Grid System Theoretical Model

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Grid System Theoretical Model

• Guanying Bu Zhiwei Xu
• Institute of Computing Technology
• China Academy of Sciences

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• 1. Introduction
• 1.1 Current grid research state
• 1.2 Aim of our work
• a theoretical model suits to describe and
  formalize grid systems
• 1.3 Formalization concerns
• How to represent a local function of grid system
• How to represent a global function of grid system
  
• How could this model be used in the research and
  practical work of grid system

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• 2. Grid System Theoretical Model

Fig. 1 Node Distribution Graph of NHPCE
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• 2.1 Informal Definition of Grid
• Definition 1 A grid system is a
  high-performance computing and information
  service environment with single system image,
  which is composed of geographically distributed
  computational resources connected by high-speed
  networks, aims to realize resource share, support
  concurrent/remote access to the computational
  resources.
• 2.2 Asynchronous Automaton
• Definition 2 An asynchronous automaton A,
  which we also call simply an automaton, consist
  of five components
• sig(A), a signature, a triple consisting of
  three disjoint sets of actions the input
  actions, in(S), the output actions, out(S), and
  the internal actions, int(S).
• states(A), a (not necessarily finite) set of
  states
• start(A), a nonempty subset of states(A) known
  as the start states or initial states
• trans(A), a state-transition relation, where
  trans(A) ? states(A)acts(sig(A))states(A)
  this must have the property that for every state
  s and every input action ?, there is a transition
  or step (s, ?, s)? trans(A)

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• tasks(A), a task partition, which is an
  equivalence relation on local(sig(A)) having at
  most countably many equivalence classes

Fig 2. A process automaton
Fig 3. A channel automaton
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• Example 1. Channel automaton
• As an example of an automaton, consider a
  communication channel automaton Ci,j. Let P be a
  fixed message alphabet. First we give the
  signature, sig(Ci,j). Here and elsewhere, we use
  the convention that if we do not mention a
  signature component (usually, the internal
  actions), then that set of actions is empty.
• Signature
• Input Output
• send(p)i,j, p?P receive(p)i,j, p?P
• The states, states(Ci,j), and the start states,
  start(Ci,j), are most conveniently described in
  terms of a list of state variables and their
  initial values.
• States
• queue, a FIFO queue of elements of P, initially
  empty
• Transitions
• send(p)i,j receive(p)i,j
• Effect Precondition
• add p to queue p is first on queue
• Effect
• remove first element of queue

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• Tasks
• receive(p)i,jp?P
•  
• Example 2. Process automaton
• As a second example of an automaton, consider a
  process automaton Pi. This automaton has the
  external interface described below. Here, V is a
  fixed value set,null is a special value not in
  V,and f is a fixed function,f Vn ? V.
• Signature
• Input Output
• init(v)i , v?V decide(v)i , v?V
• receive(v)j,i , v?V, 1?j?n, j?i send(v)i,j
  , v?V, 1?j?n, j?i
• The states and start states are as follows
• States
• val,a vector indexed by 1, , nof elements in
  V ?null,
• all initially nullTransitions init(v)i ,
  v?V receive(v)j,i , v?V
• Effect Effect
• val(i) ? v val(j) ? v

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• send(v)i,j , v?V
• Precondition
• val(i) v
• Effect
• none
•  
• decide(v)i , v?V
• Precondition
• for all j, 1?j?n
• val(j) ? null
• vf(val(1), , val(n))
• Effect
• none
• Tasks
• for every j?i
• send(v)i,j , v?V
• decide(v)i , v?V

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• 2.3 Composition Operation on Automaton
• 2.3.1 Compatible signature
• Formally, we define a countable collection
  Sii?I of signatures to be compatible if for all
  i, j?I, i?j, all of the following hold
•      A) int(Si) n acts(Sj) ?
•     B) out(Si) n out(Sj) ?
•      C) No action is contained in
  infinitely many sets acts(Si)
• We say that a collection of automata is
  compatible if their signature are compatible.
• 2.3.2 Composition Operation of Signatures
  and automata
• 1) The composition S?i?I Si of a countable
  compatible collection of signatures Sii?I is
  defined to be the signature with
•         A) out(S) ?i?I out(Si)
•         B) int(S) ?i?I int(Si)
•         C) in(S) ?i?I in(Si) ?
  ?i?I out(Si)
• 2) Now the composition A?i?I Ai of a countable
  compatible collection of signatures Aii?I can
  be defined. It is the automaton defined as
  follows

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•    A) sig(A) ?i?I sig(Ai)
•         B) states(A) ?i?I states(Ai)
•         C) start(A) ?i?I start(Ai)
•         D) trans(A) is the set of triples
  (s, p, s) such that, for all i?I, if
  p?acts(Ai), then (si, p, si) ?trans(Ai)
  otherwise sisi
•         E) tasks(A) ?i?I tasks(Ai)
• 2.3.3 Execution and Trace of Automaton
• Let A be a automaton
• 1) Execution and Trace
• Execution An execution fragment of A is either
  a finite sequence, s0, p1, s1, p2, , pr, sr,
  or an infinite sequence, s0, p1, s1, p2, , pr,
  sr, , of alternating states and actions of A
  such that (sk, pk1, sk1)is a transition of A
  for every k?0. An execution fragment beginning
  with a start state is called an execution.
• Trace The trace of an execution ? of A, denote
  by trace(?), is the subsequence of ? consisting
  of all the external actions.
• We say that ? is a trace of A if ? is the trace
  of an execution of A. We denote the set of
  traces of A by traces(A).

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• 2) Restriction
• Given an execution, ?s0, p1, s1, , of A, let
  ?Ai be the sequence obtained by deleting each
  pair pr, sr for which pr is not an action of Ai
  and replacing each remaining sr by (sr)i, that
  is, automaton Ais piece of the state sr. Also,
  given a trace ? of A (or, more generally, any
  sequence of actions), let ?Ai be the
  subsequence of ? consisting of all the actions
  of Ai in ?.
• 2.3.4 Three Theorems
• Theorem 1 Let Aii?I be a compatible
  collection of automata and let A ?i?I Ai
•      (1) If ??execs(A),then for every i?I ,
  ?Ai?execs(Ai)?
•      (2) If ??traces(A),then for every i?I ,
  ?Ai?traces(Ai)?
• Theorem 2 Let Aii?I be a compatible
  collection of automata and let A?i?I Ai. Suppose
  ?i is an execution of Ai for every i?I, and
  suppose ? is a sequence of actions in ext(A) such
  that ?Aitrace(?i) for every i?I. Then there is
  an execution ? of A such that ?trace(?) and
  ?i?Ai for every i?I.

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• Theorem 3 Let Aii?I be a compatible
  collection of automata and let A?i?I Ai. Suppose
  ? is a sequence of actions in ext(A). If
  ?Aitrace(?i) for every i?I, then ??traces(A).
• 3. An Application of This Model in Grid System
• GridBFSi Automaton
• Signature
• Input
• receive(ack)j,i, j?nbrs
• receive(m)j,i, j?nbrs,m?N
• Internal
• cleanupi
• Output
• send(ack)i,j, j?nbrs
• send(m)j,i, j?nbrs, m?N
• donei
• States
• dist ? N?8, initially 0 if ii0 8 otherwise

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• status ? idle, source, non-source, initially
  source if ii0 idle otherwise
• s_parent ? nbrs?null, initially null
• parent ? nbrs?null, initially null
• for every j?nbrs
• send-buffer(j), a FIFO queue of ack
  messages, initially empty
• deficit(j) ? N, initially 0
• send(j), a FIFO queue of elements of N,
  initially containing the single element 0 if
  ii0, else empty
• Transitions
• receive(m)j,i,m?N
• Effect
• if m1lt dist then
• dist m1
• parent j
• for all k?nbrs-j
• add dist to send(k)
• if status idle then
• status non-source
• parent j
• else add ack to send-buffer(j)

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• receive(ack)j,i
• Effect
• deficit(j) deficit(j) - 1
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• send(m)i,j,m?N
• Precondition
• m is first on send (j)
• Effect
• remove first element of send (j)
• deficit(j) deficit(j) 1
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• send(ack)i,j
• Precondition
• ack is first on send-buffer(j)
• Effect
• remove first element of send-buffer(j)
• cleanupi
• Precondition

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• deficit(k) 0
• Effect
• add ack to send-buffer(s_parent)
• status idle
• s_parent null
•  
• donei
• Precondition
• status source
• for all k?nbrs
• deficit(k) 0
• Effect
• status idle
• Tasks
• donei
• cleanupi
• for every j?nbrs
• send(ack)i,j
• send(m)i,j

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• Theorem 4 In any fair execution of the
  GridBFS algorithm, the system eventually
  stabilize to a state in which the parent
  variables represent a breadth-first spanning
  tree.
• 4. Conclusion
• This model is suitful for describing the
  asynchronous characteristic of grid systems
• This model can formalize the local and global
  functions of grid systems, and reflect their
  relationship
• This model is good at proving the properties of
  grid systems

Thank You