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<!DOCTYPE html>
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<section data-markdown
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<textarea data-template>
## scheduling | **priority**-based schedulers
## **Design of Autonomous Systems**
### csci 6907/4907-Section 86
### Prof. **Sibin Mohan**
---
so far...
---
so far...we have seen **static** schedulers
### _e.g.,_ cyclic executives
---
### _e.g.,_ cyclic executives
clearly have disadvantages
---
### _e.g.,_ cyclic executives
clearly have disadvantages...lack of,
1. flexibility
2. scalability
3. priority
4. resource management
---
### _e.g.,_ cyclic executives
clearly have disadvantages...lack of,
1. flexibility
2. scalability
3. priority
4. resource management
**frames** → alleviate _some_ of them...more required
---
### other static priority schemes
- [weighted round robin](https://par.nsf.gov/servlets/purl/10383232)
- table-scheduling+dispatch
---
### other static priority schemes
- [weighted round robin](https://par.nsf.gov/servlets/purl/10383232)
- table-scheduling+dispatch
while better, all suffer from similar problems
---
so, what is the solution?
---
**assign priorities**
---
**assign priorities** → as jobs **arrive**
---
**assign priorities** → as jobs **arrive**
scheduler → **dispatches**/**schedules** job accordingly
---
scheduler → **dispatches**/**schedules** job accordingly
- highest priority → schedule _right away_
- not highest priority → inserted into ready queue
---
what do we need?
---
what do we need?
1. an **online** scheduler → always available to make scheduling decisions
---
what do we need?
1. an **online** scheduler → always available to make scheduling decisions
2. a **priority assignment policy** → so jobs can be scheduled correctly
---
but first, let's revisit the **task model**
---
## **task model** | updates
- task set → $n$ **periodic** tasks, $\tau = {\tau_1, \tau_2...\tau_n}$
---
## **task model** | updates
- task set → $n$ **periodic** tasks, $\tau = {\tau_1, \tau_2...\tau_n}$
- deadline **equal to** period, _i.e.,_ $T=D$
---
## **task model** | updates
- task set → $n$ **periodic** tasks, $\tau = {\tau_1, \tau_2...\tau_n}$
- deadline **equal to** period, _i.e.,_ $T=D$
- all tasks **independant** → no precedence constraints
---
## **task model** | updates
- task set → $n$ **periodic** tasks, $\tau = {\tau_1, \tau_2...\tau_n}$
- deadline **equal to** period, _i.e.,_ $T=D$
- all tasks **independant** → no precedence constraints
- tasks **cannot suspend** themselves (or others)
---
## **task model** | updates
- task set → $n$ **periodic** tasks, $\tau = {\tau_1, \tau_2...\tau_n}$
- deadline **equal to** period, _i.e.,_ $T=D$
- all tasks **independant** → no precedence constraints
- tasks **cannot suspend** themselves (or others)
- tasks **preemptible** by OS → followed by highest priority task
---
## **task model** | updates
- task set → $n$ **periodic** tasks, $\tau = {\tau_1, \tau_2...\tau_n}$
- deadline **equal to** period, _i.e.,_ $T=D$
- all tasks **independant** → no precedence constraints
- tasks **cannot suspend** themselves (or others)
- tasks **preemptible** by OS → followed by highest priority task
- **bounded** execution time → wcet ($c_1, c_2, ... c_n$)
---
## **task model** | updates
- task set → $n$ **periodic** tasks, $\tau = {\tau_1, \tau_2...\tau_n}$
- deadline **equal to** period, _i.e.,_ $T=D$
- all tasks **independant** → no precedence constraints
- tasks **cannot suspend** themselves (or others)
- tasks **preemptible** by OS → followed by highest priority task
- **bounded** execution time → wcet ($c_1, c_2, ... c_n$)
- tasks placed into ready queue) **as soon as they arrive**
---
## **task model** | updates
- task set → $n$ **periodic** tasks, $\tau = {\tau_1, \tau_2...\tau_n}$
- deadline **equal to** period, _i.e.,_ $T=D$
- all tasks **independant** → no precedence constraints
- tasks **cannot suspend** themselves (or others)
- tasks **preemptible** by OS → followed by highest priority task
- **bounded** execution time → wcet ($c_1, c_2, ... c_n$)
- tasks placed into ready queue) **as soon as they arrive**
- kernel overheads (_e.g.,_ context switches) → **zero**
---
## **task model** | updates
may overly simplifying...
---
## **task model** | updates
may overly simplifying...
### helps develop → fundamental results
---
### **two** classes of online schedulers
---
### **two** classes of online schedulers
| priority assignment | algorithms |
|---------------------|------------|
| **static** | [Rate-Monotonic](#rate-monotonic-scheduler-rm) (RM), Deadline-Monotonic |
---
### **two** classes of online schedulers
| priority assignment | algorithms |
|---------------------|------------|
| **static** | [Rate-Monotonic](#rate-monotonic-scheduler-rm) (RM), Deadline-Monotonic |
| **dynamic** | [Earliest-Deadline First](#earliest-deadline-first-edf), Least-Slack Time (LST) |
||
---
Liu and Layland, [Scheduling Algorithms for Multiprogramming in a Hard- Real-Time Environment](https://dl.acm.org/doi/10.1145/321738.321743), Journal of ACM, 1973
---
## Rate-Monotonic Scheduler (RM)
---
## Rate-Monotonic Scheduler (RM)
priority assignment → task **period**
---
## Rate-Monotonic Scheduler (RM)
priority assignment → task **period**
|period| priority |
|------|----------|
| shorter | ↑|
| longer | ↓|
||
---
consider followin example:
|task|c|T|
|----|--|----|
| $\tau_1$ | 2| 4 |
| $\tau_2$ | 2| 5 |
| $\tau_3$ | 1| 10 |
||
---
consider followin example:
|task|c|T|
|----|--|----|
| $\tau_1$ | 2| 4 |
| $\tau_2$ | 2| 5 |
| $\tau_3$ | 1| 10 |
||
based on RM → $\tau_1 > \tau_2 > \tau_3$
---
consider followin example:
|task|c|T|
|----|--|----|
| $\tau_1$ | 2| 4 |
| $\tau_2$ | 2| 5 |
| $\tau_3$ | 1| 10 |
||
based on RM → $\tau_1 > \tau_2 > \tau_3$
since, $T_1 < T_2 < T_3$
---
consider following example:
|task|c|T|
|----|--|----|
| $\tau_1$ | 2| 4 |
| $\tau_2$ | 2| 5 |
| $\tau_3$ | 1| 10 |
||
based on RM → $\tau_1 > \tau_2 > \tau_3$
since, $T_1 < T_2 < T_3$
is this task set **schedulable**?
---
let's use our utilization test
$$U = \sum_{i=1}^{n} \frac{c_i}{T_i} \le 1 $$
---
let's use our utilization test
$$U = \sum_{i=1}^{n} \frac{c_i}{T_i} \le 1 $$
$$ U = \frac{1}{2} + \frac{1}{4} + \frac{2}{6} $$
---
let's use our utilization test
$$U = \sum_{i=1}^{n} \frac{c_i}{T_i} \le 1 $$
$$ U = \frac{1}{2} + \frac{1}{4} + \frac{2}{6} $$
$$ U = 0.5 + 0.4 + 0.1 $$
---
let's use our utilization test
$$U = \sum_{i=1}^{n} \frac{c_i}{T_i} \le 1 $$
$$ U = \frac{1}{2} + \frac{1}{4} + \frac{2}{6} $$
$$ U = 0.5 + 0.4 + 0.1 $$
$$ U = 1.0~ \checkmark $$
---
so, it should be **schedulable**...let's plot the schedule
---
<img src="img/scheduling/rm_unschedulable/pngs/rm_unsched1.png" width="1500" >
---
<img src="img/scheduling/rm_unschedulable/pngs/rm_unsched2.png" width="1500" >
---
<img src="img/scheduling/rm_unschedulable/pngs/rm_unsched3.png" width="1500" >
---
<img src="img/scheduling/rm_unschedulable/pngs/rm_unsched4.png" width="1500" >
---
<img src="img/scheduling/rm_unschedulable/pngs/rm_unsched5.png" width="1500" >
---
<img src="img/scheduling/rm_unschedulable/pngs/rm_unsched6.png" width="1500" >
**note**: first jobs of all three → released together
---
<img src="img/scheduling/rm_unschedulable/pngs/rm_unsched7.png" width="1500" >
priorities → $\tau_1 > \tau_2 > \tau_3$
---
<img src="img/scheduling/rm_unschedulable/pngs/rm_unsched8.png" width="1500" >
priorities → $\tau_1 > \tau_2 > \tau_3$
---
<img src="img/scheduling/rm_unschedulable/pngs/rm_unsched9.png" width="1500" >
priorities → $\tau_1 > \tau_2 > \tau_3$
---
<img src="img/scheduling/rm_unschedulable/pngs/rm_unsched10.png" width="1500" >
priorities → $\tau_1 > \tau_2 > \tau_3$
---
<img src="img/scheduling/rm_unschedulable/pngs/rm_unsched11.png" width="1500" >
priorities → $\tau_1 > \tau_2 > \tau_3$
---
<img src="img/scheduling/rm_unschedulable/pngs/rm.final.png" width="1500" >
<font style="background-color:orange">$\tau_3$ misses deadline!</font>
---
<font style="background-color:orange">$\tau_3$ misses deadline!</font>
but, it **passed** the schedulability check?
<img src="img/scheduling/confused_math.gif" width="700">
---
$\tau_3$ will **never** meet its deadline!
---
### hyperperiod
**lcm** of all task periods
---
### hyperperiod
**lcm** of all task periods
jobs repeat → after **each** hyperperiod
---
### hyperperiod
if we have **one** hyperperiod (HP)...
<img src="img/scheduling/hyperperiod_single.png" width="1500">
[not same task set from example]
---
### hyperperiod
if we have **one** hyperperiod (HP)...
it repeats **exactly** forever!
<img src="img/scheduling/hyperperiod_single.png" width="1500">
<img src="img/scheduling/hyperperiod_all.png" width="1500">
---
coming back to our example
|||
|---|---|
|<img src="img/scheduling/rm_unschedulable/pngs/rm.final.png" width="700" >| $HP = LCM (4, 5, 10) = \textbf{20}$|
||
---
coming back to our example
|||
|---|---|
|<img src="img/scheduling/rm_unschedulable/pngs/rm.final.png" width="700" >| $HP = LCM (4, 5, 10) = \textbf{20}$|
||
$\tau_3$ misses deadline at $t=10$, well within HP!
---
$\tau_3$ misses deadline at $t=10$, well within HP!
### it will **never** meet its deadlines!
---
well, that's a bummer...our utilization test isn't very useful!
---
well, that's a bummer...our utilization test isn't very useful!
except for trivial case → $ U > 1$
---
need a **better** test!
---
thankfully, these folks figured it out...
Liu and Layland, [Scheduling Algorithms for Multiprogramming in a Hard- Real-Time Environment](https://dl.acm.org/doi/10.1145/321738.321743), Journal of ACM, 1973
---
main issue → RM is a **static** priority assignment
---
main issue → RM is a **static** priority assignment
automatically **restricts** some schedules
---
automatically **restricts** some schedules
### **theoretical limit** on utilization
---
automatically **restricts** some schedules
### **theoretical limit** on utilization
based on number of tasks ($n$)
---
### utilization **upper bound** ($U_{ub}$)
---
### utilization **upper bound** ($U_{ub}$)
$$ U_{ub} = n.(2^{\frac{1}{n}} -1) $$
where $n$ → number of tasks
---
### utilization **upper bound** ($U_{ub}$)
so, for the task set to be **schedulable**,
$$ U = \sum_{i=1}^n \frac{c_i}{T_i} \le U_{ub} $$
<br>
where $n$ → number of tasks
---
### utilization **upper bound** ($U_{ub}$)
so, for the task set to be **schedulable**,
$$ U = \sum_{i=1}^n \frac{c_i}{T_i} \le n.(2^{\frac{1}{n}} -1) $$
<br>
where $n$ → number of tasks
---
### utilization **upper bound** ($U_{ub}$)
if $U < U_{ub} $ → task set is **schedulable**
---
### utilization **upper bound** ($U_{ub}$)
if $U < U_{ub} $ → task set is **schedulable**
(**necessary** but **not** sufficient)
---
### $U_{ub}$ vs $n$
$$ U \le n.(2^{\frac{1}{n}} -1) $$
---
### $U_{ub}$ vs $n$
$$ U \le n.(2^{\frac{1}{n}} -1) $$
as $n$ increases → $U_{ub}$ will **decrease**
---
as $n$ increases → $U_{ub}$ will **decrease**
but, **how much**?
---
let's check using the simulation-plotter...
---
from the plotter, we see that...
$$n = 3, \quad
U_{ub} \approx 0.78
$$
---
$$n = 3, \quad
U_{ub} \approx 0.78
$$
task set from example, $U = 1$!
**unschedulable!**
---
$U_{ub}$ for different numbers of tasks:
<img src="img/scheduling/rm_util_bounds.png" width="1100">
---
$U_{ub}$ for different numbers of tasks:
|||
|-----|------|
| <img src="img/scheduling/rm_util_bounds.png" width="700"> | ↑ $n$, ↓ $U_{ub}$|
---
$U_{ub}$ for different numbers of tasks:
|||
|-----|------|
| <img src="img/scheduling/rm_util_bounds.png" width="700"> | ↑ $n$, ↓ $U_{ub}$|
but how **low does** it go?
---
let's check using the simulation-plotter...again
---
interesting...the values seem to _tend_ to a certain value!
<img src="img/scheduling/rm_util_bound_100.png" width="1100">
---
interesting...the values seem to _tend_ to a certain value!
<img src="img/scheduling/rm_util_bound_100.png" width="1100">
with increasing $n$, $U_{ub}$ → <scb>0.69</scb>
---
for any RM-priority-assigned real-time system,
if <scb>U < 0.69</scb> → **schedulable**!
---
let's revisit our example from the cyclic executive...
<img src="img/scheduling/cyclic/cyclic6.svg" width="1500">
---
add some **period** information to it...
|task|c|T|
|----|--|----|
| $\tau_1$ | 1| **4** |
| $\tau_2$ | 2| **6** |
| $\tau_3$ | 3| **12** |
||
---
add some **period** information to it...
|task|c|T|
|----|--|----|
| $\tau_1$ | 1| **4** |
| $\tau_2$ | 2| **6** |
| $\tau_3$ | 3| **12** |
||
$$U = \frac{1}{4} + \frac{2}{6} + \frac{3}{12} $$
---
$$U = \frac{1}{4} + \frac{2}{6} + \frac{3}{12} $$
$$U = 0.25 + 0.33 + 0.25 $$
---
$$U = \frac{1}{4} + \frac{2}{6} + \frac{3}{12} $$
$$U = 0.25 + 0.33 + 0.25 $$
$$U \approx 0.83 $$
---
$$U \approx 0.83 $$
schedulable?
---
schedulable?
|||
|-----|------|
| <img src="img/scheduling/rm_util_bounds.png" width="700"> | $$U \approx 0.83 $$|
---
schedulable?
|||
|-----|------|
| <img src="img/scheduling/rm_util_bounds.png" width="700"> | $$U \approx 0.83 $$ $n=3$ → $U < \textbf{0.78}$|
---
|||
|-----|------|
| <img src="img/scheduling/rm_util_bounds.png" width="700"> | $$U \approx 0.83 $$ $n=3$ → $U < \textbf{0.78}$|
**not** schedulable!
---
**not** schedulable...let's plot the schedule
---
<img src="img/scheduling/rm_schedulable_example/pngs/rm2.png" width="1500" >
---
<img src="img/scheduling/rm_schedulable_example/pngs/rm4.png" width="1500" >
---
<img src="img/scheduling/rm_schedulable_example/pngs/rm6.png" width="1500" >
---
<img src="img/scheduling/rm_schedulable_example/pngs/rm8.png" width="1500" >
---
<img src="img/scheduling/rm_schedulable_example/pngs/rm9.png" width="1500" >
---
<img src="img/scheduling/rm_schedulable_example/pngs/rm10.png" width="1500" >
---
<img src="img/scheduling/rm_schedulable_example/pngs/rm11.png" width="1500" >
---
<img src="img/scheduling/rm_schedulable_example/pngs/rm12.png" width="1500" >
---
<img src="img/scheduling/rm_schedulable_example/pngs/rm13.png" width="1500" >
---
<img src="img/scheduling/rm_schedulable_example/pngs/rm14.png" width="1500" >
---
<img src="img/scheduling/rm_schedulable_example/pngs/rm15.png" width="1500" >
---
<img src="img/scheduling/rm_schedulable_example/pngs/rm.final.png" width="1500" >
<font style="background-color:orange"><b>all</b> tasks meet deadlines!</font>
---
what is going on?
---
remember the _necessary but **not** sufficient_ bit?
---
remember the _necessary but **not** sufficient_ bit?
|result of test|schedulable?|
|--------------|------------|
| pass, _i.e.,_ $U_{ub} < 0.69$ | yes |
| fail, $U_{ub} > 0.69$ | **unsure**|
||
---
remember the _necessary but **not** sufficient_ bit?
|result of test|schedulable?|
|--------------|------------|
| pass, _i.e.,_ $U_{ub} < 0.69$ | yes |
| fail, $U_{ub} > 0.69$ | **unsure**|
||
as we see → can still contruct **schedulable** task sets
---
need a _better_ test → **exact** analysis
---
### Response Time Analysis (RTA)
---
### Response Time Analysis (RTA)
- **all** jobs of task complete before their deadlines → **schedulable**
---
### Response Time Analysis (RTA)
- **all** jobs of task complete before their deadlines → **schedulable**
- caveat → account for **interference** from _higher priority_ jobs
---
### Response Time Analysis (RTA) | details
---
### Response Time Analysis (RTA) | details
1. **worst-case** response time of $\tau_i$,
$$R_i = c_i + I_i$$
---
### Response Time Analysis (RTA) | details
1. **worst-case** response time of $\tau_i$,
$$R_i = c_i + I_i$$
$I_i$ → **interference** faced by $\tau_i$'s job
from **all** higher prioriy jobs until then
---
### Response Time Analysis (RTA) | details
2. for _each_ higher priority job, $\tau_j$,
number of _jobs_ during time interval $R_i$ = $\left\lceil \frac{R_i}{T_j} \right\rceil$
Note:
Since _each period_ of task $\tau_j$ results in a new job being released. Since RM gives higher priority to shorter periods, those released jobs will execute ahead of the current teak, $\tau_i$
---
### Response Time Analysis (RTA) | details
3. $\lceil {R_i}/{T_j} \rceil$ number of $\tau_j$'s jobs **before** $\tau_i$
---
### Response Time Analysis (RTA) | details
3. $\lceil {R_i}/{T_j} \rceil$ number of $\tau_j$'s jobs **before** $\tau_i$
interference caused by **all** of them:
$$I_j = \left\lceil \frac{R_i}{T_j} \right\rceil .\ c_j$$
---
### Response Time Analysis (RTA) | details
4. the **total** interference,
$$I = \sum_{j\in hp(i)}\left\lceil \frac{R_i}{T_j} \right\rceil .\ c_j $$
Note:
the **total** interference then, is the sum of the individual interference by each of the higher priority jobs, _i.e.,_
---
### Response Time Analysis (RTA) | details
5. **response time** for $\tau_i$
$$R_i = c_i + I_i$$
Note:
Combine its own WCET
---
### Response Time Analysis (RTA) | details
5. **response time** for $\tau_i$
$$R_i = c_i + I_i$$
$$ R_i = c_i + \sum_{j\in hp(i)}\left\lceil \frac{R_i}{T_j} \right\rceil .\ c_j $$
---
### Response Time Analysis (RTA) | details
6. at **each** stage,