SelfStabilization

SelfStabilization - Self Stabilization CS553 Distributed...

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Self Stabilization CS553 Distributed Algorithms Prof. Ajay Kshemkalyani by Islam Ismailov & Mohamed M. Ali
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Introduction There is a possibility for a distributed system to go into an illegitimate state , for example, if a message is lost. Self-stabilization : regardless of initial state, system is guaranteed to converge to a legitimate state in a bounded amount of time without any outside intervention. Problem : nodes do not have a global memory that they can access instantaneoulsy.
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System Model An abstract computer model: state machine. A distributed system model comprises of a set of n state machines called processors that communicate with each other, which can be represented as a graph. Message passing communication model: queue(s) Q ij, for messages from P i to P j System configuration is set of states, and message queues. In any case it is assumed that the topology remains connected, i.e., there exists a path between any two nodes.
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Definition States satisfying P are called legitimate states and those not satisfying P are called illegitimate states. A system S is self-stabilizing with respect to predicate P if it satisfies the properties of closure and convergence
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Dijkstra's self-stabilizing token ring system When a machine has a privilege, it is able to change its current state, which is referred to as a move . A legitimate state must satisfy the following constraints: There must be at least one privilege in the system (liveness or no deadlock). Every move from a legal state must again put the system into a legal state (closure). During an infinite execution, each machine should enjoy a privilege an infinite number of times (no starvation). Given any two legal states, there is a series of moves that change one legal state to the other (reachability). Dijkstra considered a legitimate (or legal) state as one in which exactly one machine enjoys the privilege.
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Dijkstra's system (contd) For any machine, we use symbols S, L, and R to denote its own state, state of the left neighbor and state of the right neighbor on the ring. The exceptional machine: If L = S then S = (S+ 1) mod K; All other machines: If L = S then S = L;
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Dijkstra's system (contd) Note that a privilege of a machine is ability to change its current state on a Boolean predicate that consists of its current state and the states of its neighbors. When a machine has a privilege, it is able to change its current state, which is referred to as a move. Second solution (K = 3) The bottom machine, machine 0: If (S+1) mod 3 = R then S = (S−1) mod 3; The top machine, machine n−1 : If L = R and (L+1) mod 3 = S then S = (L+1) mod 3; The other machines: If (S+1) mod 3 = L then S = L;
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Example
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Systems with less than three states per node
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Ghosh system (contd)
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Uniform vs Non-uniform From the examples of the preceding section, we notice that at least one of the machines (exceptional machine) had a privilege and executed steps that
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SelfStabilization - Self Stabilization CS553 Distributed...

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