critical message is T
4
, that can be retransmitted twice.
The retransmission of the messages causes a prolon-
gation of the processing time that is depicted in levels
on the vertical axis. The top level of each message
represents its WCET (the worst case execution time),
i.e. the time that it takes to transmit the message un-
der the most pessimistic conditions. The prolonga-
tions are compensated by skipping less critical mes-
sages. With this mechanism, the successful transmis-
sion of highly critical messages is guaranteed while
in the average case runtime scenario the resource (i.e.
communication bus) is efficiently utilized.
Scheduling of safety-critical non-preemptive mes-
sages on this time-triggered network can be modeled
as the scheduling problem 1|mc = L,mu|C
max
(Han-
zalek et al., 2016). It represents the scheduling prob-
lem with one resource (a communication channel in
the network) with non-preemptive mixed-criticality
tasks with maximum L criticality levels, mu stands
for the match-up of the execution scenario. The cri-
terion is to minimize the maximal completion time
C
max
.
A solution of the scheduling problem is given by
a schedule that switches to the higher criticality level
when a prolongation of a task occurs. After its suc-
cessful completion, it matches-up with the original
schedule. The trade-off between the safe and efficient
schedules is achieved by skipping less critical mes-
sages when the prolongation of a more critical one
takes place.
1.2 Paper Contribution and Outline
In this paper, we solve the scheduling problem of
message retransmission in time-triggered environ-
ments. The objective is to find a non-preemptive static
schedule that accounts for unforeseen message re-
transmissions while minimizing the length occupied
by time-triggered communication. The uncertainty
about the processing time is modeled using an ab-
straction based on F-shaped tasks. We show the rela-
tion between F-shaped tasks and the underlying prob-
ability distribution functions. Furthermore, we show
a new complexity result that establishes the member-
ship of the considered problem into AP X complex-
ity class, and we provide an approximation algorithm.
We study the characterization of the set of optimal so-
lutions for the problem with two criticality levels. Fi-
nally, we propose efficient exact algorithms for prob-
lems with two and three criticality levels, which solve
instances with up to 200 tasks, beating the best-known
method by a large margin.
The rest of the paper is organized as follows. In
Sec. 2 we survey the related work. In Sec. 3 we show
the relation between F-shaped tasks and discretization
of cumulative probability distribution functions. In
Sec. 5 we prove approximability of the problem. In
Sec. 6 and 7 we show properties of the problem with
two and three criticality levels and we propose effi-
cient exact algorithms. Finally, in Sec. 8 we present
computational results on the sythetic data as well as
on the data inspired by a real-life embedded system
of our industrial partner.
2 RELATED WORK
The exhaustive survey on mixed-criticality in real-
time systems is presented by (Burns and Davis, 2013).
This research is traditionally concentrated around
event-triggered approach to scheduling. In the sem-
inal paper (Vestal, 2007) proposed a method that as-
sumes different WCETs (the worst case execution
time) obtained for discrete levels of assurance. Apart
from this proposition, the paper presents modified
preemptive fixed priority schedulability analysis al-
gorithms. However, the preemptive model is not suit-
able for communication protocols, and it significantly
changes the scheduling problem. (Baruah et al., 2010)
formulated the basic model of mixed-criticality sys-
tems. They study MC schedulability problem with
two criticality levels under special restrictive cases in
the event-triggered environment. (Theis et al., 2013)
argued that mixed-criticality shall be pursued in time-
triggered systems. (Baruah and Fohler, 2011)’s ap-
proach in the time-triggered environment assumed
preemptive tasks with up to two criticality levels.
It makes it unsuitable for communication protocols
since the preemption would be costly. (Hanzalek
et al., 2016) proposed the problem of non-preemptive
mixed-criticality match-up scheduling motivated by
scheduling messages on a highly used communica-
tion channel. They showed how a schedule with F-
shaped tasks can be used to deal with a task disruption
by skipping less critical tasks. They provide the rela-
tive order MILP model for 1|r
j
,
˜
d
j
,mc = L,mu|C
max
scheduling problem, but it can deal with instances
with only about 20 messages.
The concept of match-up scheduling was intro-
duced by (Bean et al., 1991). In a case of a disruption,
the goal is to construct a new schedule that matches
the original one at some point in the future. This con-
cept is mostly studied in the context of manufacturing
problems (Qi et al., 2006).
Taking broader perspective, the problem can be
viewed as a case of robust and stochastic optimiza-
tion due to uncertainty about transmission times while
satisfying safety requirements. (Bertsimas et al.,
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