ), and it is assumed to be independent of othe jobs of the same tas and those of othe tass The scheduling policy we assume is a geneal pioitydiven one that assigns each job Γ a static pioity and * Thoughout this pape we use a calligaphic typeface to denote andom vaiables, lie C, R, etc schedules jobs accoding to this pioity The schedule guaantees that the unning job is the one with the highest pioity among the eady jobs We ae not concened with the policy used to assign pioities to jobs, as long as they ae assigned in a deteministic way This model includes well-nown policies such as Rate Monotonic (RM), Deadline Monotonic (DM) and Ealiest Deadline Fist (EDF) Fo fixed-pioity policies, we will use P i to denote the pioity assigned to tas τ i The esponse time of the job Γ will be epesented by R This is a andom vaiable, which can tae diffeent values with diffeent pobabilities In the next section we outline a pocedue which allows us to find the pobability of occuence of each possible esponse time fo a given job, ie the pobability function (PF) of the esponse time: f R () = P{R =} Fom the job esponse time PFs, the esponse time PF fo any tas can be obtained as the aveage of the esponse time PFs of the jobs belonging to that tas The tas esponse time PF povides the analyst with significant infomation about the stochastic behavio of the system In paticula, it can be used to compute the pobability of deadline misses fo each tas The deadline miss pobability DMP τi of tas τ i can be computed fom its esponse time PF as follows: DMP τi = P{R τi >D i } = 1 P{R τi D i } 4 Stochastic analysis 41 Oveview To compute the esponse time PF of each tas, we have to now the esponse time PFs of the jobs belonging to it Howeve, since the numbe of jobs geneated by a tas may be infinite, it is not possible to conside all of them fo computation of its esponse time PF To addess this poblem, we obseve that the aival patten of jobs within a hypepeiod is epeated fo all the othe hypepeiods Thus, if some stochastic egulaity is found at the hypepeiod level, we can estict ou analysis to a single hypepeiod, and say that the deived job esponse time PFs ae applicable fo othe hypepeiods In this case, the esponse time PF f Rτi ( ) of tas τ i is epesented by the aveage of the esponse time PFs of all the jobs fom the tas in the hypepeiod That is, f Rτi () = 1 m i m f R () (1) i j=1 whee m i = T /T i is the numbe of jobs fom τ i eleased in a hypepeiod of length T To addess stochastic egulaity in hypepeiods, we fist define the P-level baclog obseved at time t as the sum of the emaining execution times of all the jobs that have

4 pioities highe than o equal to P and ae not completed up to the time t This andom quantity is denoted by W P t Then, we focus on the P-level baclog obseved at the beginning of each hypepeiod, denoted by B P = W P T, and investigate the stochastic pocess defined as the sequence of andom vaiables {B P 1,B P 2,,B P,} We pove that this stochastic pocess is a Maov chain In addition, a stationay distibution of the P-level baclog B P exists as long as the stability condition wheeby the aveage system utilization is less than one is met By deiving the Maov matix that gives the tansition pobability P{(B P = t 1 ) (B P +1 = t 2 ) } fo any two states t 1 and t 2, we compute the exact stationay P-level baclog PF f B P ( ), obseved at the beginning of the hypepeiod * Once the stationay P-level baclog PF f B P( ) is given, the stationay P-level baclog PF obseved at any time within the hypepeiod can easily be calculated using the method explained in [5, 8] Basically, by using two simple opeations called convolution and shining, the P-level baclog PF f ( ) at any time W P t t (> t) can be calculated fom the P-level baclog PF f W P( ) at time t Fo example, t let us assume a simple scenaio in which a job with a pioity highe than o equal to P is eleased at time t and thee is no futhe elease of jobs with a pioity highe than o equal to P in the inteval [t, t ) In this case, the P-level baclog PF obseved immediately afte the elease of the job, ie, f W P ( ), is obtained by pefoming convolution between t+ f W P( ) and the execution time PF of the job Then, the t P-level baclog PF f W P t ( ) at time t is obtained by shining f W P t+ ( ), that is, shifting f W P t+ ( ) to the left (t t) units and accumulating in the oigin all the pobability values defined fo the non-positive time values These opeations ae gaphically shown though an example in Figue 1 Theefoe, by iteatively applying convolution and shining to the stationay P-level baclog PF obseved at the beginning of the hypepeiod, we can compute the stationay baclog PFs fo all the jobs with pioity P, obseved at thei elease times In ode to compute the esponse time of a job, it is necessay to now the job-level baclog, defined as the baclog due to jobs with pioities highe than o equal to the pioity of that job, obseved at a given time (usually the elease time of the job) It is clea that, unde a fixed-pioity scheduling policy, the job-level baclog coincides with the P-level baclog, P being the pioity of the job unde consideation Howeve, fo dynamic-pioity policies such as EDF the method fo obtaining the job-level baclog is diffeent In Section 44 we will deal with the geneal case * It is possible to define a diffeent hypepeiod length fo each pioity level P by computing the LCM only fo tass with pioity highe than o equal to P This would educe the computational cost of the method, but we will not use this appoach in the text, in the inteest of claity 1/9 f W P t 3/9 2/9 1/18 3/18 1/18 1/18 f C w 1/3 1/3 1/3 (a) At time t, just befoe the elease of the job (the execution time PF fo that job is shown in the box) f W P f t C 10/ t t 1/27 3/27 6/27 11/545/27 5/54 2/27 1/54 1/54 1/54 1/54 (b) At time t, just afte the elease of the job (convolution) f W P t 11/54 5/27 5/54 2/27 1/54 1/54 1/54 1/ (c) At time t = t + 6 (shining) Figue 1 Example of the P-level baclog PF at two diffeent times, sepaated by 6 units of time Fo now it is sufficient to now that the job-level baclog can be obtained fom the system-level baclog, defined as the baclog due to all the jobs eleased befoe a given time (this can be egaded as a 0-level baclog) In the geneal case, then, the job-level baclog is diffeent fom the P-level baclog, so we need to intoduce a new notation to diffeentiate between them We will epesent by V the job-level baclog of the job Γ pesent at its elease time Note that in the paticula case of fixed-pioity tass, (such as RM) we can say that V = W P i λ, P i being the pioity of the tas τ i to which the job belongs Afte the job-level baclog PFs of all the jobs in the hypepeiod have been computed, we compute thei esponse time PFs Fo each job Γ, the esponse time PF can easily be calculated, since the esponse time R is defined by the following equation R = V + C i + C (2) Γ,l H whee H is the set of all the jobs that may peempt Γ, ie, w w

5 the set of jobs eleased afte time λ with a pioity highe than that of job Γ Thus, the stationay esponse time PF f R ( ) of Γ can be computed fom the stationay job-level baclog PF f V ( ) and the execution time PF f Ci ( ) of Γ and the execution time PFs of the highe-pioity jobs that peempt Γ The esponse time PF is computed as follows Let Γ be the -th job in H (we assume H to be odeed by the elease times), and λ the time that has elapsed between the elease of Γ and that of Γ Fist, we calculate the esponse time PF not consideing any possible peemptions by the highepioity jobs by convolving the job-level baclog PF f V ( ) and the execution time PF f Ci ( ) of the job Γ Then, iteatively, we calculate the esponse time PF that eflects all possible peemptions by Γ 1, Γ 2,, Γ, by convolving the execution time PFs of Γ s, in tun ( = 1,2,3,), into the esponse time PF of Γ At each step, say, the convolution is applied to the tail pat of the esponse time PF being calculated, which is defined in the ange (λ, ), since the highe-pioity job Γ may affect Γ only if Γ executes up to the elease time of Γ In othe wods, at step, the esponse time PF of Γ is calculated by (1) splitting the esponse time PF obtained at step 1 into the tail pat defined fo the ange (λ, ) and the emaining head pat, (2) convolving the tail pat and the execution time PF C of Γ, and finally (3) meging the head pat and the new tail pat esulting fom the convolution Figue 2 shows this pocess gaphically, fo the -th highe-pioity job This pocess can also be moe fomally expessed though the following notation: let f I ( ) be a patial PF defined as the pat of the PF f ( ) which taes values in the inteval I, as follows { f I f () if I () = 0 othewise Also, let R <> be a andom vaiable descibing the esponse time of job Γ that eflects all possible peemptions by the highe-pioity jobs Γ 1,Γ 2,, Γ that can peempt Γ (λ 1 λ 2 λ ) Then the esponse time PF f R <> can be calculated fom the esponse time PF f R < 1> the execution time PF f C ( ) of Γ as follows: f R <> () = f [0,λ ] R i, < 1> j () + ( f (λ, ) R < 1> f C )() ( ) ( ) and Howeve, in ode to obtain the deadline miss pobability fo each job Γ, it is not necessay to calculate the complete esponse time PF, because P{R >D i } = 1 P{R D i }, and the pobability P{R D i } can be computed fom the pat of the esponse time PF defined in the ange [0, D i ] If, at step, we find that D i λ in the calculation of the esponse time PF, the calculation pocess can be stopped, f [0,λ ] R < 1> 1/4 1/4 1/8 f R < 1> 1/4 1/ /8 1/4 1/ f (λ, ) R < 1> 1/4 λ 1/ (a) Splitting the esponse time PF obtained in step 1 f [0,λ ] R < 1> 1/4 1/4 1/8 1/ f R <> 1/4 1/4 f (λ, ) R < 1> 1/8 1/4 f C (b) Constuction of the esponse time PF fo step Figue 2 Response time PF calculation and the exact deadline miss pobability can be computed Moeove, the calculation pocess can also be stopped when the esponse time PF obtained at some step eveals that the job Γ has a null pobability of being unning up to the elease time of the next highe-pioity job Γ +1 (ie, f (λ, ) ( ) is zeo) R <> In the following subsection, we will descibe how to compute the stationay P-level (o system-level) baclog PF obseved at the beginning of the hypepeiod, and then explain how to deal with dynamic-pioity systems such as EDF in the job-level baclog PF computation In the following desciption, note that, wheneve we omit the supescipt P in B P, a pioity level P is implicit in the notation 42 Maovian modeling and stability The baclog at the beginning of the -th hypepeiod is a andom vaiable, whose distibution, in geneal, is diffeent fo each So, the sequence of andom vaiables {B } is a andom pocess, which we will call the baclog pocess We will show that this pocess is a Maov chain The PF of B can be expessed in tems of the PF of

6 B 1 using conditional pobabilities: P{B = j} = P{B 1 =i}p{b = j B 1 =i} (3) i Moeove, in ou case, all hypepeiods eceive the same sequence of jobs, in the same ode, so the conditional pobabilities P{B = j B 1 =i} do not depend on That is to say, P{B = j B 1 =i} = P{B 1 = j B 0 =i} Then, the PF of B depends only on the PF of B 1, and not on those of B 2,B 3, This is the memoyless popety of a Maov chain In this way, we can wite Equation (3) in matix fom: b = Pb 1 (4) whee b is the PF of B in the fom of an infinite column vecto [ P{B =0},P{B =1}, ], and P is the infinite Maov matix, whose elements we will call b j (i), defined as: P() = b j (i) = P{B 1 =i B 0 = j} Unde stability conditions, the Maov chain is egodic, and in this case the PFs of the andom vaiables {B } convege towads a single stationay distibution as appoaches infinity The stability condition depends on the aveage system utilization Ū, which is defined as: Ū = N i=1 C i T i (5) C i being the expected value of the execution time fo the tas τ i Wheneve Ū < 1, the system is stable, and the sequence of andom vaiables B s conveges towads a single andom vaiable, whose PF is a vecto usually denoted by π This vecto can be obtained as the unique solution of the equation π = Pπ, with the additional estiction of π i = 1 The condition Ū < 1 assues the stability and convegence of the system, and thus the existence and uniqueness of π This fact can be poved using Maov theoy and difting conditions, but it is simple to appoximate ou model by a queue with a single seve A well-nown esult of queueing theoy (see fo example [10]) is that the system is stable if ρ < 1, ρ being a paamete of the queueing model, which can be shown to be equivalent to ou Ū In ode to obtain the steady state PF of the baclog, the equation π = Pπ needs to be solved This, in geneal, is not an easy tas, because the matix P is infinite Howeve, due to the natue of ou model, the matix has a egula stuctue, which will allow us to povide a geneal method to obtain an analytical expession fo π Let us show the egulaity pesent in P Each column P(, j) in the matix P is the PF of the baclog at the end of the fist hypepeiod, if the initial baclog was j Then, each column j in P can be calculated by setting the initial baclog equal to j, and using the algoithm pesented in Section 41 to obtain the PF of the baclog at the end of the fist hypepeiod Fo each diffeent initial baclog value, we will expect to obtain a diffeent PF fo the final baclog, and so we will expect each column in P to be diffeent Howeve, thee exists an initial baclog value, which we will call, fom which the PF of the final baclog is always the same, only shifted one position to the ight This means that afte column, all the columns in P epeat the same values, only shifted down by one position The geneal fom of matix P is thus: P = b 0 (0) b 1 (0) b 2 (0) b (0) b 0 (1) b 1 (1) b 2 (1) b (1) b (0) 0 0 b 0 (2) b 1 (2) b 2 (2) b (2) b (1) b (0) 0 b (2) b (1) b (2) b 0 (m ) b 1 (m ) b 2 (m ) b (m ) b (m ) b (m ) In paticula, is the maximum possible value of the idle time in any hypepeiod It can be calculated as: = T +W min Ci min (6) i whee W min is the baclog at the end of the fist hypepeiod in which all the jobs ae eleased, assuming that the initial baclog is zeo and all the jobs equie thei minimum execution time W min is usually zeo, unless most of the wo is concentated at the end of the hypepeiod Then, the -th column of the Maov matix P epesents the PF of the final baclog in the case of the initial baclog being equal to the maximum possible idle time In this case, the whole hypepeiod is busy, and so column of P will be the convolution of the execution time PFs of all the jobs eleased in the hypepeiod, shifted (T ) units to the left An analogous easoning applies when the initial baclog is ( + 1), ( + 2), and so on, because in all these cases the whole hypepeiod is a busy peiod This is the eason fo the egulaity of matix P fom column onwads Moeove, m is the index of the last non-zeo element of column in P, and thus epesents the maximum possible baclog which could appea when the initial baclog is If the initial baclog is less than, the final baclog will be less than m, and this is why the fist columns have zeos fom ow m onwads

7 43 Solution appoaches Once the Maov matix P has been deived, we can compute the stationay P-level (o system-level) baclog PF obseved at the beginning of the hypepeiod with eithe an analytical o a numeical method The analytical method gives the exact solution fo the stationay P-level baclog PF while the numeical method gives appoximated solutions In the analytical method, we diffeentiate between the case whee U max < 1 fom geneal cases, since in this special case the P-level baclog PF is equal fo all hypepeiods, so thee is no need to pefom the Maov pocess modeling descibed in the pevious subsection In this case, we compute the exact solution without deiving the Maov matix Fo the geneal case, on the othe hand, we compute the exact solution by deiving a finite set of equations that can completely descibe the infinite stationay baclog distibution To educe the computational oveheads equied to compute the exact solution, we also intoduce some appoximation methods, which mae a tade-off between analysis accuacy and computational oveheads 431 Exact solution when U max 1 We will define the maximum system utilization U max, as the total utilization of the system calculated using the wostcase execution times of the jobs: U max = N i=1 C max i T i (7) In the paticula case whee U max < 1, the maximum amount of wo geneated in a hypepeiod will not exceed the hypepeiod length In this case, the baclog obseved at the end of the hypepeiod cannot incease without bounds In fact, it can be poved that the baclog pesent at the end of the fist hypepeiod in which all the tass wee eleased at least once, will be epeated at the end of any subsequent hypepeiod In paticula, if at time t = 0 all the tass ae eleased in-phase, then no baclog will be pesent at the end of the fist hypepeiod, and thus all subsequent hypepeiods will stat with a zeo initial baclog In these paticula cases, the stationay PF of the baclog obseved at the end of the hypepeiods can be obtained by simple calculation of the baclog PF at the end of the fist hypepeiod in which all the tass ae eleased at least once Moeove, the baclog PF obtained in this case has a finite numbe of points Note that when U max < 1, the system can be analyzed using classical esponse time analysis Using the wost-case execution times of all the tass, the wost-case esponse time can be calculated, and compaing this esponse time with each deadline, the feasibility of the system can be detemined Howeve, if a pofile of the execution times is Tas T i Pioity C i τ 1 4 High {1,2} with equal pobability τ 2 6 Low {2,3,4} with pob 02, 03 and 05 Table 1 A system example with U max > 1 used instead of a single wost-case value, the pobability of deadline misses can be found, and, if this pobability is small enough fo its application, the system could still be feasible An example of the benefits of a statistical appoach to the poblem is pesented in [5] 432 Exact solution fo the geneal case Taing advantage of the egula stuctue of P, we pesent a method fo finding the complete stationay PF of the baclog, denoted by π Since this distibution has an infinite numbe of points, what we will obtain is the exact value of some stating points, and then the expession in closed fom fo the est of the points The geneal fom of π consists of a set of initial points, whose values depend on the paametes of the system, followed by an infinite tail which appoaches zeo in a exponential way Actually, the tail is a sum of exponential functions, whose paametes depend only on convolution of all the execution times fo all the jobs in the system We call this solution analytical in the sense that a closed fom of the solution is found Howeve, to obtain this solution, the method equies the oots of a polynomial to be found, and some numeical methods will be equied nevetheless In ode to mae the method moe undestandable, we will intoduce an example system and solve it by hand, giving indications about how the example can be genealized, instead of pesenting a fomal development of the geneal solution Let us conside the system shown in Table 1 Fo this system, the hypepeiod is 12 Tas τ 1 is eleased thee times, and τ 2 twice within the hypepeiod We will obtain the stationay distibution fo the lowpioity-level baclog, ie, the PF of the baclog pesent at the beginning of any hypepeiod in the distant futue To do so, we need to constuct the Maov matix P This is done by computing the PF of the baclog at time T = 12, fo diffeent initial baclogs 0,1,2, up to Each of these PFs will be a column in P In this example, the maximum possible idle time in a peiod,, is 5 The esulting Maov

8 matix is: P = m As expected, the numbes in column ae epeated in the following columns, shifted one position down The index of the last non-zeo element in the -th column is what we called m in Section 42, and its value is 7 in this example Matix P always pesents anothe egulaity: fom ow m + 1 onwads, the values in each ow ae the same, only shifted one position to the ight Moeove, these values ae the same as the coefficients of the -th column In effect, fo this example this ind of egulaity stats fom ow 2, but in geneal it is only guaanteed fo ow (m + 1) Equation π = Pπ can be developed and gives ise to an infinite set of equations with infinite unnowns, which ae the component π i s of π This infinite set of equations can be divided into two subsets: a subset compising the fist (m +1) equations, and a second subset made up of the emaining equations The last subset defines a ecuence elation between the components of π In ou example, the fist (m + 1) equations ae: π 0 = 08375π π π π π π 5 π 7 = π π π π π π π π 12 This maes a system of 8 equations with 13 unnowns In geneal, thee will be (m + 1) equations with (m + + 1) unnowns The emaining equations, povided by ows m + 1 and following, have a geneal fom, due to thei egulaity The geneal fom, fo j m + 1 = 8, is π j = π j π j π j π j π j π j π j π j+5 Finding π j+5 we obtain the following ecuence elationship, which holds fo j 8: π j+5 = 625π j π j π j 535π j π j π j+3 6π j+4 (8) The ecuence elation of Equation (8) can be put in matix fom: whee Q j+1 = AQ j j 8 Q j = (π j 2,π j 1,π j,π j+1,π j+2,π j+3,π j+4 ) A = The elements of the last ow of matix A ae the coefficients of Equation (8) Note that they ae easily obtained fom = the -th column of P Using the matix fom, it is easy to see that Q 9 = AQ 8, Q 10 = A 2 Q 8, and in geneal, fo n 8, Q n = A n 8 Q 8 Since A is diagonalizable, we can wite A = V 1 DV, whee V 1 is the matix whose columns ae v 1,v 2,,v 7, the eigenvectos of A; D is the diagonal matix whose elements ae λ 1,λ 2,,λ 7, the eigenvalues of A, and V is the invese matix of V 1 Once diagonalized, the opeation A n 8 is easy to compute, leading to the equation: Q n = V 1 D n 8 VQ 8 whee = C 1 λ1 n 8 v 1 +C 2 λ2 n 8 v 2 +C 3 λ3 n 8 v 3 +C 4 λ4 n 8 v 4 (9) +C 5 λ5 n 8 v 5 +C 6 λ6 n 8 v 6 +C 7 λ7 n 8 v 7 (C 1 C 2 C 3 C 4 C 5 C 6 C 7 ) = V (π 6 π 7 π 8 π 9 π 10 π 11 π 12 ) The chaacteistic polynomial of A is vey easy to find, even without obtaining A, diectly fom the elements of column in P It can be poved that its geneal fom is: f (λ) = ( m b (i)λ m i ) λ m = 0 (10) i=0 Note that the degee of this polynomial is m, which can be poved to be equal to T (U max U min ) +W min In ou example, the oots of this polynomial ae λ 1 = ( i), λ 2 = ( i), λ 3 = ( i), λ 4 = ( i), λ 5 = 1, λ 6 = and λ 7 = Some of these have modulus geate than o equal to one Looing at Equation (9) it can be seen that, if all C i 0, then π n as n, due to these eigenvalues, and the sum of the components of π will be infinite This is not the case, because the Maov

9 chain is positive and thus the stationay solution has to be summable This implies that the coefficients C 1, C 2, C 3, C 4 and C 5 which multiply these eigenvalues in Equation (9), must be equal to zeo This condition gives ise to five new equations In geneal, in the case of stability, the polynomial in Equation (10) has oots with modulus geate than o equal to 1, and so it always povides additional equations This fact can be poved by applying Rouche s Theoem To summaize, we now have a linea system with 13 equations, 8 fom the fist m ows of P and 5 fom the condition of some C i s being zeo, and 13 unnowns (π 0,,π 12 ) Nevetheless, the eade can chec that the equation deived fom C 5 = 0 is a linea combination of the othes, and can be emoved In the geneal case, the equation deived fom the eigenvalue 1, is always a linea combination of the othes Following the descibed method, we will end up with a system of (m + ) equations and (m + + 1) unnowns Since the numbe of unnowns is one moe than the numbe of equations, we can put each of the fist 12 components as a linea function of the fist component π 0 We will not wite these expessions hee fo the sae of bevity The coefficient C i s ae also a linea function of π 0 In this example we obtain C 6 = π 0, C 7 = π 0 (the emaining C i s ae zeo, guaanteeing the summability of π) If we use these values in Equation (9) we find the following geneal expession fo any component of π, valid fo n 8 Q n = π 0 v 6 (03474) n π 0 v 7 ( 01325) n 8 (11) Note that v 6 and v 7 ae two eigenvectos, which wee calculated diectly fom matix A So, the only unnown in the above equation is π 0 Fo any value of π 0, we obtain a complete vecto π, which has the popety π = Pπ Howeve, only one of these possible vectos is a PF (has a sum equal to 1) So, as a final condition, we impose i=0 π i = 1, and fom this we can detemine the equied value fo π 0 This equation is easy to solve, despite the infinite summation, because the expession fo π i is a convegent sum of exponentials, fo i > 6 Solving this sum and equating it to 1, we find the value of π 0 fo this example, which is π 0 = Fom this, all the emaining values of π can be computed The fist 13 components ae obtained diectly fom the system of equations, and thei values ae shown in the last column of Table 2, ounded to the sixth decimal The emaining components ae calculated fom the fomula given in Equation (11), which, afte substituting the values of π 0, v 6 and v 7, and taing one of the components of vecto Q n, leads to: π n = 10 4( n ( 01325) n 6) valid fo n > 6 B0 B1 B2 B3 B5 B10 B20 B Table 2 Baclog PF convegence 433 Appoximations Tuncation of matix P One possible appoximation technique to compute the stationay system-level baclog PF is to tuncate the Maov matix P to a sufficiently lage squae matix P, which was fist intoduced in [1] Assuming that the infinite stationay distibution π can be modeled with a finite vecto π = [π 0,π 1,π 2,,π n], we can lead to the following equation fom π = Pπ: π = P π whee P is an (n+1)-by-(n+1) matix consisting of the component P() s (0 n) of the Maov matix P The esulting equation is an eigenvecto poblem, fom which we can calculate the appoximated solution π with a numeical method Among the calculated eigenvectos, we can choose as the appoximated solution an eigenvecto whose coesponding eigenvalue is equal to o sufficiently close to 1 In ode to obtain a good appoximation of the stationay distibution π, the tuncation point n should be inceased as much as possible, which maes the eigenvalue convege to 1 Note that, by choosing an appopiate tuncation point, we can achieve a tade-off between analysis accuacy and the computational oveheads equied to solve the coesponding eigenvecto poblem The choice of a convenient tuncation point is an open issue Iteative appoximation Anothe appoximation technique to obtain the stationay system-level baclog PF, which does not equie deivation of the Maov matix P, is the simple iteation of the algoithm which computes the PF of the baclog at the end of the hypepeiod Fo the example pesented in Section 432, the esults of successive iteations ae shown in Table 2 It can be seen that each point of the PF conveges towads the analytical solution (given in the last column) It can be poved that the convegence is geometically egodic Howeve, it is not nown in advance how many iteations will be necessay to mae the baclog PF close enough to the stationay distibution It is clea that the ate of convegence depends on how close Ū is to 1, becoming slowe as Ū appoaches 1 Howeve, we

10 have not yet found a bound, in tems of Ū, of the numbe of iteations equied by the iteative method Anothe impotant point to conside is the effect of intoducing zeo as the initial baclog fo the system In the steady state the initial baclog is a andom vaiable which can tae non-zeo values Thus, the esponse times in the steady state will be wose than those calculated fo the fist hypepeiod Indeed, they will be wose than that calculated fo any hypepeiod Using zeo as the initial baclog will lead to optimistic pobabilities of deadline misses, so design decisions based on the iteative method should be taen caefully τ 1 τ 2 Γ 1,1 0 Γ 2,1 0 Γ 1,2 Γ 1, Γ 2,2 50 Γ 1,4 90 Γ 2,3 Γ 1, Figue 3 A tas set example Hypepeiod Γ 1,3 44 Extension to dynamic-pioity systems Γ 1,1 Γ 2,1 Γ 1,2 Γ 2,2 Γ 1,4 Γ 2,3 Γ 1,5 Γ 1,1 Γ 2,1 Although dynamic-pioity systems seem consideably diffeent fom fixed-pioity systems, thee exists one similaity The similaity is that, in a hypepeiod, thee always exists at least one job that always taes the system-level baclog W λ obseved at its elease time λ as its joblevel baclog V, ie, V = W λ Such a job is called a gound job, and has a lowe pioity than all the jobs eleased befoe its elease time λ Thus, once the stationay system-level baclog PF f B 0( ) obseved at the beginning of the hypepeiod is given, the stationay job-level baclog PF f V ( ) of evey gound job Γ can be calculated as explained in Section 41, by iteatively applying convolution and shining to f B 0( ) fo each job eleased between the beginning of the hypepeiod and the elease time λ of the job Γ (fo a fixed-pioity system, this statement means that the stationay job-level baclog PF of evey job with pioity P, which is consideed a gound job at the pioity level P, can be calculated fom f B P( )) Theefoe, the diffeence between dynamic-pioity and fixed-pioity systems lies in how to compute the job-level baclog PFs of non-gound jobs Fo fixed-pioity systems, this poblem is solved by consideing the highe pioity level that each non-gound job belongs to Since any job classified as a non-gound job at a lowe pioity level is bound to become a gound job at the pioity level that is equal to the pioity of the job, its job-level baclog PF can be calculated fom the stationay system-level baclog PF obtained at that pioity level Howeve, fo dynamicpioity systems, we can avoid such an iteative analysis, and compute the job-level baclog PF of the non-gound job fom that of a peceding gound job The peceding gound job is the last gound job that is eleased befoe the non-gound job and has a highe pioity, which is called the base gound job fo the non-gound job Since the job-level baclog PF of any gound job can be calculated fom the system-level baclog PF f B 0( ) obseved at the beginning of the hypepeiod, this means that the job-level baclog PFs of all the jobs including the non-gound jobs in the hy- W λ1,1 W λ2,1 W λ1,2 W λ2,2 W λ1,3 W λ1,4 W λ2,3 W λ1,5 W λ1,1 W λ2,1 Figue 4 Gound jobs and non-gound jobs in a dynamic-pioity system pepeiod can be calculated fom the system-level baclog PF f B 0( ) Fo example, conside the tas set shown in Figue 3 This tas set consists of two tass τ 1 and τ 2 The peiods of τ 1 and τ 2 ae 30 and 50, espectively, and the elative deadline D i of each tas is equal to T i The phase Φ i s of both tass ae 0 In this example, thee ae seven gound jobs Γ 1,1, Γ 2,1, Γ 1,2, Γ 2,2, Γ 1,4, Γ 2,3, and Γ 1,5, and one non-gound job Γ 1,3, as shown in Figue 4 If the stationay system-level baclog PF f B 0( ) obseved at the beginning of the hype-peiod is given, the stationay job-level baclog PFs of all the gound jobs can be calculated by applying convolution and shining to f B 0( ) That is, f V1,1 (w) = f Wλ1,1 (w) = f B 0(w), f V2,1 (w) = f Wλ2,1 (w) = ( f C1,1 f V1,1 )(w), and so on Howeve, fo the non-gound job Γ 1,3, the stationay job-level baclog PF f V1,3 ( ) is not equal to the stationay system-level baclog PF f Wλ1,3 ( ) obseved at its elease time λ 1,3 The stationay system-level baclog PF f Wλ1,3 ( ) includes the contibution fom a lowe-pioity job Γ 2,2 that must be excluded fo the job-level baclog PF f V1,3 ( ) In this case, we obseve that the job-level baclog PF f V1,3 ( ) of the non-gound job Γ 1,3 can be calculated fom the system-level (o job-level) baclog PF of the gound job Γ 1,2 that pecedes Γ 1,3 and has a highe pioity than Γ 1,3 (ie, Γ 1,2 is the base gound job of the non-gound job Γ 1,3 ) It can be seen that the job-level baclog V 1,3 of Γ 1,3 depends only on the system-level baclog W λ1,2 of Γ 1,2 and the execution time of Γ 1,2 Thus, we calculate the stationay job-level baclog PF f V1,3 ( ) of the non-gound job Γ 1,3 fom that of its base gound job Γ 1,2 by applying convolution and shining to the system-level baclog

11 PF f Wλ1,2 ( ) of the base gound job Γ 1,2 while ignoing the lowe-pioity gound job Γ 2,2 This appoach can be genealized to compute the stationay job-level baclog PF of an abitay non-gound job Γ as follows Theoem 1 Fo any non-gound job Γ, the stationay job-level baclog PF f V ( ) can be computed fom the stationay system-level baclog PF f Wλ,l ( ) of its base gound job, (ie, the last gound job Γ,l that pecedes Γ and has a pioity highe than that of Γ ) by iteatively applying convolution and shining to f Wλ,l ( ) only fo all the nongound jobs that ae eleased in the time inteval (λ,l, λ ] and have pioities highe than that of Γ Note that, in a system scheduled by EDF, we can find the base gound job fo any non-gound job, since thee always exists a gound job that has an ealie deadline than the non-gound job Fo a geneal dynamic-pioity system, we developed a systematic appoach to finding the base gound job fo any non-gound job and computed the joblevel baclog PF of the non-gound job in a systematic way, but we do not include it hee fo the sae of bevity Fo moe infomation, efe to [8] 5 Expeimental esults In this section, we give expeimental esults obtained by the poposed stochastic analysis Fist, we compae the esults obtained fom the poposed analysis method (both the exact solution and the appoximated solution based on the Maov matix tuncation) with those obtained by Stochastic Time Demand Analysis (STDA) [7, 6] using the example given in [6] Secondly, we compae the esults obtained using the tuncation method with those obtained fom simulations while vaying the aveage system utilization and the numbe of tass 51 Compaison with STDA To compae the poposed analysis method with STDA, we used the same tas sets given in [6], which ae shown in Table 3 All thee tas sets in Table 3 ae assumed to be scheduled by RM and have the same tas paametes except fo the execution time PFs of tass given by a unifom PF that anges fom Ci min to Ci max (Ci max fo tas τ i Fo the tas sets, the simulation esults and the analytical esults given by STDA ae copied fom [6] and compaed with the esults given by the poposed stochastic analysis, using the exact solution and the appoximation obtained by the Maov matix tuncation The esults in Table 4 show that thee is a significant diffeence between the deadline miss pobability given by STDA and the one obtained by the poposed method, which is almost identical to the simulation esult This esults fom the citical instant = 2 C i C min i ) assumption made in STDA The negative impact on the degadation of analysis accuacy inceases with an incease in the (maximum) system utilization In the case of tas τ 2 in S 2, the deadline miss pobability given by STDA is moe than six times the one given by the poposed method S τ i T i D i Ci min C i Ci max τ S τ τ S τ τ S τ Table 3 Tas sets in [6] Ū U max S τ i simulation STDA Exact analysis Appoximation S 1 τ ± τ ± S 2 τ ± τ ± S 3 τ ± τ ± Table 4 Results fo the tas sets in Table 3 52 Sensitivity to the aveage system utilization and the numbe of tass To assess the sensitivity of analysis accuacy to the aveage system utilization, we pefomed expeiments with tas sets consisting of thee and five tass while vaying the aveage system utilization fom 070 to 096 Fo each expeiment, we also consideed both RM and EDF scheduling to investigate the effect of the scheduling policy on analysis accuacy The esults (which we do not detail hee due to space limit), showed that the appoximated solutions fom the Maov matix tuncation is almost equal to the simulation esults fo most cases In the case of the tas sets consisting of thee tass, the eo between the appoximated solution and the simulation esult of the deadline miss pobability fo a tas anges fom 0 to 0006 fo RM, and fom 0 to 0008 fo EDF In the case of the tas sets consisting of five tass, the eo anges fom 0 to 0015 fo RM, and fom 0 to 0028 fo EDF Accoding to the esults, the eo inceases as the aveage system utilization inceases Fo example, in the case whee the tas set consisting of five tass ae scheduled by EDF, the eo was 0, 0002, 0016, 0028 when the aveage system utilization was 07, 08, 09, 096, espectively The inaccuacy is due to the Maov matix tuncation, which convets the Maov matix with an infinite dimension to one with a finite dimension When the stationay system-level baclog PF we ae tying to obtain has a non-negligible tail distibution beyond the tuncation point, the Maov matix tuncation inevitably intoduces appoximation eos in the analysis To educe these eos, it is necessay to peseve the tail distibution as much