Fixed Priority Scheduling with Limited Priority Level

1
Fixed Priority Scheduling with Limited Priority
Level

• Yi-Lin Tsai
• Department of Computer Science and
  InformationEngineering,
• National Taiwan University

2
Introduction

• Some systems, such as network, usually support
  small number of priorities.
• If natural priority levels gt system priority
  levels
• more than one task must be grouped into the same
  system priority.
• can result in significantly reduced schedulable
  utilization levels

3
Introduction (cont.)

• This paper introduces
• NS schedulability test with limited priorities
• to test whether a task set is schedulable under a
  system with limited priorities.
• Degree of schedulable saturation
• to evaluate the impact of task groupings when
  priority levels are limited.

4
Background

• Sufficient Schedulability Condition with
  Unlimited Priorities
• If
• gt the task set is schedulable.

5
Background (cont.)

• Sufficient Schedulability Condition with Limited
  Priorities
• ln(2G) 1 G when G gt ½
• U
• G when G ½
• The system has a priority grid L0, L1, L2,.,
  Lk, where each Li is assigned a task period
• Lk is the largest period in the task set and L0
  is the smallest period
• If a task has a period T such that Li-1 lt T
  Li, it is assigned priority Li-1

6
Background (cont.)

• Relative schedulability
• ln(2/r) 1 1/r/ln2 when r
  lt 2
• Rs
  r 1/G
• 1/(r ln2) when r 2

7
Background (cont.)

• NS Schedulability Condition with Unlimited
  Priorities
• The cumulative work arrived from priority levels
  1 to i in the time interval 0, t is given by
• If , the task set will meet its
  deadlines if

8
Mapping Natural Priorities to Priority Groups

• There is a surjective mapping between tasks and
  priority groups
• Each group contains at least one task.
• If task ti gk and task tj gl, and k ! l
  then
• If P(ti) gt P(tj) then it must be the case that k
  lt l,
• If P(ti) lt P(tj) then it must be the case that k
  gt l, or
• If P(ti) P(tj) the either k 1 l or l 1
  k
• The total number of ways n task can be arranged
  into m priority groups is

9
NS Schedulability Condition with Limited
Priorities

• If ti arrived at time 0, the total worst case
  cumulative work that must be completed before ti
  completes execution at some time interval 0, t
  consists of
• Higher priority work that arrives in the time
  interval 0, t from priority groups 1 to k-1
• Other work within priority group k which has
  arrived by time 0
• gt

10
NS Schedulability Condition with Limited
Priorities (cont.)

• If Di Ti, then ti will meet all its deadline if
• gt the task set will meet its deadlines if

11
Degree of Schedulable Saturation

• where
• where
• If Smax 1, then the task set is schedulable.

12
Degree of Schedulable Saturation (cont.)

• A particular scheduling situation is said better
  if it results in a small Smax
• Given that Smax Sj for some 1 j n, then tj
  is called the limiting task.
• Smax is monotonically nondecreasing when either
  the demands of each task increase or the number
  of tasks increases.

13
Sporadic Server vs. Smax

• The largest capacity server with period Ts which
  can be added to a task set with n tasks that
  maintains schedulability is given by
• And
• gt

14
Sporadic Server vs. Smax (cont.)

• Then
• The limiting task, for which Si Smax, has an
  associated limiting time t, at which
• If kTs t, k 1,2,, then (1 Smax)Ts Cs

15
Sporadic Server vs. Smax (cont.)

• (1 Smax)Ts Cs
• Smax provides an indication of the amount of high
  priority work that can be added at or above the
  priority of the limiting task
• Minimizing Smax maxizes the capacity of the
  sporadic server

16
Relative Schedulability

• Relative schedulability, Rs, is defined as
  follows
• Rs Smax/Smax if Smax 1
• Rs 0 otherwise
• A scheduling approach is preferable if it
  minimizes Smax
• Finding the best mapping is an exponental search
  problem.

17
Case Study Multimedia Task Set

• Smax 0.6805

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Case Study Multimedia Task Set

• Minimum Smax 0.6805

19
Case Study Multimedia Task Set
20
Case Study Multimedia Task Set

• Assume tj is the limiting task with unlimited
  priorities. A task set will suffer no loss in
  schedulable utilization (Smax Smax), if
• The limiting task when the system has limited
  priority levels is also tj, and
• tj is either in a priority group by itself or is
  the lowest natural priority task in its group