source: doc/theses/thierry_delisle_PhD/thesis/text/core.tex

\chapter{Scheduling Core}\label{core}

Before discussing scheduling in general, where it is important to address systems that are changing states, this document discusses scheduling in a somewhat ideal scenario, where the system has reached a steady state.
For this purpose, a steady state is loosely defined as a state where there are always \ats ready to run and the system has the resources necessary to accomplish the work, \eg, enough workers.
In short, the system is neither overloaded nor underloaded.

It is important to discuss the steady state first because it is the easiest case to handle and, relatedly, the case in which the best performance is to be expected.
As such, when the system is either overloaded or underloaded, a common approach is to try to adapt the system to this new \gls{load} and return to the steady state, \eg, by adding or removing workers.
Therefore, flaws in scheduling the steady state tend to be pervasive in all states.

\section{Design Goals}
As with most of the design decisions behind \CFA, an important goal is to match the expectation of the programmer according to their execution mental model.
To match expectations, the design must offer the programmer sufficient guarantees so that, as long as they respect the execution mental model, the system also respects this model.

For threading, a simple and common execution mental model is the ``ideal multitasking CPU'':

\begin{displayquote}[Linux CFS\cite{MAN:linux/cfs}]
        {[The]} ``ideal multi-tasking CPU'' is a (non-existent  :-)) CPU that has 100\% physical power and which can run each task at precise equal speed, in parallel, each at [an equal fraction of the] speed.  For example: if there are 2 running tasks, then it runs each at 50\% physical power --- i.e., actually in parallel.
        \label{q:LinuxCFS}
\end{displayquote}

Applied to \ats, this model states that every ready \at immediately runs in parallel with all other ready \ats. While a strict implementation of this model is not feasible, programmers still have expectations about scheduling that come from this model.

In general, the expectation at the centre of this model is that ready \ats do not interfere with each other but simply share the hardware.
This assumption makes it easier to reason about threading because ready \ats can be thought of in isolation and the effect of the scheduler can be virtually ignored.
This expectation of \at independence means the scheduler is expected to offer two features:
\begin{enumerate}
        \item A fairness guarantee: a \at that is ready to run is not prevented by another thread indefinitely, \ie, starvation freedom. This is discussed further in the next section.
        \item A performance goal: given a \at that wants to start running, other threads wanting to do the same do not interfere with it.
\end{enumerate}

The performance goal, the lack of interference among threads, is only desired up to a point.
Ideally, the cost of running and blocking should be constant regardless of contention, but the goal is considered satisfied if the cost is not \emph{too high} with or without contention.
How much is an acceptable cost is obviously highly variable.
For this document, the performance experimentation attempts to show the cost of scheduling is at worst equivalent to existing algorithms used in popular languages.
This demonstration can be made by comparing applications built in \CFA to applications built with other languages or other models.
Recall programmer expectation is that the impact of the scheduler can be ignored.
Therefore, if the cost of scheduling is competitive with other popular languages, the goal is considered satisfied.
More precisely the scheduler should be:
\begin{itemize}
        \item As fast as other schedulers without any fairness guarantee.
        \item Faster than other schedulers that have equal or stronger fairness guarantees.
\end{itemize}

\subsection{Fairness Goals}
For this work, fairness is considered to have two strongly related requirements:

\paragraph{Starvation freedom} means as long as at least one \proc continues to dequeue \ats, all ready \ats should be able to run eventually, \ie, eventual progress.
Starvation freedom can be bounded or unbounded.
In the bounded case, all \ats should be able to run within a fix bound, relative to its own enqueue.
Whereas unbounded starvation freedom only requires the \at to eventually run.
The \CFA scheduler aims to guarantee unbounded starvation freedom.
In any running system, a \proc can stop dequeuing \ats if it starts running a \at that never blocks.
Without preemption, traditional work-stealing schedulers do not have starvation freedom, bounded or unbounded.
Now, this requirement raises the question, what about preemption?
Generally speaking, preemption happens on the timescale of several milliseconds, which brings us to the next requirement: ``fast'' load balancing.

\paragraph{Fast load balancing} means that while eventual progress is guaranteed, it is important to mention on which timescale this progress is expected to happen.
Indeed, while a scheduler with bounded starvation freedom is beyond the scope of this work, offering a good expected bound in the mathematical sense~\cite{wiki:expected} is desirable.
The expected bound on starvation freedom should be tighter than what preemption normally allows.
For interactive applications that need to run at 60, 90 or 120 frames per second, \ats having to wait milliseconds to run are effectively starved.
Therefore load-balancing should be done at a faster pace: one that is expected to detect starvation at the microsecond scale.

\subsection{Fairness vs Scheduler Locality} \label{fairnessvlocal}
An important performance factor in modern architectures is cache locality.
Waiting for data at lower levels or not present in the cache can have a major impact on performance.
Having multiple \glspl{hthrd} writing to the same cache lines also leads to cache lines that must be waited on.
It is therefore preferable to divide data among each \gls{hthrd}\footnote{This partitioning can be an explicit division up front or using data structures where different \glspl{hthrd} are naturally routed to different cache lines.}.

For a scheduler, having good locality, \ie, having the data local to each \gls{hthrd}, generally conflicts with fairness.
Indeed, good locality often requires avoiding the movement of cache lines, while fairness requires dynamically moving a \at, and as a consequence cache lines, to a \gls{hthrd} that is currently available.
Note that this section discusses \emph{internal locality}, \ie, the locality of the data used by the scheduler, versus \emph{external locality}, \ie, how scheduling affects the locality of the application's data.
External locality is a much more complicated subject and is discussed in the next section.

However, I claim that in practice it is possible to strike a balance between fairness and performance because these requirements do not necessarily overlap temporally.
Figure~\ref{fig:fair} shows a visual representation of this effect.
As mentioned, some unfairness is acceptable; for example, once the bounded starvation guarantee is met, additional fairness will not satisfy it \emph{more}.
Inversely, once a \at's data is evicted from cache, its locality cannot worsen.
Therefore it is desirable to have an algorithm that prioritizes cache locality as long as the fairness guarantee is also satisfied.

\begin{figure}
        \centering
        \input{fairness.pstex_t}
        \vspace*{-10pt}
        \caption[Fairness vs Locality graph]{Rule of thumb Fairness vs Locality graph \smallskip\newline The importance of Fairness and Locality while a ready \at awaits running is shown as the time the ready \at waits increases (Ready Time) the chances that its data is still in cache decreases (Locality).
        At the same time, the need for fairness increases since other \ats may have the chance to run many times, breaking the fairness model.
        Since the actual values and curves of this graph can be highly variable, the graph is an idealized representation of the two opposing goals.}
        \label{fig:fair}
\end{figure}

\subsection{Performance Challenges}\label{pref:challenge}
While there exists a multitude of potential scheduling algorithms, they generally always have to contend with the same performance challenges.
Since these challenges are recurring themes in the design of a scheduler it is relevant to describe them here before looking at the scheduler's design.

\subsubsection{Latency}
The most basic performance metric of a scheduler is scheduling latency.
This measures the how long it takes for a \at to run once scheduled, including the cost of scheduling itself.
This measure include both the sequential cost of the operation itself, both also the scalability.

\subsubsection{Scalability}
Given a large number of \procs and an even larger number of \ats, scalability measures how fast \procs can enqueue and dequeue \ats relative to the available parallelism.
One could expect that doubling the number of \procs would double the rate at which \ats are dequeued, but contention on the internal data structure of the scheduler can diminish the improvements.
While the ready queue itself can be sharded to alleviate the main source of contention, auxiliary scheduling features, \eg counting ready \ats, can also be sources of contention.
In the Chapter~\ref{microbench}, scalability is measured as $\# procs \times \frac{ns}{ops}$, \ie, number of \procs times total time over total operations.
Since the total number of operation should scale with the number of \procs, this gives a measure how much each additional \proc affects the other \procs.

\subsubsection{Migration Cost}
Another important source of scheduling latency is \glslink{atmig}{migration}.
A \at migrates if it executes on two different \procs consecutively, which is the process discussed in \ref{fairnessvlocal}.
Migrations can have many different causes, but in certain programs, it can be impossible to limit migration.
Chapter~\ref{microbench} has a benchmark where any \at can potentially unblock any other \at, which can lead to \ats migrating frequently.
Hence, it is important to design the internal data structures of the scheduler to limit any latency penalty from migrations.


\section{Inspirations}
In general, a na\"{i}ve \glsxtrshort{fifo} ready-queue does not scale with increased parallelism from \glspl{hthrd}, resulting in decreased performance.
The problem is a single point of contention when adding/removing \ats.
As is shown in the evaluation sections, most production schedulers do scale when adding \glspl{hthrd}.
The solution to this problem is to shard the ready queue: create multiple \emph{sub-queues} forming the logical ready-queue.
The sub-queues are accessed by multiple \glspl{hthrd} without the need for communication.

Before going into the design of \CFA's scheduler, it is relevant to discuss two sharding solutions that served as the inspiration for the scheduler in this thesis.

\subsection{Work-Stealing}

As mentioned in \ref{existing:workstealing}, a popular sharding approach for the ready queue is work-stealing.
In this approach, each \gls{proc} has its own local sub-queue and \glspl{proc} only access each other's sub-queue if they run out of work on their local ready-queue.
The interesting aspect of work stealing manifests itself in the steady-state scheduling case, \ie all \glspl{proc} have work and no load balancing is needed.
In this case, work stealing is close to optimal scheduling latency: it can achieve perfect locality and have no contention.
On the other hand, work-stealing only attempts to do load-balancing when a \gls{proc} runs out of work.
This means that the scheduler never balances unfair loads unless they result in a \gls{proc} running out of work.
Chapter~\ref{microbench} shows that, in pathological cases, work stealing can lead to unbounded starvation.

Based on these observations, the conclusion is that a \emph{perfect} scheduler should behave similarly to work-stealing in the steady-state case, \ie, avoid migrations in well balanced workloads, but load balance proactively when the need arises.

\subsection{Relaxed-FIFO}
A different scheduling approach is the ``relaxed-FIFO'' queue, as in \cite{alistarh2018relaxed}.
This approach forgoes any ownership between \gls{proc} and sub-queue, and simply creates a pool of sub-queues from which \glspl{proc} pick.
Scheduling is performed as follows:
\begin{itemize}
\item
All sub-queues are protected by TryLocks.
\item
Timestamps are added to each element of a sub-queue.
\item
A \gls{proc} randomly tests sub-queues until it has acquired one or two queues, referred to as \newterm{randomly picking} or \newterm{randomly helping}.
\item
If two queues are acquired, the older of the two \ats is dequeued from the front of the acquired queues.
\item
Otherwise, the \at from the single queue is dequeued.
\end{itemize}
The result is a queue that has both good scalability and sufficient fairness.
The lack of ownership ensures that as long as one \gls{proc} is still able to repeatedly dequeue elements, it is unlikely any element will delay longer than any other element.
This guarantee contrasts with work-stealing, where a \gls{proc} with a long sub-queue results in unfairness for its \ats in comparison to a \gls{proc} with a short sub-queue.
This unfairness persists until a \gls{proc} runs out of work and steals.

An important aspect of this scheme's fairness approach is that the timestamps make it possible to evaluate how long elements have been in the queue.
However, \glspl{proc} eagerly search for these older elements instead of focusing on specific queues, which negatively affects locality.

While this scheme has good fairness, its performance can be improved.
Wide sharding is generally desired, \eg at least 4 queues per \proc, and randomly picking non-empty queues is difficult when there are few ready \ats.
The next sections describe improvements I made to this existing algorithm.
However, ultimately the ``relaxed-FIFO'' queue is not used as the basis of the \CFA scheduler.

\section{Relaxed-FIFO++}
The inherent fairness and decent performance with many \ats make the relaxed-FIFO queue a good candidate to form the basis of a new scheduler.
The problem case is workloads where the number of \ats is barely greater than the number of \procs.
In these situations, the wide sharding of the ready queue means most of its sub-queues are empty.
Furthermore, the non-empty sub-queues are unlikely to hold more than one item.
The consequence is that a random dequeue operation is likely to pick an empty sub-queue, resulting in an unbounded number of selections.
This state is generally unstable: each sub-queue is likely to frequently toggle between being empty and nonempty.
Indeed, when the number of \ats is \emph{equal} to the number of \procs, every pop operation is expected to empty a sub-queue and every push is expected to add to an empty sub-queue.
In the worst case, a check of the sub-queues sees all are empty or full.

As this is the most obvious challenge, it is worth addressing first.
The seemingly obvious solution is to supplement each sharded sub-queue with data that indicates whether the queue is empty/nonempty.
This simplifies finding nonempty queues, \ie ready \glspl{at}.
The sharded data can be organized in different forms, \eg a bitmask or a binary tree that tracks the nonempty sub-queues, using a bit or a node per sub-queue, respectively.
Specifically, many modern architectures have powerful bitmask manipulation instructions, and, searching a binary tree has good Big-O complexity.
However, precisely tracking nonempty sub-queues is problematic.
The reason is that the sub-queues are initially sharded with a width presumably chosen to avoid contention.
However, tracking which ready queue is nonempty is only useful if the tracking data is dense, \ie tracks whether multiple sub-queues are empty.
Otherwise, it does not provide useful information because reading this new data structure risks being as costly as simply picking a sub-queue at random.
But if the tracking mechanism \emph{is} denser than the shared sub-queues, then constant updates invariably create a new source of contention.
Early experiments with this approach showed that randomly picking, even with low success rates, is often faster than bit manipulations or tree walks.

The exception to this rule is using local tracking.
If each \proc locally keeps track of empty sub-queues, then this can be done with a very dense data structure without introducing a new source of contention.
However, the consequence of local tracking is that the information is incomplete.
Each \proc is only aware of the last state it saw about each sub-queue so this information quickly becomes stale.
Even on systems with low \gls{hthrd} count, \eg 4 or 8, this approach can quickly lead to the local information being no better than the random pick.
This result is due in part to the cost of maintaining information and its poor quality.

However, using a very low-cost but inaccurate approach for local tracking can still be beneficial.
If the local tracking is no more costly than a random pick, then \emph{any} improvement to the success rate, however low it is, leads to a performance benefit.
This suggests the following approach:

\subsection{Dynamic Entropy}\cite{xkcd:dynamicentropy}
The Relaxed-FIFO approach can be made to handle the case of mostly empty sub-queues by tweaking the \glsxtrlong{prng} that drives the random picking of sub-queues.
The \glsxtrshort{prng} state can be seen as containing a list of all the future sub-queues that will be accessed.
While this concept is not particularly useful on its own, the consequence is that if the \glsxtrshort{prng} algorithm can be run \emph{backwards}, then the state also contains a list of all the sub-queues that were accessed.
Luckily, bidirectional \glsxtrshort{prng} algorithms do exist, \eg some Linear Congruential Generators~\cite{wiki:lcg} support running the algorithm backwards while offering good quality and performance.
This particular \glsxtrshort{prng} can be used as follows:
\begin{itemize}
\item
Each \proc maintains two \glsxtrshort{prng} states, referred to as $F$ and $B$.
\item
When a \proc attempts to dequeue a \at, it picks a sub-queue by running its $B$ backwards.
\item
When a \proc attempts to enqueue a \at, it runs its $F$ forward picking a sub-queue to enqueue to.
If the enqueue is successful, state of its $B$ is overwritten with the content of its $F$.
\end{itemize}
The result is that each \proc tends to dequeue \ats that it has itself enqueued.
When most sub-queues are empty, this technique increases the odds of finding \ats at a very low cost, while also offering an improvement on locality in many cases.

My own tests showed this approach performs better than relaxed-FIFO in many cases.
However, it is still not competitive with work-stealing algorithms.
The fundamental problem is that the randomness limits how much locality the scheduler offers.
This becomes problematic both because the scheduler is likely to get cache misses on internal data structures and because migrations become frequent.
Therefore, the attempt to modify the relaxed-FIFO algorithm to behave more like work stealing did not pan out.
The alternative is to do it the other way around.

\section{Work Stealing++}\label{helping}
To add stronger fairness guarantees to work stealing a few changes are needed.
First, the relaxed-FIFO algorithm has fundamentally better fairness because each \proc always monitors all sub-queues.
Therefore, the work-stealing algorithm must be prepended with some monitoring.
Before attempting to dequeue from a \proc's sub-queue, the \proc must make some effort to ensure other sub-queues are not being neglected.
To make this possible, \procs must be able to determine which \at has been on the ready queue the longest.
Second, the relaxed-FIFO approach uses timestamps, denoted TS, for each \at to make this possible.
Theses timestamps can be added to work stealing.

\begin{figure}
        \centering
        \input{base.pstex_t}
        \caption[Base \CFA design]{Base \CFA design \smallskip\newline It uses a pool of sub-queues, with a sharding of two sub-queue per \proc.
        Each \gls{proc} can access all of the sub-queues.
        Each \at is timestamped when enqueued.}
        \label{fig:base}
\end{figure}

Figure~\ref{fig:base} shows the algorithm structure.
This structure is similar to classic work-stealing except the sub-queues are placed in an array so \procs can access them in constant time.
Sharding can be adjusted based on contention.
As an optimization, the timestamp of a \at is stored in the \at in front of it, so the first TS is in the array and the last \at has no TS.
This organization keeps the highly accessed front TSs directly in the array.
When a \proc attempts to dequeue a \at, it first picks a random remote sub-queue and compares its timestamp to the timestamps of its local sub-queue(s).
The oldest waiting of the compared \ats is dequeued.
In this document, picking from a remote sub-queue in this fashion is referred to as ``helping''.

The timestamps are measured using the CPU's hardware timestamp counters~\cite{wiki:rdtsc}.
These provide a 64-bit counter that tracks the number of cycles since the CPU was powered on.
Assuming the CPU runs at less than 5 GHz, this means that the 64-bit counter takes over a century before overflowing.
This is true even on 32-bit CPUs, where the counter is generally still 64-bit.
However, on many architectures, the instructions to read the counter do not have any particular ordering guarantees.
Since the counter does not depend on any data in the cpu pipeline, this means there is significant flexibility for the instruction to be read out of order, which limites the accuracy to a window of code.
Finally, another issue that can come up with timestamp counters is synchronization between \glspl{hthrd}.
This appears to be mostly a historical concern, as recent CPU offer more synchronization guarantees.
For example, Intel supports "Invariant TSC" \cite[\S~17.15.1]{MAN:inteldev} which is guaranteed to be synchronized across \glspl{hthrd}.

However, this na\"ive implementation has performance problems.
First, it is necessary to avoid helping when it does not improve fairness.
Random effects like cache misses and preemption can add unpredictable but short bursts of latency but do not warrant the cost of helping.
These bursts can cause increased migrations, at which point this same locality problems as in the relaxed-FIFO approach start to appear.

\begin{figure}
        \centering
        \input{base_avg.pstex_t}
        \caption[\CFA design with Moving Average]{\CFA design with Moving Average \smallskip\newline A moving average is added to each sub-queue.}
        \label{fig:base-ma}
\end{figure}

A simple solution to this problem is to use an exponential moving average\cite{wiki:ma} (MA) instead of a raw timestamp, as shown in Figure~\ref{fig:base-ma}.
Note that this is more complex than it can appear because the \at at the head of a sub-queue is still waiting, so its wait time has not ended.
Therefore, the exponential moving average is an average of how long each dequeued \at has waited.
To compare sub-queues, the timestamp at the head must be compared to the current time, yielding the best-case wait time for the \at at the head of the queue.
This new waiting is averaged with the stored average.
To further limit \glslink{atmig}{migrations}, a bias can be added to a local sub-queue, where a remote sub-queue is helped only if its moving average is more than $X$ times the local sub-queue's average.
Tests for this approach indicate the precise values for the weight of the moving average and the bias are not important, \ie weights and biases of similar \emph{magnitudes} have similar effects.

With these additions to work stealing, scheduling can satisfy the starvation freedom guarantee while suffering much less from unnecessary migrations than the relaxed-FIFO approach.
Unfortunately, the work to achieve fairness has a performance cost, especially when the workload is inherently fair, and hence, there is only short-term unfairness or no starvation.
The problem is that the constant polling, \ie reads, of remote sub-queues generally entails cache misses because the TSs are constantly being updated.
To make things worse, remote sub-queues that are very active, \ie \ats are frequently enqueued and dequeued from them, lead to higher chances that polling will incur a cache-miss.
Conversely, the active sub-queues do not benefit much from helping since starvation is already a non-issue.
This puts this algorithm in the awkward situation of paying for a largely unnecessary cost.
The good news is that this problem can be mitigated.

\subsection{Redundant Timestamps}\label{relaxedtimes}
The problem with polling remote sub-queues is that correctness is critical.
There must be a consensus among \procs on which sub-queues hold which \ats, as the \ats are in constant motion.
Furthermore, since timestamps are used for fairness, it is critical that the oldest \ats eventually be recognized as such.
However, when deciding if a remote sub-queue is worth polling, correctness is less of a problem.
Since the only requirement is that a sub-queue is eventually polled, some data staleness is acceptable.
This leads to a situation where stale timestamps are only problematic in some cases.
Furthermore, stale timestamps can be desirable since lower freshness requirements mean fewer cache invalidations.

Figure~\ref{fig:base-ts2} shows a solution with a second array containing a copy of the timestamps and average.
This copy is updated \emph{after} the sub-queue's critical sections using relaxed atomics.
\Glspl{proc} now check if polling is needed by comparing the copy of the remote timestamp instead of the actual timestamp.
The result is that since there is no fencing, the writes can be buffered in the hardware and cause fewer cache invalidations.

\begin{figure}
        \centering
        \input{base_ts2.pstex_t}
        \caption[\CFA design with Redundant Timestamps]{\CFA design with Redundant Timestamps \smallskip\newline This design uses an array containing a copy of the timestamps.
        These timestamps are written-to with relaxed atomics, so there is no order among concurrent memory accesses, leading to fewer cache invalidations.}
        \label{fig:base-ts2}
\end{figure}

The correctness argument is somewhat subtle.
The data used for deciding whether or not to poll a queue can be stale as long as it does not cause starvation.
Therefore, it is acceptable if stale data makes queues appear older than they are, but appearing fresher can be a problem.
For the timestamps, this means it is acceptable to miss writes to the timestamp since they make the head \at look older.
For the moving average, as long as the operations are just atomic reads/writes, the average is guaranteed to yield a value that is between the oldest and newest values written.
Therefore, this unprotected read of the timestamp and average satisfies the limited correctness that is required.

With redundant timestamps, this scheduling algorithm achieves both the fairness and performance requirements on most machines.
The problem is that the cost of polling and helping is not necessarily consistent across each \gls{hthrd}.
For example on machines with multiple CPUs, cache misses can be satisfied from the caches on the same (local) CPU, or by the caches on a different (remote) CPU.
Cache misses satisfied by a remote CPU have significantly higher latency than from the local CPU.
However, these delays are not specific to systems with multiple CPUs.
Depending on the cache structure, cache misses can have different latency on the same CPU, \eg the AMD EPYC 7662 CPUs used in Chapter~\ref{microbench}.

\begin{figure}
        \centering
        \input{cache-share.pstex_t}
        \caption[CPU design with wide L3 sharing]{CPU design with wide L3 sharing \smallskip\newline A CPU with 4 cores, where caches L1 and L2 are private to each core, and the L3 cache is shared across all cores.}
        \label{fig:cache-share}

        \vspace{25pt}

        \input{cache-noshare.pstex_t}
        \caption[CPU design with a narrower L3 sharing]{CPU design with a narrow L3 sharing \smallskip\newline A CPU with 4 cores, where caches L1 and L2 are private to each core, and the L3 cache is shared across a pair of cores.}
        \label{fig:cache-noshare}
\end{figure}

Figures~\ref{fig:cache-share} and~\ref{fig:cache-noshare} show two different cache topologies that highlight this difference.
In Figure~\ref{fig:cache-share}, all cache misses are either private to a CPU or shared with another CPU.
This means that latency due to cache misses is fairly consistent.
In contrast, in Figure~\ref{fig:cache-noshare}, misses in the L2 cache can be satisfied by either instance of the L3 cache.
However, the memory-access latency to the remote L3 is higher than the memory-access latency to the local L3.
The impact of these different designs on this algorithm is that scheduling only scales well on architectures with the L3 cache shared across many \glspl{hthrd}, similar to Figure~\ref{fig:cache-share}, and less well on architectures with many L3 cache instances and less sharing, similar to Figure~\ref{fig:cache-noshare}.
Hence, as the number of L3 instances grows, so too does the chance that the random helping causes significant cache latency.
The solution is for the scheduler to be aware of the cache topology.

\subsection{Per CPU Sharding}
Building a scheduler that is cache aware poses two main challenges: discovering the cache topology and matching \procs to this cache structure.
Unfortunately, there is no portable way to discover cache topology, and it is outside the scope of this thesis to solve this problem.
This work uses the cache topology information from Linux's @/sys/devices/system/cpu@ directory.
This leaves the challenge of matching \procs to cache structure, or more precisely, identifying which sub-queues of the ready queue are local to which subcomponents of the cache structure.
Once a match is generated, the helping algorithm is changed to add bias so that \procs more often help sub-queues local to the same cache substructure.\footnote{
Note that like other biases mentioned in this section, the actual bias value does not appear to need precise tuning beyond the order of magnitude.}

The simplest approach for mapping sub-queues to cache structure is to statically tie sub-queues to CPUs.
Instead of having each sub-queue local to a specific \proc, the system is initialized with sub-queues for each hardware hyperthread/core up front.
Then \procs dequeue and enqueue by first asking which CPU id they are executing on, to identify which sub-queues are the local ones.
\Glspl{proc} can get the CPU id from @sched_getcpu@ or @librseq@.

This approach solves the performance problems on systems with topologies with narrow L3 caches, similar to Figure \ref{fig:cache-noshare}.
However, it can still cause some subtle fairness problems in systems with few \procs and many \glspl{hthrd}.
In this case, the large number of sub-queues and the bias against sub-queues tied to different cache substructures make it unlikely that every sub-queue is picked.
To make things worse, the small number of \procs means that few helping attempts are made.
This combination of low selection and few helping attempts allow a \at to become stranded on a sub-queue for a long time until it gets randomly helped.
On a system with 2 \procs, 256 \glspl{hthrd}, and a 100:1 bias, it can take multiple seconds for a \at to get dequeued from a remote queue.
In this scenario, where each \proc attempts to help on 50\% of dequeues, the probability that a remote sub-queue gets help is $\frac{1}{51200}$ and follows a geometric distribution.
Therefore the probability of the remote sub-queue gets help within the next 100'000 dequeues is only 85\%.
Assuming dequeues happen every 100ns, there is still 15\% chance a \at could starve for more than 10ms and a 1\% chance the \at starves for 33.33ms, the maximum latency tolerated for interactive applications.
If few \glspl{hthrd} share each cache instance, the probability that a \at is on a remote sub-queue becomes high.
Therefore, a more dynamic match of sub-queues to cache instances is needed.

\subsection{Topological Work Stealing}
\label{s:TopologicalWorkStealing}
The approach used in the \CFA scheduler is to have per-\proc sub-queues, but have an explicit data structure to track which cache substructure each sub-queue is tied to.
This tracking requires some finesse, because reading this data structure must lead to fewer cache misses than not having the data structure in the first place.
A key element, however, is that, like the timestamps for helping, reading the cache instance mapping only needs to give the correct result \emph{often enough}.
Therefore the algorithm can be built as follows: before enqueueing or dequeuing a \at, a \proc queries the CPU id and the corresponding cache instance.
Since sub-queues are tied to \procs, a \proc can then update the cache instance mapped to the local sub-queue(s).
To avoid unnecessary cache line invalidation, the map is only written-to if the mapping changes.

This scheduler is used in the remainder of the thesis for managing CPU execution, but additional scheduling is needed to handle long-term blocking and unblocking, such as I/O.