on budgeting and quality - rtns 2018 · on budgeting and quality with an application to...

On Budgeting and QualityWith an Application to Safety-Critical Real-Time

Systems

Bader Alahmad & Sathish Gopalakrishnan

The University of British Columbia

VM1

$

VM2

$$$$

VM3

$$$

Budgeting vs. Quality

Goal: Maintain a certain QoS level for the long-term operation of the system with high confidence

modulo perhaps initial transient behavior

Task Model

Task Model

» Recurrent tasks with given periods

Task Model


» Task has random execution-time demand

Task Model



» Available budget is random variable

Task Model




» and not necessarily independent

Task Model




» and not necessarily independent

» Bad situation: , central quantity:

Problem Description

: Fraction of first jobs where budget is insufficient

Problem Description


» The process is iid, sampled at job release times

Problem Description



» Given:

Problem Description



» Given:

» QoS level : Tolerable fraction of jobs that might demand more than available budget

Problem Description



» Given:


» Confidence parameter

Problem Description



» Given:


» Confidence parameter

» Question: Is there integer s.t. jobs of the task?

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n (# jobs)



m

What's in it for the system designer?

Resource Dimensioning

If we know the execution-time requirement , how should the budget look like so that for some , for all

?


» Recast. If we let : Derive a bound on so that if , then QoS requirement satisfied.



» Then any budget satisfying is okay




» We have a sufficient range




» We have a sufficient range

» The larger the , the more the flexibility the system designer has in allocating resources (budgets)

Large Deviation bounds


» Requirement: Strongest possible bound



» decays exponentially quickly in




» Gives the largest possible sufficient bound on




» Gives the largest possible sufficient bound on

» Requirement made precise: Is there strictly positive function (that might depend on ) such that

Chernoff Theorem

Let be independent random variables such that always lies in the interval . Define , and let . Then for any ,

»

»

»

»

»

»

»

»

»

»

»

»

»

»

»

»

»

»

»

» If , then setting gives

!

»

»

»

»

»

» If , then setting gives

!

»

Our Large Deviation Bound

Upper bound on -value

Let . If , then

Observe: for every , and as

System-wide QoSDependent Tasks

Dependent tasks & System-wide QoS


» Given:


» Given:

» System-wide confidence ;


» Given:


» Task QoS levels ;


» Given:



» "limiting" number of job releases of task , ;


» Given:




» That at most tasks are dependent


» Given:





» Bad event for task : , where


» Given:






» Question: Under what condition(s) is for all releases of every task task , ?

Lovász Local Lemma (LLL)

Let be a sequence of events such that each event occurs with probability at most and such that each event is

independent of all the other events except for at most of them.

If , then

System-wide QoS bounds



» Question: Under what condition(s) is for all ?




» By LLL and LD bound, when




» By LLL and LD bound, when

» When , if , then it's sufficient that

Application to safety-critical systems

Monitors + Isochronous execution



» Main task is high quality but unpredictable with execution-time demand




» Monitor is lower-quality and with deterministic WCET (to be determined)




» Monitor is lower-quality and with deterministic WCET (to be determined)

» Here

Isochronous Execution

J1 (9) J2 (6) J1 J2

J2 (6)J2J3 J3 (5)

J1 (9) J1

⇡1 ⇡2 ⇡3 ⇡4 ⇡50

2

6

14

Two problems to be solved


» Problem1: Derive lower bounds on monitor WCETs so that each task achieves the desired QoS



» Problem 2: Derive upper bounds on monitor WCETs so that all tasks meet their hard deadlines under isochronous execution while achieving QoS requirements (from Problem 1).

Problem 1: Monitor WCET lower bounds



» Solution: It is sufficient that for every



» Solution: It is sufficient that for every

» Is there a feasible isochronous schedule of the tasks with these monitor WCETs?

Problem 2: Feasible Monitor WCET Upper Bounds


» Derive upper bounds on monitor WCETs so that all tasks meet their hard deadlines under isochronous execution while achieving QoS requirements (from Problem 1).


» Derive upper bounds on monitor WCETs so that all tasks meet their hard deadlines under isochronous execution while achieving QoS requirements (from Problem 1).

» Recall: Given monitor WCETs , a feasible isochronous schedule exists if the optimal solution to the LP below is such that :


» One possible solution:


» One possible solution:

» Gives feasible range for task 's monitor WCET if instance is feasible

Do we always need monitors?

» Task may not need a monitor if its demand is unbounded

on budgeting and quality - rtns 2018 · on budgeting and quality with an application to...

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