Logical Clocks

In distributed systems, it is frequently unnecessary to know when something happened in a global context. Instead, it is only necessary to know the order of events. And often times it is only necessary to know the order of certain events, such as those that are visible to a particular host.

Very soon we will learn about timestamp based mutual exclusion -- this application makes good use of logical time. By contrast, synchronized physical clocks are expensive to maintain and inherently inaccurate. As a result, they are a poor choice of tools for this type of work.

Instead a logical clock is a better choice. The purpose of a logical clock is not necessarily to maintain the same notion of time as a reliable watch. Instead, it is to keep track of information pertaining to the order of events.

Lamport's Algorithm

Lamport's Algorithm provides one way of ensuring a consistent logical time among many hosts. Let's begin discussing it by defining a very important relationship among events, the happens-before relationship.

def'n: happens-before

• When comparing events on the same host, if event a occurs before event b then a happens-before b.
• If one host receives a message sent by another host, the send happens-before the receive
• If x occurs on P₁ and y occurs on P₂ and P₁ and P₂ have not exchanged messages then X and Y are said to be concurrent. If this is the case, we can't infer anything about the order of event x and event y. Please note, that this is only true if x and y don't exchange messages at all, even indirectly via third (or several) parties.
• The relationship is transitive: if a happens before b, and b happens before c, then a happens before c.

We use this relationship to define our clock. Instead of a clock that keeps real time, it is basically a simple counter used to label events in a way that shows the happens-before relationship among them. Here are the rules that are used for updating the value of the logical clock on a host:

• the counter is incremented before each event.
• in the case of a send, the counter is incremented, and then the message is sent. The message should carry the new (incremented) timestamp.
• in the case of a receive, the proper action depends on the value of the timestamp in the message. If the message has a higher timestamp than the receiver, the receiver's logical clock adopts the value sent with the message. In either case, the receiver's logical clock is incremented and the message is said to have been received at the new (incremented) clock value. This ensures that the messages is received after it was sent and after prior events on the receiving host.

Total Ordering with Lamport's Algorithm

It is important to note that as we have described it, Lamport's algorithm does not provide a total ordering. It allows two events occuring on two different hosts to occur at the same time (remember our discussion of concurrent events?). This later could be discovered if messages are subsequently exchanged by these hosts. To address this problem, we can break these ties using the host id. Admittedly this technique for breaking ties breaks them in an arbitrary order than may not be consistent with real world time, but this might not matter. It ensures that timestamps are unique. It is consistent and ordered, so it might be useful in avoiding deadlock or livelock. It might also be useful in uniquely naming events.

Now we have the following:

• if a occurs before b on the same host Lamport_Timestamp(a) < Lamport_Timestamp(b)
• Lamport_Timestamp(send_x < Lamport_Timestamp(recv_x)
• Lamport_Timestamp(a) != Lamport_Timestamp(b), if a and b are not the same event

An Example of Lamport Logical Time

Concurrency Control: An Introduction

We discussed concurrency and concurrency control in 15-213, and beat the topics to death in 15-410. So, right now, I'm just going to give the "Nickel tour".

Life is simple if we all have our own kingdom. Everything is ours. There's never a reason to share. No compromises ever need to be made. We get what we want, whenever we want it.

But, in the real world, this isn't always the case. Often time -- like it or not -- we have to share. And, how we share is particularly important. If we don't do it well, feelings get hurt and things get broken.

The situation is more-or-less the same for software. In many cases, different things are going on and these things need to share certain resources. We call these different, independent, things threads of control, which includes traditional threads, traditional processes, and other models for work-in-progress. A resource is an abstraction for anything that we might have to share. Another way of viewing a resource is an abstraction for any reason to wait. Some resources might be associated with physical things, such as printers, terminals, the user, or the network. Other reasons to wait are more abstract resources -- pieces of software that model objects not present in the physical world.

When we've got the potential for more than one thread of control to be chasing the same resource, we need to have some way of resolving contention. In other words, we need some way to figure out who gets the resource and when. Not only do we need such a policy, but we also need some mechanism to enforce it.

Concurrency Control: Mutual Exclusion and Other Policies

The policy that is used to divy up a recource among its contenders depends largely on the resource and the application. From the perspective of the resource, it might onyl be possible for one user to use it at a time. Or, it might be possible for a certain fixed number of users to access it at a time. Or, the number of users might depend on the character of the users. We played with all sorts of possibilities in 15-412.

From the perspective of prioritizing the users, there might be all sorts of different scenarios. Perhaps certain users are more important than others and, as a consequence, should be given priority. Or, perhaps it is more efficient for users to be scheduled in a certain order. Or perhaps fairness is our goal and users should be scheduled in exactly the same order as the request is made. Or, perhaps it doesn't matter, and any order will do.

Although the potential policies are endless, we are going to focus on only one of them -- the most important in practice. We are going to focus on finding a mechanism to implement mutual exclusion without priority. Mutually exclusive access to a resource implies that all of the users agree (hence mutual) that the use of the resource by one exludes the concurrenct use by another (hence exclusion). Given a mechanism to implement mutual exclusion, we can actually construct much more complex systems.

Characteristics of a Solution

We often like to say that, for a policy and corresponding mechanism, to ensure mutual exclusion in a meaningful and useful way, it must ensure three things: mutual exclusion, progress, and bounded wait.

The first of these three is meaningful on its face: If more than one user can access the resource at the same time, we are clearly not ensuring correct concurrency control.

The second, progress, though it takes a bit of explaining, is very common sense. A good solution to concurrency control allows the resource to be used, if available. It is not meaningful to declare, "We're implementing mutual exclusion -- no one can use the resource." Satisfying the progress requirement means that, if the resource is available, and if there is at least one user, someone should get the resource.

The third characteristic also makes sense: If someone wants a resource, they should eventually get it. Again, its not quite fair to declare, "Sure, we're implementing mutual exclusion -- only process 0 can use the resource. This condition means that, among other things, deadlocks and live locks should not occur.

Mutual Exclusion: Why Re-Invent the Wheel

Okay, great, so we're building distributed systems. Why re-invent the wheel? In 15-213 and 15-412, we discussed many techniques. The least common denominator of which includes simple mutexes and semaphores.

The problems with these tools in the context of distributed systems is that they included shared state. In the case of a simple mutex, a boolean variable. In the case of a semaphore, a shared queue. These, in turn imply some type of shared memory.

The problem with this is that shared memory requires synchronization, a type of concurrency control. The problem is that each of the multiple parties must see the same queue and value before any decisions can be made. This type of synchronization is exactly what we are trying to construct.

As a result, we are going to construct a new set of tools. By comparison, they'll seem a bit crude. But, in practice -- they work. And, we can build more convenient tools from them (at some expense), if we so desire.

Base Case: Centralized Approach

Although centralized approaches have their standard collection of shortcomings, including scalability, fault-tolerance, and accessibility they provide a useful starting point for discussions. So we'll begin by discussion a centralized approach to ensuring mutual exclusion for a critical section:

• A Central Coordinator is needed.
  • This coordinator can be appointed or elected
  • The coordinator is responsible for granting requests to enter the critical section
  • It ensures that only one thread is in the critical section at a time.
  • If the critical section is in use, the request is enqueued otherwise, the request is immediately granted
  • Enqueued requests are granted when the critical section becomes available
• Want into the CS?
  • Ask the coordinator & wait for permission.
• Done with the critical section?
  • Tell the coordinator
  • The coordinator will then let the next thread in, if any.
• Good Traits
  • It does guarantee mutual exclusion
  • It only requires 3 messages per critical section entry (request, permission, done)
• Shortcomings
  • The coordinator dies...then what?
  • Thread dies in the critical section
  • In either of the above cases, it is hard to tell what has happened
  • Standard centralized problems

Leases

When possible, especially in distributed environments, which are inherently failure-prone, we don't want to give a user a permanant right to a resource. The user might die or become inaccessible, in which case the whole system stops.

Instead, we prefer to grant renewable leases with liberal terms. The basic idea is that we give the resource to the user only for a limited amount of time. Once this time has passed, the user needs to renew the lease in order to maintain access to the shared resource. Within the last ten years or so, almost all mutual exclusion and resource allocation systems have taken this approach, which is expecially well suited for centralized approaches.

The amount of time for the lease should be long enough that it isn't affected by reasonable drift among synchronized physical clocks. But, it should be short enough that the time wasted after the end of the task and before the lease expires is minimal. It is also possible to allow the user to relinquish a lease early.

The other problem is enforcement -- the user must be unable to access the resource after the credential expires. There are basically two ways of doing this. The leasing agent can tell the resource of the leasee and the term, or the leasor can give the leasee a copy of the lease to present to the resource.

In either case, cryptography is needed to ensure that the parties are who they claim to be and that the lease's content is not altered. We'll discuss how this can be accomplished in more detail later. But, for now, let me just offer that it is often done using public key cryptography.

This can be used to authenticate the parties, such as the leasor the leasee, or the resource, and it can also be used to make the lease unalterable by the leasee.

Timestamp Approach (Lamport)

Another approach to mutual exclusion involves sending messages to all nodes, and ordering requests using Lamport logical time. The first such approach was described by Lamport. It requires that ties be broken using host id (or similar value) to esnure that there is a total ordering among events.

This approach is based on the notion of a global priority queue of requests for the critical section. This queue is ordered by the logical time of the request. Unlike the central algorithm we discussed as the "base case", this approach calls for each node to maintain a copy of this queue. The copies are maintained in a consistent way using a request-reply protocol.

When a node wants access to the critical section, it sends a REQUEST message to every other node. This message can be sent via a multicast or a collection of unicasts. This message contains the logical time of the request. When a participant receives this request, it adds it to its priority queue and sends a REPLY message to the requesting node.

The requesting node takes no action until it receives all of the replies. This ensures that the request has been entered into all of the queues, and that, at least with respect to this request, the queues are consistent. Once it receives all of the replies, the request is free to go -- once its turn arrives.

If the critical section is available (the queue was previosuly empty), the request can go as soon as the last REPLY is received. If the critical section is in use, the request must wait.

When a node exits the critical section, it removes itself from its own queue and sends a RELEASE message to every other participant, perhaps by multicast. This message directs these nodes to remove the now-completed request for the critical section from their queues. It also directs them to "peek" at their queue.

If the first request in a hosts queue is itself, it enters the critical section. Otherwise, it does nothing. A host can enter the critical section if it is at the head of its own queue, because the REPLY ensures that it will be at the head of every other node's queue.

The RELEASE message does not need an ACK or a REPLY, because it does not matter if its arrival is delayed. Since we are assuming a reliable unicast or multicast, the RELEASE will eventually reach each participant. We don't care if it arrives late -- this doesn't break the correctness of the algorithm. In the worst case, it is delayed in its arrival to the next requestor to enter the critical section. In this case, the critical section will go unused until the RELEASE arrives and is processed by the host. In the other cases, it delays the host in "peeking" at the queue, but this is without consequence -- the delayed host wasn't going to enter the critical section, anyway.

But wait! Why do we need the REPLY to the REQUEST, then? Can't we just get rid of that. Well, not exactly. The problem is that a reliable protocol guarantees that a message will eventually arrive at its destination, but makes no guarantees about when. The protocol may retransmit the information many, many times, over many, many timeout periods, before successfully delivering the message.

In the case of the RELEASE message, timing is not critical. But this is not the case for the REQUEST message. The REQUEST message must be received, before the requesting node can enter the critical section. This is the only way of ensuring that all nodes will see the same head node, should a RELEASE message arrive. Otherwise, two different hosts could look at their queues, determine that they are at the head, and enter the critical section -- disaster. This disaster could be detected after-the-fact when the belated REQUEST arrives -- but this too late since mutual exclusion has already been violated.

This approach requires 3(N - 1) messages per request: REQUEST, REPLY, and RELEASE must be sent to every other node. It isn't very fault-tolerant. Even a single failed host can disable the system -- it can't REPLY.

Timestamp Approach (Ricarti and Agrawala)

The Lamport approach described above was improved by Ricarti and Agrawala. Ricarti and Agrawala observed that the REPLY and RELEASE messages could be combined. This is achieved by having the process that is currently within the critical section delay its REPLY until it exists the critical section. In order to do this, each process must queue REQUESTs while within the critical section.

In many respect, this change converts this approach from a "global queue" approach to a "voting" approach. A node requests entry to the critical section and enters the critical section as soon as it has received an OK (REPLY) vote from every other node.

The details of this approach follow:

Requestor Request

1. Build a message
2. Send message to all participants

Paritipants

1. If not in CS and don't want in, reply OK
2. If in CS, enqueue request
3. If not in CS, but want into the CS, and the requestor's time is lower, reply OK (messages crossed, requestor was first)
4. If not in CS, but want into the CS, and the requestor's time is greater, enqueue request (messages crossed, participant was first)

Exit

1. On exit from CS, reply OK to everyone on queue (and dequeue each)

Requestor Entry

1. Once received OK from everyone, enter CS

This approach requires 2*(n - 1) messages, that is one message to and from everyone except self. This is an (n - 1) improvement over Lamport's approach.

But it fails to address the more serious limitation -- fault tolerance. Even a single failure can disable the entire system. Both timestamp approaches require more messages than a centralized approach -- and have lower fault tolerance. The centralized approach provides one single point of failure (SPF). These timestamp approaches have N SPFs.

In truth, it is doubful that we would every want to use either approach. In practice, centralized coordinators and ring approaches are the workhorses. Centralized coordinators can be made more fault tolerant using coordinator election (comming soon).

But these timestamp approaches are the most distributed -- they involve every host in every decision. They also illustrate some important examples of global state, logical time, &c -- and so they are a valuable part of this (and any) distributed systems course.

Mutual Exclusion: Voting

Last class we discussed the Ricarti and Agrawala approach to ensuring mutual exclusion. It was much like asking hosts to vote about who can enter the critical section and allowing access only upon unanimous consent. But is unanimous consent necessary? Can't we get away with a simple majority since two hosts can't concurrently win a majority of the votes.

In a simple form, it might operate similarly to a democratic election:

When entry into the critical section is desired:

1. Ask permission from all other participants via a multicast, broadcast, or collection of individual messages
2. Wait until more than 50% respond "OK"
3. Enter the critical section

When a request from another participant to enter the critical section is received:

1. If you haven't already voted, vote "OK."
2. Otherwise enqueue the request.

When a participant exits the critical section:

1. It sends RELEASE to those participants that voted for it.

When a participant receives RELEASE from the elected host:

1. It dequeues the next request (if any) and votes for it with an "OK."

Ties and Breaking Ties

So far, this approach is looking nice, but it does have a problem: ties. Imagine the case such that no processor gets a majority of the votes. Consider, for example, what would happen if each of three processors got 1/3 of the votes. Ouch!

Ties can, in fact, be broken at a somewhat high cost. If we use Lamport time with total ordering via hostid, no two messages will have concurrent time stamps. Messages that would otherwise be concurrent are ordered by hostid.

Recall that a host votes for a candidate as long as it has no outstanding votes. This becomes problematic if its vote turns out to be premature. This occurs if it votes for one candidate to later receive a request, bearing an earlier timestamp, from another candidate.

At this point, one of two things might be occuring. The system might be making progress -- the "wrong" host might have gotten more than 50% of the votes. If this is the case, we don't care. It might not be fair, but it is an edge case.

Another possibility is that no host has yet received a majority of the votes. If this is the case, it could be because of deadlock. It might be that each candidate got the same number of votes. This is the case that requires mitigation.

So, upon discovering that it voted for the "wrong" candidate, a host needs to determine which of these two situations is the case. It sends an INQUIRE message to the candidate for who it voted. If this candidate won the election, it can just ignore the INQUIRE and RELEASE normally when done. But, if it hasn't yet entered the critical section, it gives back the vote and signals this by sending back a RELINQUISH. Upon receipt of the RELINQUISH, the voter is free to vote for the preceding request.

Analysis and Looking Forward

This approach certain has some nice attributes. It does, in fact, guarantee mutual exclusion. And, it can allow a host to enter the critical section even if 1/2 of the hosts are down or unreachable.

But it has non-trivial costs. Nominally, it takes 3 messages per entry to the critical section (request, vote, release), about the same as a timestamp approach. And, in the event that votes arrive in exactly the wrong order, an INQUIRE-RELINQUISH pair of messages can occur for each host.

What we need is a way to reduce the number of hosts involved in making decisions. This way, fewer hosts need to vote, and fewer hosts need to reorganize thier votes in the event of a misvote.

Mutual Exclusion: Voting Districts

In order to address to reduce the number of messages required to win an election we are going to organize the participating systems into voting districts called coteries (pronounced, "koh-tarz" or "koh-tErz"), such that winning an election within a single district implies winning the election across all districts.

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Coteries is a political term that suggests a closed, somewhat intimate, and conspiring collection of actors (persons, states, trade organizations, unions, &c), e.g. a "Boy's Club".

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This can be accomplished by requiring that elections within any district be won by unanimous vote and then Gerrymandering each processor's district to ensure that all districts intersect. Since the subset of processors that are members of more than one district can't vote twice, they ensure that only one of the districts can gain a unanimous vote.

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Gerrymandering is a term that was coined by Federalists in the Massachusetts election of 1812. Governor Elbridge Gerry, a Republican, won a very narrow victory over his Federalist rival in the election of 1810. In order to improve their party's chances in the election of 1812, he and his Republican conspirators in the legislator redrew the electoral districts in an attempt to concentrate much of the Federalist vote into very few districts, while creating narrow, but majority, Republican support in the others.

The resulting districts were very irregular in shape. One Federalist commented that one among the new districts looked like a salamander. Another among his cohorts corrected him and declared that it was, in fact, a "Gerrymander." The term Gerrymandering, used to describe the process of contriving political districts to affect the outcome of an election, was born.

Incidentally, it didn't work and the Republicans lost the election. He was subsequently appointed as Vice-President of the U.S. He served in that role for two years. Since that time both federal law and judge-made law have made Gerrymandering illegal.

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The method of Gerrymandering disticts that we'll study was developed by Maekawa and published in 1985. Using this method, processor's are organized into a grid. Each processor's voting district contains all processors on the same row as the processor and all processors on the same column. That is to say that the voting district of a particular processor are all of those systems that form a perpendicular cross through the processor within the grid. Given N nodes, 2*SQRT(n) - 1 nodes will compose each voting district.

Using this approach, any pair of voting districts will intersect via at least one node, so two disticts cannot be one unanimously at the same time.

The voting district of processor 7

Here's what a node does, if it wants to enter the critical section:

• Send a REQUEST to every member of its district
• Wait until every member of its district votes YES
• Enter the critical section
• Upon exit from the CS, send RELEASE to each member of its district.

If a node gets a REQUEST, it does the following:

• If it has already voted in an outstanding election (it voted, but hasn't received a corresponding RELEASE), enqueue the request.
• Otherwise send YES

If a node gets a RELEASE:

• Dequeue oldest request from its queue, if any. Send a YES vote to this node, if any.

As we saw with simple majority voting last class, this approach can deadlock if requests arrive in a different order at different voters. This can allow different voters within overlapping districts to vote for different candidates. In particular, it can allow for a "split" between the two voters that are the overlap between two districts.

Fortunately, we can use the same approach we discussed last class to recover from this situation if it becomes problematic:

• A node records Lamport time w/total ordering before it sends a request. It sends this time with the request to all members of its district (the same time).
• Each voter uses a priority queue based on the time of the request.
• If a node receives a request with a time-stamp more older than the timestamp of a request for which it already voted, but for which it has not received a RELEASE, it attempts to cancel its vote. It does this by sending the candidate an INQUIRE.

  If this node hasn't won the election, it forgets about our vote and sends us a RELINQUISH. Once we receive the RELINQUISEH, we vote for the older request and enqueue the candidate for which we originally voted.

  If the candidate was already in the CS, no harm was done -- deadlock did not actually occur. When it goes out, we can vote for the other candidate. In this case, the processors may not have entered the CS in FIFO order, but that's okay -- deadlock didn't happen.

This approach requires about 3*(2*SQRT(N)-1) messages -- much nicer than 3*N messages. But it is not very fault tolerant, since a unanimous victory is required within a district. (Some failure can be tolerated, since failures outside of a district don't affect a node).

Token Passing Approaches

At this point, we've considered several different ways of approaching mutual exclusion: a centralized approach, a couple of timestamp approaches, a voting approach, and voting districts. Another approach is to create a special message, known as a token, which represents the right to access the critical section, and to pass this around among the hosts. The host which is in possesion can access the shared resource -- the others cannot. Think of it as the key to the gas station's bathroom. Since there is only on key, mutual exclusion is ensured.

Token Ring Approach

The first among these techniques is perhaps the simplest -- and certainly among the most frequently used in practice: token ring.

With this approach, every system knows its successor. The token moves from system to system through the list. Each system holds the token until it is done with the CS, and then passes it to its successor.

We can add fault tolerance to this approach if every host knows the mapping for all systems in the ring. If a successor dies, then the successor's successor, and successor's successor's successor, and so on can be tried. A host assumes that a system has failed if it cannot accept the token.

What happens if a system dies with the token? If there is a known time-out period, the origin machine can regenerate the token and start circulating it again. Depending on the nature of the CS, this could be dangerous, because multiple tokens could exist. If only one has access to the resource, this might be a problem.

The number of messages required per request is very interesting. Under high contention, the number is very, very low -- as low as one. If every system wants entry to the CS, each message will yield another entry. But if no one wants access to the CS, messages will occur for no reason.

But in general, we are more concerned about traffic when congestion is high. That makes this algorithm particularly interesting. It is especially interesting in real-time systems, because the worst-case behavior is well-bound and easily computed.