Advanced Hierarchical Event-Stream Model

Transcription

1 Advanced Hierarchica Event-Stream Mode Karsten Abers, Frank Bodmann and Frank Somka Embedded Systems / Rea-ime Systems, Um University name.surname}@uni-um.de Abstract Anayzing future distributed rea-time systems, automotive and avionic systems, is requiring compositiona hard rea-time anaysis techniques. We known estabished techniques as SymA/S and the rea-time cacuus are candidates soving the mentioned probem. However both techniques use quite simpe event modes. SymA/S is based on discrete events the rea-time cacuus on continuous functions. Such simpe modes has been choosen because of the computationa compexity of the considered mathematica operations required for rea-time anaysis. Advances in approximation techniques are aowing the consideration of more expressive descriptions of events. n this paper such a new expressive event mode and its anaysis agorithm are described. t integrates the modes of both techniques. t is aso possibe in this modue to integrate an approximative rea-time anaysis into the event mode. his aows to propagate the approximation through the anaysis of a distributed system eading to a much more efficient anaysis. 1. MOVAON he modue-based design processes make it possibe to hande the compexity in software and hardware design. Systems are buid using a set of cosed modues. hese modues can be designed and deveoped separatey. Modues have ony designated interfaces and connections to other modues of their set. he purpose of moduarisation is to spit the chaenging job of designing the whoe system into mutipe smaer jobs, aowing the reuse of modues in different designs or to incude P components of third-party vendors. Every modue-based design concept requires a we defined interface-concept for connecting the modues. Deveoping reatime systems requires for this interface-concept to cover aso the rea-time aspects of the modues. A concept for the reatime anaysis is required to hande the modues separaty and aows a propagation of the rea-time anaysis resuts through the system. t is necessary to propagate the resuts of the reatime anaysis of the different modues in an abstract way. he goba anaysis is buid by connecting the oca anayses of the singe modues. herefore it is essientia to have an expressive and efficient interface describing the infuence in timing of one modue to the next modue. One aspect of this interface is the timing description of events which are produced by one modue to trigger the next foowing modue. Another aspect is the computation capacity that remains for ower priority modues eft over by the higher priority ones. Consider for exampe a network packet processor as shown in figure 1. he singe packages are processed by chains of tasks τ which can be ocated on different processing eements P. he processing eements P can be processors, dedicated hardware or the communication network. he events triggering the different tasks are equa to the packages P 1 τ 1 τ 2 τ 3 s p 3 S 1 S 2 S P 2 Figure 1. τ 4 τ 5 s p 2 S 4 S P 3 τ 6 τ 7 τ 8 s p 3 S 6 S 7 S Network processor exampe fowing through the network. Each processing unit P uses a fixed-priority scheduing and the task τ on each unit are sorted by their priority eve. Each task τ has, as avaiabe capacity, the capacity S eft over by the tasks τ with a higher priority ocated on the same processing unit. he purpose of this paper is to provide an efficient and fexibe approach for the rea-time anaysis of such a moduarized system. herefore is a powerfu and sufficient event mode for describing the different time interfaces for the different aspects is necessary. 2. RELAED WORK he most advanced approach for the rea-time anaysis of such a moduare network is the rea-time cacuus by hiee et a. [4], [13]. t is based on the network cacuus approach defined by Cruz [5] and Parekh and Gaager [9]. he event pattern is modeed by an sub-additive upper and super-additive ower arriva curve α u f (Δt) and α f (Δt) deivering for every Δt the maximum number of events or the minimum, respectivy. he service curves βr u (Δt) and βr (Δt) mode the upper and ower bound of the computationa requirements which can be handed by the resource during Δt. he rea-time cacuus provides equations to cacuate the outgoing arriva and service curves out of the incoming curves of a task. o evauate the modification equations independenty from each other, a good finit description for the curves is needed. he compexity of the equations depends directy on the compexity of this description. n [8] and [4] an approximation with a fixed degree of exactness for the arriva and service curves was proposed in which each curve is described by three straight ine segments. One segment describes the initia offset or arriva time, one an initia burst and one the ong time rate.

2 Δt,a, k θ =(,a) ˆ ˆθ =(,a,,g, ˆ ˆθ ) s ϒ(Δt, ˆ), Ψ(Δt, ˆ) (Δt, ˆ), B interva period, offset, imitation number of test intervas event stream event eement hierachica event stream hierachica event stream eement separation point event bound function, demand bound function interva bound function, busy period Events Figure 2. abe LS OF SYMBOLS Exampe Event Stream As outined in [3] this approach is too simpified to be suitabe for compex systems. No suitabe description for the function is known so far. n this paper we wi propose a mode for the curves having a seectabe approximation error. A tradeoff between this degree of accuracy and the necessary effort for the anaysis becomes possibe. SymA/S [11],[12] is another approach for the moduarized rea-time anaysis. he idea was to provide a set of interfaces which can connect different event modes. herefore the different modues can use different event modes for anaysis. Unfortunaty, the event modes for which interfaces are provided are quite simpe. n [11] an event mode covering a these modes was described. he probem of these modes is that mutipe bursts or bursts with different minimum separation times cannot be handed. However in [10] a rea-time anaysis probem was formuated, which can t be soved by SymA/S and the reatime cacuus by each technique excusivy. o sove it, it is necessary to integrate the modes of both techniques into one powerfu new mode. he event stream mode proposed by Gresser [7] with its extension the hierachica event stream mode proposed by Abers et a. [1] can mode systems with a kinds of bursts efficienty. he probem is that it can ony mode discrete events and not the continious service function as needed for the rea-time cacuus Event stream mode For the event stream mode a system is described by a set of communicating tasks τ. Each task is assigned to one resource ρ. τ =(ˆ,c,d) is given by the worst-case execution time c τ, the deadine d τ and an event pattern ˆ τ triggering the tasks activations. he key question is to find a good mode for the event pattern ˆ. For rea-time anaysis this mode has to describe the worst-case densities of a possibe event patterns. hey ead to the worst-case demand on computation time. Comparing these worst-case demands with the avaiabe computation time ime j t Figure 3. Exampe event streams ([6]) aows to predict the scheduabiity of a system. he event stream mode gives an efficient genera notation for the event bound function. Definition 1: ([7], [2], [1]) he event bound function ϒ(Δt, ) gives an upper bound on the number of events occuring within any interva Δt. Lemma 1: ([7]) he event bound function is a subadditive function, that means for each interva Δt, ΔJ: ϒ(Δt + Δt,) ϒ(Δt,)+ϒ(Δt,) Proof: he events in Δt + Δt have to occure either in Δt or in Δt. Definition 2: An event stream is a set of event eements θ =(,a) given by a period and an offset a. 1 = (6,0),(6,1),(6,3)} (figure 2) describes three events requiring at east an interva Δt = 3 to occure, two of them have a minimum distance of one time unit. ˆ 1 is repeated with a period of 6. n cases where the worst-case density of events is unknown for a concrete system an upper bound can be used for the event stream. he mode can describe any event sequence. Ony those event sequences for which the condition of sub-additivity hods are vaid event streams. Lemma 2: ([7]) he event bound function for an event sequence and an interva is given by: ϒ(Δt, )= θ Δt a θ Δt aθ θ + 1 Proof: See [2] t is a monotonic non-decreasing function. A arger intervaength cannot ead to a smaer number of events. n figure 3 some exampes for event streams can be found. he first one 5 = (,0)} has a stricty periodic stimuus with a period. he second exampe 6 = (,0), (, j)} shows a periodic stimuus in which the singe events can jitter within a jitter interva of size j. n the third exampe 7 =(,0), (,0), (,0), (,t) } three events occur at the same time and the fourth occurs after a time t. his pattern is repeated with a period of. Event streams can describe a these exampes in an easy and intuitive way. he offset vaue of the first event eement is aways zero as this vaue modes the shortest interva in which one singe event can occur. For the rea-time anaysis for this mode et us first repeate the demand bound function definition for the event streams: Ψ(Δt,Γ)= τ Γ ϒ(Δt d τ, τ )c τ = Δt aθ d τ Γ τ θ τ + 1 c θ τ Δt a θ +d τ

3 Costs Figure 4. ψ (Δ t,τ,κ) ψ(δ t,τ) } cτ }κ c τ Approximated event stream eement anaysis agorithms, it aso aows to propagate the approximation together with the event modes through the distributed system eading to an efficient, fexibe and powerfu anaysis methodoogy for distributed rea-time systems. he new mode can, of course, aso mode the service functions of the reatime cacuus in the same fexibe way and aows therefore the integration of the discrete event mode of SymA/S with the continuous service functions. Let θ be an event eement beonging to the event stream which beongs to the task τ. he demand bound function aows a scheduabiity anaysis for singe processor systems by testing Δt : Ψ(Δt, Γ) C (Δt). Often an ideaized capacity function C with C (Δt)= Δt is assumed. For an efficient anaysis an approximation is necessary 2.2. Approximation of event streams Definition 3: ([2]) Approximated event-bound-function Let k be a chosen number of steps which shoud be considered exacty. Let Δt θ,k = d τ + a θ + k. We ca ϒ ϒ(Δtθ,k,θ)+ c τ (Δt,θ,τ,k)= (Δt Δt) Δt > Δt θ θ,k ϒ(Δt,θ) Δt Δt θ,k the approximated event bound function for task τ. he function is shown in figure 4. he first k events are evauated exacty, the remaining events are approximated using the specific utiization U θ = c τ θ. he interesting point of this θ function is that the error can be bounded to ε θ,k = 1 k and therefore does ony depend on the chooseabe number of steps, and is independent of the concrete vaues of the parameters of the tasks. he compete approximated demand-bound-function is Ψ (Δt,Γ,k)= τ Γ θ τ Ψ (Δt,θ,k) and has the same error. he hierachica event stream mode [1] extends the event stream mode and aows a more efficient description of bursts. n this mode an event eement describes the arriva not for just one periodic event but of a compete set of periodic events. his set of events can be aso modeed by an event sequence having a imitation in the number of events generated by this event sequence. One imit of this mode is that it can ony describe discrete events. For the approximation it woud be appropriate for the mode to be capabe to describe aso the continuous part of the approximated event bound function. 3. CONRBUON n this paper we wi present an event mode covering both, the discrete event mode of SymA/S and the continuous functions of the rea-time cacuus. t makes the eegant and tighter description of event bursts compared to the SymA/S approach possibe and aows a tighter modeing of the continuous function of the rea-time cacuus by integrating an approximation with a chooseabe degree of exactness into the mode. his does not ony ead to more fexibe and simper 4. MODEL We wi define the hierachica event sequence first. he hierarchica event stream is ony a speciaised hierachica event sequence fufiing the condition of sub-additivity and can therefore be described by the same mode. Definition 4: A hierachica event sequence ˆ = ˆθ} consists of a set of hierarchica event eements ˆθ each describing a pattern of events or of demand which is repeated periodicay. he hierarchica event eements are described by: ˆθ =(,a,,g, ˆ) where θ is the period, a θ is the offset, θ is a imitation of the number of events or the amount of demand generated by this eement during one period, G ˆθ and ˆ ˆθ are the time pattern how the events respectivey the demand is generated. he gradient G ˆθ describing a constanty growing set of events, gives the number of events occurring within one time unit. A vaue G ˆθ = 1 means that after one time unit one event has occured, after two time units two events and so on. he gradient aows modeing approximated event streams as we as modeing the capacity of resources. Both cases can be described by a number of events which occurs respectivey can be processed within one time unit. ˆ ˆθ is again a hierarchica event stream (chid event stream) which is recursivey embedded in ˆθ. Condition 1: Either ˆ ˆθ = /0 or G ˆθ = 0. Due to this condition it is not necessary to distribute the imitation between the gradient and the sub-eement. his simpifies the anaysis without restricting the modeing capabiities. he arriva of the first event occurs after a time units and at a +, a + 2, a + 3,..., a + i (i N) the other events occurs. Definition 5: A hierarchica event stream fufis for every Δt,Δt the condition ϒ(Δt + Δt, ˆ) ϒ(Δt, ˆ)+ϒ(Δt, ˆ) n the foowing we wi give a few exampes to show the usage and the possibiities of the new mode. A simpe periodic event sequence with period 5 can be modeed by : ˆ 1 = (5,0,1,0,e)} Lemma 3: Let be an event stream with = θ 1,...,θ n }. can be modeed by ˆ = ˆθ 1,..., ˆθ n } with ˆθ i = ( θi,a θi,1,, /0 ) Proof: Each of the hierarchica event eements generates exacty one event at each of its periods foowing the pattern of the corresponding event eement. ˆ 1 approximated after 10 events woud be modeed by: ˆ 10 (,0,10,0,(5,2,1,0,e)}),(,47,, =

4 Events Figure 5. Exampe for overapping events of different periods 20 Note that 47 = 2 +(10 1) 5 is the point in time in which the ast reguar event occurs and therefore the start of the approximation. One singe event is modeed by ˆ 2 = (,0,1,. A gradient of woud ead to an infinite number of events but due to the imitation ony one event is generated. An event bound function requiring constanty 0.75 time units processor time within each time unit can be described by ˆθ 2 =(,0,,0.75, /0 ). With the recursivey embedded event sequence any possibe pattern of events within a burst can be modeed. he pattern consists of a imited set of events repeated by the period of the parent hierarchica event eement. For exampe a burst of five events in which the events have an intra-arriva rate of 2 time units which is repeated after 50 time units can be modeed by ˆ 3 = (50,0,5,0,(2,0,1, )}. he chid event stream can contain grand-chid event streams. For exampe if ˆ 3 is used ony for time units and than a break of time units is required woud be modeed by ˆ 4 = (2000,0,100,0, ˆ 3 )}. he ength Δt ˆθ of the interva for which the imitation of ˆθ is reached can be cacuated using a interva bound function (x, ˆ)=min(Δt x = ϒ(Δt, ˆ)) which is the inverse function to the event bound function ( (, /0 )=0): Δt ˆθ = (, ˆ ˆθ )+ ˆθ G ˆθ Note that this cacuation requires the condition of the mode that either G ˆθ = 0 or ˆ ˆθ = /0 and that the cacuation of the interva bound function requires the distribution of ˆθ on the eements of ˆ ˆθ Assumptions and Condition For the anaysis it is usefu to restrict the mode to event sequences having no overapping periods. Consider for exampe (figure 5) ˆθ 5 = (28,0,15,0,(3,0,1, )}. he imitation interva Δt ˆθ 6 has the ength Δt ˆθ 6 =(15 1) 3 = 42. he first period [0,42] and the second period [28,70] of the event sequence eement overap. Condition 2: (Separation Condition) ˆθ fufis the separation condition if the interva in which events are generated by G ˆθ or ˆ ˆθ is equa or smaer than its period ˆθ : ( ˆθ, ˆ ˆθ )+ ˆθ G ˆθ ˆθ or ˆθ ϒ(ˆθ, ˆθ ˆ ˆθ )+ G ˆθ he condition 2 does not reduce the space of event patterns that can be modeed by a hierarchica event sequence. Lemma 4: A hierarchica event sequence eement ˆθ that does not meet the separation condition can be exchanged with ( a set of event sequence eements ˆθ 1,..., ˆθ k with k = ˆθ, ˆθ) and ˆθ i =(kˆθ,(i 1)ˆθ + a ˆθ, ˆθ,G ˆθ, ˆ ˆθ ). Proof: he proof is obvious and therefore skipped. ˆθ Figure 6. Hierarchica event sequence ˆ 6 ˆ 5 can be transferred into ˆ 5 meeting the separation condition: ˆ 5 = (56,0,15,0,(3,0,1, ),(56,28,15,0,(3,0,1, )} he separation condition prohibits events of different event sequence eements to overap. We aso do not aow recursion, so no event eement can be the chid of itsef (or a subsequent chid eement) Hierarchica Event Bound Function he event bound function cacuates the maximum number of events generated by ˆ within Δt. Lemma 5: Hierarchica Event Bound Function ϒ(Δt, ): Let for any Δt, define mod(δt,) =Δt Δt and ϒ(Δt, /0 )=0. Let ϒ(Δt, ˆ)= ˆθ ˆ ϒ(Δt, ˆθ) and Δt a ˆθ ˆθ ˆθ =,G ˆθ = Δt a ˆθ + 1 ˆθ ˆθ ˆθ,G ˆθ = min( ˆθ,(Δt a ˆθ )G ˆθ ϒ(Δt, ˆθ)= +ϒ(Δt a ˆθ, ˆ ˆθ )) ˆθ =,G ˆθ Δt a ˆθ ˆθ + min( ˆθ, ˆθ mod(δt a ˆθ,ˆθ )G ˆθ +ϒ(mod(Δt a ˆθ,ˆθ ), ˆ ˆθ )) ˆθ,G ˆθ Proof: Due to the separation condition it is aways possibe to incude the( maximum ) aowed number of events Δt a for competed periods ˆθ ˆθ ˆθ. Ony the ast incompete fraction of a period has to be considered separatey (min(...)). his remaining interva is given by subtracting a compete periods, and the offset a from the interva Δt ( ) mod(δt a ˆθ,ˆθ. For the chid event stream, the number of events is cacuated by using the same function with now the remaining interva and the new embedded event sequence. n case of the gradient the number of events is simpy G ˆθ Δt. he imitation bounds both vaues due to the separation condition. ndependenty of the hierarchica eve of an event sequence eement it is considered ony once during the cacuation for one interva. his aows bounding the compexity of the cacuation. Exampe 1: ˆ 6 = (20,6,10,0,(3,0,2,1. ϒ(Δt, ˆ 7 ) is shown in figure 6. ϒ(33, ˆ 6 ) is given by 27 ϒ(33, ˆ 6 )= ˆθ ˆθ + min( ˆθ,mod(27,ˆθ )G ˆθ + ϒ(mod(27,ˆθ ), ˆ ˆθ ))

5 = min(10,0 + ϒ(7, ˆ 20 ˆθ )) = 10 + min(10,ϒ(7, ˆ ˆθ )) ϒ(7, ˆ ˆθ )=ϒ(7, ˆθ 7 )= 2 + min(2,mod(7,3) 1 + 0) =4 + 1 = 5 3 ϒ(33, ˆ 6 )=10 + min(10,5)= Reduction and Normaization n the foowing we wi reduce event streams to a norma form. he hierarchica event stream mode aows severa different description for the same event pattern. For exampe an event stream ˆ = (100,0,22,0, ˆ a )} with ˆ a = (7,0,3,, /0 ),(5,3,2, can be rewritten as ˆ = (100,0,12,0, ˆθ a,1 ), (100,0,8,0, ˆθ a,2 ),(100,23,2, with ˆθ a,1 =(7,0,3,, /0 ) and ˆθ a,2 =(5,3,2,, /0 ). Lemma 6: An event stream ˆ a = ( a,a a, a,0, ˆ a)} with a chid eement ˆ a = ( 1,a 1, 1,G 1, ˆ 1 ),...,( k,a k, k,g k, ˆ k )} can be transferred into an equivaent event stream ˆ b with ˆ b = ˆθ a,1, ˆθ a,2,..., ˆθ a,n, ˆθ a,x } having ony chid event sequences with one eement where ˆθ b,i =(,a,ϒ(δt a, ˆθ a,i),0, ˆθ a,i) Δt a = im( ( a, ˆ a) ε) ε 0 ε>0 ˆθ a,x =(, ( a, ˆ a ), a ˆθ ˆ a ϒ(Δt a, ˆθ a,i ),, /0 ) Proof: We have to distribute the imitation a on the eements of the chid event sequence. First we have to find the interva Δt for which the imitation of the parent eement a is reached by the chid event sequence ˆ a. Δt is given by ( a, ˆ a ). We have to cacuate the costs required for each of the chid event sequence eements for Δt. t is given by ϒ(Δt, ˆθ i ). he probem is that severa eements can have a gradient of exacty at the end of Δt. n this situation the sum of ϒ(Δt, ˆθ) may exceed the aowed imitation a of the parent eement. he tota costs is bounded by the goba imitation a rather than the imitations i. o take this effect into account we excude the costs occurring exacty at the end of Δt for each hierarchica event eement and we hande these costs seperatey modeing them with the hierarchica event eement ˆθ a,x. o do so we cacuate the imitation not by ϒ(Δt, ˆθ i ) but by ϒ(Δt ε, ˆθ i ) where ε is an infinity sma vaue excuding ony costs occurring at the end of Δt exacty. his aows a better comparison between different hierarchica event streams Capacity Function he proposed hierarchica event stream mode can aso mode the capacity of processing eements and aows to describe systems with fuctuating capacity over the time. n the standard case a processor can hande one time unit execution time during one time unit rea time. For many resources the capacity is not constant. he reasons for a fuctuating capacity can be for exampe operation-system tasks or variabe processor speeds due to energy constraints Costs Costs 2000 a) 100 c) Figure Costs t Costs Exampe service bound functions Assuming the capacity as constant aso does not support a moduarization of the anaysis. his is especiay needed for hierarchica scheduing approaches. Consider for exampe a fixed priority scheduing. n a moduar approach each priority eve gets the capacity eft over by the previous priority eve as avaiabe capacity. he remaining capacity can be cacuated step-wise for each priority eve taking ony the remaining capacities of the next higher priority eve into account. Such an approach is ony possibe with a mode that can describe the eft-over capacities exacty. Definition 6: he service function β (Δt,ρ) gives the minimum amount of processing time that is avaiabe for processing tasks in any interva of size Δt for a specific resource ρ for each interva Δt. t can aso be modeed with the hierarchica event sequence mode. he service function is superadditiv and fufis the inequation β (Δt + Δt ) β (Δt)+β(Δt ) for a Δt,Δt. he definition matches the service curves of the rea-time cacuus. We propose to use the hierarchica event stream mode as an expicit description for service curves. n the foowing we wi show, with a few exampes, how to mode fuctuating service functions with the hierarchica event streams. he constant capacity, as shown in 7 a) can be modeed by: β basic = (,0,,1 Bocking the service for a certain time t (figure 7 b) is done by: β bock = (,t,,1 A constanty growing service curve in which the service is bocked periodicay every 100 time units for 5 time units (for exampe by a task of the operating system): β bock = (100,5,95,1 (figure 7 c) ) he service for a processor that can hande ony time units with fu speed and than time units with haf speed (figure 7 d)): β vary = (2000,,500, 2 1, /0 ),(2000,0,,1 hese are ony a few exampes for the possibiities of the new mode Operations n the foowing we wi introduce some operations on hierarchica event sequences and streams. Lemma 7: (+ operation) f ˆ C = ˆ A + ˆ B than for each interva Δt the equation ϒ(Δt, ˆ C )=ϒ(Δt, ˆ A )+ϒ(Δt, ˆ B ) is d) b) 2000

6 true. t can be cacuated by the union ˆ C = ˆ A ˆ B. t is aso necessary to shift vaues. Lemma 8: ( shift-operation) We have ϒ(Δt, ˆ ϒ(Δt t, ˆ) Δt t )= 0 ese if ˆ contains and ony contains for each eement ˆθ ˆ an ˆθ ˆ with ˆθ =(ˆθ,a ˆθ + t, ˆθ,G ˆθ, ˆ ˆθ ). he shift operation ( ) ϒ(Δt, ˆ )=ϒ(Δt +t, ˆ) is defined in a simiar way with ˆθ =(ˆθ,a ˆθ t, ˆθ,G ˆθ, ˆ ˆθ ). his operation (, ) is associative with the (+) operation so we have ( ˆ A + ˆ B ) t =(ˆ A t)+( ˆ B t) and ( ˆ A + ˆ B ) t =(ˆ A t) + ( ˆ B t). For ( ˆ t) v we can write aso ˆ (t + v). o scae the event stream by a cost vaue is for exampe necessary to integration of the worst-case execution times. Lemma 9: Let ˆ = c ˆ. hen for each interva Δt: ϒ(Δt, ˆ )=cϒ(δt, ˆ) if the chid set of ˆ contains and ony contains for each eement ˆθ of the chid set of ˆ an eement ˆθ ˆ having ˆθ =(ˆθ,a ˆθ,c ˆθ,cG ˆθ,c ˆ ˆθ ). Proof: We do the proof for the add-operation: ϒ(Δt, ˆ C )=ϒ(Δt, ˆ A )+ϒ(Δt, ˆ B ) = ϒ(Δt, ˆθ)+ ϒ(Δt, ˆθ ) ˆθ ˆ A ˆθ ˆ B = ˆθ ˆ A ˆ B ϒ(Δt, ˆθ)=ϒ(Δt, ˆ A ˆ B ) he other proofs can be done in a simiar way Utiization Lemma 10: he utiization U Γ of a task set in which the event generation patterns are described by hierarchica event streams is given by (( τ Γ)Λ( ˆθ ˆ τ ) ( ˆθ ˆθ = )): ( ) n U Γ = τ Γ ˆθ ˆθ ˆ τ + τ Γ ˆθ ˆθ ˆ τ U ˆ + G ˆθ ˆθ τ ˆθ = ˆθ = Note that event-eements with an infinite period and a finite imitation do not contribute to the utiization. 5. SCHEDULABLY ESS For the scheduabiity tests of uni-processor system using the hierarchica event stream mode anaysis, we can integrate the approximation and the avaiabe capacity into the anaysis Scheduabiity tests for dynamic priority systems A system schedued with EDF is feasibe if for a intervas Δt the demand bound function does not exceed the service function Ψ(Δt) C (Δt, ρ). Both, the demand bound and the service function can be described by and cacuated out of hierarchica event streams. his eads to the test τ Γ ˆθ ˆ τ ϒ(Δt d τ, ˆθ)c τ C (Δt,ρ). he anaysis can be done using the approximation as proposed in [2]. For the exact anaysis an upper bound for Δt, a maximum test interva is required to imit the run-time of the test. For the hierarchica event stream mode one maximum test interva avaiabe is the busy period Response-time cacuation for static priority scheduing n the foowing we wi show how a worst-case response time anaysis for scheduing with static priorities can be performed with the new mode. he request bound function Φ cacuates the amount of computation time of a higher priority task that can interfere and therefore deays a ower-priority task within an interva Δt. n contrary to the event bound function the request bound function does ony contain the events of the start, not the events of the end point of the interva: Φ(Δt, τ)= im (ϒ(Δ, τ )c τ ) Δ Δt 0 Δ<Δt For the hierarchica mode it is ony necessary to hande the cases Δt = 0 differenty than in the cacuation of the event bound function: Φ(Δt,Γ)= τ Γ c τ ˆθ ˆ τ Φ(Δt, ˆθ,τ) with Δt a ˆθ ˆθ ˆθ ˆθ = Φ(Δt, ˆθ,τ)= 0 Δt a ˆθ 0 Δt a ˆθ ˆθ ˆθ + min( ˆθ,G ˆθ (Δt a ˆθ + Φ(mod(Δt a ˆθ,ˆθ ), ˆ ˆθ )) ese With this function it is possibe to cacuate the worst-case response times for the tasks: Lemma 11: Let τ be schedued with fixed priorities and Γ hp(τ) containing a task with a higher priority than τ. he response time r(τ i,1 ) for the first event of τ i is given by: r(τ i,1 )=min(δt C (Δt) c τ + Φ(Δt,Γ hp(τ) )) he vaue for Δt can be cacuated by a fix-point iteration starting with Δt = c τ. o cacuate the maximum response time it is necessary to do the cacuation for a events within the busy period. he busy period of a task set is the maximum interva in which the resource is competey busy, so in which does not exists ide time for the resource: B(Γ) =min(δt C (Δt) Φ(Δt,Γ)) Lemma 12: he worst-case response time of τ can be found in the busy period of any task set containing τ and Γ hp(τ). t is the maximum response time of a r(j,τ) where: r(j, τ)= min (Δt C (J + Δt) ϒ(J)c τ + Φ(J + Δt,Γ hp(τ) )) 0 Δt< r(τ)=max 0 J B(Γ) (r(j,τ) J is ess or equa than the busy period (J B(Γ)). his minimum response time has to be ower than the deadine of the task. 6. APPROXMAON o imit the number of test intervas and therefore the computationa compexity we integrate the approximation approach of [2]. We can now integrate the approximation directy into the mode. We aow the approximation of an event eement to start after the necessary number of test intervas are reached gobay for this eement, independenty in which period of the parent event eement this happens. n case that the event eement ˆθ is a chid eement of another (parent) event eement ˆθ we have to distinguish for ˆθ between those periods in which ˆθ is evauated exacty and those in which ˆθ

7 is approximated. o do this it is necessary to spit ˆθ at the ast exacty considered interva of ˆθ Case simpe sequence with gradient Let us consider first a simpe hierarchica event eement: ˆθ = (,a,,g ˆθ k is the approximative counter-part for ˆθ starting with the approximation after k exacty considered test intervas. ˆθ k is modeed by: ˆθ k = (,0, A,0, ˆθ),(,a A,,G, /0 ),(,a B,, with A = k, a A = a + k, a B = a A + G. For the specia case with G = we have a A = a B. Exampe 2: Let us, for exampe consider ˆ = (10,0,3, 1 2. he approximation ˆ 5 for ˆ after k = 5 exacty considered test intervas is given by ˆ 5 = (,0,15,0,(10,0,3, 1 2 ),(,50,3, 1 3 2, /0 ),(,56,, 10 where A = 5 3 = 15, a A = = 50, a B = = 56. We can simpify this exampe to: 2 ˆ 5 = (,0,18,0,(10,0,3, 2 1),(,56,, 10 3 Consider another exampe ˆ = (10, 2, 3,. he approximation ˆ 10 is given by: ˆ 10 = (, 0,30,0,(10,2,3,,(,103,3,, /0 ),(,103,, 10 3 or: ˆ 10 = (,0,33,0,(10,2,3,,(,103,, Approximation of one-eve chid eement Let us consider a hierarchica event sequence with one chid eement: ˆθ =(,a,,0, ˆθ ), ˆθ =(,a,,g, /0 ) ˆ k is given in this case by: ˆ k = (,0, A,0, ˆθ ),(,a A,k A,0,(,a,,G, /0 ), (,a + G,, ),(,a B,x,, /0 ),(,a B,, he first eement of ˆ k modes the part in which the chideement ˆθ is considered exacty. n case that the first possibe approximation interva for ˆθ occures within the first period of ˆθ, we have to start the approximation within this first period of ˆθ. Otherwise it woud not be possibe to find a reasonabe bound for the number of considered test intervas for ˆθ. So ˆθ depends on whether k or > k. We have ˆθ ˆθ k = (,0,,0, ˆθ k )} > k ˆθ k = (,0,k ˆθ,0, ˆθ),(,kˆθ, ˆθ,G ˆθ, /0 ),(,kˆθ + ˆθ,, ˆθ G ˆθ ˆθ he cacuation of A, B, a A, a B and a C are done as foows: k k A = > k k + a k a A = + a > k a B = k + a + a he approximation of ˆθ can be done by an eement ˆθ k with a gradient G ˆθ k =. When starting finay the approximation of ˆθ a cost-offset x is required to ensure that the approximated function ϒ(Δt, ˆθ k ) is aways equa or higher than the exact function ϒ(Δt, ˆθ). Costs Figure 8. x y y o c gradient period y imitation Case ˆθ approximated, ˆθ not approximated Figure 8 outines this situation. his cost-offset is necessary as a new period of the parent eement spits the approximation of the chid eement. he cacuation of x can be done as foows: x = y ( x = 1 y ) y gives the interva between the start of the chid eement ˆθ and the point in time in which the imitation of ˆθ is reached. he reaching of the imitation is cacuated using the approximative description of the chid eements of ˆθ with the seperate consideration of every first event of ˆθ. For a simpe chid eement ˆθ = (,a,,0, ˆθ )} with ˆθ = (,a,, this vaue y is given by (y a ) ( )= y = + a = + a Hence for x: x = 2 + a Exampe 3: Let us consider the exampe hierarchica event sequence: ˆ=(80,2,16,0, ˆ )}, ˆ =(10,2,3,,/0 )} For the approximation ˆ 10 we get the vaues: k 10 3 A = = 16 = k 10 3 a A = + a = = a B = k + a = = 802 y = + a = = ( x = 1 y ) ( = ) = ˆ 10 = (,0,32,0,(80,2,16,0,(10,2,3, )}), (,162,128,0,(,2,3,, /0 ),(,2,, 3, /0 ), 80 (80,2,13, ,(,802,6.9333,, /0 ),(,802,, 80

8 6.3. Approximation of n-eve chid eement Let us consider the foowing hierarchica event eement with two eves of chid eements ˆθ = (,a,,0, ˆθ )}, ˆθ = (,a,,0, ˆθ )}, ˆθ = (,a,,g. We consider the approximation ˆθ k. ˆθ k is given by ˆθ k = (,0, A,0, ˆθ 1),(,a A, B,0, ˆθ 2), (,a B, C,0,(,a,x,, /0 ),(,a, x, ), (,a C,x,,0),(,a C,, ˆθ 1 depends on whether k or > k. We have ˆθ k ˆθ 1 = (,0,,0, ˆθ k )} > k ˆθ k = (,0,k ˆθ,0, ˆθ),(,kˆθ, ˆθ,G ˆθ, /0 ),(,kˆθ + ˆθ,, ˆθ G ˆθ ˆθ ˆθ 2 depends on whether k or > k. We have ˆθ /0 k 2 = (,a,,g,0),(,a + G,, > k he cacuation of A, a A and B : k k A = B = > k k A k 0 > k C = k ( A + B ) k + a + a k a A = a B = + a + a > k k k a C = k + a > k he cacuation of x is the same as the cacuation for x in the previous section. We have y = + a x = ( y he cacuation of x and y is simiar but using the approximation of ˆθ. We have ) (y a) ( )= x y = x + a ( ) y x = Note that when setting x = the cacuation of x and y on the one side and x and y on the other side are the same. herefore the proposed description for ˆ k can be generaized to hande event sequences with n-eve chid event sequences. he cacuation is visuaized in figure 8. Exampe 4: Let us consider the exampe hierarchica event sequence: ˆ = (,10,100,0, ˆ )}, ˆ = (80,2,16,0, ˆ )}, ˆ = (10,2,3,. For an approximation ˆ 10 in which k = 10 test intervas are considered exacty we get the vaues: y = 16 3 ( 3 ) + 2 = ( 10 ) x =16 = y = ( ) + 2 = ( x = 100 ) = ˆ 10 = (,0,100,0, ˆ 10 2,1 ),(,1012,100,0,(,2,3,, /0 ), (,2,, 3 3, /0 ),(80,2,13, ),(,2010,800,0, (,2,6.9333,, /0 ),(,2,, , /0 ),(,2, , 16 ˆ 10 2,1 80 ),(,10010, ,, /0 ),(,10010,, 100 = (,0,32,0,(80,2,16,0,(10,2,3, )}), (,162,3,, /0 ),(,162,, 3 3, /0 ),(80,162,13, Approximation of eement with severa chid eements A hierarchica event sequence with severa chid eements can be transferred into a normaized hierarchica event sequence in which each event sequence eement has ony one chid eement. Each eement matches one of the previous pattern and can therefore be approximated. he overa approximation of the event sequence is than ony a merge of the singe eements Required number of test intervas n those cases in which the approximation of the chid eement starts within the competion of the first period of the parent eement we cannot postpone it unti the first period of the parent. t woud not be possibe to bound the number of test intervas for the chid hierarchica event eement. Exampe 5: Consider the foowing exampe: ˆθ 10 = 0,0,4000,0, ˆθ 11 }}, ˆθ 11 = 10,0,5,, /0 } Postponing the approximation of the chid up to the end of the first period of the parent woud cost 3000 additiona test intervas. We can sti find a simpe bound on the required number of test intervas. For those cases in which the approximation does not start within the first period, the number of test intervas for one period of the parent event eement has to be ess than the approximation bound k. Otherwise the approximation woud be aowed somewhere within the first period. herefore the maximum number of test intervas we have to additionay consider due to the postponing is bounded aso by k, so a tota bound of 2k.

9 6.6. Spitting points he spitting points are the points in which the parent eement is spitted to destinguish between the non-approximated and the approximated part of one of its chid eements. n genera, the parent eement is spitted at the first of its competed period which is greater than the first possibe approximation interva of the chid eement. Each eement can require as many spitting points as its tota chid-set has members. he tota chid-set contains its chidren, the chidren of its chidren and so on. he parent chain contains the parent eement of an eement, the parent of the parent eement and so on. For reason of simpification we consider ony normaized hierarchica event sequences, in which each ˆθ can ony have one direct chid eement at most. Let ˆθ 1 be the owest-eve chid eement and ˆθ n be the highest eve parent eement. he spitting point for an eement ˆθ i is determined by the upper-most member ˆθ j of a parent chain for which the first possibe approximation interva for k exacty considered test intervas t ˆθ i,k of ˆθ i is arger than the end of the first competed period of ˆθ j. his first compete period is given by a ˆθ j + ˆθ j, so t ˆθ i > a ˆθ j + ˆθ j. he spitting point is the first start of a new period of ˆθ t after t ˆθ, so s k i, j = min(δt Δt = a t i + k ti Δt t ˆθ j,k ) t is necessary to spit each eement of the parent-chid chain between ˆθ t and ˆθ c at this point. A members of the parent chain of ˆθ ti, which are of cause aso member of the parent chain of ˆθ i, are spitted at their first period instead, so j > t s i, j = a ˆθ j + ˆθ j n genera we get a matrix of possibe spitting points: Lemma 13: (Spitting points) Let ˆθ 1,.., ˆθ n be a set of hierarchica event eements with ˆθ 1 =( 1,a 1, 1,G 1, /0 ) and ˆθ i = i,a i, i,0, ˆθ i 1 ) for 0 < i n. Let s k i, j be the spitting points for eement j on the event eement ˆ i with the minimum number of k test-intervas considered exacty for ˆθ j. Let t j,k denote the first possibe approximated test interva of ˆθ j after k exact test intervas. s k i, j can be cacuated: s k i, j = min(x x = a i + y i,y N,x t j,k ) s k i, j = s k i, j s k i, j < a i+1 + i+1 s k i+1, j ese s k i,0 = a i s k n, j = sk n, j Proof: he first competed period of the hierarchica event eement ˆθ i after the first possibe approximation start for the hierarchica event eement ˆθ j k gives the potentia spitting point s k i, j. he resuting spitting point s i, j is ony in those cases identica to the potentia spitting point s k i, j in which either ˆθ i is the top-eve parent eement (i = n) or s k i, j is smaer than the end of the first period of the parent eement ˆθ i+1. n a other cases, the competion point s k i, j is identica to the corresponding competion point of the parent eement of ˆθ i, s i+1, j, which can again be identica to the spitting points of the (i + 1)-th parent eement and so on. We can cacuate the approximated hierarchica event streams using these spitting points. Lemma 14: Let us consider a chain of hierarchica event streams ˆ 1,..., ˆ n with ˆ j = (ˆθ j,a ˆθ j, ˆθ j,0, ˆ j+1 )} and j<n ˆ n = (ˆθ j,a ˆθ j, ˆθ j,g ˆθ n. he approximated event eements are given by the foowing equations (s 0, j = 0): ˆ k j = ˆθ i, j i + j n s i, j s i, j 1 } ˆ j, j+1 ˆ k i,i = (,s i, j 1,x ˆθ i,, /0 ),(,s i, j 1,, ˆθ i ˆθ i ˆ k i,i+1 = (,s i,i,x ˆθ i,, /0 ),(,s i,i,, ˆθ j ˆθ j (,s i, j 1, s i, j s i, j 1 ˆθ ˆθ i,0, s i, j s i+1, j i ˆθ i, j = (ˆθ i,0, ˆθ i,g ˆθ i, ˆ i 1, j (ˆθ i,a ˆθ i, ˆθ i,g ˆθ i, ˆ i 1, j s i, j = s i+1, j ˆ ˆθ i, j i, j = } s i+1, j s i+1, j+1 ˆθ i, j } ˆ i, j+1 s i+1, j = s i+1, j+1 ( ) x ˆθ i = ˆθ i 1 y i, j ˆθ i y ˆθ i = ˆθ i x ˆθ i 1 ˆθ i 1 x ˆθ 1 = ˆθ 1 ˆθ i 1 + a ˆθ i 1 Proof: Ony for those spitting points s k i, j being different from their predecessor spitting point s k i, j 1 a hierarchica event eement can be constructed. he other spitting points woud ead to eements generating no events. For the construction of the eement we have to distinguish, whether the spitting point is identica to the corresponding spitting point of the parent eement or whether it is a new vaue on its own. n the first case (s k i, j = sk i+1, j ), the imitation is simpy inherited from the parent eement, in the second case (s k i, j sk i+1, j ), the imitation has to be cacuated by distributing the previous imitation on the new parts. Note that sk i, j sk i, j 1 N by definition and therefore ˆθ i the imitation of the new parts are mutipe of the imitation of the singe eements. he emma summarizes (and simpifies) the resuts of the previous sections. Each eement of the top-parent event sequence and therefore each chain of eements can be considered seperatey. 7. EXAMPLE Exampe 6: Fig. 9 shows the advanced approximation for the event bound function of the event stream ˆ 7 = (20,0,10,0,(2,0,2,, /0 ))} and compares it with the description by SymA/S and by the rea-time cacuus. For SymA/S we have used an execution time of 2, a period of 4, a jitter of 10 and a minimum distance between two events of 2 time units. he ines of SymA/S and the rea-time cacuus are neary identica with the exception that SymA/S modes

10 Costs origina event bound function, exact case of new mode approximate event bound funon, new mode Sympta/S rea time cacuus Figure 9. nterva ength Approximated hierarchica event bound function discrete events. he ine for the new mode in is exact form is aways equa or beow both other ines and in its approximated form it is beow and the beginning and than equa to the rea-time cacuus curve. he degree of approximation is freey seectabe. Note, that the event discrete modeing of the SymA/S approach requires additiona effort for the anayis. he event stream consists of bursts with five events. he advanced approximated event stream with an approximation after three events is has the foowing separation points: s 1,0 = 0, s 1,1 = 20, s 2,0 = 0, s 2,1 = 4, s 2,2 = 60. For x and y we have the vaues:y = +a = = 8, 2 x = ( 1 y ) ( ) = = 6 t is given by the foowing description: ˆ 3 7 = (,0,10,0,(20,0,6,0,(2,0,2,, /0 )),(,10,4,1),(,20,12,0, (20,0,2,, /0 ),(20,0,8, 20 2 ), (,60,6,, /0 ),(,60,, 1 2 Such a description imits the maximum number of test intervas for each hierarchica event eement separatey. n the exampe five test intervas for the chid eement and four test intervas for the parent eement are required. 8. CONCLUSON n this work we presented a new advanced event mode especiay suitabe for the modeing of distributed systems. Such a system consists of severa tasks bound on different processing eements and triggering each other. o divide the probem of rea-time anaysis of the whoe system to a probem of reatime anaysis of the singe tasks, a mode efficienty describing the densities of the events triggering the tasks (incoming events) and those events generated by the tasks to trigger other tasks (outgoing events) was required. Additionay, a mode for the capacity of the processing eements avaiabe for the tasks was necessary. his is especiay compicated in the case with a higher priority task aready having used up a part of the capacity. n this paper we proposed a unified mode for a of this. Additionay this mode is capabe to introduce approximations into the description of the event densities which guarantees a fast evauation as we as an upper bound on the approximation error. he new mode integrates the efficient modeing of periodic and aperiodic events, burst of events in various kinds, approximated event streams and the origina and the remaining capacites of processors in one singe mode. t can be seen as an expicit description for the arriva, service and capacity curves of the rea-time cacuus having the necessary modeing capabiies for them. We have presented the rea-time anaysis for this mode for both, systems with dynamic or static priorities. n future we wi show the concrete integration of this mode in the rea-time cacuus. Remark 1: his work was funded by the Deutsche Forschungsgemeinschaft (DFG) under grand SL 47/3-1. REFERENCES [1] K. Abers, F. Bodmann, and F. Somka. Hierachica event streams and event dependency graphs. n Proceedings of the 18th Euromicro Conference on Rea-ime Systems (ECRS 06), pages , [2] K. Abers and F. Somka. An event stream driven approximation for the anaysis of rea-time systems. n EEE Proceedings of the 16th Euromicro Conference on Rea-ime Systems, pages , Catania, [3] K. Abers and F. Somka. Efficient feasibiity anaysis for rea-time systems with edf-scheduing. n Proceedings of the Design Automation and est Conference in Europa (DAE 05), pages , [4] S. Chakraborty, S. Künzi, and L. hiee. Performance evauation of network processor architectures: Combining simuation with anaytica estimations. Computer Networks, 41(5): , [5] R.L. Cruz. A cacuus for network deay. n EEE ransactions on nformation heory, voume 37, pages , [6] K. Gresser. Echtzeitnachweis ereignisgesteuerter Reazeitsysteme. Dissertation, Düssedorf, [7] K. Gresser. An event mode for deadine verification of hard rea-time systems. n Proceedings of the 5th Euromicro Workshop on Rea-ime Systems, [8] S. Künzi. Efficient Design Space Exporation for Embedded Systems. PhD thesis, EH Zürich No , [9] A.K. Parekh and R.G.Gaager. A generaized processor sharing approach to fow contro in integrated service networks. n EEE/ACM ransactions on Networking, voume 1, pages , [10] S. Perathoner, E. Wander, L. hiee, A. Hamann, S. Schiecker, R. Henia, R. Racu, R. Ernst, and M. Gonzáez Harbour. nfuence of different system abstractions on the performance anaysis of distributed rea-time systems. n EMSOF 2007, pages EEE Computer Society Press, [11] K. Richter. Compositiona Scheduing Anaysis Using Standart Event Modes. Dissertation, U Braunschweig, [12] K. Richter and R. Ernst. Event mode interfaces for heterogeneous system anaysis. n Proceedings of the Design Automation and est Conference in Europe (DAE 02), [13] L. hiee, S. Chakraborty, M. Gries, and S. Künzi. Design space exporation for the network processor architectures. n 1st Workshop on Network Processors at the 8th nternationa Symposium for High Performance Computer Architectures, 2002.