Title: Non-Preemptive Scheduling Policy Design for Tasks with Stochastic Execution Times*
1Non-Preemptive Scheduling Policy Design for Tasks
with Stochastic Execution Times
- Chris Gill
- Associate Professor
- Department of Computer Science and Engineering
- Washington University, St. Louis, MO, USA
- cdgill_at_cse.wustl.edu
The University of Pennsylvania Monday November
23, 2009
Research supported by NSF grants CNS-0716764
(Cybertrust) and CCF-0448562 (CAREER) and driven
by numerous contributions from doctoral students
Robert Glaubius and Terry Tidwell undergraduates
Braden Sidoti, David Pilla, Justin Meden, and
Cameron Cross and Prof. William D. Smart
2Washington University in St. Louis
3Dept. of Computer Science and Engineering
- 24 faculty members and 71 Ph.D. students working
in - real-time and embedded systems, robotics,
graphics, HCI, AI, bioinformatics, networking,
high-performance architectures, chip
multi-processors, mobile computing, sensor
networks, distributed systems, optimization - PhD students are fully funded, and we emphasize
individual mentorship and interdisciplinary work - Recent graduates are on faculty at U. Mass,
UT-Austin, Rochester, RIT, CMU, Michigan St.,
and UNC-Charlotte - Graduate study application deadline for Fall 2010
is January 15 http//www.cse.wustl.edu
4Motivation
- Systems are increasingly being designed to
interact with the physical world - This trend offers compelling new research
challenges that motivate our work - Consider for example the domain of mobile robotics
my name is
Lewis Media and Machines Laboratory Washington
University in St. Louis
5Motivation
- As in many other systems, resources must be
shared among competing tasks - Fail-safe modes may reduce consequences of
resource-induced timing failures, but precise
scheduling matters - The physical properties of some resources
motivate new models and techniques
my name is
Lewis Media and Machines Laboratory Washington
University in St. Louis
6Motivation
- For example, sharing a camera between navigation
and surveying tasks - (1) in general doesnt allow efficient
preemption - (2) involves stochastically distributed
durations - Other scenarios also raise scalability questions,
e.g., multi-robot heterogeneous real-time data
transmission
Lewis Media and Machines Laboratory Washington
University in St. Louis
7System Model Assumptions
- To begin, time is modeled as being discrete
- E.g., some multiple of the Linux jiffy is the
time quantum - Separate tasks require a shared resource
- Access is mutually exclusive (a task binds the
resource) - Binding durations are independent and
non-preemptive - Each tasks distribution of durations can be
known - Each task is always available to run
- Goal precise resource allocation among the tasks
- E.g., 21 utilization share targets for tasks A
vs B - Need a deterministic scheduling policy (decides
which task gets the resource when) that best fits
that goal
8Towards Optimal Policies
- A Markov decision process (MDP) is a 4-tuple
(X,A,C,T) that matches our system model well - X a finite set of states (e.g., utilizations of
8 vs. 17 quanta) - A the set of actions (giving resource to a
particular task) - C cost function for taking an action in a state
- T transition function (probability of moving
from one state to another state based on the
action chosen) - Solving the MDP gives a policy that maps each
state to an action to minimize long term expected
costs - However, to do that we need a finite set of
states
9Share Aware Scheduling
- A system state cumulative resource usage of each
task - Dispatching a task moves the system
stochastically through the state space according
to that tasks duration
(8,17)
10Share Aware Scheduling
- Utilization target induces a ray ?u??0 through
the state space - Encode each states goodness (relative to the
share) as a cost - Require that costs grow with distance from
utilization ray
?u
u(1/3,2/3)
11Transition Structure
- Transitions are state-independent
- I.e., relative distribution over successor states
is the same in each state
12Cost Structure
- States along same line parallel to the
utilization ray have equal cost
13Equivalence Classes
- Transition and cost structure thus induce
equivalence classes - Equivalent states have the same optimal long-term
cost and policy!
14Periodicity
- Periodic structure allows us to represent each
equivalence class with a single exemplar
15Wrapping the State Model
- Remove all but one exemplar from each equivalence
class - Actions and costs remain unchanged
- Remap any dangling transitions (to removed
states) to the corresponding exemplar
(0,0)
16Truncating the State Model
- Inexpensive states are nearer the utilization
target - Good policies should keep costs small
- Can truncate the state space by bounding sizes of
costs considered
17Bounding the State Model
- Map any dangling transitions produced by
truncation, to a high-cost absorbing state - This guarantees that we will be able to find
bounded-cost policies if they exist - Bounded costs also guarantee bounded deviation
from the resource share (precision)
18A Scheduling Policy Design Approach
- Iteratively increase the bounds and re-solve the
resulting MDP - As the bounds increase, the bounded model
solution converges towards the optimal wrapped
model policy
19Automating Model Discovery
- ESPI Expanding State Policy Iteration
- 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
20Policy Evaluation Envelope
- Enumerate states reachable from the initial state
- Explore state space breadth-first under the
current policy, starting from the initial state
(0,0)
21Policy Improvement Envelope
- Consider alternative actions
- Close under the current policy using
breadth-first expansion - Evaluate and improve the policy within this
envelope
22ESPI Termination
- As long as the initial policy has finite closure,
each ESPI iteration terminates (this is satisfied
by starting with the heuristic policy that always
runs the most underutilized task) - Policy strictly improves at each iteration
- Anecdotally, ESPI terminates on all of the task
scheduling MDPs to which we have applied it
23Comparing Design Methods
- Policy performance is shown normalized and
centered on the ESPI solution data - Larger bounded state models yield the ESPI
solution
24What 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 wont apply directly to
larger task sets
25What 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
26Two-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?
27Time Horizons Suggest a Generalization
Htx x1x2xnt
?u
(0,0,2)
?u
(0,2,0)
H0
H1
(0,0)
(2,0,0)
H0
H1
H2
H3
H4
H2
28Three-task MDP Policy
t 10
t 20
t 30
- Action partitions meet along a decision ray that
is parallel to the utilization ray - Action partitions are roughly cone-shaped
29Parameterizing 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
a1
a2
a3
30Conic Policy Parameters
- Decision offset d
- Action vectors a1,a2,,an
- Sufficient to partition each time horizon into n
regions - Allows good policy parameters to be found through
local search
31Comparing Policies
- Policy found by ESPI (for small numbers of
tasks) - pESPI(x) chooses action at state x per solved
MDP - Simple heuristics (for all numbers of tasks)
- punderused(x) runs the most underutilized task
- pgreedy(x) minimizes immediate cost from state
x - Conic approach (for all numbers of tasks)
- pconic(x) selects action with best aligned
action vector -
32Policy Comparison on a 4 Task Problem
- Task durations random histograms over 2,32
- 100 iterations of Monte Carlo conic parameter
search - ESPI outperforms, conic eventually approximates
well
33Policy Comparison on a Ten Task Problem
Repeated the same experiment for 10 tasks ESPI is
omitted (intractable here) Conic outperforms
greedy underutilized heuristics
34Comparison with Varying s of Tasks
100 independent problems for each (avg, 95
conf) ESPI only tractable through all 2 and 3
task cases Conic approximates ESPI, then
outperforms others
35Conclusions
- We have developed new techniques for designing
non-preemptive scheduling policies for tasks with
stochastic resource usage durations - MDP-based methods provide good approximations to
optimal policies for 2 or 3 tasks - Conic policy performance is competitive with ESPI
for smaller problems, and for larger problems
improves on underutilized and greedy policies - Future work will focus on applying and evaluating
our results in different cyber-physical systems,
and on extending them further in design and
verification
36For Further Information
- 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 - 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, Tech. Report WUCSE-2009-71, 2009 (Under
Review for RTAS 2010) - R. Glaubius, T. Tidwell, C. Gill, and W.D. Smart,
Scheduling Design with Unknown Execution Time
Distributions or Modes. Tech. Report
WUCSE-2009-15, 2009 - T. Tidwell, R. Glaubius, C. Gill, and W.D. Smart,
Scheduling for Reliable Execution in Autonomic
Systems, ATC 2008 - C. Gill, W.D. Smart, T. Tidwell, and R. Glaubius,
Scheduling as a Learned Art, OSPERT, 2008 - Project web site http//www.cse.wustl.edu/cdgill
/Cybertrust/
37Thank you!
- Chris Gill
- Associate Professor of Computer Science and
Engineering
38Appendix Comparison to EDF Scheduling
- Earliest-Deadline-First (EDF) scheduling
- Enforces timeliness by meeting task deadlines.
- Not share aware.
- We introduce deadlines as a function of
worst-case execution time. - Miss rate is a function of deadline tightness.
39Appendix Varying Temporal Resolution
40Appendix Stable Conic Policies
(0,0,t)
- Guaranteed that stable conic policies exist.
- For example, set each action vector to point
opposite its corresponding vertex. - Induces a vector field that stochastically orbits
the decision ray.
(t,0,0)
(0,t,0)
41Appendix Stable Conic Policies
(0,0,t)
- Guaranteed that stable conic policies exist.
- For example, set each action vector to point
opposite its corresponding vertex. - Induces a vector field that stochastically orbits
the decision ray.
(t,0,0)
(0,t,0)
42Appendix More Tasks Implies Higher Cost
- Simple problem Fair-share scheduling of n
deterministic tasks with unit duration - Trajectories under round robin scheduling
- 2 tasks Ec(x) 1/2.
- Trajectory (0,0)?(1,0)?(1,1)?(0,0)
- Costs c(0,0)0 c(1,0)1.
- 3 tasks Ec(x) 8/9.
- Trajectory (0,0,0)?(1,0,0)?(1,1,0)?(1,1,1)?(0,0,0
) - Costs c(0,0,0)0 c(1,0,0)4/3 c(1,1,0)4/3
- n tasks Ec(x) (n1)(n-1)/(3n)
43Appendix Share Complexity