A Partitioning Algorithm

1
A Partitioning Algorithm

• Let Ti be a task in T such that Pi ? Pj for all
  tasks Ti in T
• S1 Ti, T T - Ti, For i2..g, Si ?
• For i2..g
• For every task Tl in T
• maxTl gcd(S1 ? Tl) S w over all Tk in (S1
  ? Tl)
• For j2..i-1
• maxTl maxmaxTl , gcd(Sj ? Tl) S w over
  all Tk in (Sj ? Tl)
• Let Tl be a task in T such that maxTl ? maxTk
  for every Tk in T
• Si Tl, T T - Tl
• For every Ti in T
• For j1..g
• SlackSj gcd(Sj ? Tl) S w over all Tk in
  (Sj ? Tl)
• Add Ti to the subset Sj for which SlackSj is
  maximum

2
Scheduling for Horizontal Striping

• Assume wi stripe units for supervideo Vi is
  stored on disk 0
• The Problem
• Schedule non-preemptible tasks T1Tr with periods
  P1Pr on q processors each task Ti has wi
  subtasks, with compute time of 1 interval
  between consecutive subtasks is m
• If a task has P4, w3 and m6 is it schedulable?
• Condition for subtasks of a single task Ti to be
  schedulable on a single processor
• For all 1 ? j, k ? wi, j ? k, ((j-1).m) mod Pi ?
  ((k-1).m) mod Pi
• Why?
• What is the corresponding algorithm to test
  schedulability?

3
Testing for Schedulability

• Procedure is_schedulable(T1..Tn,m)
• Let ggcd(P1..Pn)
• Take g time units 0..(g-1) that are not assigned
• Sort the tasks in decreasing order of wi
• For each task Ti in the sorted order
• j0
• while (j lt g)
• if for k0(wi 1), time unit (jk.m) mod g is
  free
• for k0(wi 1), assign time unit (jk.m) mod g
  to Ti
• exit while
• else jj1
• If jg return FAIL
• return (Ti , x) 1?? i ?? n and the first
  subtask of Ti is assigned to time unit x

4
Scheduling for Horizontal Striping

• Algorithm schedule_tasks_2 Ozden 97
• Ris_schedulable(T1Tn,m)
• If RFAIL
• Let gcd(P1...Pn,)g
• If g1, return FAIL
• Let T1Tn be tasks with periods P1/gPn/g and
  w1wn subtasks.
• Partition the tasks into g subsets S1Sg.
• For i1..g, let Ri schedule_tasks_2(Si,m/g)
• If for any i Ri FAIL, return FAIL, else return
• (Ti, x.g(j-1)) 1 ? j ? g and (Ti,x) ? Rj
• Return R

5
Example

• Tasks T1, T2, T3 with periods 6,8,8 have two
  subtasks each. The interval between them is 4.
• Try the algorithm.