Alina Weffers-Albu, [e-mail]

1
On a Theory of Media Processing Systems
Behaviour, with Applications

• M.A. Weffers-Albu, J.J. Lukkien, E.F.M. Steffens,
  P.D.V. v.d. Stok

2
Contents

• Domain introduction
• Relevant issues
• Related work
• Approach
• Model
• Conclusions

3
Domain introduction

• Media Processing Applications systems with
  real-time requirements
• Execute on physical platform (CPU, memory, bus)

4

Domain introductionMedia Processing Applications
5

Domain introductionMedia Processing Applications

Physical Platform
6

Domain introductionMedia Processing Applications

Physical Platform
7

Domain introductionMedia Processing Applications
8
Relevant Issues

• When constructing / modifying an application
• How much resources does the application need?
• Will the application meet its timing
  requirements?
• What is the minimum of the resources needed such
  that timing requirements are met?
• What can one do to reduce the resource needs of
  the application?
• What can be done to improve the measure in which
  the application meets timing requirements?
• Need to predict, control the overall behavior of
  the application

9
Relevant Issues

• Calculate / Optimize
• response time of tasks, chain,
• necessary and sufficient buffer capacities
• number of context switches (cost associated)
• Need model for dynamic behavior of system.

10
Related work

• Classical RT theory does not help directly
• Mainly periodic independent tasks
• Tasks with precedence constraints
• Previous work in domain
• Estimations for some parameters (Lowet)
• Reduce the problem to an equivalent one where
  tasks could be treated as independent periodic
  (Goddard, Groba).
• Our approach
• Calculate parameters rather than estimate
• Uses the fact that tasks are dependent rather
  than reduce it into a classical RT problem
• Contribution of paper
• foundations of theory for engineers to reason
  rigorously about system behavior and associated
  resource needs.

11
Approach

• First step single linear chain executing in
  cooperative environment
• Express the execution of the system (chain) as a
  trace of the atomic actions taken by each
  component.
• Analyze the system in terms of time and behaviour
  as a function of
• the choice of the atomic action order in the
  components
• the channel properties (e.g., the capacity)
• the priority assignment
• the timing assignment.

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ApproachBasic concepts

• Component program example C a b c d
• Component alphabet set of actions taken by the
  component (A(C)a,b,c,d).
• Alphabets disjoint ? k?l, A(Ck) ? A(Cl) ?.
• Trace of a component sequence of actions of a
  component, recording its execution.
• Tr(C) (a b c d)
• Trace prefix Tr(C)(abcd), Pref(Tr(C)) (a),
  (a b), (a b c), (a b c d)
• Prefix State Pref(Tr(C)) St(Tr(C))
• Parallel composition of components arbitrary
  interleaving of actions
• Ck Cl, with Ck a b and Cl c d
• Tr(CkCl) (a b c d), (a c b d), (c a d b), (c
  d a b), (a c d b), (c a b d)

13
ApproachBasic concepts

• Projection of a trace on an alphabet (t?A) trace
  obtained by removing all actions not in A while
  maintaining the order given in t.
• t(abcd), A(Ck) a, b, t?A (ab)
• (t,a) nb. of occurences of action a in trace
  t.
• Ci while (true) do
• receive( fqi-1, p)
• receive( eqi, q)
• process_fct(p,q)
• send( eqi-1, p)
• send( fqi, q)

Tr(Ci) (fqi-1? eqi? ci eqi-1! fqi!)?
for 1 ? i ? n.
14
Approach

• Express the execution of the system (chain) as
  trace(s) of the atomic actions taken by each
  component.

Impose predicates until obtain trace(s) that
specifies the system behavior.
Tpc ? Tcc ? Til
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Arbitrary interleavings traceset (Til)

• A A(Ci),
• Til Tr( Ci )
• Til s s consists of elements of A ? s?A(Ci)?
  Tr(Ci), ?i, i 1..n

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Introducing channel constraints

• Scc imposes that no buffer is allowed to overflow
• or underflow.
• Scc(t) (? s?Pref(t), fqi, eqi (1 i lt n)
• 0 (t,fqi!) - (t,fqi?) Cap( fqi) ?
• 0 (t,eqi!) - (t,eqi?) Cap( eqi )).
• Tcc t?Til Scc(t) channel consistent traces.

17
Introducing channel constraintsDefinitions
ready and blocked actions

• Tcc constrained set
• Given s from St(Tcc), a component Ci is
  ready-to-run in state s whenfor an action a in
  A(Ci) such that sa? A(Ci) is a state of Tr(Ci) we
  have that sa ? St(Tcc). Also, a is called a
  ready action of Ci in state s.
• sa ? A(Ci) ? sa ? St(Tcc) ? a is a ready action
  in state s
• If sa is not an element of St(Tcc) it is not
  possible to execute a in the constrained set in
  which case we say that Ci is blocked at a in
  state s of Tcc.
• sa ? A(Ci) ? sa ? St(Tcc) ? a is a blocked
  action in state s

18
Introducing channel constraintsProperties

• For component Ci (1lt i n), blocking at eqi-1!
• is not possible.
• For component Ci (1lt i n), blocking at fqi! is
  not possible.

19
Introducing precedence order constraints

• Priority function returning for each component
  a
• unique natural number.
• The assignation is fixed during the entire
  execution of
• the system.
• Scp(t) (? s?Pref(t), ai u?A? t saiu ? ai?
  A(Ci) P(Comp(ai)) max P(C)).
• Limiting Tcc according to Scp gives Tpc, the
  priority consistent traces Tpc ? t ? Tcc
  Scp(t).

C ?RR(s)
20
Introducing precedence order constraintsPropertie
s, Lemmas

• Tpc has precisely one element. Unique trace in
  Tpc denoted by ?.
• Let Ci be such that (? j jlti P(Cj) gt P(Ci))
  and consider a state s of ? such that the next
  action after s in Tpc is one of A(Ci). Then Cj b
  eqj? in s of Tcc for all jlti. (a)
• Let Ci be such that (? j jgti P(Cj) gt P(Ci))
  and consider a state s of ? such that the next
  action after s in Tpc is one of A(Ci). Then Cj b
  fqj-1? in s of Tcc for all jgti. (b)

a).
b).
? j j lti P(Cj) gt P(Ci)
? j j gti P(Cj) gt P(Ci)
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Stable phase characterizationTheorems,
Properties, Corollaries

• Theorem. The pipeline system assumes a
  repetitive behavior
• after a finite initialization phase. The complete
  behavior is
• characterized by
• ? tinit (fqm-1? eqm? cm eqm-1! tL fqm!
  tR)?
• Notation.
• tinit trace recording the initialization
  phase of the system execution.
• tstable (fqm-1? eqm? cm eqm-1! tL fqm! tR)?.
• tstable - trace recording the stable phase of
  the system execution.

22
Stable phase characterizationTheorems,
Properties, Corollaries

• Theorem. The pipeline system assumes a
  repetitive behavior
• after a finite initialization phase. The complete
  behavior is
• characterized by
• ? tinit (fqm-1? eqm? cm eqm-1! tL fqm!
  tR)?
• Notation.
• tinit trace recording the initialization
  phase of the system execution.
• tstable (fqm-1? eqm? cm eqm-1! tL fqm! tR)?.
• tstable - trace recording the stable phase of
  the system execution.

23
Stable phase characterizationTheorems,
Properties, Corollaries

• The length of the initialization phase in number
  of
• actions is
• The length of the initialization phase can be
  reduced by decreasing m or the capacity of the of
  the queues in the system.
• The stable phase starts when Cm executes for the
  first time.
• When Cm executes, ? i 1 i lt m,
  L(fqi)Cap(fqi)-1 ? L(eqi)0 ? ? i m i lt n,
  L(fqi) 0 ? L(eqi)Cap(eqi).

(m-1).
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Stable phase characterizationTheorems,
Properties, Corollaries

• Minimum NCS
• The minimum number of context switches (NCS)
  during the initialization phase is achieved when
  m1.
• b. The minimum NCS during one iteration of
  tstable can be achieved either when
• i. P(C1) min(P(Ci))

i 1..n
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Stable phase characterizationTheorems,
Properties, Corollaries

• or when
• ii. P(Cn)ltP(C1)ltP(C2)ltltP(Cn-1) with
• ?i1 iltn-1, Cap(fqi)2.

26
Stable phase characterizationTheorems,
Properties, Corollaries

• Tradeoff
• The priority assignment suggested by Corollary 10
  - ii implies that all context switches caused by
  preemptions were eliminated, at the cost of an
  initialization phase.
• The priority assignment suggested by Corollary 10
  - ii. achieves a minimum NCS at the cost of more
  memory needed (? i 1 i lt n-1, Cap(fqi)2)
• Optimizing memory needs.
• The minimum queue capacity necessary and
  sufficient for each of the queues in the chain is
  1.

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Timing aspects

• Definitions
• ? A x x Til ? that for each action
  of a component Ci returns the computation time
  needed to execute the action.
• Schedule function ? A x x Til ? of the
  concurrent execution of Ci, i1..n processes on a
  processor. Returns the start time for each action
  of each component Ci within the concurrent
  execution of all components.

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Timing aspects

• Definitions
• Response time Ri,k of component Ci (1? i ? n)
  that processes packet k is
• Ri,k ?(fqi !k) ? (fqi !k) - ?(fqi-1?k).
• Response time of the chain (RTC) CN as the time
  needed by all components in the chain to process
  the kth packet
• RTCk ?(fqN!k) ? (fqN!k) - ?(fq0?k).
• NAk is the number of actions occurring from
  arrival of packet k until the packet leaves the
  chain.

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Timing aspects

• NAk is minimal for all packets k when P(C1) min
  P(Ci)
• RTCk is minimal for all packets k when P(C1)
  min P(Ci)

i 1..n
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Conclusions

• Presented a model for the dynamic behavior of
  linear processing chains
• Expressed the execution of a chain as a trace
  that can be calculated at design time.
• Formally proven that the trace becomes repetitive
  after an initial finite prefix.
• Approach allows the calculation and optimization
  of
• The initial phase length
• The necessary and sufficient capacities for the
  queues
• Number of context switches
• Number of actions between input/output of a
  packet (component, chain)
• Response time for a packet (component, chain)