\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 \glspl{thrd} 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.

I believe 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 load and return to the steady state, \eg, by adding or removing workers. Therefore, flaws in scheduling the steady state can 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 multi-tasking CPU'' :

\begin{displayquote}[Linux CFS\cit{https://www.kernel.org/doc/Documentation/scheduler/sched-design-CFS.txt}]
	{[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 tasks running, then it runs each at 50\% physical power --- i.e., actually in parallel.
	\label{q:LinuxCFS}
\end{displayquote}

Applied to threads, this model states that every ready \gls{thrd} immediately runs in parallel with all other ready \glspl{thrd}. 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 center of this model is that ready \glspl{thrd} do not interfere with each other but simply share the hardware. This assumption makes it easier to reason about threading because ready \glspl{thrd} can be thought of in isolation and the effect of the scheduler can be virtually ignored. This expectation of \gls{thrd} independence means the scheduler is expected to offer two guarantees:
\begin{enumerate}
	\item A fairness guarantee: a \gls{thrd} that is ready to run is not prevented by another thread.
	\item A performance guarantee: a \gls{thrd} that wants to start or stop running is not prevented by other threads wanting to do the same.
\end{enumerate}

It is important to note that these guarantees are expected only up to a point. \Glspl{thrd} that are ready to run should not be prevented to do so, but they still share the limited hardware resources. Therefore, the guarantee is considered respected if a \gls{thrd} gets access to a \emph{fair share} of the hardware resources, even if that share is very small.

Similarly the performance guarantee, the lack of interference among threads, is only relevant up to a point. Ideally, the cost of running and blocking should be constant regardless of contention, but the guarantee 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 equivalent to or lower than other popular languages, I consider the guarantee achieved.

More precisely the scheduler should be:
\begin{itemize}
	\item As fast as other schedulers that are less fair.
	\item Faster than other schedulers that have equal or better fairness.
\end{itemize}

\subsection{Fairness vs Scheduler Locality}
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\footnote{This section discusses \emph{internal locality}, \ie, the locality of the data used by the scheduler versus \emph{external locality}, \ie, how the data used by the application is affected by scheduling. External locality is a much more complicated subject and is discussed in part~\ref{Evaluation} on evaluation.}, \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 \gls{thrd}, and as consequence cache lines, to a \gls{hthrd} that is currently available.

However, I claim that in practice it is possible to strike a balance between fairness and performance because these goals do not necessarily overlap temporally, where Figure~\ref{fig:fair} shows a visual representation of this behaviour. As mentioned, some unfairness is acceptable; therefore it is desirable to have an algorithm that prioritizes cache locality as long as thread delay does not exceed the execution mental-model.

\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 \gls{thrd} awaits running is shown as the time the ready \gls{thrd} waits increases, Ready Time, the chances that its data is still in cache, Locality, decreases. At the same time, the need for fairness increases since other \glspl{thrd} 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}

\section{Design}
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 adding/removing \glspl{thrd} is a single point of contention. As 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 sub-ready-queues that multiple \glspl{hthrd} can access and modify without interfering.

Before going into the design of \CFA's scheduler proper, I want to discuss two sharding solutions which served as the inspiration scheduler in this thesis.

\subsection{Work-Stealing}

As I mentioned in \ref{existing:workstealing}, a popular pattern shard the ready-queue is work-stealing. As mentionned, in this pattern each \gls{proc} has its own ready-queue and \glspl{proc} only access each other's ready-queue if they run out of work.
The interesting aspect of workstealing happen in easier scheduling cases, \ie enough work for everyone but no more and no load balancing needed. In these cases, work-stealing is close to optimal scheduling: it can achieve perfect locality and have no contention.
On the other hand, work-stealing schedulers only attempt to do load-balancing when a \gls{proc} runs out of work.
This means that the scheduler may never balance unfairness that does not result in a \gls{proc} running out of work.
Chapter~\ref{microbench} shows that in pathological cases this problem can lead to indefinite starvation.


Based on these observation, I conclude that \emph{perfect} scheduler should behave very similarly to work-stealing in the easy cases, but should have more proactive load-balancing if the need arises.

\subsection{Relaxed-Fifo}
An entirely different scheme is to create a ``relaxed-FIFO'' queue as in \todo{cite Trevor's paper}. This approach forgos any ownership between \gls{proc} and ready-queue, and simply creates a pool of ready-queues from which the \glspl{proc} can pick from.
\Glspl{proc} choose ready-queus at random, but timestamps are added to all elements of the queue and dequeues are done by picking two queues and dequeing the oldest element.
The result is a queue that has both decent scalability and sufficient fairness.
The lack of ownership means that as long as one \gls{proc} is still able to repeatedly dequeue elements, it is unlikely that any element will stay on the queue for much longer than any other element.
This contrasts with work-stealing, where \emph{any} \gls{proc} busy for an extended period of time results in all the elements on its local queue to have to wait. Unless another \gls{proc} runs out of work.

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

While the fairness, of this scheme is good, it does suffer in terms of performance.
It requires very wide sharding, \eg at least 4 queues per \gls{hthrd}, and the randomness means locality can suffer significantly and finding non-empty queues can be difficult.

\section{\CFA}
The \CFA is effectively attempting to merge these two approaches, keeping the best of both.
It is based on the
\begin{figure}
	\centering
	\input{base.pstex_t}
	\caption[Base \CFA design]{Base \CFA design \smallskip\newline A list of sub-ready queues offers the sharding, two per \glspl{proc}. However, \glspl{proc} can access any of the sub-queues.}
	\label{fig:base}
\end{figure}



% The common solution to the single point of contention is to shard the ready-queue so each \gls{hthrd} can access the ready-queue without contention, increasing performance.

% \subsection{Sharding} \label{sec:sharding}
% An interesting approach to sharding a queue is presented in \cit{Trevors paper}. This algorithm presents a queue with a relaxed \glsxtrshort{fifo} guarantee using an array of strictly \glsxtrshort{fifo} sublists as shown in Figure~\ref{fig:base}. Each \emph{cell} of the array has a timestamp for the last operation and a pointer to a linked-list with a lock. Each node in the list is marked with a timestamp indicating when it is added to the list. A push operation is done by picking a random cell, acquiring the list lock, and pushing to the list. If the cell is locked, the operation is simply retried on another random cell until a lock is acquired. A pop operation is done in a similar fashion except two random cells are picked. If both cells are unlocked with non-empty lists, the operation pops the node with the oldest timestamp. If one of the cells is unlocked and non-empty, the operation pops from that cell. If both cells are either locked or empty, the operation picks two new random cells and tries again.

% \begin{figure}
% 	\centering
% 	\input{base.pstex_t}
% 	\caption[Relaxed FIFO list]{Relaxed FIFO list \smallskip\newline List at the base of the scheduler: an array of strictly FIFO lists. The timestamp is in all nodes and cell arrays.}
% 	\label{fig:base}
% \end{figure}

% \subsection{Finding threads}
% Once threads have been distributed onto multiple queues, identifying empty queues becomes a problem. Indeed, if the number of \glspl{thrd} does not far exceed the number of queues, it is probable that several of the cell queues are empty. Figure~\ref{fig:empty} shows an example with 2 \glspl{thrd} running on 8 queues, where the chances of getting an empty queue is 75\% per pick, meaning two random picks yield a \gls{thrd} only half the time. This scenario leads to performance problems since picks that do not yield a \gls{thrd} are not useful and do not necessarily help make more informed guesses.

% \begin{figure}
% 	\centering
% 	\input{empty.pstex_t}
% 	\caption[``More empty'' Relaxed FIFO list]{``More empty'' Relaxed FIFO list \smallskip\newline Emptier state of the queue: the array contains many empty cells, that is strictly FIFO lists containing no elements.}
% 	\label{fig:empty}
% \end{figure}

% There are several solutions to this problem, but they ultimately all have to encode if a cell has an empty list. My results show the density and locality of this encoding is generally the dominating factor in these scheme. Classic solutions to this problem use one of three techniques to encode the information:

% \paragraph{Dense Information} Figure~\ref{fig:emptybit} shows a dense bitmask to identify the cell queues currently in use. This approach means processors can often find \glspl{thrd} in constant time, regardless of how many underlying queues are empty. Furthermore, modern x86 CPUs have extended bit manipulation instructions (BMI2) that allow searching the bitmask with very little overhead compared to the randomized selection approach for a filled ready queue, offering good performance even in cases with many empty inner queues. However, this technique has its limits: with a single word\footnote{Word refers here to however many bits can be written atomically.} bitmask, the total amount of ready-queue sharding is limited to the number of bits in the word. With a multi-word bitmask, this maximum limit can be increased arbitrarily, but the look-up time increases. Finally, a dense bitmap, either single or multi-word, causes additional contention problems that reduces performance because of cache misses after updates. This central update bottleneck also means the information in the bitmask is more often stale before a processor can use it to find an item, \ie mask read says there are available \glspl{thrd} but none on queue when the subsequent atomic check is done.

% \begin{figure}
% 	\centering
% 	\vspace*{-5pt}
% 	{\resizebox{0.75\textwidth}{!}{\input{emptybit.pstex_t}}}
% 	\vspace*{-5pt}
% 	\caption[Underloaded queue with bitmask]{Underloaded queue with bitmask indicating array cells with items.}
% 	\label{fig:emptybit}

% 	\vspace*{10pt}
% 	{\resizebox{0.75\textwidth}{!}{\input{emptytree.pstex_t}}}
% 	\vspace*{-5pt}
% 	\caption[Underloaded queue with binary search-tree]{Underloaded queue with binary search-tree indicating array cells with items.}
% 	\label{fig:emptytree}

% 	\vspace*{10pt}
% 	{\resizebox{0.95\textwidth}{!}{\input{emptytls.pstex_t}}}
% 	\vspace*{-5pt}
% 	\caption[Underloaded queue with per processor bitmask]{Underloaded queue with per processor bitmask indicating array cells with items.}
% 	\label{fig:emptytls}
% \end{figure}

% \paragraph{Sparse Information} Figure~\ref{fig:emptytree} shows an approach using a hierarchical tree data-structure to reduce contention and has been shown to work in similar cases~\cite{ellen2007snzi}. However, this approach may lead to poorer performance due to the inherent pointer chasing cost while still allowing significant contention on the nodes of the tree if the tree is shallow.

% \paragraph{Local Information} Figure~\ref{fig:emptytls} shows an approach using dense information, similar to the bitmap, but each \gls{hthrd} keeps its own independent copy. While this approach can offer good scalability \emph{and} low latency, the liveliness and discovery of the information can become a problem. This case is made worst in systems with few processors where even blind random picks can find \glspl{thrd} in a few tries.

% I built a prototype of these approaches and none of these techniques offer satisfying performance when few threads are present. All of these approach hit the same 2 problems. First, randomly picking sub-queues is very fast. That speed means any improvement to the hit rate can easily be countered by a slow-down in look-up speed, whether or not there are empty lists. Second, the array is already sharded to avoid contention bottlenecks, so any denser data structure tends to become a bottleneck. In all cases, these factors meant the best cases scenario, \ie many threads, would get worst throughput, and the worst-case scenario, few threads, would get a better hit rate, but an equivalent poor throughput. As a result I tried an entirely different approach.

% \subsection{Dynamic Entropy}\cit{https://xkcd.com/2318/}
% In the worst-case scenario there are only few \glspl{thrd} ready to run, or more precisely given $P$ \glspl{proc}\footnote{For simplicity, this assumes there is a one-to-one match between \glspl{proc} and \glspl{hthrd}.}, $T$ \glspl{thrd} and $\epsilon$ a very small number, than the worst case scenario can be represented by $T = P + \epsilon$, with $\epsilon \ll P$. It is important to note in this case that fairness is effectively irrelevant. Indeed, this case is close to \emph{actually matching} the model of the ``Ideal multi-tasking CPU'' on page \pageref{q:LinuxCFS}. In this context, it is possible to use a purely internal-locality based approach and still meet the fairness requirements. This approach simply has each \gls{proc} running a single \gls{thrd} repeatedly. Or from the shared ready-queue viewpoint, each \gls{proc} pushes to a given sub-queue and then pops from the \emph{same} subqueue. The challenge is for the the scheduler to achieve good performance in both the $T = P + \epsilon$ case and the $T \gg P$ case, without affecting the fairness guarantees in the later.

% To handle this case, I use a \glsxtrshort{prng}\todo{Fix missing long form} in a novel way. There exist \glsxtrshort{prng}s that are fast, compact and can be run forward \emph{and} backwards.  Linear congruential generators~\cite{wiki:lcg} are an example of \glsxtrshort{prng}s of such \glsxtrshort{prng}s. The novel approach is to use the ability to run backwards to ``replay'' the \glsxtrshort{prng}. The scheduler uses an exclusive \glsxtrshort{prng} instance per \gls{proc}, the random-number seed effectively starts an encoding that produces a list of all accessed subqueues, from latest to oldest. Replaying the \glsxtrshort{prng} to identify cells accessed recently and which probably have data still cached.

% The algorithm works as follows:
% \begin{itemize}
% 	\item Each \gls{proc} has two \glsxtrshort{prng} instances, $F$ and $B$.
% 	\item Push and Pop operations occur as discussed in Section~\ref{sec:sharding} with the following exceptions:
% 	\begin{itemize}
% 		\item Push operations use $F$ going forward on each try and on success $F$ is copied into $B$.
% 		\item Pop operations use $B$ going backwards on each try.
% 	\end{itemize}
% \end{itemize}

% The main benefit of this technique is that it basically respects the desired properties of Figure~\ref{fig:fair}. When looking for work, a \gls{proc} first looks at the last cell they pushed to, if any, and then move backwards through its accessed cells. As the \gls{proc} continues looking for work, $F$ moves backwards and $B$ stays in place. As a result, the relation between the two becomes weaker, which means that the probablisitic fairness of the algorithm reverts to normal. Chapter~\ref{proofs} discusses more formally the fairness guarantees of this algorithm.

% \section{Details}
