An Optimal Algorithm for Scheduling
Soft-Aperiodic Tasks
in Fixed-Priority Preemptive Systems
John P. Lehoczkyt and Sandra Ramos-Thuelj
Department of Statistics+
Department of Electrical and Computer Engineering*
Carnegie Mellon University,
Pittsburgh, PA 15213
Abstract
This paper presents a new algorithm for servicing
soft deadline aperiodic tasks in II red-time system in
which hard deadline periodic tusks are scheduled us-
ing a fixed priority algorithm.
The new algorithm
is is proved to be optimal in the sense that it provides
the shortest aperiodic response time amony ull possi-
ble aperiodic service methods. Simulation studies show
that it offers substantial performance improvements
over current approaches including the sporudic server
algorithm. Moreover, standard yueueiny formulas cun
be used to predict aperiodic response times over (I wide
range of conditions. The algorithm can be extended to
schedule hard deadline aperiodics and to eficiently re-
claim unused periodic service time when periodic tusks
have stochastic execution times.‘12
1 Introduction
In 1973, Liu and Layland [l] presented an anal-
ysis of the rate monotonic algorithm for scheduling
periodic tasks with hard deadlines. Recently, this
algorithm has gained popularity as an approach to
designing predictable real-time systems. Moreover,
the algorithm has been modified to allow for the
‘The authors wish to thank Stephen W. Gdyas for his par-
ticipation in this research effort and Jay K. Strosuider aud his
research group for their insightful comments and suggestions.
‘This research is supported in part by a grant from the Of-
fice of Naval Research under contracts N00014-84-K-0734 and
NOOO14-91-J-1304Nl73, by the Naval Ocean Systems Center un-
der contract N66001-87-C-01155, by the Federal Systems Divi-
sion of IBM Corporation under University Agreement Y-278067,
and by AT&T Bell Laboratories uuder the Cooperative He-
search Fellowship Program.
solution of many practical problems which arise in
actual real-time systems including task synchroniza-
tion, transient overload, and simultaneous scheduling
of both periodic and aperiodic tasks, among others.
The mixed task scheduling problem is important, be-
cause many real-time systems have substantial ape-
riodic task workloads. Moreover, the aperiodic tasks
may themselves have a variety of timing requirements,
ranging from hard deadlines to soft deadlines. For ex-
ample, recovery from transient failures may create an
aperiodic stream of hard deadline periodic tasks which
must be reexecuted (see Ramos-Thuel [2]).
In this paper, we reconsider the problem of jointly
scheduling hard deadline periodic tasks and aperiodic
tasks. Although the methods presented in this pa-
per apply both to hard deadline and soft deadline
aperiodic tasks, we limit our attention to the case of
scheduling soft deadline aperiodic tasks. That is, we
seek to schedule a mixture of periodic and aperiodic
tasks in such a way that all periodic task deadlines are
met and the response times for the aperiodic tasks are
as small as possible.
There are two standard approaches to this problem.
The least effective approach is to service the aperi-
odic tasks in the background of the periodic tasks (i.e.,
when the processor is idle). A better approach is to
create a periodic polling task with as large a capacity
as possible. The polling task will be run periodically,
and its capacity will be used to service aperiodic tasks.
While the polling server is faz superior to background,
the periodic polling task is not necessarily coordinated
with the aperiodic arrival process, so some aperiodic
arrivals must wait for the return of the polling task
before they can be executed. This waiting may create
unnecessarily long task response times. In addition,
110
1052-8725192 $3.00 0 1992 IEEE
the polling task may be ready but have no tasks ready
for execution, a situation that wastes the high priority
capacity of the polling task.
Recently, new approaches to the joint scheduling
problem have been developed including the sporadic
server algorithm by Sprunt [3, 41 and the deferrable
server algorithm by Strosnider [5]. Although similar
in spirit to the polling server, these algorithms allow
their capacity to be used throughout the server’s pe-
riod rather than only at the beginning. The two algo-
rithms differ in the way their capacity is replenished,
and each has its own individual schedulability analy-
sis; however, in certain circumstances, both can offer
up to an order of magnitude improvement in aperiodic
response time over the polling approach.
This paper develops a new approach to aperiodic
service, and shows that this method can offer sub-
stantial improvements over the deferrable aud spo-
radic server algorithms. The new approach, called the
slack stealing algorithm, does not create a periodic
server for aperiodic task service. Rather it creates a
passive task, referred to as the slack stealer, which
when prompted for service attempts to make time for
servicing aperiodic tasks by “stealing” all the process-
ing time it can from the periodic tasks without, caus-
ing their deadlines to be missed. This is equivalent
to “stealing slack” from the periodic tasks. Note the
similarity of this approach to cycle stealiug t8echniques
used in memory systems [6].
The slack stealer relies on the exact schedulability
conditions given by Lehoczky, Sha and Ding [7] and
Lehoczky [S] to provide the maximum possible capac-
ity for aperiodic service at the time it is needed. Sub-
stantial improvements in aperiodic task response times
will be demonstrated with the slack stealer. It will
also be shown to be optimal for the particular fixed
priority assignment chosen for the periodic tasks. In
addition, the slack stealer can be generalized to han-
dle hard deadline aperiodic tasks, and its functionality
can be efficiently augmented by a recl&~cr. A re-
claimer can cooperate with a slack stealer by making
available for aperiodic service any processing time uu-
used by the periodic tasks when they require less thau
their worst-case execution times. The slack stealing
algorithm requires a relatively large amount of calcula-
tion. Consequently, a direct implementation may not
be practical. It does, however, provide a lower bound
on aperiodic response which is attainable and a basis
for finding nearly optimal implementable algorithms.
2 F’ramework and Assumptions
Consider a real-time system with n periodic tasks,
Tl, . . . , r, . Each task, ri, has a worst-case computa-
tion requirement Ci, a period Ti, an initiation time
4; 2 0 or offset relative to some time origin, and a
deadline Di, assumed to satisfy Di 5 Ti. The pa-
rameters Ci, Ti, +i, and Di are known deterministic
quantities. We require that these tasks be scheduled
according to a fixed priority algorithm, such as the
deadline monotonic algorithm, in which tasks with
small values of D; are given relatively high priority
[‘3]. We assume that the periodic tasks are indexed in
priority order with 71 having highest priority and r,,
having lowest priority. For simplicity, we refer to those
levels as 1, . . . , n with 1 indicating highest priority and
R the lowest. The aperiodic tasks can be assigned any
priority, and we even permit them to be executed dy-
namically at different priority levels. We assume that
if an aperiodic task executes at priority level k, then it
has lower priority than any periodic task with priority
1
.‘1
k - 1 and higher priority than any periodic task
Gith priority k, k + 1, . . . ,
n. Aperiodic task execution
at priority level n + 1 is equivalent to background ex-
ecutiou.
A periodic task, say Ti, gives rise to an infinite se-
quence of jobs. The kth such job is ready at time
di + (k - 1)Ti and its Ci units of required execution
must, be completed by time di + (k - 1)Ti + Di or else
a periodic task timing fault will occur.
We next introduce the aperiodic tasks, {Jk, k 2 1).
Each aperiodic job, Jk, has an associated arrival time
Uk and a processing requirement pk. The tasks are
indexed such that 0 5 &‘k 5 (Yk+l, k 2 1. It is useful
to defiue the cumulative aperiodic workload process,
WA(t) = c Pk,
(1)
{k I aklfl
which accumulates all the aperiodic work that arrives
in the interval [0, t]. Any algorithm for scheduling
both periodic and aperiodic loads will, for any periodic
task set and aperiodic task stream { Jk, k 2 l}, create
a cumulative aperiodic execution process, c(t), giving
the cumulative time during [O,t] that aperiodic tasks
were executed. ~(1) is a continuous function which
must, necessarily satisfy E(t) 5 WA(t),t 2 0 , and we
require that the associated algorithm must meet all
periodic deadlines.
We assume aperiodic tasks are processed in FIFO
order”. The completion time of Jk, denoted Tk, is
31r~ section 4 we consider the shortest remaining processing
time queue discipline which will result in lower average aperiodic
t,wk response times.
111
given by
executed at priority level i or higher during [O,t] and
Tk = min{t 1 E(t) = &pi },
still have Tij finish by Dij.
(2)
Since we seek the largest amount of aperiodic pr*
i=l
cessing possible, and are only concerned with rij’s
deadline, the processor will be busy with level i or
and the response time of JI; , denoted &, is given by
R, = Tk - Qk.
(3)
We seek a scheduling algorithm that will minimize
& which is equivalent to minimizing Tk. Thus, we
need to find a scheduling algorithm whose associated
c(t) is the supremum or upper envelope of all possible
aperiodic execution functions that are associated with
algorithms which meet all periodic deadlines. It is
important to note that if such an upper envelope can
be found, it will lead to the minimum response time
for every aperiodic task, not just the average response
time. We will determine the upper envelope and the
associated optimal aperiodic scheduling algorithm in
the next section, under the following assumptions:
l Al: All overhead for context swapping, task
scheduling, etc., is assumed to be zero.
l A!?: Tasks are ready at the start of their period
and do not suspend themselves or synchronize
with any other task.
l AJ: Any task can be instantly preempted.
higher priority-work from 0 until the completion time
Of Tij.
We now follow the methods developed by
Lehoczky, Sha, and Ding [7] and Lehoczky [8] to de-
termine the necessary and sufficient conditions for Tij
to be schedulable.
Suppose ai is the aperiodic processing at level i
or higher during [0, t], 0 5 t 5 Dij , resulting from
some algorithm. The job rij will finish by Dij, thus
meeting its deadline, if and only if there is a time
t E [Rij, Dij] at which all ai units of aperiodic pro-
cessing and all periodic jobs of priority i or higher
ready before t, including the j jobs of ri are com-
pleted. Let Pi(t) be the periodic ready work in [O,t],
where Pi(t) = CjS: Cj * [maz(O, t - dj)/Tjl + jCi.
The total ready work in [0, t] is then defined by
wi(t) = f%(t) f%(t) + + Pi(t) Pi(t) + + h(t), h(t),
(4) (4)
where Zi(t) is the cumulative level-i inactivity in [0, t].
Thus rij will meet its deadline if and only if there ex-
ists t E [ Rij , Dij]
such that Wi(t) = t or equivalently,
Wi(t)/t = 1. This condition for the feasibility of ai
can be alternatively expressed as
min{Rij 5 t < Dij} {wi(t)/t} I l*
(5)
l A4: There is unlimited buffer space for the ape-
riodic tasks.
3 The Slack Stealing Algorithm
3.1 Formulation
If we assume that the aperiodic workload is suffi-
ciently large so that WA(t) > c(t) for any feasible E(t),
then we can increase ai by Ii(t) and the processor
will be continually busy with level i or higher priority
work up to the completion time of Tij. Equation (5)
can now be rewritten as
mini0 5 t 5 Dij} {Wilt)lt.l I l.
(6)
To determine the upper envelope on aperiodic pro-
cessing, we focus on the maxjmum amount of process-
ing possible such that all periodic deadlines are met.
Consider, for example, the jth job of ri, or Tij, which
is ready at time l&j = & + (j - 1)Ti and must be
finished by &j + Di
= Dij. During [0, Dij] the pro-
cessor may execute tasks at a priority level equal to or
greater than i, tasks at a priority level below i, or may
be idle. Under our fixed-priority system any tasks ex-
ecuted at priority level lower than i are equivalent to
being idle or inactive relative to level i, thus we refer
Given that we want to increase the aperiodic pro-
cessing time as much as possible, we define Aij to be
the largest amount of aperiodic processing possible at
level i or higher during [0, Cij], such that Cij 5 Dij
(Cij refers to the completion time of ‘ii). Thus Aij is
the largest value such that
min{o < t < oij>{( Aj + Pi(t) )/t ) = 1.
(7)
- -
A;j is well defined because the periodic task set is as-
sumed to be schedulable and the function being min-
to level-i inactivity as processor time spent on activi-
imized is piecewise continuous and decreasing. Aij is
ties with priority lower than i. Since level-i inactivity
increased until a minimum of 1 is exactly achieved.
cannot influence the schedulability of any 7i job, we
The completion time of rij, or Cij, is the smallest
seek to find the amount of aperiodic work that can be
value oft for which equality holds in Equation (7).
112
Aperiodic processing at level i or higher given by
Aij during [O,t]
will cause the processor to be con-
stantly busy, but Tij will meet its deadline. We now
need to guarantee that all jobs of ri meet their dead-
line. To ensure the schedulability of Ti, we define
Ai = Aij, Cij-1 5 t < Cij, j > 1,
(8)
where Cio = 0. The non-decreasing step function
Ai gives the largest amount of aperiodic processing
in [0, t] at priority level i or higher possible such that
the processor is constantly busy with priority level i or
higher activity but all jobs of Ti meet their deadline.
To illustrate, let us consider a task set with two
tasks, ~1 and 72, with Cl = 1, Tl = 4, D1 = 1, 41 = 0,
and C2 = 3, Tz = 6, D2 = 6, q5z = 0. Note that
the tasks follow a deadline monotonic priority order.
We restrict our attention to an interval of time [0, H],
where H is the hyperperiod of the task set, or the time
at which the distribution of periodic arrivals repeats
itself. The hyperperiod of a periodic task set is equiv-
alent to the least common multiple of the task periods
which is 12 for this example. Figure 1.a shows the pro-
cessor schedule if no aperiodic work is processed. The
non-decreasing functions Al(t) and AZ(t) are shown
in Figure 1.b. These are step functions, with jump
points corresponding to the completion times of the
jobs of Ti, or the Cij’s, and jump heights correspond-
ing to the Aij values computed by Equation (7). Note
that in this example, all jump points for ~1 are known
a-priori because Clj = Dlj , for all j 2 1. Thus, ev-
ery job of ~1 has zero slack, in the interval of time
between its arrival and completion. On the contrary,
the jobs for 5 have non-zero slack, so their execution
can be delayed by the processing of aperiodic tasks. As
a result, their exact completion times depend on the
amount of higher priority aperiodic processing done
at run-time. Although the exact completion times for
each job of 72 cannot be determined a-priori, their
best- and worst-case values are known. For instance,
job 721 in Figure 1.a will complete no earlier than time
4 and no later than time 6, so its jump point is guar-
anteed to lie somewhere in the time interval [4, 61.
For the particular case in which aperiodics consume
all aperiodic processing time possible, the jump point
for 721 is 6, as shown in Figure 1.b for AZ(t).
We next determine bounds on the aperiodic pro-
cessing at each level for which all deadlines of all pe-
riodic tasks can still be met. Let Li(t) denote the
total amount of aperiodic processing in [0, t] at prior-
ity level i, 1 5 i 5 n. For Tk to meet all deadlines, it
is necessary that
J%(i) J%(i) + + . . . . . . + + Lk(t) Lk(t) < < Ak(t), Ak(t), 15 15 k k 5 5 n. n.
(0) (0)
Tl Tl 3 3
4O 4O ' ' * * 3 3 ' ' 5.F2s 5.F2s 7 7 5 5
0 0 10 10 11 11 12 12
r, r,
, , # # / / /_ /_ r., r.,
/_ /_ /./_ /./_
012345675 012345675
0 0 IO IO
11 11 12 12
(a)
l 3 3
+31
+3 +3
* * I I
AI(~) AI(~)
,,,,I,. ,,,,I,. I I I I I I -I -I
012 012
3 3 4 4 5 5 6 6
7 7 5 5 9 9 IO IO
11 11 12 12
+l +l )
l 2) 2)
IA2" IA2"
0 0 1 1 2 2
3 3 4 4 5 5 6 6
7 7 8 8 0 0 10 10
11 11
+o, +‘: ,
+q
A%, 9
( , , ,
1 I 1
, 1 1
012 012
3 3 4 4 5 5 6 6
7 7 5 5 0 0 10 10
11 11 12 12
(b) (b)
Figure 1: Example 1. Illustrating Slack Stealer opera-
tion: (a) Processor schedule in absence of aperiodics;
(b) Functions used by the Slack Stealer
Let A*(t) = mirql 5 k 5 ,.,I AI(t) and k*(t) be the
index of the highest priority level satisfying &e(t) =
A*(t). Thus &(t) + . . . + Lkt(t)(t) can be no larger
than A*(t). If this sum does assume its maximum
value, then all n inequalities in (9) will hold. Hence,
all periodic tasks at all levels with deadlines no later
than t will meet their deadlines and the processor will
be continuously busy throughout [0, t] executing only
tasks of priority k*(t) or higher. Hence, all periodic
task deadlines before t will be met if and only if
L1(2) +. . .
+ Lk*(t)(t) 5 A*(t)
(10)
Figure 1.b illustrates the function A*(t) for our task
set example. Note that priority level 1 places the tight-
est constraint on aperiodic processing time available
in the interval [0, l), whereas in the interval [l, 12),
priority level 2 places the tightest constraint. There-
fore, k*(t) =
1 , for 0 5 t < 1, and k*(t) = 2 , for
1 5 t < 12.
We next address the question of the priority level
at which the aperiodic tasks can be executed. If
J&(t) + . . .
+ Lqt)(t) < A*(t), then one can modify
&(t)+. . .+&(t)(t) to L’,(t) = &(t)+...+&*(t)(t),
L’,(t) = . . . = L&*)(l) =
0 and still be feasible. In
other words, one can carry out all aperiodic processing
at the highest priority level without any reduction in
aperiodic capacity. Since elevating the priority level
of aperiodic processing reduces their response times,
it is optimal to service aperiodic tasks at the high-
est priority level, and the total aperiodic processing
time cannot exceed A*(t), t 2 0. It follows for the
case in which WA(t) > c(t) for all feasible schedul-
ing algorithms, that the upper envelope on aperiodic
113
processing time is given by A*(t), t 2 0.
rc
The previous analysis assumed that there was ai-
,L_-_
+3r- --------
ways a sufficiently large amount of aperiodic work to +0, ‘Y---------’
be processed such that aperiodic processing would al- ’ ’ 2 3 4 5 6 ? ’ ’ di ”
ways use ail available slack at ail levels. This is, how- +2,--
+, __--___________-_
A
ever, not the general case. There may often be times
012 012 3 3 4 4 5 5 6 6 7 7 6 6
9 9 10 10 11 11 12 12
at which aperiodic processing could be done but none
+,____--_-t2_r--
____~U___________-________.
d(A*“’
is ready. We must modify our analysis to accommo-
*o, : : , , , , , , ,
date this case and define the upper envelope. Define
012 012 34 34 5 5 6+7 6+7 6 6
, ,
9 10 10
,; ,; ,2p&=, ,2p&=, 1) 1)
C C
t~5.5 t~5.5
A(t) = cumulative aperiodic processing consumed
at any priority level during [0, t]
Zi(t) = level-i inactivity during [O,t], for 1 5 i 2 71
and t 2 0.
Here, level-i inactivity refers to the cumulative amount
of time spent processing periodic tasks of priority i+ 1
or lower or any time the processor is idle during [0, t].
Suppose we now start at time s rather than at 0
and wish to determine the maximum amount, of aperi-
odic processing possible during [s, t], t 2 s. The anal-
ysis is the same as before. For example, Aij gives the
largest possible amount of aperiodic processing at pri-
ority level i or higher that can be carried out in [0, t]
and still meet the deadline of rij. However, during
[0, s], A(s) units of processing have already been used
for aperiodic processing and Ii(s) units of level-i inac-
tive time have taken place, time which was available
for level i aperiodic processing but was not used for
that purpose. Thus the amount of time availsMe for
additional aperiodic processing at time t, Aij must be
reduced by A(S) + Z;(S). G eneraiizing Equation (8) to
an arbitrary time origin s, we define for t 2 s,
Ai(s,t) = Aij-A(s)-Zi(s), Cij_1 5 t < Cij. (11)
This quantity gives the maximum amount, of aperi-
odic processing time possible during [s, t] at level i or
higher with ail 7i deadlines still being met. The anal-
ysis is now the same as the earlier analysis with s = 0.
Specifically, define
A*(s,t) = mini1 5 i 5 n}
A($, t)
(12)
and Ap~~)(s, t) = A*(s, t) where k*(t) is the high-
est priority such task. As before, the total aperi-
odic processing during [s, t] cannot exceed A*(s, t) and
ail should be executed at the highest priority level.
The function Apct)(s, t) thus gives the upper envelope
on aperiodic processing over any interval [s, t] during
which the aperiodic workload does not vanish.
Referring back to our previous example, suppose
that no aperiodic work was ready during [0, 5.51 and
an aperiodic task, rap, requiring 2 units of computa-
tion arrives at 5.5. We now use 5.5 as the new time
Figure 2: Example 1. Illustrating Slack Stealer oper-
ation with a change in the time origin
origin and note that A(5.5) = 0, Zl(5.5) = 3.5 (corre-
sponding to the level-l inactivity during 11, 41 and [5,
5.51) and Zx(5.5) = 0.5 (
corresponding to the level-2
inactivity during [5, 5.51). The level-i inactivity values
can be visualized in Figure 1.a. Since we have changed
the time origin to 5.5, the curves from Figure 1.b must
be adjusted for t 2 5.5, to reflect the fact that some
aperiodic processing time has been lost due to inac-
tivity. Thus, the functions Al(5.5, t), Az(5.5, t), and
A”(5.5,t) are obtained according to Equations (11)
and (12). These are depicted in Figure 2. Given these
functlions, the slack stealer finds 2.5 units of processing
capacity at time 5.5 and immediately allocates 2 units
to service rap. Consequently, rap finishes at time 7.5
and leaves 0.50 units of aperiodic processing available
for any aperiodic tasks that may arrive during [5.5,
7.51. if no other aperiodic tasks arrive, the processor
will spend [7.5, 81 on 7-2, [8, 91 on ~1, [9, 11.51 on rz
and then idle during [11.5, 12).
We next define the slack stealing algorithm, an ai-
gorit,hm which achieves the maximum possible amount
of aperiodic service time subject to the constraint of
meeting all the periodic deadlines. Later we will prove
its optimaiity property using the upper envelope de-
rived in this section.
3.2 Algorithm Description
The slack stealing algorithm uses the functions
A;(t) for each task Ti, 1 5 i 5 n, in determining the
capacity that can be allocated to aperiodic service.
Because the arrival pattern of the periodic workload
repeats itself every H time units, or the task set hyper-
period, it is sufficient to compute ail Ai functions
for 0 5 t < H. Given this, we compute the jump
heights associated with each job Tij, of each task ri,
according to Equation (7), for 0 5 t < H. These jump
heights are then stored as pairs of points (i, j), where i
is the task priority, 1 5 i 5 n and i is the job number
114 114