1 synthesis of pipelined systems for the contemporaneous execution of periodic and aperiodic tasks...

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SYNTHESIS of PIPELINED SYSTEMS for the

CONTEMPORANEOUS EXECUTION of PERIODIC

and APERIODIC TASKS with HARD REAL-TIME CONSTRAINTS

Paolo PalazzariLuca Baldini

Moreno Coli

ENEA – Computing and Modeling UnitUniversity “La Sapienza” – Electronic Engineering Dep’t

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Outline of Presentation Problem statement Asynchronous events Mapping methodology Searching space Optimization by RT-PSA Algorithm Results Conclusions

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Problem Statement We want to synthesize a

synchronous pipelined system which executes both the task PSy , sustaining its throughput, and m mutually exclusive tasks PAs

1, PAs2

, …, PAs

m whose activations are randomly triggered and whose results must be produced within a prefixed time.

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Problem Statement We represent the tasks as Control Data Flow

Graphs (CDFG) G = (N, E)

N = {n1, n2, …, nN}: operations of the task

E=

(data and control dependencies)

ijjiji nnnnnn on dependent data/ctrl is ,, | , N

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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 = {RTCAs

1, RTCAs2, ..., RTCAs

m}, where RTCAs

i contains the RTC on the ith aperiodic task.

Input data for the synchronous task PSy arrive with frequency fi = 1/t, being t the period characterizing PSy.

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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 PAs

1, PAs2, …, PAs

m satisfy the constraints in RTCAs.

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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.

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Problem Statement Example of a pipeline serving the

synchronous task PSy

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DATA INTRODUCTION INTERVAL

DII = 2

0 100 200 300 400 500 600 700 800 900 1000

t (ut)

Tck = 50 ut

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Problem Statement

Sk = (k-1)Tck and Sk = kTck

In a pipeline with L stages, SL denotes the last stage.

DII = t/Tck

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Problem Statement

We assume the absence of synchronization delays due to control or data dependencies:

Throughput of the pipeline system =1/DII.

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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

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Asynchronous eventsIn red the asynchronous service requests in a pipelined system.

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t (ut) t0A1 t0A2 t0A3 t0A4 t0A5

(I0A1) (I0A2) (I0A3) (I0A4) (I0A5)

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Asynchronous eventsLike the synchronous events, we represent the asynchronous events

{PAs1, PAs

2, ..., PAs

m}

through a set of CDFG

ASG = {AsG1(NAs1,EAs

1), ... , AsGm(NAsm,EAs

m)}

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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, EAs

1)

AsG2(NAs2, EAs

2) ..……. AsGm(NAsm, EAs

m)

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Asynchronous eventsAperiodic tasks are subjected to Real-Time constraints (RTC):

As all RTC must be respected, mapping function M has to define a scheduling so that

- i 0 RTCAsi RTCAs

iiii

Li

As AsSRTC by finish must execution AsP|,

iLAs

S

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Mapping methodology

In order to develop a pipeline system implementing G

a HW resource rj = D(nj)

anda time step Sk

must be associated to each nj N

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Mapping methodology

We must determine the mapping functionM: N UR S

UR is the set of the used HW resources (each rj is replicated kj times),

p1

j

kp

2p

1p

k22

22

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k1

21

11

p21jk

j

r,...,r,r,...,r,...,r,r,r,...,r,r

UR,...,UR,URRrr

UR

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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

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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

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Mapping methodology The mapping is determined by solving

the following minimization problem:

)()()(min)'( 321 MCMCMCMCMCM

C1(M) is responsible of the fulfillment of all the RTCC2(M) minimizes the used silicon area

C3(M) minimizes the length of the pipeline.

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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.

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Mapping methodology

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DII = 2

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(I0A1) (I0A2) (I0A3) (I0A4) (I0A5)

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Mapping methodology

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Mapping methodologyIn a pipelined system with DII=1

the used resource set is maximumthe execution time of each AsGi on the pipeline is minimumA 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

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Mapping methodology

Maximum value allowed for DII, compatible with all the RTCAs

iRTCAs:

LpAsi Tck gives the minimal

execution time for AsGi

The deadline associated to AsGi is i.

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Mapping methodologyMaximum 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.

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Mapping methodologyMaximum value allowed for DII, compatible

with all the RTCAsiRTCAs (continued):

A necessary condition to match all the RTCAs

iRTCAs 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.

i DII Tck + LpAsiTck , i = 1, 2, ..., m

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Mapping methodology

Combining previous relations with a congruence condition between the period of the synchronous task (t) and the clock period (Tck), we obtain the set DIIp wich contains all the admissible DII values.

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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

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Mapping methodologySteps 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

)()()(min)'( 321 MCMCMCMCMCM

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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

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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

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Searching space

A A A A A A A A

A A A A A A A

C C C C C C C C

1 2 3 4 5 6 7 8

9 10 11 12 13 14 15 16

17 18 19 20 21 22 23

AsG1

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Searching space For DII=3, a possible mapping M is

described through the following t(M) OS1 OS2 OS3

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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

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Searching spaceAn allocation table t(M) must respect both

1. Causality condition

And the

2. Overlapping condition

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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};

is not feasible.

MtMt

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Searching space We can write the minimization

problem in terms of the cost associated to the mapping M represented by the allocation table:

)]([min)]'([ )(

MtCMtCMt

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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.

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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

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Searching spaceSearching space connection:

Theorem 2. The searching space is connected adopting the adjacency conditions.

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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

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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.

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Results

In order to illustrate the results achievable through the presented RT-PSA algorithm, we consider the following graphs

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Results

(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).

SyG

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Results

A A A A A A A A

A A A A A A A

C C C C C C C C

1 2 3 4 5 6 7 8

9 10 11 12 13 14 15 16

17 18 19 20 21 22 23

AsG1

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Results

14 A

10

B 15

J 17 18

D A 11

6

A 1

D

B

4

2 A

J

A 16

B

J

D

A A B

J

19

20

21

22

23 24

B 12 13 J

8 D 7

D

A

5

B 9

B 3

25

AsG2

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Results

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Results We have

N = NSy + NAs1 + NAs

2 + 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) = 10usT(B) = 20ut, Ar(B) = 10usT(C) = 30ut, Ar(C) = 13usT(E) = 40ut, Ar(E) = 15usTr = 5ut, Ar(mr) = 1us

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Results The input data interarrival period is t =

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}

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Results Results for DII = 3

DII = 3 Cost

Function ur LSy

Fulfilled RTC

Starting 2667.942692 37 12 1

Final 3.999681 29 20 3

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Results Results for DII = 1

DII = 1 Cost Function ur LSy Fulfilled

RTC Starting 9.554314 104 10 3

Final 7.799348 79 18 3

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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 PAs

1, PAs2

, … PAs

m subjected to real time (RT) constraints

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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.

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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.

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Remarks

[email protected]@casaccia.enea.it

1 synthesis of pipelined systems for the contemporaneous execution of periodic and aperiodic tasks...

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