Chapter 3
Task Assignment and Scheduling
3.1 Introduction
3.2 Rate monotonic analysis
3.3 Other uniprocessor scheduling
algorithms
3.4 Task assignment
3.5 Fault-tolerant scheduling
Objective : Look at techniques for allocating & sc
heduling task to ensure deadline is met
Introduction
 Real-time computing objective :
– Execute, by appropriate deadlines it’s control tasks
 Objective of Chapter:
– Techniques for allocating & scheduling tasks on processors to en
sure that deadlines are met.
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Scheduling
Scheduling
research
growth
1970
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Years
3
The allocation/scheduling problem can be
stated as follows:
 Given a set of factors affecting
allocation/scheduling
– Tasks (consumes resources)
• number of tasks, priorities
– task characteristics
• periodicity
• timing constraints
– task precedence constraints
(best described using precedence graph)
– resource requirements
– inter-task interactions
We are asked to devise a feasible
allocation / schedule on a given computer
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Precedence Graph
T1
T2
T3
T4
T6
T5
T7
T8
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Precedence Graph
 The arrows indicate which task has precedence over
the other task.
 We denote the precedence task set of task T by <(T)
that is , (T) indicates which tasks must be completed
before Y can begin.
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Precedence Graph (Cont.)








<(1) = 
<(2) = {1}
<(3) = {1}
<(4) = {1}
<(5) = {1,2,3} or {2,3}
<(6) = {1,3,4} T2
<(7) = {1,3,4,6}
<(8) = {1,3,4,6,7}
T1
T3
T4
T6
T5
T7
T8
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Precedence Graph (Cont.)
 We can also write :
i<j
to indicate that task Ti must precede task Tj.
 It can also be written as :
j>i
 The precedence operator is transitive:
i< j and j < k  i < k
 For economic representation:
Only the list of immediate ancestors in the precedence s
et:
– E.g. < (5) = {2,3} since <(2) = {1}
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Each task has :
 Each task requires resources. Eg. Process
or execution time, memory or access to a b
us
 Resources examples:
 Resources may be (depending to its usage
):
– Exclusively held by a task
 Release Time of a task- the time at which a
ll the data that are required to begin executi
ng the task are available.
 Deadline – the time at which the task must
complete its execution. (Deadline  mayb
e hard or soft).
 Relative deadline = Absolute deadline – rel
ease time
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Each task …(cont.)
 Task
– Periodic
• every Pi seconds, the constraints is that it ha
s to run exactly once every period.
• Every period is generally  Deadline
– Sporadic
• not periodic but has an upper bound on the
rate in which it has to be invoked.
• Irregular intervals
– Aperiodic
• Not periodic but has no upper bound
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 Precedence constraints
–
–
–
–
inter-task relationship
precedence graph
＜(T) : precedent-task set of task T
i ＜ j : task Ti precedes task Tj
T1
T4
T2
 Resource requirements
– exclusive
– nonexclusive
T3
T5
T6
T7
T8
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Characteristics of task assignment/scheduling
– feasible schedule
• a valid schedule by which every task completes by its
deadline
– task assignment
• in case of multiple processors
– for a set of processors P, time t, set of tasks Τ, the
schedule S is a function such that
• S: P × t  Τ
• S(i, t) : task scheduled to run on processor i at time t
– online (dynamically) vs offline scheduling
(precomputed)
– Static(doesn’t change within a mode) vs dynamic
priority algorithm
– preemptive vs nonpreemptive scheduling
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 Inter-task interactions
– inter-task communication
• synchronous
• asynchronous
– mutual exclusion problem (synchronization)
• priority inversion
• chained blocking
• deadlock
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Assignment / scheduling problems
 Most problems pertaining are more than tw
o processors must make do with heuristics.
Heuristics are motivated by the fact that uni
processor scheduling are tractable.
 Thus, multiprocessor schedule are divided
into two (2) steps:
– 1) assign tasks to processors
– 2) Run a uniprocessor schedule to sche
dule the task allocated to each processo
r. If one or more schedules cannot be fe
asible, then we must either return to the
allocation step and change the allocatio
n or declare that a schedule cannot be f
ound.
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Developing a multiprocessor schedule
Make an allocation
Schedule each
processor based on
the allocation
Change Allocation
Are all these
schedules feasible
Output schedule
Check stopping
criterion
Continue
Stop
Declare Failure
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Overview
 Uniprocessor scheduling algorithms
–
–
–
–
–
–
traditional rate-monotonic (RM)
rate-monotonic deferred server (DS)
earliest deadline first (EDF)
precedence and exclusion conditions
multiple task versions
IRIS tasks
• increased reward with increased service
– mode changes
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 Multiprocessor scheduling
–
–
–
–
–
–
utilization balancing algorithm
next-fit algorithm
bin-packing algorithm
myopic offline scheduling algorithm
focused addressing and bidding algorithm
assignment with precedence constraints
 Critical sections
 Fault-tolerant scheduling
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 Notation
–
–
–
–
–
–
–
nnumber of tasks in task set
ci
execution time of task τi
Ti
period of periodic task τi
Ii
phase of periodic task τi
di
relative deadline of task τi
Di
absolute deadline of task τi
ri release time of task τi
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Commonly Used Approaches
 Weighted round-robin approach
– tasks waiting in the FIFO queue
– a task with weight wt get wt time slices every round
– suitable for scheduling real-time traffic in high-speed
switched networks
• a switch downstream can begin to transmit an earlier portion
of the message upon receipt of the portion, without having to
wait for the arrival of the later portion
– no need for sorted priority queue  speedup of
scheduling
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 Priority-driven approach
– never leaves any resource idle intentionally
– greedy scheduling, list scheduling, work-conserving
scheduling
– most scheduling algorithms used in nonreal-time
systems are priority-driven
– preemptive vs. nonpreemptive
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 Clock-driven(time-driven) approach
– tasks and their timing constraints are known a priori
except for aperiodic tasks
– relies on hardware timers
– a static schedule
• constructed off-line
• cyclic schedule: periodic static schedule
• clock-driven schedule: cyclic schedule for hard real-time
tasks
– foreground/background approach
• foreground: interrupt-driven scheduling
• background: cyclic executive (“Big loop”)
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Foreground/Background Systems
Background
Foreground
code block
interrupt
code block
Loop
ISR
..
.
interrupt
ISR
ISR
ISR: interrupt service
routine
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A clock-driven scheduler
Input: Stored schedule (tk, τ(tk)) for k = 0, 1, …, N-1
Task SCHEDULER:
set the next decision point i and table entry k to 0;
set the timer to expire at tk;
do forever:
accept timer interrupt;
if an aperiodic job is executing, preempt the job;
current task τ = τ(tk);
increment i by 1;
compute the next table entry k = i mod N;
set the timer to expire at fl(i/N)H + tk
{ fl: floor function H: hyperperiod N: #tasks in H}
if the current task τ is an idle interval (or idle task),
let the job at the head of the aperiodic job queue execute;
else, let the task τ execute;
sleep;
end SCHEDULER
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3.2 Rate Monotonic Analysis
 Assumptions
– A1. No nonpreemptible parts in a task, and negligible
preemption cost
– A2. Resource constraint on CPU time only
– A3. No precedence constraints among tasks
– A4. All tasks periodic
– A5. Relative deadline = period
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Rate-Monotonic Scheduling(RMS)
 Overview
– rate monotonic priority
• the higher rate, the higher priority
– schedulability guaranteed if utilization rate is below a
certain limit
– for feasible schedules
• fi = 1/Ti : frequency (=rate)
• ci or Ci : execution time
n
c
i
fi 1
i 1
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3.3 Other Uniprocessor Scheduling Algorithms
 Period transformation for transient overload
– a modified form of RM scheduling
 Dynamic scheduling
– earliest deadline first scheduling
– least laxity first scheduling
 Scheduling of IRIS tasks
– imprecise computation
 Scheduling of aperiodic tasks
 Mode change
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Period Transformation
 Period transformation for transient overload
– changes the period to cope with transient overloads
(in terms of RM scheduling)
– actually, to cope with semantic criticality in RM
scheduling
– example
• tasks:
T1: T1 = 12, C1 = 4, C1+ = 7
[Ci+: worst case]
T2: T2 = 22, C2 = 10, C2+ = 14
utilization rates: average = 0.79, worst case = 1.22
• problem: if T2 is hard rt and T1 is soft (or not), how can we
guarantee T2’s deadline in case of transient overload, and
T1’s deadline in the average case?
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(continued)
• solution: boost priority of T2 by reducing its period
replace T2 by T2’:
T2’ = T2 /2, C2’ = C2 /2, C2’+ = C2 +/2
• an alternative: lower the priority of T1 by lengthening its
period
– in this case, double the value of parameters
– the new deadline must be ok
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Earliest Deadline First Scheduling
 Also know as Deadline Monotonic
 EDF scheduling
– dynamic priority based, deadline monotonic
scheduling
 Properties
– EDF is optimal for uniprocessors
– for periodic tasks with their relative deadline equal to
periods: if the total utilization of the task set is no
greater than 1, the task set can be feasibly scheduled
on a single processor by EDF.
– Allows preemptions.
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– Procedure
1. Sort task instances that require execution in time
interval [0, L] in reverse topological order.
2. Initialize the deadline of the kth instance of task Ti
to (k-1)Ti + di, if necessary
3. Revise the deadlines in reverse topological order.
4. Select the task with earliest deadline to execute
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Uniprocessor Scheduling of IRIS Tasks
 Introduction
– Not necessary to run to completion. Iterative
algorithms.
– Task of this type are known as increased reward
with increased service (IRIS)
– reward function R(x)
• typically
 0

R( x)   r ( x)
r ( o  m )

if x  m
if m  x  o  m
if x  o  m
• where r(x) is monotonically nondecreasing in x.
• m, o : execution time of the mandatory and optional parts,
respectively
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3.4 Task Assignment
 Assignment of tasks to processors
– use heuristics  cannot guarantee that an allocation
will be found that permits all task to be feasibly
scheduled.
– consider communication costs precedence of task
completion.
– Sometime an allocation algorithm uses
communication costs as part of its allocation criterion.
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Utilization-balancing algorithm
– Objective to balance processor utilization, and proceeds by
allocating the tasks one by one and selecting the least utilized
processor.
– Considers running multiple copies for fault-tolerance systems.
for each task Ti, do
allocate one copy of Ti to each of the ri least utilized
processors
update the processor allocation to account for the
allocation of Ti
end do
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A Next-fit algorithm for RM scheduling
– Used in conjunction with RM
– separation of allocation and scheduling
• simplifies the scheduler to a local one
• allocation: centralized, scheduler: distributed
– objectives:
• to partition a task set so that each partition is scheduled later
for execution on a processor by RM scheduling
• to use as few processors as possible
– task characteristics
• each task has constant period and deadline constraints
• independent, no precedence constraints
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– allocation algorithm
– n tasks
– ui : utilization factor of Ti
– Pi,j : set of tasks assigned to a processor
– Nk : number of class-k processors used so far
• tasks are divided into M classes such that
task Ti  class - k if 2
1
k +1
1
k
1  u  2 1
task Ti  class - M if 0  u  2
1
M
1
• assigns k class-k tasks to each class-k processor, keeping
1 factor
 k <
M,class-M
M > 3 processor less than ln 2
thewhere
utilization
of the
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– Algorithm Next-Fit-M
for k = 1 to M do set Nk = 1;
set i = 1;
while i <= n do
if Ti is a task from class-k, 1 <= k < M, then
assign Ti to Pk,Nk;
if Pk,Nk has currently k tasks assigned to it then
set Nk = Nk +1
endif
else (Ti is a task from class-M)
if the total utilization factors of all the tasks
assigned to PM,NM is greater than ln2-ui then
set NM = NM + 1
endif
assign Ti to PM,NM
endif
set i = i +1
endwhile
if Pk,Nk has not task assigned to it then set Nk = Nk -1
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Bin-packing assignment algorithm for EDF
– periodic independent preemptible tasks
– bin-packing problem: assign tasks such that the sum
of utilization factors does not exceed 1, and minimize
the number of processors needed
– first fit decreasing algorithm
Initialize i to 1. Set U(j) = 0, for all j.
(L : a list of tasks with their utilizations sorted in descending
order, nT : # tasks )
while i <= nT do
Let j = min{k | U(k) + u(i) <= 1}.
Assign the i-th task in L to pj
Set i = i + 1 .
L- sorted list of t
end while
ask so their utiliz
ation are in
n
on-increasing
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Myopic Offline Scheduling (MOS) Algorithm
– Offline Algorithm –given in advance arrival times, execution time
and deadline.
– Non-pre-emptive task
– Not only processor resources but also others resources such as
memory etc.
Schedule Tree
– MOS proceeds by building up a schedule tree.
– Each node represents an assignment and scheduling of a subset
of the tasks.
– The root of the schedule tree is an empty schedule.
– Each child of a node consists of a schedule of its parent node,
extended by one task.
– A leaf of this tree consist of the schedule of the entire task set.
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Myopic Offline Scheduling (MOS) Algorithm
– algorithm
i) start with an empty partial schedule
ii) determine if the current partial schedule is
strongly feasible
then proceed; else backtrack
iii) extend the current partial schedule by one task
2 Questions – w
hich one task &
when to stop
(1) apply the heuristic function to the first Nk tasks in the
task set
(2) choose the task with the smallest heuristic value to
extend the current schedule
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Develop a node if it is strongly feasible.
If not feasible, we backtrack that is we mark that node
as hopeless and then go back to its parents
Focused addressing and bidding (FAB)
– Introduction
•
•
•
•
online
distributed environment, loosely coupled
both critical and noncritical tasks
local scheduler: handles (critical) tasks arriving at a
given node
• global scheduler: schedules noncritical tasks
across processor boundary
• global state
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FAB cont.
– algorithms for global scheduling
to which node the task should be sent
• noncooperative algorithm-if enough resources for critical yes;
else no for non-critical.
• random scheduling algorithm-if a processor load is exceeding
its threshold then another processor is chosen randomly.
• focused addressing algorithm – overloaded processor checks
its surplus info. and selects a processor which it feels it is
able to process the task within its deadline. Prob: surplus info
may be outdated.
• bidding algorithm – simultaneous – lightly loaded to bid
(Request For Bids)
• flexible algorithm <-- focused addressing + bidding
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– focused addressing algorithm
• FAS: focused addressing surplus, tunable
parameter
• locally unschedulable tasks sent to the node with
the highest surplus ( > FAS)
• if no such node is found, the task is rejected
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– bidding algorithm
• first, select k nodes with sufficient surplus
k: chosen to maximize the chances of finding a
node
• a request-for-bid(RFB) message is sent to these
nodes
• those nodes that receive RFB message
– calculate a bid ( = likelihood that the task can
be guaranteed)
– send the bid to the bidder node if the bid >
minimum bid req’d
• the bidder sends the task to node that offers the
best bid
• if no good bid available, reject the task
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– symbols
• pi: a processor node with a newly arriving task that is not
locally guaranteed
• ps: a node that is selected by FA algorithm
• pt: a node that receives RFB message
– the flexible algorithm (FAB algorithm)
• pi selects k nodes with sufficient surplus
– if the largest value of the surplus > FAS
» the node with that surplus is chosen as focused
node(ps)
» pi sends the task to ps immediately
» also, pi sends in parallel a RFB message to the
remaining k-1 nodes. RFB contains info on ps
• when a node receives the RFB message
– it calculates a bid, sends the bid to ps if ps exists
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(continued)
• when the task reaches ps
– it first invokes the local scheduler and checks the feasibility
– if it succeeds, all the bids for the task will be ignored
if it fails, ps evaluates the bids, sends the task to the node
responding with the highest bid, and sends this info to pi
• in case there is no focused node, pi will handle the bidding
• if ps cannot guarantee the task and if there is no good bid
available, then corrective actions follow
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ps
(focused node)
(original node)
pi
network
bidding
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The Buddy Strategy
 Same as FAB in the sense that if the processor is overlo
aded it will try to offload some task to
lightly loaded p
rocessor.
 However, it differs in the manner in which it finds the light
ly loaded tasks:
 Each processor has 3 thresholds of loading:
– U:Under (TU), F: full (TF) and T: over (TV)
 If a processor has a transition from F/T to U it broadcast
an announcement to this effect. This broadcast is not to
all processors but to a subset and this effect is known as
a buddy effect.
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3.5 Fault-Tolerant Scheduling
 Introduction
– in case of hardware failure
– Systems have sufficient reserve capacity and
sufficiently fast failure-response mechanism.
– multiple processors with a set of periodic tasks
– multiple copies of each version of a task executed in
parallel
– the approach taken : ghost copies of tasks
• embedded into the schedule
• need not be identical to the primary copies
• the tasks concerned are those that were to have been run by
the failing processor
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 Fault-tolerant schedule
– should be able to run one or more copies of each
version (or iteration) of a task despite the failure of up
to nsust processor
– Output of each fault-tolerant processor
• has a ghost schedule + 1+ primary schedules
• makes room for ghosts by shifting primary copies.
– feasible pair of a ghost schedule and a primary
schedule
• if both schedules can be merged/adjusted to be feasible
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 Ghosts
– each version of a task must have ghost copies
scheduled on nsust distinct processors
– ghosts are conditionally transparent, only if
• two ghost copies may overlap in the schedule of a processor
if no other processor carries the copies of both tasks (that is,
if the primary copies of both tasks are not assigned to the
same processor)
• primary copies may overlap the ghosts only if there is
sufficient slack time in the schedule to continue to meet all
the deadlines
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 Algorithm FA1
Ha: assignment procedure, Hs: EDF scheduling procedure
1. Run Ha to obtain a candidate allocation of copies to
processors.
2. Run Hs for ghost and primary copies on a processor i.
• if the resulting schedule is found infeasible, return to step 1
• otherwise, record the position of the ghost copies in ghost
schedule Gi, and the position of the primary copies in
schedule Si. (the primary copies will always be schedule
according to S regardless of any ghost happen or not)
Limitation:
- primary tasks are needlessly delayed when the ghost do not
have to be executed. While all task will meet their deadlines,
it is frequently best to complete execution of the task early to
provide slack time.
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 Algorithm FA2
1. Run Ha to obtain a candidate allocation of copies to
processors.
2. Run Hs for ghost and primary copies on a processor i.
• if the resulting schedule is found infeasible, return to step 1
• otherwise, record the position of the ghost copies in ghost
schedule Gi. Assign static priorities to the primary tasks in
the order in which they finish executing.
3. Run a static-priority preemptive scheduler for primary
copies with the priorities to obtain Si
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 Example
ghosts: g4, g5, g6
primaries: h1, h2, h3
h1 h2 h3 g4 g5 g6
release time
2 5 3 0 0 9
execution time 2 2 4 2 2 2
deadline
6 8 15 5 6 12
for the primary copies of g4 and g5
case 1: they are allocated to the same processor
case 2: they are on different processors
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g4
g5
0
g6
5
10
15
ghost schedule of p if g4 and g5 cannot overlap
g4, g5
g6
0
5
10
15
ghost schedule of p if g4 and g5 can overlap
h1
0
h3
h2
h3
5
10
15
feasible primary schedule of p if g4 and g5 can overlap
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