Mapping of Periodic Tasks in Reconﬁgurable Heterogeneous Multi-core

Platforms

Aymen Gammoudi

1,2,3

, Daniel Chillet

, Mohamed Khalgui

2,4

and Adel Benzina

2,3

IRISA Lab, University of Rennes1, Rennes, France

LISI Lab, University of Carthage, Tunis, Tunisia

Tunisia Polytechnic School, University of Carthage, Tunis, Tunisia

School of Electrical and Information Engineering, Jinan University (Zhuhai Campus), Zhuhai 519070, China

Keywords:

Heterogeneous Multi-core Platform, Reconﬁguration, Task Mapping and Scheduling.

Abstract:

Multi-core Real-time Systems (MRS) powered by a battery have been adopted for a wide range of high per-

formance applications, such as mobile communication and automotive systems. A system is composed of

N dependent and periodic Operating System (OS) tasks to be assigned to p heterogeneous cores linked by a

network-on-chip (NoC). This paper deals with the problem of task allocation in MRS in such a way that the

cost of communication between cores is minimized by trying to place the dependent tasks as close as possible

to each other. The main objective is to develop a new strategy for allocating N tasks to p cores of a given

distributed system using task clustering by considering both the cost of inter task communication and that of

communication between cores. The proposed strategy guarantees that, when a task is mapped into the system

and accepted, then it is correctly executed prior to the task deadline. A novel periodic task model based on

elastic coefﬁcients is proposed to compute useful temporal parameters allowing to assign all tasks to p cores,

by minimizing the trafﬁc between cores. Experimental results reveal the effectiveness of the proposed strategy

by comparing the derived solutions with the optimal ones, obtained by solving an Integer Linear Program

(ILP).

1 INTRODUCTION

Multi-core real-time platforms typically are com-

posed of multiple processors (processing units),

memories, and a communication infrastructure. Het-

erogeneous multi-core platforms contain different

types of processing units. Therefore, the system de-

signers can take advantage of their properties when

mapping tasks to speciﬁc processor types and opti-

mize criteria such as computational performance, cost

and energy consumption (Schranzhofer et al., 2010).

Several academic and industrial studies reported in

(Wang et al., 2016), (Cecilio and Furtado, 2014),

(Li et al., 2014) have addressed the dynamic recon-

ﬁguration of real-time systems. These approaches

can be divided into two categories: manual applied

by users (Rooker et al., 2007); and automatic ap-

plied by intelligent control agents (Vrba and Marik,

2010). The most promising approach is the elastic

scheduling reported in (Wang et al., 2016), (Wang

et al., 2015), (Marinoni and Buttazzo, 2007), (But-

tazzo et al., 1998).

A multi-core real-time system, denoted MRS, can

be implemented by N reconﬁgurable real-time depen-

dent and periodic tasks to be assigned to p heteroge-

neous cores linked by a network-on-chip (NoC) (Hu

and Marculescu, 2005) in order to ensure the commu-

nication between dependent tasks. Since we consider

a heterogeneous platform, each task cannot be exe-

cuted by all cores, it can be supported only by one

or more speciﬁc cores according to system speciﬁ-

cation. Similarly to the research works reported in

(Wang et al., 2016), (Marinoni and Buttazzo, 2007),

we consider a more ﬂexible task model, in which

the tasks can operate within a given range of peri-

ods with different performances. A reconﬁguration

scenario is deﬁned in this paper as any internal or

external event that leads to the addition and/or re-

moval of periodic tasks as well as their exchanged

messages to adapt the system’s behavior to its envi-

ronment (Quadri et al., 2012). In this paper, we are

interested basically in reconﬁguration scenarios that

add new software tasks. New tasks must be assigned

to their appropriate cores by trying to place the depen-

Gammoudi, A., Chillet, D., Khalgui, M. and Benzina, A.

Mapping of Periodic Tasks in Reconﬁgurable Heterogeneous Multi-core Platforms.

DOI: 10.5220/0006698500990110

In Proceedings of the 13th International Conference on Evaluation of Novel Approaches to Software Engineering (ENASE 2018), pages 99-110

ISBN: 978-989-758-300-1

dent tasks as close as possible to each other in order

to reduce the trafﬁc on the NoC. Therefore the en-

ergy consumption, proportional to the data commu-

nication volume, will be decreased. As partitioned

scheduling converts the problem of scheduling real-

time tasks into a set of uniprocessor scheduling prob-

lems (Baruah and Goossens, 2004), we decided to se-

lect earliest deadline ﬁrst (EDF) scheduler to schedule

the local tasks for each core.

For two new added dependent tasks assigned to

different cores, a new periodic message is added au-

tomatically to the NoC. Consequently, the real-time

constraints can be not satisﬁed when a task misses

the related deadline and a message can take a long

time to arrive to its destination. Therefore, the system

MRS will be unfeasible. An MRS is feasible if and

only if it satisﬁes two constraints: (i) the deadlines

of the tasks, and (ii) the messages’ deadlines on the

communication medium. Moreover, the provided so-

lutions must be optimal real-time ones that minimize

the power consumption of system.

The general goal of this paper is to propose a new

mapping strategy based on grouping heavily commu-

nicating tasks into the same cluster. The tasks that are

grouped into the same cluster are assigned to the same

core while keeping its processor utilization coefﬁcient

less than or equal to 1 and the communication bus is

feasible. Whenever these constraints are not satisﬁed,

the proposed strategy offers two solutions (A and B)

based on the modiﬁcation of temporal parameters of

the processed tasks to meet the real-time and commu-

nication constraints. If after assigning the new task

to core C

, the real-time constraint is violated, then

the proposed strategy starts by modifying the periods

of tasks and the exchanged messages until reaching

their speciﬁc maximal periods (Solution A) in order

to decrease the processor utilization of C

. If the sec-

ond constraint is also violated after assigning one or

more new tasks, then the proposed strategy applies the

second solution (Solution B) that modiﬁes the mes-

sage’s period and the associated tasks to slow down

the communicating tasks to satisfy the communica-

tion constraint. To evaluate this strategy, we formalize

the problem as an optimization problem by using in-

teger linear programming (ILP) and compare the pro-

posed solutions with the optimal results provided by

the CPLEX solver (CPLEX, 2013). The originality of

this paper is to propose a new strategy of mapping and

scheduling tasks in reconﬁgurable architectures. The

proposed strategy looks for the near optimal place-

ment of tasks on cores after any reconﬁguration sce-

nario in order to minimize the trafﬁc on the NoC. The

proposed strategy is deterministic since it is able to

ﬁnd the suitable solutions for each situation.

The rest of this paper is organized as follows: Sec-

tion 2 reviews some related works. Section 3 presents

the system model and the used terminologies. We

formalize the considered problem in Section 4. The

proposed strategy is presented in Section 5 that deals

with the modiﬁcation of temporal parameters of the

processed tasks to guarantee the related real-time and

communication constraints after any reconﬁguration

scenario. The strategy is implemented, simulated and

analyzed in Section 6. Finally, Section 7 concludes

this work.

2 RELATED WORK

Reducing the cost of communication between cores

becomes a major concern for the high-performance

and reliability of such systems. Several academic

studies reported in (Carle et al., 2015), (Bhardwaj

and Kumar, 2013), (Tosun et al., 2009) have ad-

dressed the mapping algorithms using task clustering

by taking both inter task communication and execu-

tion costs. The basic idea of clustering based algo-

rithm in (Bhardwaj and Kumar, 2013), (Tosun et al.,

2009) is to group heavily communicating tasks into

the same cluster. The tasks that are grouped into the

same cluster are assigned to the same processor in

an effort to avoid communication costs. These algo-

rithms are interesting since the communication cost is

reduced, but the authors are not interested in the veri-

ﬁcation of the processor utilization after assigning the

tasks. In certain cases, the processor utilization ex-

ceeds 100%. In addition, they are not interested to

check the overﬂow on the communication medium.

To resolve these two problems, effective solutions

based on the modiﬁcation of WCETs, deadlines, and

periods of tasks in order to verify the processor uti-

lization after any assignment of tasks are reported in

(Gammoudi et al., 2015), (Wang et al., 2016), (Wang

et al., 2015), (Gharsellaoui et al., 2013), (Marinoni

and Buttazzo, 2007).

The work presented in (Wang et al., 2015) pro-

poses a feasible low-power dynamic reconﬁguration

of real-time systems where addition and removal of

tasks are applied at run-time. Three solutions are

presented to cut-down the energy consumption af-

ter any reconﬁguration scenario. The authors pro-

pose to prolong the periods (or reduce the WCETs)

of tasks by assigning a single value to all the tasks

in order to re-obtain the system feasibility. Recently,

the work reported in (Gammoudi et al., 2015) im-

proves these solutions by proposing a new approach

based on the modiﬁcation of task parameters with

classiﬁcation in packs by assigning a unique period

ENASE 2018 - 13th International Conference on Evaluation of Novel Approaches to Software Engineering

100

(or WCET) to all the tasks related to the same pack.

Each pack is a group of tasks having “similar” peri-

ods (or WCETs). For each reconﬁguration scenario,

speciﬁc modiﬁcations are performed on the parame-

ters of the packs and their related tasks in order to

meet the real-time and energy constraints. In the re-

search work reported in (Gammoudi et al., 2016c),

the authors develop the Recon f − Pack simulator to

evaluate and compare these two approaches by gen-

erating a large number of random systems and recon-

ﬁguration scenarios for each one. The authors show

that the total cost introduced by applying the solu-

tion presented in (Wang et al., 2015) in 88% of ran-

dom cases is largely higher than that introduced by

the method reported in (Gammoudi et al., 2015). To

show that the pack solution, proposed in (Gammoudi

et al., 2015), can be implemented in a real applica-

tion, the work reported in (Gammoudi et al., 2016a)

proposes a new reconﬁgurable middleware, named

Recon f − Middleware, that presents a plugin imple-

mented in RTLinux and describes the transition from

the pack theory to the implementation. An exten-

sion of the technique reported in (Gammoudi et al.,

2015) to homogeneous multi-core platforms is de-

scribed in (Gammoudi et al., 2016b). The authors

propose four solutions to schedule a real-time appli-

cation composed of independent periodic tasks under

energy constraints. Nevertheless, they do not consider

the scheduling of dependent tasks onto heterogeneous

platforms and they do not treat the inﬂuence of the

task mapping on the communication and therefore on

the energy consumption.

3 SYSTEM MODEL AND

ASSUMPTIONS

In this section, we present the task model and the

platform architecture that consists of a reconﬁgurable

MRS composed of N periodic OS tasks executed by

p heterogeneous cores. We also present the models

of the processor architecture and the communication

between cores.

3.1 Task Model

We suppose that MRS processes a task set Π contain-

ing N periodic tasks, i.e., Π = {τ

, τ

, ..., τ

}. Some

tasks are considered as ﬂexible, whose utilization can

be modiﬁed by changing their periods within a spec-

iﬁed range. According to (Wang et al., 2016), (Mari-

noni and Buttazzo, 2007), (Buttazzo et al., 1998), a

periodic task τ

is characterized by: (i) a release time

, (ii) a worst-case execution time (WCET) W

, (iii)

a relative deadline D

, (iv) a period T

, and (v) a max-

imum tolerable period T

max

. The relative deadline

of a periodic task is considered as a hard deadline

if its missing is unacceptable (Baruah and Goossens,

2004). According to (Liu and Layland, 1973), assum-

ing that the period is equal to the deadline for each

task, the tasks are schedulable if and only if the pro-

cessor utilization coefﬁcient is less than 1. Hence, we

apply in this paper the EDF policy to schedule the lo-

cal tasks for each core.

3.2 Platform Model

Let the considered MRS consists of a set of p het-

erogeneous cores Γ={C

, ...,C

}, interconnected

by communication links. Since we consider hetero-

geneous platforms, we suppose that each task τ

can

be executed by one or more speciﬁc cores according

to system speciﬁcation. Hence, we deﬁne the matrix

γ that represents the different possibilities to execute

tasks on cores.

i, j



1 τ

can be executed by C

0 otherwise.

In order to ensure that all the tasks are mapped on the

cores, we deﬁne the mapping matrix H where

i, j



1 if task τ

is running on core C

0 otherwise.

3.3 Communication Model

Each core runs periodic OS tasks which can exchange

messages on the NoC (Khemaissia et al., 2016), (Hu

and Marculescu, 2005). We denote in the following

by M

i, j

, the message to be exchanged between a pair

of tasks τ

and τ

. According to (Bui et al., 2012) the

communication model is based on: (i) a regular inter-

arrival time T M

i, j

, (ii) a spent time to transmit a mes-

sage W M

i, j

, (iii) an absolute deadline DM

i, j

, and (iv)

an amount of exchanged data SM

i, j

(in bits). We con-

sider that the period T M

i, j

of message M

i, j

is equal to

the period T

of the related sending task τ

(T M

i, j

If the task’s period is changed, then the periods of the

related messages will be changed implicitly.

The NoC architecture implies a communication cost

between each pair of cores depending on the distance

between them. We model each NoC architecture by a

speciﬁc cost matrix called Cost such as

Cost



if k 6= l, ∀ k, l ∈ [1..p].

0 otherwise.

where X

is the cost (e.g. energy) of sending one

bit from core C

to core C

To calculate the cost of communication between each

Mapping of Periodic Tasks in Reconﬁgurable Heterogeneous Multi-core Platforms

101

pair of cores C

and C

, it is necessary to multiply

the volume of exchanged data by Cost

. Then, the

total cost of communications (in units of cost) is given

TotalCost =

∑

k=1

∑

l=1

∑

i=1

∑

j=1

i, j

∗Cost

∗H

i,k

∗H

j,l

(1)

We suppose, in this paper, that the energy consumed

by the communication is proportional to the distance

between cores.

3.4 Processor Utilization Model

According to (Liu and Layland, 1973) the processor

utilization U

of core C

is given by

∑

i=1

∗ H

i, j

;∀ j ∈ [1..p] (2)

The core C

is feasible, if and only if

≤ 1;∀ j ∈ [1.. p] (3)

Eq. 3 is the real-time constraint.

According to the works reported in (Bui et al., 2012),

(Khemaissia et al., 2014), (Khemaissia et al., 2016),

to evaluate the communication feasibility between

two cores, the related medium is considered as a vir-

tual processor. Based on this hypothesis, the uti-

lization of the communication medium between two

cores C

and C

, denoted U

Com

), is given by

Com

) =

∑

i=1

∑

j=1

W M

i, j

T M

i, j

∗H

i,k

∗H

j,l

;∀k, l ∈ [1..p]

(4)

According to the work reported in (Khemaissia et al.,

2016), to ensure that the communication between

cores that are physically close is feasible, it is nec-

essary to check

Com

) ≤ 1; ∀k, l ∈ [1.. p] | Cost

= 1 (5)

Eq. 5 is the communication constraint.

4 PROBLEM FORMULATION

The problem, being addressed in this paper, is con-

cerned with an optimal allocation of the tasks to the

cores after any reconﬁguration scenario. An optimal

allocation is considered as one that minimizes the cost

of communication between cores such that the real-

time and communication constraints are satisﬁed. We

present, in this section, the formulation of the map-

ping problem and formalize each constraint as an op-

timization problem by using ILP.

4.1 Mapping Problem

We suppose that at time t

MRS is composed of Γ(t

) =

, ...,C

} and Π(t

) = {τ

, τ

, ..., τ

}. An MRS

is feasible if and only if it satisﬁes the two constraints

(real-time and communication constraints). We as-

sume in the following that MRS is dynamically re-

conﬁgured at time t

> t

) by adding n

tasks.

Therefore, the new implementation of tasks is Π(t

) =

{τ

, τ

, ..., τ

, τ

, ..., τ

}, with N = n

+ n

. The

new tasks provided by this reconﬁguration must be

mapped to their appropriate cores.

We consider, in this paper, that the initial mapping

at time t

is also a reconﬁguration scenario such that

Π(t

) = {∅} and Π(t

)6={∅}.

To guarantee that each task is executed at most by one

of its appropriate cores, it is necessary to satisfy

∑

j=1

i, j

∗ γ

i, j

= 1;∀i ∈ [1..N] (6)

We need to assign all the old/new tasks to their appro-

priate cores by minimizing the cost of communication

between them. This idea is formalized by











Minimize TotalCost

s.t.

≤ 1, ∀ j ∈ [1.. p] (Eq.3)

Com

) ≤ 1 | Cost

= 1 (Eq.5)

∀ k, l ∈ [1..p]

∑

j=1

i, j

∗ γ

i, j

= 1, ∀ i ∈ [1..N] (Eq.6)

where, TotalCost is calculated by Eq.1. Eq.3 satis-

ﬁes the real-time scheduling under the EDF policy

for each core, Eq.5 ensures that the communication

between the cores C

and C

is feasible and Eq.6 en-

sures that each task is executed at most by one of its

appropriate cores. The RE problem can be solved by

integer linear programming solvers such as CPLEX

(CPLEX, 2013), which is hard to be implemented in

an embedded platform since it is too complex to be

executed on-line. We note that this solver does not

produce any solution if one or more constraints are

not satisﬁed. To solve the problem RE, it is necessary

that all the constraints are satisﬁed. We suppose that

ILP is only used to compute the optimal solution to

have a comparison reference for the heuristic we will

propose later. After different reconﬁguration scenar-

ios, one or more of these constraints can be violated

and ILP cannot provide a solution for that situation.

ENASE 2018 - 13th International Conference on Evaluation of Novel Approaches to Software Engineering

102

4.2 Real-time Problem

After assigning one or more tasks on C

, the processor

utilization of C

will increase and it can be unfeasible

> 1). Since U

∑

i=1

∗ H

i, j

;∀ j ∈ [1..p], we

propose ﬁrst of all to modify the periods of the local

OS tasks in order to decrease the processor utilization.

The idea is to extend the periods of tasks until they

reach T

max

. Using the ILP model, we formalize this

problem by ∀ j ∈ [1..p] | U

> 1

Pb1 =











Minimize

∑

i=1

∆T

∗ H

i, j

s.t.

∑

i=1

+ ∆T

∗ H

i, j

≤ 1

+ ∆T

≤ T

max

, ∀ i ∈ [1..N]

where ∆T

is an integer that extends the period of the

real-time tasks. This formalization exploits the pos-

sible ﬂexibility of the periods in order to reduce the

processor utilization.

4.3 Communication Problem

After the addition of a pair of dependent tasks on two

cores, a periodic message will be added to the NoC

and should respect a corresponding deadline related

to these tasks. Then, it is necessary to verify the fea-

sibility of the communication between each pair of

cores C

and C

(k, l ∈ [1..p]). According to Eq. 5,

if U

Com

) is greater than 1, then the communi-

cation between these two cores is not feasible. In or-

der to solve this problem, we propose to modify the

period of messages exchanged between the tasks run-

ning on cores C

and C

. The idea here is to slow

down the communicating tasks to satisfy the commu-

nication constraint. We can formalize this problem by

∀ k, l ∈ [1..p] | U

Com

) > 1

Pb2 =











Minimize

∑

i=1

∑

j=1

∆T M

i, j

∗ H

i,k

∗ H

i,l

s.t.

∑

i=1

∑

j=1

W M

i, j

T M

i, j

+ ∆T M

i, j

∗ H

i,k

∗ H

i,l

≤ 1

T M

i, j

+ ∆T M

i, j

≤ T

max

, ∀ i, j ∈ [1..N]

where ∆T M

i, j

is an integer value that extends the pe-

riod of message M

i, j

(with M

i, j

Pb1 and Pb2 can be solved by CPLEX (CPLEX,

2013) that provides an optimal solution since it seeks

Figure 1: Example of application.

for each task and message the suitable and exact mod-

iﬁcation with a minimal cost. Nevertheless, it is not

reasonable to implement an ILP solver in an embed-

ded system. Therefore, we present in the next Section

some efﬁcient and implementable heuristics and we

compare them to the optimal solutions provided by

the CPLEX solver to solve these problems.

5 CONTRIBUTION

When the conﬁguration (or reconﬁguration) scenario

is applied, each new task must be assigned to its ap-

propriate core by minimizing the cost of communica-

tion. We present in the next subsection a mapping

strategy based on grouping heavily communicating

tasks into the same cluster. The tasks that are grouped

into the same cluster are assigned to the same core

such that its processor utilization coefﬁcient is less

than or equal to 1 and the communication bus is fea-

sible.

5.1 Proposed Task Allocation Algorithm

Before presenting the allocation algorithm, we intro-

duce some deﬁnitions:

SingleCore Task: any task that can be executed only

by one speciﬁc core. Example : τ

in Fig. 1.

Cluster (CL): a group of communicating tasks.

Example : τ

, τ

and τ

are grouped in CL

(Fig. 1).

Common Processor (CP): a processor that can exe-

cute all tasks of a cluster CL such that its processor

utilization is less than or equal to 1. Example : C

can

be the CP of CL

. C

and/or C

can be the CP of CL

(Fig. 1).

CPCL: we denote by CPCL each cluster CL that has

a common processor CP.

Data Trafﬁc (TD): is the total volume of data ex-

changed between tasks in CL. Example : T D(CL

) =

Mapping of Periodic Tasks in Reconﬁgurable Heterogeneous Multi-core Platforms

103

Figure 2: Global MappingTasks algorithm.

TU ∗ (

data1 + data2

) and T D(CL

) = 0 (Fig. 1).

The global structure of the proposed strategy is pre-

sented in Fig. 2 and is based on three main steps:

Step 1: Assign all the SingleCore tasks.

Step 2: Assigning the communicating tasks that have

a common processor CP:

Step 2.1: Each group of communicating tasks are

grouped in a cluster CL.

Step 2.2: Calculate the data trafﬁc (TD) for each

CL.

Step 2.3: Sort all the CL in a descending order

according to their TD.

Step 2.4: A CL that has a CP will be denoted by

CPCL. The list of all CPCLs called LCPCL.

Step 2.5: Assign the tasks of each CPCL.

Step 3: Assigning the tasks of CL as close as

possible.

Step 3.1: Take the CL that has a highest TD.

Step 3.2: Place each task on the best position (in

term of communication cost).

The detailed structure of the proposed strategy is

presented in Fig. 3. Its objective is to ﬁnd the near

optimal mapping of the tasks on cores in order to op-

timize the energy consumption while meeting tem-

poral and communication constraints. Communicat-

ing tasks are placed as close as possible. The pro-

posed strategy solves the problem formalized by RE

in sub-section 4.1. As shown in this ﬁgure, the pro-

posed strategy takes the task’s characteristics, the ma-

trix γ, the matrix of architecture Cost and the size of

exchanged messages as inputs and outputs the map-

ping of the tasks on the architecture.

After assigning the new tasks, the processor uti-

lization of certain cores will certainly increase. If

the new utilization is greater than 1, then the real-

time constraint is violated as reported in (Wang et al.,

2015), (Liu and Layland, 1973). Besides, the commu-

nication constraint can be violated after the addition

of tasks and consequently MRS will be unfeasible. To

solve these two problems that have been formalized

in 4.2 and 4.3, we present two solutions in the next

subsection.

5.2 Task Scheduling for Feasible

Reonﬁgurable Platforms

We propose two heuristics based on grouping the

tasks that have “similar” periods in several packs, de-

noted Pack, to re-obtain the system feasibility. After

a reconﬁguration, we do not calculate a new period

for each task, but assign a new one period to all tasks

of the ﬁrst pack Pack

1, j

( j ∈ [1..p]). Moreover, all

the new periods affected to the tasks of pack Pack

x, j

are multiple of the one affected to the tasks of Pk

1, j

Hence, we have only to compute the suitable value

1, j

. We show in the following solutions how the

tasks are grouped in packs.

5.2.1 Solution A: Modiﬁcation of Periods under

Real-Time Constraint

If C

( j ∈ [1..p]) is not feasible after assigning some

new tasks to it, then we propose to extend the periods

of tasks that run on C

. In order to satisfy the real-time

constraints, we assign each task to pack Pack

x, j

ac-

cording to its period. To construct the packs, we need

to ﬁnd the suitable new period Pk

1, j

that minimizes

the cost of the new solution for the whole system. We

formalize this problem by











Min

∑

i=1



(Pk

1, j

− (T

mod Pk

1, j

)) mod Pk

1, j



∗ H

i, j

s.t.

1, j

≥ Min(T

), ∀ i ∈ [1..N] | H

i, j

= 1

Once Pk

1, j

is calculated, we construct the packs of

the system tasks as follows: Pack

x, j

, x ≥ 1, includes

the tasks that have a period in the range [(x − 1) ∗

1, j

+ 1 ; x ∗ Pk

1, j

], x ∈ N

. The ﬁrst value of

x is 1 and when all tasks are affected, x is no more

ENASE 2018 - 13th International Conference on Evaluation of Novel Approaches to Software Engineering

104

Figure 3: MappingTasks algorithm.

incremented. Hence, Pack

1, j

includes the tasks that

have a period in the range [1 ; Pk

1, j

], Pack

2, j

includes the tasks that have a period in the range

[Pk

1, j

+ 1 ; 2 ∗ Pk

1, j

], etc.

To satisfy the real-time constraints under the EDF pol-

icy, it is necessary that

∑

i=1



∗ H

i, j



≤ 1. Since

the periods are now multiple of the same value Pk

1, j

we get

∑

∈Pack

1, j

+ ... +

∑

∈Pack

x, j

x.Pk

1, j

≤ 1

thus,

1, j

∗





∑

∈Pack

1, j

+ ... +

∑

∈Pack

x, j





≤ 1

then,

1, j

≥

∑

∈Pack

1, j

+ ... +

∑

∈Pack

x, j

Since the periods are integer values, then,

1, j





∑

∈Pack

1, j

+ ... +

∑

∈Pack

x, j





(7)

where Pk

1, j

is the new period affected to tasks of the

ﬁrst pack Pack

1, j

, x ∗ Pk

1, j

to the tasks of x

pack

Pack

x, j

. After adapting the tasks’ parameters running

on C

, all the tasks are executed without missing their

deadlines. Therefore, the real-time constraint can be

satisﬁed. As described in subsection 3.3, if the task’s

period is changed implicitly, then the message’s pe-

riod will be changed since T M

i, j

. For that, the pe-

riods of messages that are associated to the adapted

tasks will be changed.

5.2.2 Solution B: Modiﬁcation of Message’s

Period under Communication Constraint

For two added dependent tasks assigned to different

processors, a message is added automatically on the

NoC, we are interested in this part in the communi-

cation feasibility. If U

Com

) (∀ k, l ∈ [1..p]) is

greater than 1, then the communication between the

cores is not feasible. We propose to extend the peri-

ods of exchanged messages between the tasks running

on cores C

and C

. In order to satisfy the communi-

cation constraint, we assign each message to Pack

T M

x,k,l

according to its period. We formalize this problem by

Mapping of Periodic Tasks in Reconﬁgurable Heterogeneous Multi-core Platforms

105

∀ k, l ∈ [1..p] | U

Com

) > 1











Min

∑

i=1

∑

j=1

(Pk

T M

1,k,l

− (T M

i, j

mod Pk

T M

1,k,l

)) mod

T M

1,k,l

∗ H

i,k

∗ H

i,l

s.t.

T M

1,k,l

≥ Min(T M

i, j

), ∀ i, j ∈ [1..N]

Once Pk

T M

1,k,l

is calculated, the packs of messages are

processed such that Pack

T M

x,k,l

includes the messages

that have a period in the range [(x − 1) ∗ Pk

T M

x,k,l

1 ; x ∗ Pk

T M

1,k,l

], x ∈ N

To satisfy the communication constraint, it is neces-

sary that

∑

i=1

∑

j=1



W M

i, j

T M

i, j

∗ H

i,k

∗ H

j,l



≤ 1. Since

the periods T M

i, j

are now multiple of the same value

T M

1,k,l

, we get

∑

i, j

∈Pack

T M

1,k,l

W M

i, j

T M

1,k,l

+ ... +

∑

i, j

∈Pack

T M

x,k,l

W M

i, j

x.Pk

T M

1,k,l

≤ 1

Since the periods are integer, then,

T M

1,k,l





∑

i, j

∈Pack

T M

1,k,l

W M

i, j

+ ... +

∑

i, j

∈Pack

T M

x,k,l

W M

i, j





(8)

where Pk

T M

1,k,l

is the new period affected to the mes-

sages of the ﬁrst pack Pack

T M

1,k,l

, x ∗ Pk

T M

x,k,l

to the mes-

sages of x

pack Pack

T M

x,k,l

. After adapting the param-

eters of messages running on the communication bus

between cores C

and C

, U

Com

) will be smaller

than 1. Therefore, the communication constraint can

be satisﬁed. Since the periods of messages are modi-

ﬁed, it is necessary to adapt the periods of their send-

ing tasks.

6 EXPERIMENTAL STUDY

We present in the ﬁrst subsection a case study in

which we calculate the communication cost through

the use of the proposed strategy. In the second, we

evaluate the performance of the strategy by generat-

ing a large number of random tasks.

6.1 Implementation of the Proposed

Strategy

Example - 1 In this example, we have considered

a typical program made up of 10 executable tasks

Figure 4: Inter task communication graph of example 1 (in

kilo bits).

{τ

, τ

, ..., τ

} to be executed on a MRS having three

cores {C

}. The tasks’ characteristics are de-

scribed in Tab. 1 and the inter task communication

graph is illustrated in Fig. 4. The cores connections

matrix Cost is shown as follow:

Cost =

↓ ↓ ↓





0 1 1

1 0 1

1 1 0





← C

Table 1: Tasks’ characteristics.

Task T

max

5 6

6 8

10 10

4 6

12 22

5 16

8 21

8 32

6 14

7 12

The WCET W

of each task is presented in Tab. 2

Table 2: WCET W

of tasks.

WCET C

1 3 0

1 0 2

2 0 0

1 2 0

2 0 0

0 2 0

0 0 3

0 2 1

0 0 2

0 2 1

The matrix γ

i, j

that represents the different possi-

ENASE 2018 - 13th International Conference on Evaluation of Novel Approaches to Software Engineering

106

bilities to execute tasks on cores is

γ =

↓ ↓ ↓







1 1 0

1 0 1

1 0 0

1 1 0

1 0 0

0 1 0

0 0 1

0 1 1

0 0 1

0 1 1







← τ

Step 1 Assign the SingleCore tasks:

, τ

and τ

are SingleCore tasks and they

must be assigned.

Step 2 Assign the tasks of CPCL to their related com-

mon processor CP:

Tab. 3 shows the clusters CPCL that have common

processors CP and their data trafﬁc T D.

Table 3: CPCL and CL clusters.

Clusters Tasks TD CP

CPCL

, τ

13 C

CPCL

, τ

10 C

, τ

14 -

0 C

, C

The proposed strategy assigns CPCL

and CPCL

on C

Step 3 Assign the tasks of the CL:

We start to assign the tasks of CL

because it has the

highest T D. Since τ

and τ

are already mapped in

step 1 to C

, the proposed strategy assigns τ

to C

(same core of the related task τ

The output of the proposed strategy is the mapping

matix H

H =

↓ ↓ ↓







1 0 0

0 1 0

0 0 1

0 1 0

0 0 1

0 1 0







← τ

The total cost of communication TotalCost is 40 units

of cost.

The processor utilization of the cores after assigning

tasks is presented in Tab. 4.

Table 4: Processor utilization for each core.

Core

Processor utilization U 0.983 0.935 0.708

At a particular time, the system undergoes the ﬁrst

reconﬁguration by adding two tasks that are presented

in Tab. 5.

Table 5: Added tasks.

Task T

max

12 20

4 15

The WCET W

of each new task is presented in

Tab. 6

Table 6: WCET W

of each new task.

WCET C

0 2 0

1 0 1

γ =

↓ ↓ ↓



0 1 0

1 0 1



← τ

The new inter task communication graph is illustrated

in Fig. 5.

Figure 5: Inter task communication graph application’s ex-

ample after adding tasks (in kilo bits).

After assigning τ

on C

, since it is a SingleCore

task, the processor utilization U

= 1.102 > 1. The

proposed strategy decreases U

by applying Solution

A, i.e., U

equals to 0.933, and the new periods of

tasks running on C

are presented in Tab. 7.

Since the task τ

communicates with τ

, then the

best position to place it, is C

(same core to τ

). Thus,

the processor utilization of C

after assigning τ

will

be equal to 1.233. The proposed strategy begins by

applying Solution A to decrease the processor utiliza-

tion. After calculating the new periods, it appears that

Mapping of Periodic Tasks in Reconﬁgurable Heterogeneous Multi-core Platforms

107

Table 7: Modiﬁcation of period’s tasks running on C

Task T

max

5 16

10 32

10 12

15 20

Solution A can not increase the periods of tasks run-

ning on C

because their T

max

are not enough. Conse-

quently, the proposed strategy proposes to assign τ

to the next core C

and U

= 0.958.

Thus, the new mapping matrix H is given by

H =

↓ ↓ ↓







1 0 0

0 1 0

0 0 1

0 1 0

0 0 1

0 1 0

0 0 1







← τ

The new total cost of communication TotalCost is

equal to 40+20=60 units of cost.

6.2 Evaluation of Performance

In order to generalize the performance evaluation of

the proposed strategy, we generate randomly tasks by

using the developed tool Task −Generator presented

in (Gammoudi et al., 2016c). The proposed strategy

tries to form clusters of tasks and then allocate these

clusters to the cores. The effectiveness of the pro-

posed algorithm is compared with the optimal solu-

tion provided by the CPLEX solver (CPLEX, 2013).

For several sets of input data (N, p), the comparison

is shown in a tabular form as well as in a graphical

form. Tabs. 8 and 9 illustrated in the form of Figs. 6

and 7 respectively.

It can be observed from Fig. 6 that the values of

total cost obtained by the proposed strategy are near

to those obtained by the optimal solution, in the case,

when the number of cores is kept ﬁxed and number

of tasks are taken in increasing order. The similar ob-

servation can also be made from Fig. 7 in the case

when the number of tasks is ﬁxed and that of cores is

taken in an increased order. Thus, it is concluded that

the proposed strategy results in a near optimal cost in

both cases.

Table 8: Communication cost when number of tasks in-

creases.

Tasks N Cores p Proposed algo Optimal sol

10 4 68 66

11 4 92 88

12 4 131 126

13 4 146 138

14 4 188 177

15 4 235 219

16 4 292 280

17 4 355 336

Table 9: Communication cost when number of cores in-

creases.

Tasks N Cores p Proposed algo Optimal sol

17 4 355 336

17 5 385 358

17 6 372 359

17 7 385 361

17 8 390 372

17 9 402 386

17 10 423 399

17 11 430 415

Figure 6: Communication cost of our strategy when tasks

are in increasing order and number of cores is 4.

Figure 7: Communication cost of our strategy when cores

are in increasing order and number of tasks is 17.

7 CONCLUSIONS

Task allocation problem for reconﬁgurable multi-

core systems, using task clustering, is discussed.

ENASE 2018 - 13th International Conference on Evaluation of Novel Approaches to Software Engineering

108

A novel task model based on elastic coefﬁcients is

proposed. This is to adapt task parameters allowing

task allocation and reconﬁguration while minimizing

communication costs. The task allocation problem

is known to be NP-hard. When considering com-

munication costs, the proposed technique gives a

near optimal solution while respecting real-time

and communication constraints. Optimal solution

is obtained by formulating the problem as an ILP

problem and we compare the results of the proposed

technique with the optimal solution. After some

evaluations, we conclude that the proposed technique

results in a near optimal ones provided by the CPLEX

solver.

In a future work, we will focus on the implemen-

tation of the paper’s contribution in a real-time oper-

ating system that will be evaluated by assuming real

case studies.

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