Scheduling Policy Design for Stochastic Non-preemptive Real-

time Systems*

Chris GillProfessor of Computer Science and Engineering

Washington University, St. Louis, MO, USAcdgill@cse.wustl.edu

CSE 131 Guest LectureOctober 8, 2012

*Research supported by NSF grants CNS-0716764 (Cybertrust) and CCF-0448562 (CAREER) and driven by numerous contributions from

Drs. Robert Glaubius (PhD 2009) and Terry Tidwell (PhD 2011); undergraduates Braden Sidoti, David Pilla, Justin Meden, Eli Lasker,

Micah Wylde, Carter Bass, Cameron Cross, and Percy Fang; and Prof. William D. Smart

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Why Good Schedules Are Needed

System resources are usually limitedAdding more raises cost, weight, powerSo, there’s rarely enough to go around

Activities contend for resourcesAiming camera to find/photograph

facesAiming camera to find/avoid obstaclesHow to share camera between these?

Scheduling access to resourcesAllows resources to be sharedRaises many other interesting

questions

Lewis Media and Machines LabWashington University

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Why Good Schedules are Difficult to Find

How long an activity needs a resource may varyWe’ll focus mainly on this issue in today’s talk

Issues we’ve addressed beyond that basic problemWe may have to learn distributions of times on-lineDifferent distributions in different operating modes

Image capture times with occlusion modes

timepro

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y

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Developing a System Model: a Good Start

A system model helps capture a problem rigorously» Gives a sound basis for reasoning about the problem» Focuses attention on particular kinds of analysis

Identifies the important abstractions to work with» For example, resources, activities, and shares

Captures key assumptions about the problem» E.g., is time treated as discrete or continuous?» E.g., is data available before, during, or after run-

time?

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Basic Scheduling Problem System Model

Time is considered to be discrete» E.g., a Linux jiffy is the time quantum

Separate activities require a shared resource» Access is mutually exclusive (activity binds the

resource)» Binding intervals are independent and non-

preemptive» Each activity’s distribution of intervals is known up

front

Goal: guarantee each activity a utilization fraction» For example, 1/2 and 1/2 or 1/3 and 2/3 » Want to define a scheduling policy (decides which

activity gets the resource when) that best fits that goal

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Formal System Model Representation

A state space describes such a system model well » Circles represent different

combinations of utilizations» Lower left corner is (0,0)» Vertical transitions give

quanta to one resource» Horizontal transitions give

quanta to the other one

Dashed ray shows goal» E.g., 1/3 vs 2/3 share (x, y)

Number of dimensions is number of activities» Generalizes to 3-D, …, n-D

0,1

0,0 1,0

1,1

0,3

0,2 1,2

1,3

2,1

2,0 3,0

3,1

2,3

2,2 3,2

3,3

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Dealing with Uncertainty

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y

timepro

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y

Easy if the resource is bound for one quantum at a time» Just move closest to goal ray

However, we have a probability distribution of binding times»Multiple possibilities per action

Need to consider probable consequences of each action »Leads to our use of a Markov

Decision Process (MDP) approach

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From Binding Times to a Scheduling MDP

We model these scheduling decisions as a Markov Decision Process (MDP) over use of the resource

The MDP is given by 4-tuple: (X,A,R,T)» X: the set of resource utilization states (how much

use)» A: the set of actions (giving resource to an activity)» R: reward function for taking an action in a state

(how close to the goal ray are we likely to remain)» T: transition function (probability of moving from one

state to another state)

Want to solve MDP to obtain a locally optimal policy

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Policy Iteration Approach

Define a cost function r(x) that penalizes deviation from the target utilization ray

Start with some initial policy 0

Repeat for t=0,1,2,…Compute the value Vt(x) -- the accumulated cost of following t -- for

each state x.Obtain a new policy, t+1, by choosing the greedy action at each state.

Guaranteed to converge to the optimal policy, requires storing Vt and t in lookup tables.

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Can’t do Policy Iteration Quite Yet

Unfortunately, the state space we have is infinite

Can’t apply MDP solution techniques directly to the state space as it stands» Need to bound the state

space to solve for a policy

Our approach» Reduce the state space to a

set of equivalence classes

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Insight: State Value Equivalence

Two states co-linear along the target ray have the same cost

Also have the same relative distribution of costs over future states (independent actions)

Any two states with the same cost have the same optimal value!

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Technique: State Wrapping

This lets us collapse the equivalent states down into a set of exemplar states» Notice how arrows (successors)

wrap back into “earlier” states

Now we can add “absorbing” states to bound the space»Far enough from target

ray, best decision is clear

Now we can use policy iteration to obtain a policy

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Automating Model Discovery

ESPI: Expanding State Policy Iteration [3]

1. Start with a policy that only reaches finitely many states from (0,…,0).

E.g., always run the most underutilized task.

» Enumerate enough states to evaluate and improve that policy

» If policy can not be improved, stop» Otherwise, repeat from (2) with newly improved

policy

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What About Scalability?

MDP representation allows consistent approximation of the optimal scheduling policy

Empirically, bounded model and ESPI solutions appear to be near-optimal

However, approach scales exponentially in number of tasks so while it may be good for (e.g.) sharing an actuator, it won’t apply directly to larger task sets

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What our Policies Say about Scalability

To overcome limitations of MDP based approach, we focus attention on a restricted class of appropriate scheduling policies

Examining the policies produced by the MDP based approach gives insights into choosing (and into parameterizing) appropriate policies

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Two-task MDP Policy

Scheduling policies induce a partition on a 2-D state space with boundary parallel to the share target

Establish a decision offset d to identify the partition boundary

Sufficient in 2-D, but what about in higher dimensions?

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Time Horizons Suggest a Generalization

H0 H1 H2 H3 H4

Ht={x : x1+x2+…+xn=t}

H0

H1

H2

(0,0) (2,0,0)

(0,2,0)

(0,0,2)

u

u

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Three-task MDP Policy

Action partitions meet along a decision ray that is parallel to the utilization ray

t =10 t =20 t =30

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Parameterizing a Partition

Specify a decision offset at the intersection of partitions

Anchor action vectors at the decision offset to approximate partitions

A “conic” policy selects the action vector best aligned with the displacement between the query state and the decision offset

a1a2

a3

x

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Conic Policy Parameters

Decision offset dAction vectors a1,a2,…,an

Sufficient to partition each time horizon into n regions

Allows good policy parameters to be found through local search

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Comparing Policies

Policy found by ESPI (for small numbers of tasks)πESPI(x) – chooses action at state x per solved MDP

Simple heuristics (for all numbers of tasks)πunderused(x) – runs the most underutilized task

πgreedy(x) – minimizes immediate cost from state x

Conic approach (for all numbers of tasks)πconic(x) – selects action with best aligned action vector

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Policy Comparison on a 4 Task Problem

Task durations: random histograms over [2,32]100 iterations of Monte Carlo conic parameter

searchESPI outperforms, conic eventually approximates

well

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Policy Comparison on a Ten Task Problem

Repeated the same experiment for 10 tasksESPI is omitted (intractable here)Conic outperforms greedy & underutilized

heuristics

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Comparison with Varying #s of Tasks

100 independent problems for each # (avg, 95% conf)

ESPI only tractable through all 2 and 3 task casesConic approximates ESPI, then outperforms

others

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Expanding our Notion of Utility

Previously, utility was proximity to utilization target; now we let tasks’ utility and job availability* varytime-utility function (TUF) name

period boundarytermination time

termination timeperiod boundary

* Availability variable qi is defined over {0,1}; {0, tmi/pi }; or {0,1} tmi/pi

Time

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Utility × Execution Utility Density

A task’s time-utility function and its execution time distribution (e.g., Di(1) = Di(2) = 50%) give a distribution of utility for scheduling the task

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Actions and State Space StructureState space can be more compact here than before:

dimensions are task availability, e.g., over (q1, q2), vs. time

Can wrap the state space over the hyper-period of all tasks (e.g., D1(1) = D2(1) = 1; tm1 = p1 = 4; tm2 = p2 = 2)

Scheduling actions induce a transition structure over states (e.g., idle action = do nothing; action i = run task i)

action 2action 1idle action

time timetime

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Reachable States, Successors, RewardsStates with the same task availability and the

same relative position within the hyper-period have the same successor state and reward distributions reachable states

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Evaluation

(target sensitive)

(linear drop)

(downward step)Different TUF shapes are useful

to characterize tasks’ utilities (e.g., deadline-driven, work-ahead, jitter-sensitive cases)

We chose three representative shapes, and randomized their key parameters: ui, tmi, cpi

(we also randomized 80/20 task

load parameters: li, thi, wi)

utilitybounds

criticalpoints

terminationtimes

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How Much Better is Optimal Scheduling?

Greedy (Generic Benefit*) vs. Optimal (MDP) Utility Accrual

* P. Li, PhD Dissertation, VA Tech, 2004

2 tasks 3 tasks

5 tasks4 tasks

TUF nuances matter: e.g., work conserving approach degrades target sensitive policy

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Divergence Increases with # of Tasks

Note we can solve 5 task MDPs for periodic task sets (but even representing a policy may be expensive)

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How Should Policies be Represented?

State Action

0 a1

1 a2

2 a2

3 a1

4 a1

5 a2

6 a2

7 a2

8 ?

9 a2

Scheduling policy can be stored as a lookup table (size = # states)»Tells best action to take in

each (modeled) state How to minimize

run-time memory cost? What to do about

unexpected states? How to take advantage

of heuristics?

Policy Table

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How to minimize memory footprint?

Decision trees compactly encode tabular data Trees can be built to approximate the policy

(0, a1) (1, a2) (2, a2) (3, a1) (4, a1) (5, a2) (6, a2) (7, a2) (8, ?) (9, a2)

x < 5

x < 3

a1

Inner Nodes Contain Predicates Over State Variables

Leaf Nodes Contain Action Mappings

a2

a2

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What to do about Unexpected States?

Trees abstract structure of encoded policy State x = 8 assigned a “reasonable” action (a2)

(0, a1) (1, a2) (2, a2) (3, a1) (4, a1) (5, a2) (6, a2) (7, a2) (8, ?) (9, a2)

x < 5

x < 3

a1

Inner Nodes Contain Predicates Over State Variables

Leaf Nodes Contain Action Mappings

a2

a2

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How to Take Advantage of Heuristics?

Leaf nodes also can recommend heuristics Trade run-time cost for accuracy of encoding

(0, a1) (1, a2) (2, a2) (3, a1) (4, a1) (5, a2) (6, a2) (7, a2) (8, ?) (9, a2)

x < 5

x < 3

a1

Inner Nodes Contain Predicates Over State Variables

Leaf Nodes Contain Action Mappings

greedy(x)

a2

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Optimal Tree Size Varies

36

Size of Tree

00 20 40 60 80 100

0.1

0.2

0.3

0.4

0.5

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Comparative Performance of Trees

37

1

0.8

0.6

0.4

0.2

0

Fraction of Optimal0 0.

10.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

optimalbesttreeheuristicgreedyPseudo 0

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Publications (so far)T. Tidwell, C. Bass, E. Lasker, M. Wylde, C. Gill, and W. D. Smart, "Scalable

Utility Aware Scheduling Heuristics for Real-time Tasks with Stochastic Non-preemptive Execution Intervals," 23rd Euromicro Conference on Real-Time Systems (ECRTS'11), Porto, Portugal, July 6 - 8, 2011.

T. Tidwell, R. Glaubius, C. Gill, and W. D. Smart, "Optimizing Expected Time Utility in Cyber-Physical Systems Schedulers," 31st IEEE Real-Time Systems Symposium (RTSS '10), San Diego, CA, USA, November 30 - December 3, 2010.

R. Glaubius, T. Tidwell, C. Gill, and W.D. Smart, “Real-Time Scheduling via Reinforcement Learning”, UAI 2010

R. Glaubius, T. Tidwell, B. Sidoti, D. Pilla, J. Meden, C. Gill, and W.D. Smart, “Scalable Scheduling Policy Design for Open Soft Real-Time Systems”, RTAS 2010 (received Best Student Paper award)

R. Glaubius, T. Tidwell, C. Gill, and W.D. Smart, “Scheduling Policy Design for Autonomic Systems”, International Journal on Autonomous and Adaptive Communications Systems, 2(3):276-296, 2009

R. Glaubius, T. Tidwell, C. Gill, and W.D. Smart, “Scheduling Design and Verification for Open Soft Real-Time Systems”, RTSS 2008

T. Tidwell, R. Glaubius, C. Gill, and W.D. Smart, “Scheduling for Reliable Execution in Autonomic Systems”, ATC 2008

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Concluding Remarks

Markov Decision Process (MDP) models are useful for scheduling stochastic real-time systems»Outperform heuristic approaches (sometimes by a lot)

Approximations of resulting policies also help»Where possible, geometric approximations do very well»Otherwise, decision trees offer good trade-offs

Research is a team sport»Undergraduates on this project outnumbered doctoral

students and faculty 4:1:1»Check out the CSE Department’s summer REU program