Logical Clocks and Global State

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Logical Clocks and Global State
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Causal Relations between Events

• Typical events in a distributed system
• execution of an instruction
• sending of a message
• receipt of a message
• happened before relation ? between events A and B
  holds if
• both A B occur at the same process and A
  occurred before B
• A is the sending of a message and B is the
  receipt of the same message
• there exists event C such that A?C and C?B
• if A?B then A casually affects B
• Events A B are concurrent if neither A? nor B?A

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Lamports Logical Clocks

• each process Pi maintains a counter Ci and
  assigns to each event E a timestamp t(E) equal to
  the value of Ci
• the happened before relation between events can
  be realized if the following conditions hold
• for all events A B in Pi
• A?B implies Ci(A) lt Ci(B)
• if A is the sending of a message by Pi and B is
  the receipt of the same message by Pj then Ci(A)
  lt Cj(B)

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Lamports Logical Clocks

• assume that Pi timestamps each message M it sends
  with Ci
• The ? relationship can be achieved if the
  counters Ci are updated as follows
• between any two successive events at Pi, Ci is
  incremented by a positive value d
• any process Pj, upon receiving a message with
  timestamp t, sets Cj max(Cj, td)
• ? is a partial temporal ordering of events which
  could be augmented to a total temporal ordering
  of events by using a total ordering of processes
• Theorem. A?B implies that C(A) lt C(B) but the
  reverse is false

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Vector Clocks (Fidge and Mattern)

• Assume N processes Pi with each process having a
  counter Ci that is a vector of N simple counters
• Cii is the value of Pis Lamport clock
• Cij is Pis best guess of the value of Pjs
  Lamport clock
• Cij indicates the time of the last event at Pj
  that is known (by Pi) to have happened before the
  time Cii at Pi
• The vector clock at Pi is updated as follows
• Cii is incremented by dgt0 between any two
  consecutive events at Pi
• upon receiving a message with vector timestamp T
  at Pi
• Cij max(Cij, Tj) for all j1,2,,N

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Vector Clocks Temporal Ordering of Events

• For any two vector clocks
• Cii gt Cji (why?)
• Compare vector clocks component-wise and define
• Ci Cj if Cik Cjk for k1,2,,N
• Ci lt Cj if Cik lt Cjk for k1,2,..,N
• Ci lt Cj if Ci lt Cj and Ci not equal to Cj
• two events A B are concurrent if neither C(A)
  lt C(B) nor C(B) lt C(A)
• Theorem. A?B if and only if C(A) lt C(B)
• Vector clocks provide us with a total temporal
  ordering of events

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Causal Ordering of Messages

• Problem
• order the Send and Receive of messages such that
• Send(M1) ? Send(M2) implies Receive(M1)
  ?Receive(M2)
• for any two messages M1 and M2
• Applications replica management, monitoring
  distributed computations, simplifying distributed
  algorithms, etc
• Solution idea
• upon arrival of a message at a process, buffer
  (delay delivery) the message until the message
  immediately preceding it is delivered

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Birman-Schiper-Stephenson Protocol

• Assumes broadcast communication channels that do
  not loose or corrupt messages
• Use vector clocks to count messages (i.e. set
  d1)
• Pi upon receiving a message M with timestamp T
  from Pj buffers the message until
• Pi has received all messages send by Pj before
  sending M
• Cij Tj-1
• Pi received all messages that Pj received before
  sending M
• Cik gt Tk, k1,2,..,N, k ltgt j
• Schipper-Eggli-Sandoz solves the problem without
  broadcast channels

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Global State

• Channels can not record their state
• Definitions
• LSi local state at Si (as well the event of
  recording this local state)
• send(M) and rec(M) the send and receive events of
  a message M from Si to Sj
• time(E) the timestamp of event E
• send(M) in LSi iff time(send(M)) lt time(LSi)
• rec(M) in LSj iff time(rec(M)) lt time(LSj)
• transit(LSi, LSj) set of messages M from Si to
  Sj such that send(M) in LSi and rec(M) not in LSj
• inconsistent(LSi,LSj) set of messages M from Si
  to Sj such that rec(M) in LSj and send(M) not in
  LSi

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Consistent Global State

• Global state of a system with N sites consists of
  the set of the local states LSi, i1,2,,N
• A global state is
• consistent if and only if for every pair of local
  states LSi, LSj there are no inconsistent
  messages
• inconsistent(LSi,LSj) empty
• transitless iff transit(LSi,LSj) empty for
  every pair of local states LSi, LSj
• strongly consistent if it is consistent and
  transitless

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Chandy-Lamport Protocol

• Assumes FIFO communication channels
• recording of global state uses a special message
  (the marker)
• markers delineate the messages in a FIFO channel
  that need to be included in the local state
  recorded at the receiving end of the channel
• whenever a process wants to initiate recording of
  global state it creates a new marker
• more than one process can initiate global state
  recording

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Chandy-Lamport Protocol

• Rule 1 P sends the marker
• P records its local state (together with the
  state of its channels)
• P sends the marker to all outgoing channels on
  which the marker has not been sent yet before
  sending anymore messages on these channels
• Rule 2 P receives the marker along channel C
• If P has not recorded its state yet then
• record the state of C as empty and follow Rule 1
• else
• record the state of C as the sequence of messages
  received between time(local-state-of-P) and
  time(marker-received)

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Properties of Chandy-Lamport Global State

• Recorded global state may be phantom
• So global state when protocol starts
• Sf global state when protocol completes
• Sr global state recorded by the protocol
• E sequence of actions that take the system from
  So to Sf
• Theorem. There exists a permutation E of E such
  that a prefix of E takes the system from So to
  Sr and the remaining actions in E take the
  system from Sr to So
• Global state recorded by Chandy-Lamports
  protocol is useful in inferences about persistent
  system properties

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Huangs Termination Detection Protocol

• Processes are idle or active
• idle processes become active by receiving a
  computation message
• protocol messages are control messages
• one process is the controlling agent Pc and
  monitors the system
• initially all processes, but the controlling
  agent, are idle
• the controlling agent has a weight 1 and all
  other processes have weight 0
• each active process has weight gt 0

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Huangs Termination Detection Protocol

• Messages carry a weight
• B(dw) computation message with weight dwgt0
• C(dw) control message with weight dwgt0
• Protocol
• Any active process Pi may initiate a computation
  at a process Pj by selecting dwgt0, setting W(Pi)
  W(Pi) - dw, and then sending B(dw) to Pj
• Any process Pj upon receiving B(dw) sets W(Pj)
  W(Pj)dw
• An active process Pi becomes idle by sending
  C(W(Pi)) to the controlling agent Pc and setting
  W(Pi)0
• Controlling agent Pc upon receiving C(dw) sets
  W(Pc) W(Pc) dw
• If W(Pc)1 the computation has terminated

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Correctness of Huangs Protocol

• Correctness
• the sum of the weights among
• active processes
• messages in transit
• controlling agent
• is always equal to 1
• total weight among idle processes is 0
• detects all terminations correctly in finite time
  as long as
• messages are not lost or corrupted
• message delays are finite