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SYNTHESIS of PIPELINED SYSTEMS for the CONTEMPORANEOUS EXECUTION of PERIODIC and APERIODIC TASKS with HARD REAL-TIME CONSTRAINTS Paolo Palazzari Luca Baldini Moreno Coli ENEA – Computing and Modeling Unit University “La Sapienza” – Electronic Engineering Dep’t 1 Outline of Presentation Problem statement Asynchronous events Mapping methodology Searching space Optimization by RT-PSA Algorithm Results Conclusions 2 Outline of Presentation Problem statement Asynchronous events Mapping methodology Searching space Optimization by RT-PSA Algorithm Results Conclusions 3 Problem Statement We want to synthesize a synchronous pipelined system which executes both the task PSy , sustaining its throughput, and m mutually exclusive tasks PAs1, PAs2, …, PAsm whose activations are randomly triggered and whose results must be produced within a prefixed time. 4 Problem Statement We represent the tasks as Control Data Flow Graphs (CDFG) G = (N, E) N = {n1, n2, …, nN}: operations of the task E= ni , n j | ni , n j N, n j is data/ctrl dependent on ni (data and control dependencies) 5 Problem Statement Aperiodic tasks, characterized by random execution requests and called asynchronous to mark the difference with the synchronous nature of periodic tasks, are subjected to Real-Time constraints (RTC), collected in the set RTCAs = {RTCAs1, RTCAs2, ..., RTCAsm}, where RTCAsi contains the RTC on the ith aperiodic task. Input data for the synchronous task PSy arrive with frequency fi = 1/Dt, being Dt the period characterizing PSy. 6 Problem Statement We present a method to determine The target architecture: a (nearly) minimum set of HW devices to execute all the tasks (synchronous and asynchronous); The feasible mapping onto the architecture: the allocation and the scheduling on the HW resources so that PSy is executed sustaining its prefixed throughput and all the mutually exclusive asynchronous tasks PAs1, PAs2, …, PAsm satisfy the constraints in RTCAs. 7 Problem Statement The adoption of a parallel system can be mandatory when Real Time Constraints are computationally demanding The iterative arrival of input data makes pipeline systems a very suitable solution for the problem. 8 Problem Statement Example of a pipeline serving the synchronous task PSy DATA INTRODUCTION Iteration 1 S1 S2 S3 S4 S5 S6 S7 S8 S9 S 10 INTERVAL Iteration 2 S1 S2 S3 S4 S5 S6 S7 S8 S9 S 10 DII = 2 Iteration 3 S1 S2 S3 S4 S5 S6 S7 S8 S9 S 10 Iteration 4 S1 S2 S3 S4 S5 S6 S7 S8 S9 S 10 S1 S2 S3 S4 S5 S6 S7 S8 S9 S 10 Iteration 5 S1 S2 S3 S4 S5 S6 S7 S8 S9 S 10 Iteration 6 t (ut) 0 100 200 300 400 500 600 700 800 900 1000 Tck = 50 ut 9 Problem Statement Sk = (k-1)Tck and Sk = kTck In a pipeline with L stages, SL denotes the last stage. DII = Dt/Tck 10 Problem Statement We assume the absence of synchronization delays due to control or data dependencies: Throughput of the pipeline system =1/DII. 11 Outline of Presentation Problem statement Asynchronous events Mapping methodology Searching space Optimization by RT-PSA Algorithm Results Conclusions 12 Asynchronous events We assume the asynchronous tasks to be mutually exclusive, i.e. the activation of only one asynchronous task can be requested between two successive activations of the periodic task 13 Asynchronous events In red the asynchronous service requests in a pipelined system. Iteration 1 S1 S2 S3 S4 S5 S6 S7 S8 S9 S 10 DII = 2 Iteration 2 S1 S2 S3 S4 S5 S6 S7 S8 S9 S 10 Iteration 3 S1 S2 S3 S4 S5 S6 S7 S8 S9 S 10 Iteration 4 S1 S2 S3 S4 S5 S6 S7 S8 S9 S 10 S1 S2 S3 S4 S5 S6 S7 S8 S9 S 10 Iteration 5 S1 S2 S3 S4 S5 S6 S7 S8 S9 S 10 Iteration 6 t0A1 t0A2 t0A3 t0A4 t0A5 t (ut) 0 100 200 300 400 500 600 700 800 900 1000 (I0A1) (I0A2) (I0A3) (I0A4) (I0A5) 14 Asynchronous events Like the synchronous events, we represent the asynchronous events {PAs1, PAs2, ..., PAsm} through a set of CDFG ASG = {AsG1(NAs1,EAs1), ... , AsGm(NAsm,EAsm)} 15 Asynchronous events We consider a unique CDFG made up by composing the graph of the periodic task with the m graphs of the aperiodic tasks: G(N, E) = SyG(NSy, ESy) AsG1(NAs1, EAs1) AsG2(NAs2, EAs2) ..……. AsGm(NAsm, EAsm) 16 Asynchronous events Aperiodic tasks are subjected to Real-Time constraints (RTC): RTC Asi i i i i S L As , D | PAs execution must finish by D As all RTC must be respected, mapping function M has to define a scheduling so that i S LAs - Di 0 RTCAsi RTCAs 17 Outline of Presentation Problem statement Asynchronous events Mapping methodology Searching space Optimization by RT-PSA Algorithm Results Conclusions 18 Mapping methodology In order to develop a pipeline system implementing G a HW resource rj = D(nj) and a time step Sk must be associated to each nj N 19 Mapping methodology We must determine the mapping function M: N UR S UR is the set of the used HW resources (each rj is replicated kj times), UR r j kj r j R UR1 ,UR2 ,...,UR p r11 , r12 ,...,r1k1 1 2 k2 1 2 kp , r2 , r2 ,...,r2 ,...,rp , rp ,...,rp 20 Mapping methodology rj = D(ni) is the HW resource on which ni will be executed S(ni) is the stage of the pipeline, or the time step, in which ni will be executed 21 Mapping methodology We search for the mapping function M’ which, for a given DII: Respects all the RTC Uses a minimum number ur of resources Gives the minimum pipeline length for the periodic task 22 Mapping methodology The mapping is determined by solving the following minimization problem: C ( M ' ) minC M C1 ( M ) C 2 ( M ) C 3 ( M ) M C (M) is responsible of the fulfillment of all the RTC 1 C2(M) minimizes the used silicon area C3(M) minimizes the length of the pipeline. 23 Mapping methodology While searching for a mapping of G, we force the response to aperiodic tasks to be synchronous with the periodic task The execution of an aperiodic task, requested at a generic time instant t0, is delayed till the next start of the pipeline of the periodic task. 24 Mapping methodology Iteration 1 S1 S2 S3 S4 S5 S6 S7 S8 S9 S 10 DII = 2 Iteration 2 S1 S2 S3 S4 S5 S6 S7 S8 S9 S 10 Iteration 3 S1 S2 S3 S4 S5 S6 S7 S8 S9 S 10 Iteration 4 S1 S2 S3 S4 S5 S6 S7 S8 S9 S 10 S1 S2 S3 S4 S5 S6 S7 S8 S9 S 10 Iteration 5 S1 S2 S3 S4 S5 S6 S7 S8 S9 S 10 Iteration 6 t0A1 t0A2 t0A3 t0A4 t0A5 t (ut) 0 100 200 300 400 500 600 700 800 900 1000 (I0A1) (I0A2) (I0A3) (I0A4) (I0A5) 25 Mapping methodology Iteration 1 S1 S2 S3 S4 S5 S6 S7 S8 S9 S 10 DII = 2 Iteration 2 S1 S2 S3 S4 S5 S6 S7 S8 S9 S 10 Iteration 3 S1 S2 S3 S4 S5 S6 S7 S8 S9 S 10 Iteration 4 S1 S2 S3 S4 S5 S6 S7 S8 S9 S 10 S1 S2 S3 S4 S5 S6 S7 S8 S9 S 10 Iteration 5 S1 S2 S3 S4 S5 S6 S7 S8 S9 S 10 Iteration 6 t0A1 t0A2 t0A3 t0A4 t0A5 t (ut) 0 100 200 300 400 500 600 700 800 900 1000 (I0A1) (I0A2) (I0A3) (I0A4) (I0A5) 26 Mapping methodology In a pipelined system with DII=1 the used resource set is maximum the execution time of each AsGi on the pipeline is minimum A lower bound for the execution time of AsGi is given by the lowest execution time of the longest path of AsGi: LpAsi is such a lower bound, expressed in number of clock cycles 27 Mapping methodology Maximum value allowed for DII, compatible with all the RTCAsiRTCAs: LpAsi Tck gives the minimal execution time for AsGi The deadline associated to AsGi is Di. 28 Mapping methodology Maximum value allowed for DII, compatible with all the RTCAsiRTCAs (continued): The request for the aperiodic task can be sensed immediately after the pipeline start, the aperiodic task will begin to be executed DIITck seconds after the request: at the next start of the pipeline. 29 Mapping methodology Maximum value allowed for DII, compatible with all the RTCAsiRTCAs (continued): A necessary condition to match all the RTCAsiRTCAs is that the lower bound of the execution time of each asynchronous task must be smaller than the associated deadline diminished by the DII, i.e. Di DII Tck + LpAsiTck , i = 1, 2, ..., m 30 Mapping methodology Combining previous relations with a congruence condition between the period of the synchronous task (Dt) and the clock period (Tck), we obtain the set DIIp wich contains all the admissible DII values. 31 Mapping methodology Steps of the Mapping methodology: A set of allowed values of DII is determined Sufficient HW resource set UR0 is determined At the end of optimization process the number of used resources ur could be less than ur0 if mutually exclusive nodes are contained in the graph 32 Mapping methodology Steps of the Mapping methodology (continued): An initial feasible mapping M0 is determined; SL0 is the last time step needed to execute P by using M0. Starting from M0, we use the Simulated Annealing algorithm to solve the minimization problem C ( M ' ) minC M C1 ( M ) C 2 ( M ) C 3 ( M ) M 33 Outline of Presentation Problem statement Asynchronous events Mapping methodology Searching space Optimization by RT-PSA Algorithm Results Conclusions 34 Searching space In order to represent a mapping function M we adopt the formalism based on the Allocation Tables t(M) t(M) is a table with ur horizontal lines and DII vertical sectors Osi with i=1,2,...,DII Each Osi contains time steps Si+kDII (k=0, 1, 2, ...) which will be overlapped during the execution of P 35 Searching space Each node is assigned to a cell of t(M), i.e. it is associated to an HW resource and to a time step. For example, we consider the 23-node graph AsG1 36 Searching space AsG1 1 2 3 4 5 6 7 8 A A A A A A A A C C C C C C C C 9 10 11 12 13 14 15 16 A A A A A A A 17 18 19 20 21 22 23 37 Searching space For DII=3, a possible mapping M is described through the following t(M) OS1 OS2 OS3 S1 S4 S7 S2 S5 S8 S3 S6 S9 A1 n1 n6 n17 A2 n2 n7 n18 A3 n3 n8 n21 A4 n4 n19 n22 A5 n5 n20 n23 C1 n15 n9 n12 C2 n16 n10 n13 C3 n11 n14 38 Searching space An allocation table t(M) must respect both 1. Causality condition And the 2. Overlapping condition 39 Searching space We define the Ω searching space over which minimization of C(M) must be performed. Ω is the space containing all the feasible allocation tables: ={t(M) | t(M) is a feasible mapping}; t M t M is not feasible. 40 Searching space We can write the minimization problem in terms of the cost associated to the mapping M represented by the allocation table: C[t (M ' )] min C[t (M )] t ( M ) 41 Searching space We solve the problem by using a Simulated Annealing (SA) algorithm SA requires the generation of a sequence of points belonging to the searching space; each point of the sequence must be close, according to a given measure criterion, to its predecessor and to its successor. 42 Searching space As consists of allocation tables, we have to generate a sequence of allocation tables t(Mi)Neigh[t(Mi-1)] being Neigh[t(M)] the set of the allocation tables adjacent to t(M) according to some adjacency criteria 43 Searching space Searching space connection: Theorem 2. The searching space is connected adopting the adjacency conditions. 44 Outline of Presentation Problem statement Asynchronous events Mapping methodology Searching space Optimization by RT-PSA Algorithm Results Conclusions 45 Optimization by RT-PSA Algorithm We start from a feasible allocation table t(M0) We entrust in the optimization algorithm to find the wanted mapping M 46 Optimization by RT-PSA Algorithm We iterate over all the allowed values of DII The final result of the whole optimization process will be the allocation table characterized by minimal cost. 47 Outline of Presentation Problem statement Asynchronous events Mapping methodology Searching space Optimization by RT-PSA Algorithm Results Conclusions 48 Results In order to illustrate the results achievable through the presented RT- PSA algorithm, we consider the following graphs 49 Results SyG (1,A), (2,B), (3,B), (4,A), (5,B), (6,C), (7,A), (8,C), (9,E), (10,A), (11,C), (12,E), (13,B), (14,C), (15,E), (16,A), (17,B), (18,E), (19,C), (20,A), (21,C), (22,E), (23,A), (24,B), (25,C), (26,B), (27,B), (28,E), (29,A), (30,B), (31,E), (32,C), (33,B), (34,A), (35,C), (36,E), (37,B), (38,A), (39,E), (40,A), (41,B), (42,A), (43,C), (44,B), (45,E), (46,E), (47,B), (48,B), (49,A), (50,C). 50 Results AsG1 1 2 3 4 5 6 7 8 A A A A A A A A C C C C C C C C 9 10 11 12 13 14 15 16 A A A A A A A 17 18 19 20 21 22 23 51 Results AsG2 52 Results AsG3 53 Results We have N = NSy + NAs1 + NAs2 + NAs3 = 50 + 23 + 25 + 28 = 126 r1 = A, r2 = B, r3 = C, r4 = E The execution times and resources areas are T(A) = 10ut, Ar(A) = 10us T(B) = 20ut, Ar(B) = 10us T(C) = 30ut, Ar(C) = 13us T(E) = 40ut, Ar(E) = 15us Tr = 5ut, Ar(mr) = 1us 54 Results The input data interarrival period is Dt = 150ut We fix the pipeline clock cycle Tck = 50ut RTC are RTCAs1 = 300ut RTCAs2 = 250ut RTCAs3 = 350ut. The set of DII possible values is DIIp = {1, 3} 55 Results Results for DII = 3 Cost Fulfilled DII = 3 ur LSy Function RTC Starting 2667.942692 37 12 1 Final 3.999681 29 20 3 56 Results Results for DII = 1 Fulfilled DII = 1 Cost Function ur LSy RTC Starting 9.554314 104 10 3 Final 7.799348 79 18 3 57 Outline of Presentation Problem statement Asynchronous events Mapping methodology Searching space Optimization by RT-PSA Algorithm Results Conclusions 58 Conclusions We presented an algorithm to optimize the mapping, into a dedicated pipeline system, of a periodic task PSy and m mutually exclusive aperiodic tasks PAs1, PAs2, … PAsm subjected to real time (RT) constraints 59 Conclusions The algorithm, while searching for a mapping which satisfies all RT constraints of the aperiodic tasks, tries to minimize the number of HW resources needed to implement the system as well the length of the schedule. The mapping optimization is formulated as a minimization problem that has been solved through the Simulated Annealing algorithm. 60 Conclusions Mappings are represented through allocation tables. The searching space, as well adjacency criteria on it and a cost function evaluating the quality of a mapping have been defined. We demonstrated that the searching space containing all the feasible mappings is connected. 61 Remarks luca.baldini@ieee.org palazzari@casaccia.enea.it 62