CPS Scheduling Policy Design with Utility and Stochastic Execution*

1. CPS Scheduling Policy Design with Utility and Stochastic Execution* Chris Gill Associate Professor Department of Computer Science and Engineering Washington University, St. Louis, MO, USA cdgill@cse.wustl.edu Georgia Tech CPS Summer School Atlanta, GA, June 23-25, 2010 *Research supported in part by NSF grants CNS-0716764 (Cybertrust) and CCF-0448562 (CAREER) and driven by numerous contributions from post-doctoral student Robert Glaubius; doctoral student Terry Tidwell; undergraduate students Braden Sidoti, David Pilla, Justin Meden, Carter Bass, Eli Lasker, Micah Wylde, and Cameron Cross; and Prof. William D. Smart

2. Washington University in St. Louis

3. Dept. of Computer Science and Engineering 24 faculty members and 70 Ph.D. students working in: real-time and embedded systems, robotics, graphics, computer vision, HCI, AI, bioinformatics, networking, high-performance architectures, chip multi-processors, mobile computing, sensor networks, 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 2011 is January 15: http://www.cse.wustl.edu

4. Why Pursue CPS Research? 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

5. Why is This Work CPS Research? 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

6. Which Problem Features are Interesting? Sharing e.g., a camera between navigation and image capture tasks (1) in general doesn’t allow efficient preemption (2) involves stochastically distributed durations Also important in general: (3) scalability (many tasks sharing such a resource); (4) task utility/availability Lewis Media and Machines Laboratory Washington University in St. Louis

7. System Model Assumptions • We model time as being discrete • E.g., based on some multiple of the Linux jiffy • States and scheduling decisions align with those quanta • Separate tasks require a shared resource • Access is mutually exclusive (a task binds the resource) • Binding durations are independent and non-preemptive • Tasks’ duration distributions are known (or learned [1]) • Each task is always available to run (relaxed in part III) • Goal: precise resource allocation among tasks [5] • E.g., 2:1utilization share targets for tasks A vs. B • Need a deterministic scheduling policy (decides which task gets the resource when) that best fits that goal

8. Part I Utilization State Spaces and Markov Decision Processes

9. Towards 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

10. Share 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 task’s duration (8,17)

11. Share Aware Scheduling u Utilization target induces a ray{u:0} through the state space Encode each state’s “goodness” (relative to the share) as a cost Require that costs grow with distance from utilization ray u=(1/3,2/3)

12. Transition Structure Transitions are state-independent I.e., relative distribution over successor states is the same in each state

13. Cost Structure States along same line parallel to the utilization ray have equal cost

14. Equivalence Classes Transition and cost structure thus induce equivalence classes Equivalent states have the same optimal long-term cost and policy!

15. Periodicity Periodic structure allows us to represent each equivalence class with a single exemplar [4]

16. Wrapping 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)

17. c(x)= c(x)= Truncating 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

18. Bounding 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)

19. A 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

20. Automating Model Discovery ESPI: Expanding State Policy Iteration [3] • 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

21. Policy 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)

22. Policy Improvement Envelope Consider alternative actions Close under the current policy using breadth-first expansion Evaluate and improve the policy within this envelope

23. ESPI 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

24. Comparing Design Methods Policy performance is shown normalized and centered on the ESPI solution data Larger bounded state models yield the ESPI solution

25. Part II Scalability and Approximation Techniques

26. 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

27. 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 [2]

28. 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?

29. Time Horizons Suggest a Generalization Ht={x : x1+x2+…+xn=t} u (0,0,2) u (0,2,0) H0 H1 (0,0) (2,0,0) H0 H1 H2 H3 H4 H2

30. Three-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

31. x 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 a2 a1 a3

32. Decision offset d Action vectors a1,a2,…,an Sufficient to partition each time horizon into nregions Allows good policy parameters to be found through local search Conic Policy Parameters

33. 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

34. Policy 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

35. Policy Comparison on a Ten Task Problem Repeated the same experiment for 10 tasks ESPI is omitted (intractable here) Conic outperforms greedy & underutilized heuristics

36. Comparison 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

37. Part III Expanding our Notions of Utility and Availability

38. Time-Utility Functions Previously, utility was proximity to utilization target; now we let tasks’ utility and job availability* vary time-utility function (TUF) name period boundary termination time termination time period boundary Time * Availability variable qi is defined over {0,1}; {0, tmi/pi }; or {0,1} tmi/pi

39. 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

40. Actions and State Space Structure State space can be more compact here than in parts I and II: 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) idle action action 1 action 2 time time time

41. Reachable States, Successors, Rewards States with the same task availability and the same relative position within the hyper-period have the same successor state and reward distributions reachable states

42. Evaluation (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) (linear drop) termination times utility bounds (target sensitive) critical points

43. How Much Better is Optimal Scheduling? Greedy (Generic Benefit*) vs. Optimal (MDP) Utility Accrual 2 tasks 3 tasks TUF nuances matter: e.g., work conserving approach degrades target sensitive policy 4 tasks 5 tasks * P. Li, PhD Dissertation, VA Tech, 2004

44. Divergence Increases with # of Tasks Note we can solve 5 task MDPs for periodic task sets (smaller state spaces; scalability is an ongoing issue)

45. Conclusions We have developed new techniques for designing non-preemptive scheduling policies for tasks with stochastic resource usage durations MDP-based methods are effective for 2 or 3 task utilization share problems (e.g., for an actuator) Conic policy performance is competitive with ESPI for smaller problems, and for larger problems improves on the underutilized and greedy policies Ongoing work is focused on identifying and evaluating important categories of time-utility functions and tailoring our approach to address their nuances

46. Publications [1] R. Glaubius, T. Tidwell, C. Gill, and W.D. Smart, “Real-Time Scheduling via Reinforcement Learning”, UAI 2010 [2] 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) [3] 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 [4] R. Glaubius, T. Tidwell, C. Gill, and W.D. Smart, “Scheduling Design and Verification for Open Soft Real-Time Systems”, RTSS 2008 [5] T. Tidwell, R. Glaubius, C. Gill, and W.D. Smart, “Scheduling for Reliable Execution in Autonomic Systems”, ATC 2008

47. Thanks, and hopeto see you at CPSWeek 2011! Chris Gill Associate Professor of Computer Science and Engineering