MULTIMEDIA AND SYSTEM DESIGN

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 i=2..g, Si = φ
• For i=2..g
• For every task Tl in T
• max[Tl] = gcd(S1 ∪ {Tl}) – Σ w over all Tk in (S1 ∪ {Tl})
• For j=2..i-1
– max[Tl] = max{max[Tl] , gcd(Sj ∪ {Tl}) – Σ w over all Tk in (Sj ∪ {Tl})}
• Let Tl be a task in T such that max[Tl] ≤ max[Tk] for every Tk in T
• Si = {Tl}, T = T - {Tl}
• For every Ti in T
• For j=1..g
• Slack[Sj] = gcd(Sj ∪ {Tl}) – Σ w over all Tk in (Sj ∪ {Tl})
• Add Ti to the subset Sj for which Slack[Sj] 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 T1…Tr with periods P1…Pr
on q processors; each task Ti has wi subtasks, with
compute time of 1; interval between consecutive subtasks
is m
• If a task has P=4, w=3 and m=6 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

3. Testing for Schedulability
• Procedure is_schedulable({T1..Tn},m)
• Let g=gcd(P1..Pn)
• Take g time units 0..(g-1) that are not assigned
• Sort the tasks in decreasing order of wi
• For each task Tiin the sorted order
• j=0
• while (j < g)
– if for k=0…(wi – 1), time unit (j+k.m) mod g is free
• for k=0…(wi – 1), assign time unit (j+k.m) mod g to Ti
• exit while
– else j=j+1
• If j=g return FAIL
• return {(Ti, x): 1 ≤ i ≤ n and the first subtask of Tiis

4. Scheduling for Horizontal
Striping
• Algorithm schedule_tasks_2 [Ozden 97]
• R=is_schedulable({T1…Tn},m)
• If R=FAIL
• Let gcd(P1...Pn,)=g
• If g=1, return FAIL
• Let T1…Tn be tasks with periods P1/g…Pn/g and w1…wn
subtasks.
• Partition the tasks into g subsets S1…Sg.
• For i=1..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.

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