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BASIC BUILDING BLOCKS

BASIC BUILDING BLOCKS. -Harit Desai. Byzantine Generals Problem. If a computer fails, it behaves in a well defined manner A component always shows a zero at the output or simply stop execution It behaves arbitrarily

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BASIC BUILDING BLOCKS

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  1. BASIC BUILDING BLOCKS -Harit Desai

  2. Byzantine Generals Problem • If a computer fails, • it behaves in a well defined manner • A component always shows a zero at the output or simply stop execution • It behaves arbitrarily • Sends totally different information to different components with which it communicates The problem of reaching an agreement in a system where components can fail in an arbitrary manner is called byzantine generals problem

  3. Interactive Consistency Problem • Each node makes decision based on the values it gets • We require all non-faulty nodes to make same decision • So, the goal is that all non-faulty nodes gets the same set of values • Hence, consensus can be achieved • But, a faulty node may send different values to different nodes

  4. Transmitter 1 1 Node j Node i 0 Transmitter 1 0 Transmitter Transmitter 0

  5. Protocols with ordinary messages • Requirements n >= 3m+1 where, n = total number of nodes m = number of faulty nodes • Assumptions about message passing system • Every message that is sent by node is delivered correctly by the message passing system to the receiver

  6. Assumptions – Continued… • The receiver of a message knows which node has sent the message • Absence of a message can be detected

  7. Interactive consistency algorithm Algorithm ICA(0) 1) The transmitter sends its value to all the other N-1 nodes. 2)Each node uses the value it receives from the transmitter or uses the default value.

  8. Algorithm ICA(m), m>0 1) The transmitter its value to all the other n-1 nodes. 2)Let Vi be the value the node i receives from the transmitter, or else be the default value if it receives no value sends.Node i acts as the transmitter in algorithm ICA(m-1) to send the value Vi to each of the other n-2 nodes 3)For each node i, let Vj be the value received by by the node j (j != i). Node i uses the value majority(V1, v2,….,Vn-1).

  9. Protocol with signed messages • Algorithm SM(m) Initially Vi = null 1) The transmitter signs its value and sends to all other nodes. 2) For each i : (a) If a node i receives a message of the form v from the transmitter then (a1)it sets Vi to {v}, and (a2)it sends the message v:0:i to every other node.

  10. continued……. (b) If node i receives a message of the form v:0:j1:j2: …. :jk and v is not in Vi, then (b1) it adds v to Vi, and (b2) if k<m it sends the message v:0:j1:j2: …. :jk:i to every node other than j1, j2,….,jk. 3)For each i: when node i will receive no more messages, it considers the final value as choice(Vi)

  11. Clock synchronization • Problems • clocks of different nodes have different times and may be running at different speeds. • communication will induce delay between sending and receiving of the message. • networks delays can vary. • clocks may be faulty(dual-faced).

  12. Requirements of clock synchronization • for a nonfaulty clock Ci |dCi/dt –1| < $ where $ is of the order of 10e-5 • at any time ,the value of all the nonfaulty processors’ clocks must be approximately equal |Ci(t) – Cj(t)| <= b b = constant • there is a small bound by which a nonfaulty clock is changed during resynchronization.

  13. Synchronization protocols • Deterministic protocols • clock synchronization conditions and bounds are guaranteed. • but, they require some assumption about message delays. • Probabilistic protocols • does not require any assumptions about message delays. • but guarantees precision only with a probability.

  14. Deterministic Clock Synchronization • all clocks are initially synchronized to approximately the same value. |Ci(t0) – Cj(t0)| < b • each process can communicate directly with any other process. • if a process sends a message at real-time t and received at t’, then message delay is t’- t. message is delivered in [&-e, &+e] time, for fixed & and e, with &>e.

  15. Th algo works in rounds… • The ith rounds is triggered when the clock reaches Ti. • When process j reaches Ti, it broadcasts a message containing Ti. • It also collects ith round messages for a bounded amount of time and records their arrival times according to its local clock. • This waiting period is to ensure that correct process will send a message in this waiting period.

  16. Boundedwaitingtime • Process j ‘s clock reaches Ti. • Process k’sclock will reach Ti within a time b • At this time k will broadcast Ti to all processes • Message delay = &+e • J receives k’s message at (b+&+e) after it own clock reaches the value Ti. • Clock rates may differ by $ from real time. • So, the bounded time, within which j should receive the message of k containing Ti is (1+$)(b+&+e) • Once this time is elapsed , the process must have received messages from all non-faulty processes.

  17. A process then calculates averaging function from the set of arrival times. • By this averaging function it switches it logical clock to new value. • It then waits for t time to execute next round.(Ti+t). • Averaging function: there are atmost f faulty process, so the averaging function discards the top f and bottom f values from the set of values and then it takes the mid-point of the remaining values.

  18. algorithm • $ = bound on clock drift. • b = bound on how far apart the clocks are initially • & , e = bounds on message delays. • t = period between rounds. • CORR = correction variable , initially empty • ARR = array containing arrival of most recent messages. • NOW = represents the current logical clock time. • reduce = function on array and returns middle values. • mid = returns mid-point of a set of values.

  19. Do forever /* in case messages are received before the process reaches Ti */ while u = (m,k) do ARR[k] = NOW /*fall out of loop when u = START or TIMER and begin round */ T := NOW broadcast(T) set-timer(T +(1 + $)(b + & + e) while u = (m,k) do ARR[k] = NOW /*fall out of loop when u =TIMER; end round */ AV := mid(reduce (ARR) ADJ := T + & - AV CORR := CORR + ADJ set-timer(T + t) Enddo

  20. Probabilistic clock synchronization • Assumes that dual-faced clocks do not exist. • Clocks are assumed to be correct • Message delay are unbounded but there is minimum delay (min) that exists. • To read the clock of process i ,process j sends a message to i.When I receives this message, it replies with a message(T). • Round trip delay for receiving i’s clock,as measured by j’s clock is 2D. • Process j can make some estimation of the time at node i.

  21. Let t be the real-time when j receives a reply from i and 2d be the real-time round trip delay. • The time of receipt of message, according to I’s clock has tobe more than T +min(1-$),where $ is the bound on the drift of the clocks. • Maximum delay of the return message is 2d – min. • Clock time maximum delay is ( 2d – min )(1 + $). • Also, 2d <= 2D(1+$). • Hence maximum clock time delay is • 2D(1+$)(1+$)-min(1+$) = 2D(1+2$)-min(1+$) • So, j can infer that at time j receives the message from i , i’s clock is in the range : • [T+min(1+$) , T+2D(1+2$-min(1+$)]

  22. Now, j has to select its value in the interval as i’s clock value. • The error is minimised if mid-point value is selected. • Maximum error possible E =D(1+2$)-min. • Maximum error value E can be taken as the precision with which j can read the value of the i’s clock. • Shorter the round trip delay ,better is the precision of reading a clock. • If a process j wants to read the clock of another process i with specified precision e, it must discard all reading attempts in which the round trip delay is greater than 2U, where U = (1-2$)(e+min).

  23. Stable storage • Computer system has some stable storage whose contents are preserved despite failures. • Failures that these techniques cannot handle… • Transientfailures: these cause the disk to behave unpredictably for short period of time. • Bad sector: page becomes corrupted, and data stored in it cannot be read. • Controller failure: the disk controller fails. • Disk failure: entire disk becomes unreadable.

  24. Undesirable results of a read(a) operation are: • Soft read error: page a is good ,but read returns bad.this situation may not persist for long ,and is caused by transient failures. • Persistent read error: page a is good,but read returns bad, and successive reads also returns bad.(bad sector) • Undetected error: page a is bad but returns good ,or page a is good but returns different data. • Undesirable results of a write(a,d)operation are: • Null write: page a is unchanged. • Bad write: page a becomes (bad,a).

  25. Decay events Corruption: A page goes from (good , d) to (bad , d). Revival: A page goes from (bad , d) to (good , d). Undetectederror:A page changes from(s,d) to (s,d’) with d<>d’.

  26. Implementation • Using one disk: • CarefulRead: read is performed repeatedly until it returns the status good, or the page cannot be read after certain amount of tries. • CarefulWrite: performs a write followed by read until read returns the status good. • But, this cannot take care of decay events. Stable storage is represented as by an ordered pair of disk pages. • StableRead: performs a CarefulRead from one of the paired pages, and if the result is bad, performs a CarefulRead from the other. • StableWrite:performs CarefulWrite to one of the representative pages first.When the operation is completed, it performs a CarefulWrite to the other page.

  27. This takes care of the decay events. • A crash during StableWrite may cause two pages to differ. To handle this, cleanup operation is performed. Do a CarefulRead from each of the two representative pages If both return good and same data then Do nothing Elseif one returns badthen Do a CarefulWrite of data from good page to bad page. Elseif both return good, but different data then choose either one of the page and do a CarefulWrite of its data to the other page

  28. Disk shadowing • It is technique for maintaining a set of identical disk images on separate disk devices. • Primary purpose is to increase reliability amd availability. • Consider a case of two disks(mirrored disks) • Total failure occur only if both disk fail. MTTFm = MTTF/2 * MTTF/MTTR

  29. Redundant Arrays of Disks • Data is spread over multiple disks using “bit-interleaving”. • Bit-interleaving provides high I/O performance. • But are not reliable,since the failure of any disk can cause entire data to become unavailable. • So, disks are partitioned in groups. • Each group has some data disk and some check disk. • Number of check disk depends on the coding technique used. • Say, check disk stores parity then one one disk is required.

  30. Failure of RAID occurs only if more than one disk fails. • Assume that a RAID consists of only one group of disks, with G data disks and C check disks. • If failure and repair of disks are exponential distributed, then mean time to failure of the group = MTTF/(G+C) * (MTTF/(G+C-1))/MTTR

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