Quasi Partitioned Scheduling

Optimal and Adaptive
Multiprocessor
Real-Time Scheduling:
The Quasi-Partitioned Approach
Γ = { { E.Massa, G.Lima, P.Regnier }, { G.Levin, S.Brandt } }
Real-Time Systems
Andrei Petrov
Padua, October 13th, 2015

Outline
+ Motivation
+ QPS Algorithm
+ Evaluation
+ Conclusion
2

Introduction
3
QPS
#optimal
#partitioned proportionate fairness
#preserves space & time locality of tasks
#gentle off-line bin-packing

Motivation
4
QPS
#on-line adaptation
#efficient switch mechanism P-EDF <--> G-Scheduling Rules
#low preemption and migration overhead

QPS Algorithm
5
Off-line phase On-line phase
❏ Major/minor execution
partitions
❏ Allocates processors to
execution sets
❏ Generates the schedule
❏ Manages the
activation/deactivation of
QPS servers
Quasi-Partition
cornerstone in dealing
with sporadic tasks

QPS Algorithm /1
6
System Model
❏ Γ = { i
:(Ri
, Di
) } for i=1,..,n
sporadic set
❏ independent, implicit-
deadlines tasks
❏ Ri
≤ 1 execution rate
❏ Di
infinite set of deadlines
❏ m identical processors
❏ migrations and preemptions
allowed
❏ A job J:(r,c,d)∈ i
❏ J.r∈{ 0 U Di
}
❏ J.d = min{d∈D, d>J.r}
❏ J.c = Ri
*(J.d-J.r)

QPS Algorithm /2
7
Definition
❏ A server S is a scheduling
abstraction
❏ acts like a proxy for a
collection of client
tasks/servers
❏ schedules its clients
according to the EDF policy
❏ Fixed-Rate EDF Server S:(RS
,DS
)
❏ RS
= Σ ∈T
R( )
❏ RS
≤ 1
❏ DS
= ⋃
∈T
D( )

QPS Algorithm /3
Definition
❏ Quasi-partition Q(Γ,m)
is a partition of Γ on m processors s.t.
1. | Q(Γ,m)|≤m
➢ restricts the cardinality of execution sets to the
number of available processors
2. ∀P∈Q(Γ,m), 0<R(P)<2
➢ Each element P in Q(Γ,m) is
○ (if R(P) <=1) a minor execution set
○ (if R(P) >1) a major execution set
➢ P at maximum requires two-processors for
being correctly scheduled
3. ∀P∈Q(Γ,m), ∀σ∈P,
R(P)>1 => R(σ)>R(P)-1
➢ P is a major execution
➢ the P’s extra ratio must be
less than the ration of any
other server σ in P
➢ keystone for the QPS’s
flexibility
N.B.
A graphical example follows
8

QPS Algorithm /4
9
➢ are given
2 processors
and a task set P
P
1
:(0.6)
3
:(0.8)
4
:(0.2)
2
:(0.3)
➔ R(P)=1.9≤2
◆ 1st cond. is OK
◆ 2 execution sets
Proc1
and Proc2
◆ max(capacity( Proci∈{1,2}
))=1
3
0
1
2
◆
3
Proc1
Proc2
➔ R(P)=1.9>1
◆ 2nd condition is OK
◆ more than 1 processor is required
to schedule P
➔ Let P = {PA
, PB
} such that
◆ PA
={ 3
, 4
}, PB
={ 1
, 2
}
◆ R(PA
)=1, R(PB
)=0.9
● no excess ratio,fully partitioned
● the slack 1-R(PB
) may be
an idle task on Proc2
➔ if modifying the 4
’s ratio to 0.3 then the
processor system becomes 100% utilized
◆ the 3rd cond. holds
◆ the excess rate (red color) is less than
the rate of all tasks in Proc1 execution
set
● otherwise a better re-allocation
of tasks exists
4
3
1
Example

QPS Algorithm /5
10
❏ QPS servers
➔ P is a major execution set with rate R(P)
=1+x
➔ P={PA
, PB
} is a bi-partition
➔ σA
:(R(PA
)-x, PA
)
σB
:(R(PB
)-x, PB
)
σM
:(x, P)
σS
:(x, P)
Dedicated servers
Master/Slave
servers
Excess ratio
➢ At any time t, all QPS servers
associated with P share the same
deadline D(P,t)
➢ Each QPS server σ
is a Fixed-Rate EDF Server
Definitions

QPS Algorithm /6
11
❏ Γ = { i
:(2/3,3) } for i=1,
2,3
❏ two-processors
❏ P1
= { 1
, 2
}, P2
= { 3
}
❏ R(P1
)= 4/3 , R(P2
)= 2/3
❏ [2,3] time interval denotes
parallel execution
Fixed-Rate Servers
Master/Slave
servers

QPS Algorithm /7
12
Off-line phase
❏ Γ = { σi
} for i = 1,2,...,5
a server set, three-
processors
❏ R(σi
) = 0.6
❏ Proc.1 and Proc.2 dedicated
processors
❏ Proc.3 shared processor
➢ σ6
,σ7
external servers reserve computer
capacity for exceeding parts
Processor Hierarchy
Lemma IV.1.
Any server set Γ0
with ceiling(R(Γ0
))≤m will be
allocated no more than m processors

QPS Algorithm /8
13
On-line phase Scheduling
❏ servers and tasks are scheduled according to
the following rules:
❏ visit each processor in revers order
to that of its allocation
❏ select via EDF the highest priority
server/task
❏ if a master server is selected,
❏ select also its slave associated
server
❏ for all selected servers
❏ select all highest priority client
❏ dispatch all selected tasks to execute

QPS Algorithm /9
14
On-line phase Scheduling
❏ Γ ={ 1
:(6,15), 2
:(12,30),
3
:(5,10), 4
:(3.5,5)}
set of sporadic tasks
❏ to be scheduled by QPS
servers on two-processors
❏ all tasks release their first
job at time 0
❏ the second job of 3
arrives at time t=16
whereas the other tasks
behave periodically
❏ Q(Γ,2)={P1
,P2
} with P1
={ 1
, 2
, 3
} and P2
={ 4
}
❏ R(P1
)=1.3>1 ⟹ σA
:(0.1, { 1
}), σB
:(0.6, { 2
, 3
}),
σM
:(0.3, P1
), σS
:(0.3, P1
)

QPS Correctness /1
15
Assumptions Time Definitions
❏ Δ = [t,t*
) time interval such that
❏ P is major execution set
❏ Δ is complete EDF interval if
Manager activates P at time t, next
activates QPS mode at t*
❏ Δ is complete QPS interval if all tasks
in P are active
❏ Δ is a QPS job interval if some task in
P releases a job at time t and has the
t*
=D(Γ,t)
QPS
The Partitoner
The Manager The Dispatcher

QPS Correctness /2
16
Theorem V.1
QPS produces a valid schedule for a set of implicit-deadline sporadic tasks Γ on
m ≥ ceiling(R(Γ)) identical processors
Proof: Let P be a major execution set
Lemma V.1.
Consider a complete QPS interval Δ. If the master server σM
in charge of P is scheduled on its
shared processor so that it meets all its deadlines in Δ, then the other three QPS servers will
also meet theirs on P’s dedicated processor
Lemma V.2.
The individual tasks and servers in P will meet all their deadlines provided that the master server
in charge of P meets its deadlines

Implementation
17
QPS
implemented 1
on top of LITMUSRT
1 Compagnin D., Mezzetti E., Vardanega T., “Experimental evaluation of optimal schedulers
based on partitioned proportionate fairness ” ECRTS15
#off-line decisions may influence run-time performance
#RUN vs QPS comparison
#empirical evaluation

Evaluation /1
18
QPS RUN1
U-EDF2
1 Regnier P., Lima G., Massa E., Levin G., Brandt S., “RUN: Optimal Multiprocessor Real-Time
Scheduling via Reduction to Uniprocessor”, 2011 32nd IEEE Real-Time Systems Symposium
2 Nelissen G., Berten V., Nelis V., Goossens J., Milojevic D., “U-EDF: An Unfair but Optimal
Multiprocessor Scheduling Algorithm for Sporadic Tasks”, 2012 ECRTS
#optimal scheduling algorithms
#the performance of the algorithms was assessed via simulation
#QPS’s performance is influenced by the “processor hierarchy”
tasks running on kth
processor may migrate
to any of m-k
processors

Evaluation /2
19
QPS RUN U-EDFDeveloping intuition about run-time overhead
❏ given a set of
❏ m processors
❏ m+1 periodic tasks that fully utilize the m processors
❏ the average hierarchy size is “(m-1)/2”
❏ for instance:
❏ if m=5 then 2 is the average hierarchy size
❏ m+1=5+1=6 periodic tasks
❏ in the worst case the hierarchy levels are as
many as the number of available m processors
(i.e. 6 processor levels )
❏ bin over-packing propagation phenomenon

Evaluation /3
20
QPS RUN U-EDF
Continuing the case study
❏ whenever the size of the task set grows up
❏ for example from m+1 tasks to 2m
❏ the run-time overhead drops down due of
❏ more lighter tasks
❏ more linear task set partitioning
❏ EDF mode which behaves like a
Partitioned EDF

Conclusion
21
❏ presents itself in a fashionable way against other
state-of-the-art global schedulers
★ while showing a manageable run-time overhead
❏ outperforms similar schedulers
★ when the processor hierarchy size is gently low
★ in presence of a fully partitioned task system (i.e P-EDF bevavior)
➔ off-line phase may influence at run-time significant overhead
◆ inter-server coordination my be quite not trivial
❏ depicts a simple scheduling abstraction model based on servers
★ dynamic adaptation as a function of system load variations
★ parallel execution needs are assured by master/slave relation
QPS

Future Work
22
❏ is related with the extension of QPS to broader problem constraints
❏ arbitrarily deadlines, heterogeneous multiprocessors, etc
❏ different adaptation strategies to extend the current master/slave abstraction model
to enhance less expensive inter-processor communication mechanism
❏ improve the quasi-partioning implementation
❏ extend it with different implementations
❏ evaluate the performance differences between them

Quasi Partitioned Scheduling

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