Transcript No Slide Title

Real Time Operating Systems
Schedulability Part 2
Course originally developed by
Maj Ron Smith
7/21/2015
Dr Alain Beaulieu
1
Outline
 Review of RMA and Utilization Bound Testing
 Revisiting Critical Instances
 Time Demand Analysis
 Schedulability & Aperiodic Tasks
 Deadline Monotonic Priority Algorithm
 Schedulability & Bandwidth Preserving Servers
 Deferrable Server Schedulability
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Utilization based schedulability tests
 pros:
 usually involves a “simple” computational test
 mathematically rigorous
 Seminal paper
 cons:
 not always conclusive
 sufficient but not necessary
 only valid if the workload model and the underlying
assumptions of the test are consistent
 can lead to false confidence
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Alternatives
 When utilization based tests are inconclusive, we can
always construct a timing graph of the system
 construct graph over the interval [0, max {pi}]
 if all tasks meet their deadlines within their first period,
the system is schedulable
 assumes that all tasks are in phase
 the problem with this approach is that it will be impractical
for most large systems and is not easily automated
 As experienced in the first homework
 alternative: time-demand analysis
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Revisiting Critical Instants
 Recall that a critical instant is defined as the instant
in time that a job is released such that it has the
maximum response time of all jobs in the task
 think of it as the worst possible time (busiest) to release on
the processor
 Theorem: in a fixed priority system (where every job in
a task completes before another job in the same task releases),
the critical instant for Ti occurs when it releases at
the same time as a job in each higher priority task
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Critical Instants
 Example 1:
 T= { (3,1), (5, 1), (10, 2) }
T1
1
T2
1
1
1
1
1
T3
1
0
1
2
1
1
4
6
8
10
12
 The critical instant for tasks T2 and T3 is 0
In fact for in phase tasks, a critical instant always exists at t = 0
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Critical Instants
 Example 2:
 T= { (1,3,1,3), (5, 1), (10, 2) }
T1
1
T2
1
1
1
1
1
T3
1
0
1
2
1
1
4
6
8
10
 The critical instant for tasks T2 and T3 is 10
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Time-Demand Analysis (Di ≤ pi)
 Under time-demand analysis we:
 predict the worst case response time for each task going
from highest priority to lowest
 then compare each task’s response time to its deadline
 if all worst case response times are less than their
respective deadlines, the system is schedulable
 Recall that a tasks response time is its completion
time minus its release time
This technique is sometimes referred to as response time analysis
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Worst-Case Response Time
 Alternatively, the response time of a task is defined to
be the sum of its own worst case computation time and
its maximum interference ->
wi = ei + Ii
(1)
where Ii is the maximum interference* that task i can experience in
any time interval [t, wi )
*The condition for maximum interference obviously occurs
when all higher priority tasks are released at the same time
as task i (in other words, at a critical instant)
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Counting Releases
 Consider task i and a higher priority task j
 now, the number of releases of task j in time interval 0
to wi can be derived as follows:
# of releases j =  wi/pj 
(2)
[ 0, wi )
ceiling function
i
40
j
5
40
0
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# of releases = 2
40
pj
100
wi
pi
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Determining Interference
 From this, the maximum interference of task j on task
i in interval 0 to wi is given by:
max interference (task j) =  wi/pj  ej
[ 0, wi )
(3)
 but there may be other higher priority tasks,
therefore:
Ii =
  wi/pj  ej
(4)
jhp(i)
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Calculating Response Time
 Substituting equation (4) into (1) gives the general
expression for response time:
wi = ei +   wi/pj  ej
(5)
jhp(i)
 Note the following issue (problem):
 wi is on both sides of this “strange” equation
 Before we attempt to solve this equation, let’s first
see if we can understand it
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Understanding Response Time
T = { (100, 20), (150, 30), (250, 60) }
w3 = e3 +  w3/p1  e1 +  w3/p2  e2
= 60 + 2(20) + 1(30) {by viewing the graph}
= 130
T1
w3 = 130
T2
T3
0
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100
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Solving the Response Time Equation
 Essentially, form a recursive relationship with
equation (5) and solve iteratively:
win+1 = ei +   win / pj  ej
(6)
jhp(i)
where initial (seed) value wi0 = ei
 The algorithm then becomes:
solve for successive values of win+1
if win+1 = win , then solution found -> wi = win
if win+1 > Di , then task i cannot meet its deadline
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Response Time Analysis Example
Given T = { (7,3), (12, 3), (20,5) }
 Use RMA scheduling
 Apply the utilization based schedulabilty test
 If necessary, use time-demand analysis to determine
whether the system is schedulable
exercise to be completed in class
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Schedulability & Asynchronous Tasks
 The simple task model that we have been able to deal
with thus far is restrictive in several ways. Not being
able to handle asynchronous tasks is a major
restriction.
 One approach is to make asynchronous tasks
resemble periodic tasks
 consider that an asynchronous task’s minimum inter-arrival
time can be treated like a period, T
 with just this assumption one can use RMA to extend to both
types of tasks, BUT
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Deadline Monotonic Priority Ordering
 The simple task model assumption that D=p is also
unrealistic for most asynchronous tasks
 often, an asynchronous task will occur infrequently (large
inter-arrival time) but must be serviced quickly -> D < p
 therefore priority assignment based upon p will usually not
satisfy the requirement to meet D
 The deadline monotonic priority ordering (DMPO)
scheme is introduced as follows:
 the shorter the task deadline, the higher the priority
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DMPO Example
Task
1
2
3
4
ei
3
3
4
3
pi
20
15
10
20
Di
5
7
10
20
i
1
2
3
4
wi *
3
6
10
20
 * Response times are calculated using the methods
presented earlier using equation (6)
Note that a rate monotonic ordering would not
schedule these tasks
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Schedulability & BP Servers
 Recall that a bandwidth preserving server is a type
of polling server that permits the budget of the
server to be preserved until later in the period
should an asynchronous job arrive
 We described two general classes of bandwidth
preserving server:
 deferrable server
 sporadic server
How do we determine the schedulability of the system in
the presence of bandwidth preserving servers?
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Recall: Deferrable Servers
 A deferrable server is a bandwidth preserving server
with period ps and execution budget es which
 preserves its budget if idle, and
 replenishes its budget fully every period
 Budgets are not allowed to accumulate from period to
period
 Since the deferrable server is purely periodic, it can
be assigned priority along with all other periodic
tasks
 … and, for reasons that will now be explained, we must
ensure that the priority of the deferrable server is higher
than all other periodic tasks
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Recall the Deferrable Server Example
 The schedule for this system is given below:
T= {(2, 3.5, 1.5, 3.5), (6, 0.5) }
TDS = (3,1)
A1 = (2.5, 2)
Note:  = {TDS, T1, T2}
TDS
1/2
T1
1
1/2
1/2
T2
1
1/2
1
1/2
0
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1/2
2
2.5
4
5.5
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Critical Instants &
Systems with Deferrable Servers
 In order to use time demand analysis on a system with
a deferrable server, a critical instant, t0, must be
found as follows:
 at t0 a job of task i is released
 a job in every higher priority task is also released at t0
 a (higher priority) deferrable server has:
 a full budget es at t0
 an asynchronous job is released at t0,
 a full asynchronous backlog exists for an extended time (beyond
wi), and
 the next replenishment is at t0 + es
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Critical Instants in Systems
with Deferrable Servers
 To understand the critical instant in the presence of
a deferrable server, study the graph below
 note how the worst case release time for any periodic task
occurs when TDS has (and uses) its full budget just prior to
getting a full replenishment
TDS
es
es
es
es
T1
T2
t0
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es
Time-Demand Analysis with
Deferrable Servers (Di ≤ pi)
 Understanding this formula:
wi = ei + es + (wi-es) / ps es +  wi/pk ek
khp(i), excluding TDS
execution of deferrable
server immediately
following critical instant
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Dr Alain Beaulieu
Time-Demand Analysis with
Deferrable Servers (Di ≤ pi)
 Understanding this formula:
wi = ei + es + (wi-es) / ps es +  wi/pk ek
khp(i), excluding TDS
interference of the deferrable
server, TDS, in the interval
(t0+es , wi]
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Dr Alain Beaulieu
Time-Demand Analysis with
Deferrable Servers (Di ≤ pi)
 Understanding this formula:
wi = ei + es + (wi-es) / ps es +  wi/pk ek
khp(i), excluding TDS
interference of all k higher
priority tasks, excluding the
deferrable server, TDS
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Exercise
 Draw the schedule for the system below
 (go out far enough in time to include as a minimum the
completion of the first job in each periodic task and all
aperiodic jobs)
 Determine the response time of A1
 Determine the worst case response time of all
periodic tasks
 Is the system schedulable?
 T={ (5, 1.5), (7, 2) }
 TDS = ( 4, 1) , A = {(0.5, 2.5), (2.5, 0.25)}
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Dr Alain Beaulieu
Exercise Solution
 draw the schedule for the system below
T={ (5, 1.5), (7, 2) }
TDS = ( 4, 1) , A = {(0.5, 2.5), (2.5, 0.25)}
Tasks
TDS
A1
A1
A1
A
2
T1
T2
0
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4
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t
Exercise Solution
 determine the response time of A1
wA1 = 8.5 - 0.5 = 8
wA2 = 8.75 - 2.5 = 6.25
 determine the worst case response time of all
periodic tasks
w11
w12
= e1 + es + (w10-es) / ps es
where w10 = 1.5
= 1.5 + 1 + (1.5 -1) / 4 (1) = 3.5
= 2.5 + (3.5 -1) / 4 (1)= 3.5
therefore w1 = 3.5  D1
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(worst-case)
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Exercise Solution
 determine the worst case response time of all
periodic tasks (continued)
w21
w22
w23
= e2 + es + (w20-es) / ps es + (w20) / p1 e1
= 2 + 1 + (2 -1) / 4 (1) + 2 / 5 (1.5) = 5.5
= 3 + (5.5 -1) / 4 (1) + 5.5 / 5 (1.5) = 8
= 3 + (8 -1) / 4 (1) + 8/ 5 (1.5)
=8
therefore w2 = 8 > D2
w20 = 2
(worst-case)
 is the system schedulable?
No since all tasks do not meet their deadlines under worst-case
conditions, we can not guarantee the system schedulable
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Dr Alain Beaulieu
References
[1] Liu, J.W.S., “Real-Time Systems”, Prentice-Hall, 2000.
[2] Burns, A. and Wellings, A., “Real-Time Systems and
Programming Languages”, Chapter 13, Addison Wesley,
1997
[3] TimeSys Corp, “The Concise Handbook of Real-Time
Systems”, Version 1.0, 1999
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Dr Alain Beaulieu