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Dynamic Multiple-Period Reconfiguration of
Real-Time Scheduling Based on Timed DES
Supervisory Control
Xi Wang, ZhiWu Li, W. M. Wonham
Version
Post-print/accepted manuscript
Citation
(published version)
Xi Wang, Zhiwu Li, W. M. Wonham. Dynamic multiple-period
reconfiguration of real-time scheduling based on timed DES
supervisory control. IEEE Trans. on Industrial Informatics 12(1)
February 2016, pp.101-111.
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1
Dynamic Multiple-Period Reconfiguration of
Real-Time Scheduling Based on Timed DES
Supervisory Control
Abstract—Based on the supervisory control theory (SCT)1
of timed discrete-event systems (TDES), this study presents a2
dynamic reconfiguration technique for real-time scheduling of3
real-time systems running on uni-processors. A new formalism is4
developed to assign periodic tasks with multiple-periods. By im-5
plementing SCT, a real-time system (RTS) is dynamically recon-6
figured when its initial safe execution sequence set is empty. Dur-7
ing the reconfiguration process, based on the multiple-periods, the8
supervisor proposes different safe execution sequences. Two real-9
world examples illustrate that the presented approach provides10
an increased number of safe execution sequences as compared11
to the earliest-deadline-first (EDF) scheduling algorithm.12
Index Terms—Real-time system, timed discrete-event system,13
supervisory control, dynamic reconfiguration, non-preemptive14
scheduling.15
I. INTRODUCTION16
In [1], Liu and Layand define a periodic task model with17
a deadline equal to its period, which we refer to as the Liu-18
Layand (LL) model. Thereafter, Nassor and Bres propose a19
new task model in [2], with a deadline less than or equal20
to its period, which we refer to as the Nassor-Bres (NB)21
model. Currently, the most widely-used scheduling algorithms22
for hard periodic real-time systems (RTS) running on a uni-23
processor are fixed priority (FP) scheduling and earliest-24
deadline-first (EDF) scheduling algorithms [1]. Moreover, an25
RTS can be scheduled in a preemptive or non-preemptive26
mode [3], [4]. In real-time scheduling theory, these widely27
applied algorithms provide at most one schedulable sequence28
for an RTS to meet the hard deadlines. For non-preemptive29
scheduling of an RTS that executes the NB model tasks,30
Chen and Wonham [5] propose a timed discrete-event system31
(TDES)-based task model, which we refer to as the Chen-32
Wonham (CW) model, and a real-time scheduling technique.33
Based on supervisory control theory (SCT), all safe execution34
sequences are generated by the TDES supervisor, from which35
the user chooses a preferred sequence to schedule the RTS.36
The RTS is claimed to be non-schedulable if the supervisor37
is empty. Based on the LL model and SCT, a priority-38
based and preemptive real-time scheduling policy and a task39
model, which we refer to as the Janarthanan-Gohari-Saffar40
(JGS) model, are proposed by Janarthanan et al. in [6]. The41
work in [5] and [6] is a significant improvement over real-42
time scheduling. However, the authors did not reconfigure the43
system in case of non-schedulability.44
In [7]–[10], an elastic period task model is proposed to45
handle the overload of an RTS by decreasing the task processor46
utilization. Moreover, the supremal controller found by SCT47
provides the RTS with all the safe execution sequences [5].48
Building on the two latter studies, we present a new modeling 49
technique to endow the real-time tasks represented by the CW 50
and JGS models with multiple-periods. To handle the overload 51
of an RTS, SCT is utilized to find all the possible solutions 52
based on different periods of each task. For each solution, all 53
the safe execution sequences are provided. 54
Dynamic reconfiguration in the present study consists of 55
two steps: 1) the initial model of each task is assigned with 56
the shortest period (the highest processor utilization), and by 57
utilizing SCT, all the RTS’ safe execution sequences (if any) 58
are found; 2) for the purpose of reconfiguring the RTS in case 59
of non-schedulability, this study reconfigures the RTS’ com- 60
posite task model by assigning to the tasks multiple-periods. 61
The multiple-period provides multiple processor utilization for 62
each task. Thereafter, a processor utilization interval for the 63
RTS is obtained. SCT is utilized again to find all the safe 64
execution sequences (possible reconfiguration scenarios) in the 65
predefined processor utilization interval. If the supervisor is 66
still empty, we claim that the RTS is non-schedulable. Two 67
real-world examples are implemented in this study. The results 68
illustrate that, in the dynamic reconfiguration approach, the 69
presented method finds a set of safe execution sequences. 70
The rest of this paper is structured as follows. The state 71
of the art is reviewed in Section II. Section III presents the 72
terminology used throughout the paper. The multiple-period 73
TDES model for RTS is defined in Section IV. Section V re- 74
ports methodologies of supervisory control and reconfiguration 75
of RTS. A real-world example is implemented in Section VI 76
to verify the supervisory control and reconfiguration. Further 77
relevant issues are discussed in Section VII. Conclusions are 78
provided in Section VIII. 79
II. STATE OF THE ART 80
In a periodic RTS, a permanent overload condition occurs 81
if the processor utilization is greater than one [11]. In this 82
case, the RTS needs to be reconfigured. In recent years several 83
academic and industrial studies [12]–[14] have addressed the 84
dynamic reconfiguration of RTS. These approaches can be 85
divided into two categories: manual, applied by users [15], 86
and automatic, applied by intelligent control agents [16]. 87
The most widely used overload management approaches are: 88
elastic scheduling [7]–[10] and job skipping [17]. In real-time 89
scheduling, effective solutions for reconfiguration based on 90
sensitivity approach of worst-case execution times (WCET), 91
deadlines, and periods of tasks are reviewed in [18]. These 92
solutions are utilized to reconfigure the RTS scheduled by 93
FP real-time scheduling. There is no reconfiguration result 94
2
based on the sensitivity approach to reconfigure the tasks’95
periods for dynamic-priority real-time scheduling [18]. Based96
on SCT, this study presents a new dynamic reconfiguration97
technique to reconfigure the RTS when they are claimed98
to be non-schedulable under the approaches in [5] or [6].99
Unlike traditional real-time scheduling and reconfiguration100
via the calculation of processor utilization, processor demand101
[19], and on-line monitoring to provide one safe execution102
sequence, an off-line technique is presented: in a predefined103
processor utilization interval, based on SCT, all the safe104
execution sequences (possible reconfiguration scenarios) are105
found.106
III. CONCEPTS AND TERMINOLOGY107
A. Preliminaries on TDES108
In the language-based Ramadge-Wonham (RW) framework109
[20], [21], a finite discrete-event system (DES) is represented110
by a state machine G = (Q, Σ, δ, q
0
, Q
m
), where Q is the111
state set, Σ is the event set, δ: Q × Σ → Q is the (partial)112
state transition function, q
0
is the initial state, and Q
m
is113
the marker state set satisfying Q
m
⊆ Q. Let Σ
+
(resp., ǫ)114
denote the set of all finite sequences over Σ (resp., empty115
string). We have Σ
∗
= Σ
+
∪ {ǫ}. A plant and a specification116
are represented by G and S, respectively. In [22], by adjoin-117
ing to the RW framework time bounds on the transitions,118
G starts from an (untimed) activity transition graph (ATG)119
G
act
= (A, Σ
act
, δ
act
, a
0
, A
m
) with Σ := Σ
act
˙
∪{tick}. The120
elements of the activity set A are “activities”, denoted by a.121
Σ
act
is partitioned into two subsets, Σ
act
= Σ
spe
˙
∪Σ
rem
,122
where Σ
spe
(resp. Σ
rem
) is the prospective (resp. remote)123
event set with finite (resp. infinite) upper time bounds [21].124
By defining the timer interval for σ, represented by T
σ
, to be125
[0, u
σ
] or [0 , l
σ
] for σ ∈ Σ
spe
and σ ∈ Σ
rem
, respectively,126
the initial state is q
0
:= (a
0
, { t
σ0
|σ ∈ Σ
act
}), where t
σ0
127
equals u
σ
or l
σ
for a prospective or remote state, respectively.128
The marker state set is Q
m
⊆ A
m
×
Q
{T
σ
|σ ∈ Σ
act
}.129
Thus a TDES is represented by G = (Q, Σ, δ, q
0
, Q
m
). An130
event σ ∈ Σ
act
is enabled at q if δ
act
(a, σ) is defined,131
written δ
act
(a, σ)!; it is eligible if its transition δ(q, σ) is also132
defined, i.e., δ(q, σ )!. The closed behavior of G is the language133
L(G) := {s ∈ Σ
∗
|δ(q
0
, s)!}. In addition, the marked behavior134
of G is L
m
(G) := {s ∈ L(G)|δ(q
0
, s) ∈ Q
m
}. G is non-135
blocking if L
m
(G) satisfies L
m
(G) = L(G), where L
m
(G)136
is the (prefix) closure of L(G). In a TDES plant G, the eligible137
event set Elig
G
(s) ⊆ Σ at a state q corresponding to a string138
s ∈ L(G) is defined by El ig
G
(s) := {σ ∈ Σ|sσ ∈ L(G)}.139
For an arbitrary language K ⊆ L(G), let s ∈ K, Elig
K
(s) :=140
{σ ∈ Σ|sσ ∈ K}. The set of all controllable sublanguages of141
K is denoted by C(K); this family is nonempty (the empty142
set belongs) and is closed under arbitrary set unions. Hence, a143
unique supremal (i.e., largest) element exists, and is denoted by144
supC(K). Considering a specification language E ⊆ Σ
∗
, there145
exists an optimal monolithic supervisor S. Its closed behavior146
is L(S) = L
m
(S), where L
m
(S) is the marked behavior147
represented by L
m
(S) =supC(E ∩ L
m
(G)) ⊆ L
m
(G).148
B. Synchronous product 149
Suppose that we have a set of generators G
i
with i ∈ n = 150
{1, 2, . . . , n}. In accordance with [21], the behavior of G
i
is 151
represented by language L
i
. Synchronous product [21] is a 152
standard way to combine several DES into a single and more 153
complex one. Suppose that we have two languages L
1
⊆ Σ
∗
1
154
and L
2
⊆ Σ
∗
2
with Σ = Σ
1
∪ Σ
2
. The natural projection 155
P
i
: Σ
∗
→ Σ
∗
i
is defined by 156
• P
i
(ǫ) = ǫ, 157
• P
i
(σ) =
ǫ, if σ /∈ Σ
i
σ, if σ ∈ Σ
i
, 158
• P
i
(sσ) = P
i
(s)P
i
(σ), s ∈ Σ
∗
, σ ∈ Σ. 159
The inverse image function of P
i
is 160
P
−1
i
: P wr(Σ
∗
i
) → P wr(Σ
∗
). 161
For H ⊆ Σ
∗
i
, 162
P
−1
i
(H) := {s ∈ Σ
∗
|P
i
(s) ∈ H}. 163
The synchronous product of L
1
and L
2
, denoted by L
1
||L
2
, 164
is defined as 165
L
1
||L
2
:= P
−1
1
L
1
∩ P
−1
2
L
2
. 166
C. System Model 167
Suppose that a periodic RTS S processes n tasks, i.e., S = 168
{τ
1
, τ
2
, . . . , τ
n
}, i ∈ n. Assume also that this set contains 169
at least one task with a multiple-period, namely one having 170
a lower and upper (non-negative integral time) bound. The 171
execution model of such a system is a set of tasks processed 172
in a uni-processor, in which a task τ
i
is described by τ
i
= 173
(R
i
, C
i
, D
i
, T
i
) with 174
• release time R
i
, 175
• WCET C
i
, 176
• hard deadline D
i
, and 177
• multiple-period T
i
. 178
An RTS is a synchronous system [19] in case all the processed 179
tasks are released at the same time, namely R
i
= 0. In this 180
research the RTS is synchronous. A deadline is hard if its 181
violation is unacceptable. A multiple-period is a period set 182
containing several possible periods: the lower bound (i.e., 183
shortest one) is represented by T
i
min
, and the upper bound 184
(i.e., longest one) is represented by T
i
max
. Thus, we have 185
T
i
= [T
i
min
, T
i
max
]. 186
During the real-time scheduling process, for task τ
i
, only 187
one period T satisfying T
i
min
≤ T ≤ T
i
max
is selected in 188
each scheduling period. The processor utilization U
i
of task 189
τ
i
is calculated by 190
U
i
= C
i
/T.
191
The total processor utilization of S is U
S
=
n
P
i=1
U
i
. An RTS S 192
is not schedulable in case U
S
> 1 [11]. 193
Task τ
i
consists of an infinite sequence of jobs J
i,j
= 194
(r
i,j
, C
i
, d
i,j
, p
i,j
) repeated periodically. The absolute dead- 195
line d
i,j
denotes the global clock time at which the execution 196
of J
i,j
must be completed. Similarly, we define the absolute 197
3
release time (resp. period) r
i,j
(resp. p
i,j
) to mean the global198
clock time at which τ
i
must be released (resp. start the199
next period). The subscript “i, j” of J
i,j
represents the j-th200
execution of task τ
i
. For each j, J
i,j
requests the processor at201
global clock time r
i,j
. Moreover, the execution of J
i,j
takes C
i
202
ticks, which must be completed no later than d
i,j
. The absolute203
deadline d
i,j
occurs no later than the absolute period p
i,j
. The204
EDF scheduling algorithm [1] assigns the priority of each job205
based on the absolute deadlines: the earlier the deadline, the206
higher is the job’s priority. The EDF scheduling algorithm207
can be utilized to schedule RTS. At each time unit, the job208
with the highest priority enters the processor. If the execution209
of a job is allowed to be preempted by other jobs before its210
execution finishes, the scheduling is preemptive; otherwise, it211
is non-preemptive.212
IV. TDES MODEL FOR REAL-TIME SYSTEMS213
A. CW Model214
The CW model [5] represents a real-time periodic task τ
i
=215
(R
i
, C
i
, D
i
, T
i
), i ∈ n, with D
i
≤ T
i
, by a TDES G
i
=216
(Q
i
, Σ
i
, δ
i
, q
0i
, Q
mi
). As depicted in Fig. 1, the corresponding217
ATG part is G
act
= (A
i
, Σ
acti
, δ
acti
, a
0i
, A
mi
) with218
• A
i
= {Y
i
, I
i
, W
i
},219
• Σ
acti
= {γ
i
, α
i
, β
i
},220
• δ
acti
: A
acti
× Σ
acti
→ A
acti
with221
– δ
acti
(Y
i
, γ
i
) = I
i
,222
– δ
acti
(I
i
, α
i
) = W
i
, and223
– δ
acti
(W
i
, β
i
) = Y
i
.224
• a
0i
= Y
i
, and225
• A
mi
= {Y
i
}.226
i
J
i
Y
i
I
i
W
i
D
i
E
Fig. 1: ATG of a real-time task.
States Y
i
, I
i
, and W
i
represent that task τ
i
is at states delay,227
idle, and work, respectively. The events in the alphabet Σ
i
are228
• γ
i
: the event that τ
i
is released,229
• α
i
: the execution of τ
i
is started, and230
• β
i
: the execution of τ
i
is finished.231
Event α
i
is controllable and events γ
i
and β
i
are uncontrol-232
lable. Moreover, all the events in Σ
acti
are forcible. Suppose233
that, after enabling, events γ
i
, α
i
, and β
i
should wait for234
t
γ
i
, t
α
i
, and t
β
i
ticks, respectively, until they are eligible to235
occur. Thus, t
α
i
is the time at which τ
i
starts its execution.236
Furthermore, in the CW model, t
β
i
= C
i
. A CW model has237
the following two features: 1) γ
i
signals that after r
i,1
, τ
i
238
will release at every T
i
ticks periodically; and 2) β
i
must239
occur before τ
i
is released again. The time interval between240
the occurrences of events β
i
and γ
i
is the remaining time of 241
the current period, which decreases along with the increase of 242
t
α
i
. Hence, in two adjacent periods, the values of t
γ
i
could 243
be different. Formally, 244
• γ
i
has time bounds
[0, 0],
if τ
i
releases at r
1,1
[T
i
− t
α
i
− t
β
i
, T
i
− t
α
i
− t
β
i
],
if (∀j > 1) τ
i
releases at r
i,j
, 245
• α
i
has time bounds [0, D
i
− t
β
i
], and 246
• β
i
has time bounds [t
β
i
, t
β
i
]. 247
B. JGS Model 248
Another TDES real-time task model, the JGS model pro- 249
posed in [6], can be utilized to preemptively schedule periodic 250
tasks τ
i
satisfying D
i
= T
i
. The scheduling is priority-based. 251
The general TDES models for the WCET and the period of 252
each task are represented by the two TDES generators shown 253
in Figs. 2 and 3, respectively, in which Σ = Σ
1
∪Σ
2
∪· · ·∪Σ
n
, 254
and Σ
t
= Σ ∪ { t}. The event set Σ
i
for τ
i
is composed of 255
• a
i
: the arrival of task τ
i
, 256
• c
i
: the execution of task τ
i
, and 257
• e
i
: the execution of the last time unit of task τ
i
. 258
Event a
i
is uncontrollable while events c
i
and e
i
are control- 259
lable. Moreover, all the events in the alphabet Σ
i
are forcible. 260
t
i
a
\66
t
i
i
e
i
c
t
\66
t
i
\
6
6
t
i
Fig. 2: JGS WCET model.
i
a
\{ }6
i
a
t
\{ }6
i
a \{ }
6
i
a
t
\{ }6
i
a
t
Fig. 3: JGS period model.
C. Comparison between CW and JGS Models 261
Several differences between CW and JGS models are shown 262
in Table I, in which Y and N represent “yes” and “no”, 263
respectively. 264
TABLE I: CW model v.s. JGS model
Model D ≤ T priority preemption
CW Y N N
JGS N Y Y
Both CW and JGS models have their advantages and dis- 265
advantages. Thus they can be utilized to model different RTS. 266
The CW model can be utilized to model an RTS executing 267
4
a set of periodic tasks with deadlines less than or equal to268
their corresponding periods. However, priority-based schedul-269
ing and preemptive scheduling cannot be accommodated by270
the CW model. On the contrary, the JGS model can only271
be utilized to model an RTS executing a set of tasks with272
deadlines equal to their periods. Moreover, in the JGS model,273
priority-based scheduling and preemptive scheduling of real-274
time tasks are addressed. Users can choose different models275
to solve different real-time scheduling problems.276
D. TDES Model for Multiple-Period Tasks277
The elastic task model in [7]–[10] assigns a lower and an278
upper period bound for each task to dynamically reconfigure279
an RTS. At each time, the reconfiguration of each task’s280
period is assigned a value between the two bounds T
i
min
and281
T
i
max
. Consequently, the processor utilization U
i
of an elastic282
periodic task has a lower bound U
i
min
and an upper bound283
U
i
max
. Formally, we have284
U
i
= [U
i
min
, U
i
max
]285
with U
i
min
= C
i
/T
i
max
and U
i
max
= C
i
/T
i
min
. The system286
processor utilization is287
U
S
= [U
min
, U
max
]288
with U
min
=
n
P
i=1
U
i
min
and U
max
=
n
P
i=1
U
i
max
. In the289
interval [U
min
, U
max
], there may exist multiple safe execution290
sequences (reconfiguration scenarios) that correspond to dif-291
ferent processor utilizations. Moreover, SCT [21] is utilized to292
find the supremal controllable sublanguages, i.e., it is possible293
to provide multiple reconfiguration scenarios for each task.294
Building on the elastic task model and SCT, we present a new295
model that provides all the possible periods for each task; the296
supervisor provides all the safe execution sequences (possible297
reconfiguration scenarios) simultaneously. Users choose any298
scenario to reconfigure the RTS dynamically.299
A regular periodic task with a fixed period is considered300
as a multiple-period task τ
i
with T
i
min
= T
i
max
. With a301
regular task, the reconfiguration of its period would affect its302
utilization, which is not allowed. On the other hand, SCT is303
utilized to provide all the possible scheduling paths based on304
different periods (utilizations).305
1) Multiple-period CW (MCW) model:306
In this study, the MCW model is depicted in Fig. 4, in which307
y
0
is the initial state, and {y
min
, y
min+1
, . . . , y
max−1
, y
0
}308
is the marker state set. Each marker state represents that309
τ
i
has finished the current execution of J
i,j
and is ready310
for the release of J
i,j+1
. State y
0
represents that job J
i,j
311
finishes its operation at T
i
max
or has never been invoked. States312
y
min
, y
min+1
, and y
max−1
represent that job J
i,j
finishes its313
operation at times T
i
min
, T
i
min+1
, and T
i
max−1
, respectively.314
On the occurrence of α
i
, τ
i
starts the processing of the315
current job. After event tick occurs C
i
times, the execution of316
τ
i
is completed. The next occurrence of event γ
i
drives τ
i
into317
the next execution period. Formally,318
i
E
i
D
tt
tt
tt
tt
t
t
t
t
i
J
i
E
i
D
i
E
i
D
i
E
i
D
0
y
i
C
i
C
i
C
i
C
t
tt
t
max 1
y
min
y
min 1
y
i
J
i
J
i
J
Fig. 4: General TTG model for MCW tasks.
• γ
i
has time bounds
[0, 0],
if τ
i
releases at r
i,1
[T
i
min
− t
α
i
− t
β
i
, T
i
max
− t
α
i
− t
β
i
],
if (∀j > 1) τ
i
releases at r
i,j
,319
• α
i
has time bounds [0, min {D
i
, T
i
min
} − t
β
i
], and 320
• β
i
has time bounds [t
β
i
, t
β
i
]. 321
Remarks: 322
1. Initially, a task with T
i
= T
i
min
plays the role of the 323
task proposed in [5]. In this case, task τ
i
always stays at the 324
highest processor utilization. If the RTS is non-schedulable, 325
the multiple-period model with T
i
= [T
i
min
, T
i
max
] is utilized 326
to provide all the possibilities to compress the processor 327
utilization. 328
2. For example, a non-reconfigurable periodic task τ
1
is 329
defined as τ
1
= (0, 1, 4, [5, 5]). The processor utilization of 330
task τ
1
is fixed to be U
1
= 1/5. The TTG model G
1
for task 331
τ
1
is depicted in Fig. 5. For the events in Σ
act1
, 332
• γ
1
has time bounds
[0, 0],
if τ
1
releases at r
1,1
[4 − t
α
1
, 4 − t
α
1
],
if (∀j > 1) τ
1
releases at r
1,j
, 333
• α
1
has time bounds [0, 4 − t
β
1
], and 334
• β
1
has time bounds [1, 1]. 335
1
E
1
D
t
t
t
t
t
t
t
t
1
J
1
E
1
D
1
E
1
D
1
E
1
D
t
t
t t
Fig. 5: MCW TDES G
1
.
