quoracle 0.0.4

A library for modelling, analyzing, and optimizing quorum systems

Quoracle

Quoracle is a library for constructing and analyzing read-write quorum systems. Run pip install quoracle and then follow along with the tutorial below to get started. You can find more information on Quoracle in our PaPoC '21 paper.

Tutorial

Quorum Systems

Given a set of nodes X, a read-write quorum system is a pair (R, W) where

1. R is a set of subsets of X called read quorums,
2. W is a set of subsets of X called write quorums, and
3. every read quorum intersects every write quorum.

quoracle allows us to construct and analyze arbitrary read-write quorum systems. First, we import the library.

from quoracle import *

Next, we specify the nodes in our quorum system. Our nodes can be strings, integers, IP addresses, anything!

>>> a = Node('a')
>>> b = Node('b')
>>> c = Node('c')
>>> d = Node('d')
>>> e = Node('e')
>>> f = Node('f')

Now, we construct a two by three grid of nodes. Every row is read quorum, and one element from every row is a write quorum. Note that when we construct a quorum system, we only have to specify the set of read quorums. The library figures out the optimal set of write quorums automatically.

>>> grid = QuorumSystem(reads=a*b*c + d*e*f)

This next code snippet prints out the read quorums.

>>> for r in grid.read_quorums():
...     print(r)
{'a', 'b', 'c'}
{'d', 'e', 'f'}

And this next code snippet prints out the write quorums.

>>> for w in grid.write_quorums():
...     print(w)
{'a', 'd'}
{'a', 'e'}
{'a', 'f'}
{'b', 'd'}
{'b', 'e'}
{'b', 'f'}
{'c', 'd'}
{'c', 'e'}
{'c', 'f'}

Alternatively, we can construct a quorum system be specifying the write quorums.

>>> QuorumSystem(writes=(a + b + c) * (d + e + f))

Or, we can specify both the read and write quorums.

>>> QuorumSystem(reads=a*b*c + d*e*f, writes=(a + b + c) * (d + e + f))

But, remember that every read quorum must intersect every write quorum. If we try to construct a quorum system with non-overlapping quorums, an exception will be thrown.

>>> QuorumSystem(reads=a+b+c, writes=d+e+f)
Traceback (most recent call last):
...
ValueError: Not all read quorums intersect all write quorums

We can check whether a given set is a read or write quorum. Note that any superset of a quorum is also considered a quorum.

>>> grid.is_read_quorum({'a', 'b', 'c'})
True
>>> grid.is_read_quorum({'a', 'b', 'c', 'd'})
True
>>> grid.is_read_quorum({'a', 'b', 'd'})
False
>>>
>>> grid.is_write_quorum({'a', 'd'})
True
>>> grid.is_write_quorum({'a', 'd', 'd'})
True
>>> grid.is_write_quorum({'a', 'b'})
False

Resilience

The read resilience of our quorum system is the largest number f such that despite the failure of any f nodes, we still have at least one read quorum. Write resilience is defined similarly, and resilience is the minimum of read and write resilience.

Here, we print out the read resilience, write resilience, and resilience of our grid quorum system. We can fail any one node and still have a read quorum, but if we fail one node from each row, we eliminate every read quorum, so the read resilience is 1. Similarly, we can fail any two nodes and still have a write quorum, but if we fail one node from every column, we eliminate every write quorum, so our write resilience is 1. The resilience is the minimum of 1 and 2, which is 1.

>>> grid.read_resilience()
1
>>> grid.write_resilience()
2
>>> grid.resilience()
1

Strategies

A strategy is a discrete probability distribution over the set of read and write quorums. A strategy gives us a way to pick quorums at random. We'll see how to construct optimal strategies in a second, but for now, we'll construct a strategy by hand. To do so, we have to provide a probability distribution over the read quorums and a probability distribution over the write quorums. Here, we'll pick the top row twice as often as the bottom row, and we'll pick each column uniformly at random. Note that when we specify a probability distribution, we don't have to provide exact probabilities. We can simply pass in weights, and the library will automatically normalize the weights into a valid probability distribution.

>>> # The read quorum strategy.
>>> sigma_r = {
...     frozenset({'a', 'b', 'c'}): 2.,
...     frozenset({'d', 'e', 'f'}): 1.,
... }
>>>
>>> # The write quorum strategy.
>>> sigma_w = {
...     frozenset({'a', 'd'}): 1.,
...     frozenset({'b', 'e'}): 1.,
...     frozenset({'c', 'f'}): 1.,
... }
>>> strategy = grid.make_strategy(sigma_r, sigma_w)

Once we have a strategy, we can use it to sample read and write quorums. Here, we expect get_read_quorum to return the top row twice as often as the bottom row, and we expect get_write_quorum to return every column uniformly at random.

>>> strategy.get_read_quorum()
{'a', 'b', 'c'}
>>> strategy.get_read_quorum()
{'a', 'b', 'c'}
>>> strategy.get_read_quorum()
{'d', 'e', 'f'}
>>> strategy.get_write_quorum()
{'b', 'e'}
>>> strategy.get_write_quorum()
{'c', 'f'}
>>> strategy.get_write_quorum()
{'b', 'e'}
>>> strategy.get_write_quorum()
{'a', 'd'}

Load and Capacity

Typically in a distributed system, a read quorum of nodes is contacted to perform a read, and a write quorum of nodes is contacted to perform a write. Assume we have a workload with a read fraction fr of reads and a write fraction fw = 1 - fr of writes. Given a strategy, the load of a node is the probability that the node is selected by the strategy. The load of a strategy is the load of the most heavily loaded node. The load of a quorum system is the load of the optimal strategy, i.e. the strategy that achieves the lowest load. The most heavily loaded node in a quorum system is a throughput bottleneck, so the lower the load the better.

Let's calculate the load of our strategy assuming a 100% read workload (i.e. a workload with a read fraction of 1).

• The load of a is 2/3 because the read quorum {a, b, c} is chosen 2/3 of the time.
• The load of b is 2/3 because the read quorum {a, b, c} is chosen 2/3 of the time.
• The load of c is 2/3 because the read quorum {a, b, c} is chosen 2/3 of the time.
• The load of d is 1/3 because the read quorum {d, e, f} is chosen 2/3 of the time.
• The load of e is 1/3 because the read quorum {d, e, f} is chosen 2/3 of the time.
• The load of f is 1/3 because the read quorum {d, e, f} is chosen 2/3 of the time.

The largest node load is 2/3, so our strategy has a load of 2/3. Rather than calculating load by hand, we can simply call the load function.

>>> strategy.load(read_fraction=1)
0.6666666666666666

Now let's calculate the load of our strategy assuming a 100% write workload. Again, we calculate the load on every node.

• The load of a is 1/3 because the write quorum {a, d} is chosen 1/3 of the time.
• The load of b is 1/3 because the write quorum {b, e} is chosen 1/3 of the time.
• The load of c is 1/3 because the write quorum {c, f} is chosen 1/3 of the time.
• The load of d is 1/3 because the write quorum {a, d} is chosen 1/3 of the time.
• The load of e is 1/3 because the write quorum {b, e} is chosen 1/3 of the time.
• The load of f is 1/3 because the write quorum {c, f} is chosen 1/3 of the time.

The largest node load is 1/3, so our strategy has a load of 1/3. Again, rather than calculating load by hand, we can simply call the load function. Note that we can pass in a read_fraction or write_fraction but not both.

>>> strategy.load(write_fraction=1)
0.3333333333333333

Now let's calculate the load of our strategy on a 25% read and 75% write workload.

• The load of a is 0.25 * 2/3 + 0.75 * 1/3 = 5/12 because 25% of the time we perform a read and select the read quorum {a, b, c} with 2/3 probability and 75% of the time, we perform a write and select the write quorum {a, d} with 1/3 probability.
• The load of b is 0.25 * 2/3 + 0.75 * 1/3 = 5/12 because 25% of the time we perform a read and select the read quorum {a, b, c} with 2/3 probability and 75% of the time, we perform a write and select the write quorum {b, e} with 1/3 probability.
• The load of c is 0.25 * 2/3 + 0.75 * 1/3 = 5/12 because 25% of the time we perform a read and select the read quorum {a, b, c} with 2/3 probability and 75% of the time, we perform a write and select the write quorum {c, f} with 1/3 probability.
• The load of d is 0.25 * 1/3 + 0.75 * 1/3 = 1/3 because 25% of the time we perform a read and select the read quorum {d, e, f} with 2/3 probability and 75% of the time, we perform a write and select the write quorum {a, d} with 1/3 probability.
• The load of e is 0.25 * 1/3 + 0.75 * 1/3 = 1/3 because 25% of the time we perform a read and select the read quorum {d, e, f} with 2/3 probability and 75% of the time, we perform a write and select the write quorum {b, e} with 1/3 probability.
• The load of f is 0.25 * 1/3 + 0.75 * 1/3 = 1/3 because 25% of the time we perform a read and select the read quorum {d, e, f} with 2/3 probability and 75% of the time, we perform a write and select the write quorum {c, f} with 1/3 probability.

The largest node load is 5/12, so our strategy has a load of 5/12. At this point, you can see that calculating load by hand is extremely tedious. We could have skipped all that work and called load instead!

>>> strategy.load(read_fraction=0.25)
0.41666666666666663

We can also compute the load on every node.

>>> print(strategy.node_load(a, read_fraction=0.25))
0.41666666666666663
>>> print(strategy.node_load(b, read_fraction=0.25))
0.41666666666666663
>>> print(strategy.node_load(c, read_fraction=0.25))
0.41666666666666663
>>> print(strategy.node_load(d, read_fraction=0.25))
0.3333333333333333
>>> print(strategy.node_load(e, read_fraction=0.25))
0.3333333333333333
>>> print(strategy.node_load(f, read_fraction=0.25))
0.3333333333333333

Our strategy has a load of 5/12 on a 25% read workload, but what about the quorum system? The quorum system does not have a load of 5/12 because our strategy is not optimal. We can call the strategy function to compute the optimal strategy automatically.

>>> strategy = grid.strategy(read_fraction=0.25)
>>> strategy
Strategy(reads={('a', 'b', 'c'): 0.5, ('d', 'e', 'f'): 0.5}, writes={('a', 'f'): 0.33333333, ('b', 'e'): 0.33333333, ('c', 'd'): 0.33333333})
>>> strategy.load(read_fraction=0.25))
0.3749999975

Here, we see that the optimal strategy picks all rows and all columns uniformly. This strategy has a load of 3/8 on the 25% read workload. Since this strategy is optimal, that means our quorum system also has a load of 3/8 on a 25% workload.

We can also query this strategy's load on other workloads as well. Note that this strategy is optimal for a read fraction of 25%, but it may not be optimal for other read fractions.

>>> strategy.load(read_fraction=0)
0.33333333
>>> strategy.load(read_fraction=0.5)
0.416666665
>>> strategy.load(read_fraction=1)
0.5

We can also use a quorum system's load function. The code snippet below is a shorthand for grid.strategy(read_fraction=0.25).load(read_fraction=0.25).

>>> grid.load(read_fraction=0.25)
0.3749999975

The capacity of strategy or quorum is simply the inverse of the load. Our quorum system has a load of 3/8 on a 25% read workload, so it has a capacity of 8/3.

>>> grid.capacity(read_fraction=0.25)
2.6666666844444444

The capacity of a quorum system is proportional to the maximum throughput that it can achieve before a node becomes bottlenecked. Here, if every node could process 100 commands per second, then our quorum system could process 800/3 commands per second.

Workload Distributions

In the real world, we don't often have a workload with a fixed read fraction. Workloads change over time. Instead of specifying a fixed read fraction, we can provide a discrete probability distribution of read fractions. Here, we say that the read fraction is 10% half the time and 75% half the time. strategy will return the strategy that minimizes the expected load according to this distribution.

>>> distribution = {0.1: 1, 0.75: 1}
>>> strategy = grid.strategy(read_fraction=distribution)
>>> strategy.load(read_fraction=distribution)
0.40416666474999996

Heterogeneous Node

In the real world, not all nodes are equal. We often run distributed systems on heterogeneous hardware, so some nodes might be faster than others. To model this, we instantiate every node with its capacity. Here, nodes a, c, and e can process 1000 commands per second, while nodes b, d, and f can only process 500 requests per second.

>>> a = Node('a', capacity=1000)
>>> b = Node('b', capacity=500)
>>> c = Node('c', capacity=1000)
>>> d = Node('d', capacity=500)
>>> e = Node('e', capacity=1000)
>>> f = Node('f', capacity=500)

Now, the definition of capacity becomes much simpler. The capacity of a quorum system is simply the maximum throughput that it can achieve. The load can be interpreted as the inverse of the capacity. Here, our quorum system is capable of processing 1333 commands per second for a workload of 75% reads.

>>> grid = QuorumSystem(reads=a*b*c + d*e*f)
>>> strategy = grid.strategy(read_fraction=0.75)
>>> strategy.load(read_fraction=0.75)
0.00075
>>> strategy.capacity(read_fraction=0.75)
1333.3333333333333

Nodes might also process reads and writes at different speeds. We can specify the peak read and write throughput of every node separately. Here, we assume reads are ten times as fast as writes.

>>> a = Node('a', write_capacity=1000, read_capacity=10000)
>>> b = Node('b', write_capacity=500, read_capacity=5000)
>>> c = Node('c', write_capacity=1000, read_capacity=10000)
>>> d = Node('d', write_capacity=500, read_capacity=5000)
>>> e = Node('e', write_capacity=1000, read_capacity=10000)
>>> f = Node('f', write_capacity=500, read_capacity=5000)

With 100% reads, our quorum system can process 10,000 commands per second. This throughput decreases as we increase the fraction of writes.

>>> grid = QuorumSystem(reads=a*b*c + d*e*f)
>>> grid.capacity(read_fraction=1)
10000.0
>>> grid.capacity(read_fraction=0.5)
3913.043450018904
>>> grid.capacity(read_fraction=0)
2000.0

f-resilient Strategies

Another real world complication is the fact that machines sometimes fail and are sometimes slow. If we contact a quorum of nodes, some of them may fail, and we'll get stuck waiting to hear back from them. Or, some of them may be stragglers, and we'll wait longer than we'd like. We can address this problem by contacting more than the bare minimum number of nodes.

Formally, we say a read quorum (or write quorum) q is f-resilient if despite the failure of any f nodes, q still forms a read quorum (or write quorum). A strategy is f-resilient if it only selects f-resilient quorums. By default, strategy returns 0-resilient quorums. We can pass in the f argument to get more resilient strategies.

>>> strategy = grid.strategy(read_fraction=0.5, f=1)

These sets are quorums even if 1 machine fails.

>>> strategy.get_read_quorum()
{'b', 'f', 'e', 'd', 'a', 'c'}
>>> strategy.get_write_quorum()
{'b', 'd', 'a', 'e'}

Note that as we increase resilience, quorums get larger, and we decrease capacity. On a 100% write workload, our grid quorum system has a 0-resilient capacity of 2000 commands per second, but a 1-resilient capacity of 1000 commands per second.

>>> grid.capacity(write_fraction=1, f=0)
2000.0
>>> grid.capacity(write_fraction=1, f=1)
1000.0

Also note that not all quorum systems are equally as resilient. In the next code snippet, we construct a "write 2, read 3" quorum system using the choose function. For this quorum system, every set of 2 nodes is a write quorum, and every set of 3 nodes is a read quorum. This quorum system has a 0-resilient capacity of 2000 (the same as the grid), but a 1-resilient capacity of 1333 (higher than the grid).

>>> write2 = QuorumSystem(writes=choose(2, [a, b, c, d, e]))
>>> write2.capacity(write_fraction=1, f=0)
2000.0
>>> write2.capacity(write_fraction=1, f=1)
1333.3333333333333

Latency

In the real world, not all nodes are equally as far away. Some are close and some are far. To address this, we associate every node with a latency, i.e. the time the required to contact the node. We model this in quoracle by assigning each node a latency, represented as a datetime.timedelta. Here, nodes a, b, c, d, e, and f in our grid have latencies of 1, 2, 3, 4, 5, and 6 seconds.

>>> import datetime
>>>
>>> def seconds(x: int) -> datetime.timedelta:
>>>     return datetime.timedelta(seconds=x)
>>>
>>> a = Node('a', write_capacity=1000, read_capacity=10000, latency=seconds(1))
>>> b = Node('b', write_capacity=500, read_capacity=5000, latency=seconds(2))
>>> c = Node('c', write_capacity=1000, read_capacity=10000, latency=seconds(3))
>>> d = Node('d', write_capacity=500, read_capacity=5000, latency=seconds(4))
>>> e = Node('e', write_capacity=1000, read_capacity=10000, latency=seconds(5))
>>> f = Node('f', write_capacity=500, read_capacity=5000, latency=seconds(6))
>>> grid = QuorumSystem(reads=a*b*c + d*e*f)

The latency of a quorum q is the time required to form a quorum of responses after contacting every node in q. For example, the read quorum {a, b, c} has a latency of three seconds. It takes 1 second to hear back from a, another second to hear back from b, and then a final second to hear back from c. The write quorum {a, b, d, f} has a latency of 4 seconds. It takes 1 second to hear back from a, another second to hear back from b, and then another 2 seconds to hear back from d. The set {a, b, d} is a write quorum, so the latency of this quorum is 4 seconds. Note that we didn't have to wait to hear back from f in order to form a quorum.

The latency of a strategy is the expected latency of the quorums that it chooses. The latency of a quorum system is the latency of the latency-optimal strategy. We can use the strategy function to find a latency-optimal strategy by passing in the value "latency" to the optimize flag.

>>> sigma = grid.strategy(read_fraction=0.5, optimize='latency')
>>> sigma
Strategy(reads={('a', 'b', 'c'): 1.0}, writes={('c', 'd'): 1.0})

We can find the latency of this strategy by calling the latency function.

>>> sigma.latency(read_fraction=1)
0:00:03
>>> sigma.latency(read_fraction=0)
0:00:04
>>> sigma.latency(read_fraction=0.5)
0:00:03.500000

As with capacity, we can call the latency function on our quorum system directly. In the follow code snippet grid.latency(read_fraction=0.5, optimize='latency') is a shorthand for grid.strategy(read_fraction=0.5, optimize='latency').latency(read_fraction=0.5).

>>> grid.latency(read_fraction=0.5, optimize='latency')
0:00:03.500000

Note that finding the latency-optimal strategy is trivial. The latency-optimal strategy always selects the read and write quorum with the smallest latencies. However, things get complicated when we start optimizing for capacity and latency at the same time. When we call the strategy function with optimize='latency', we can pass in a constraint on the maximum allowable load using the load_limit argument. For example, in the code snippet below, we find the latency-optimal strategy with a capacity of at least 1,500.

>>> sigma = grid.strategy(read_fraction=0.5,
...                      optimize='latency',
...                      load_limit=1/1500)
>>> sigma
Strategy(reads={('a', 'b', 'c'): 1.0}, writes={('a', 'd'): 0.66666667, ('c', 'e'): 0.33333333})
>>> sigma.capacity(read_fraction=0.5)
1499.9999925
>>> sigma.latency(read_fraction=0.5)
0:00:03.666667

This strategy always picks the read quorum {a, b, c}, and picks the write quorum {a, d} twice as often as write quorum {c, e}. It achieves our desired capacity of 1,500 commands per second (ignoring rounding errors) and has a latency of 3.66 seconds. We can also find a load-optimal strategy with a latency constraint.

>>> sigma = grid.strategy(read_fraction=0.5,
...                       optimize='load',
...                       latency_limit=seconds(4))
>>> sigma
Strategy(reads={('a', 'b', 'c'): 0.98870056, ('d', 'e', 'f'): 0.011299435}, writes={('a', 'd'): 0.19548023, ('a', 'f'): 0.22429379, ('b', 'd'): 0.062711864, ('b', 'e'): 0.097740113, ('c', 'e'): 0.41977401})
>>> sigma.capacity(read_fraction=0.5)
3856.2090893331633
>>> sigma.latency(read_fraction=0.5)
0:00:04.000001

This strategy is rather complicated and would be hard to find by hand. It has a capacity of 3856 commands per second and achieves our latency constraint of 4 seconds.

Be careful when specifying constraints. If the constraints cannot be met, a NoStrategyFound exception is raised.

>>> grid.strategy(read_fraction=0.5,
...               optimize='load',
...               latency_limit=seconds(1))
Traceback (most recent call last):
...
quoracle.quorum_system.NoStrategyFoundError: no strategy satisfies the given constraints

Network Load

Another useful metric is network load. When a protocol performs a read, it has to send messages to every node in a read quorum, and when a protocol performs a write, it has to send messages to every node in a write quorum. The bigger the quorums, the more messages are sent over the network. The network load of a quorum is simply the size of the quorum, the network load of a strategy is the expected network load of the quorums it chooses, and the network load of a quorum system is the network load of the network load-optimal strategy.

We can find network load optimal-strategies using the strategy function by passing in "network" to the optimize flag. We can also specify constraints on load and latency. In general, using the strategy function, we can pick one of load, latency, or network load to optimize and specify constraints on the other two metrics.

>>> sigma = grid.strategy(read_fraction=0.5, optimize='network')
>>> sigma
Strategy(reads={('a', 'b', 'c'): 1.0}, writes={('c', 'f'): 1.0})
>>> sigma.network_load(read_fraction=0.5)
2.5
>>> grid.network_load(read_fraction=0.5, optimize='network')
2.5
>>> sigma = grid.strategy(read_fraction=0.5,
...                       optimize='network',
...                       load_limit=1/2000,
...                       latency_limit=seconds(4))

Search

Finding good quorum systems by hand is hard. quoracle includes a heuristic based search procedure that tries to find quorum systems that are optimal with respect a target metric and set of constraints. For example, lets try to find a quorum system

• that has resilience 1,
• that is 1-resilient load optimal for a 75% read workload,
• that has a latency of most 4 seconds, and
• that has a network load of at most 4.

Because the number of quorum systems is enormous, the search procedure can take a very, very long time. We pass in a timeout to the search procedure to limit how long it takes. If the timeout expires, search returns the most optimal quorum system that it found so far.

### Search
>>> qs, sigma = search(nodes=[a, b, c, d, e, f],
...                    resilience=1,
...                    f=1,
...                    read_fraction=0.75,
...                    optimize='load',
...                    latency_limit=seconds(4),
...                    network_limit=4,
...                    timeout=seconds(60))
>>> qs
QuorumSystem(reads=choose3(a, c, e, (b + d + f)), writes=choose2(a, c, e, (b * d * f)))
>>> sigma
Strategy(reads={('a', 'c', 'e', 'f'): 0.33333333, ('a', 'b', 'c', 'e'): 0.33333333, ('a', 'c', 'd', 'e'): 0.33333333}, writes={('a', 'b', 'c', 'd', 'f'): 0.15714286, ('b', 'c', 'd', 'e', 'f'): 0.15714286, ('a', 'c', 'e'): 0.52857143, ('a', 'b', 'd', 'e', 'f'): 0.15714286})
>>> sigma.capacity(read_fraction=0.75)
3499.9999536250007
>>> sigma.latency(read_fraction=0.75)
0:00:03.907143
>>> sigma.network_load(read_fraction=0.75)
3.9857142674999997

Here, the search procedure returns the quorum system choose(3, [a, c, e, b+d+f]) with a capacity of 3500 commands per second and with latency and network load close to the limits specified.

Publishing To pypi

First, bump the version in setup.py. Then, run the following where $VERSION is the current version in setup.py.

python -m unittest
python -m build
python -m twine upload dist/quoracle-$VERSION*