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Jan 14, 2021, 12:23:14 PM (3 years ago)
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Thierry Delisle <tdelisle@…>
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Merge branch 'master' of plg.uwaterloo.ca:software/cfa/cfa-cc

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  • doc/theses/thierry_delisle_PhD/thesis/text/core.tex

    r4468a70 r342af53  
    11\chapter{Scheduling Core}\label{core}
    22
    3 Before discussing scheduling in general, where it is important to address systems that are changing states, this document discusses scheduling in a somewhat ideal scenerio, 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 ressources necessary to accomplish the work, \eg, enough workers. In short, the system is neither overloaded nor underloaded.
     3Before 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.
    44
    5 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 the new load and return to the steady state, \eg, adding or removing workers. Flaws in the scheduling when the system is in the steady state can therefore to be pervasive in all states.
     5I 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.
    66
    77\section{Design Goals}
    8 As with most of the design decisions behind \CFA, an important goal is to match the expectation of the programmer, according to their probable mental model. To match these expectations, the design must offer the programmers sufficient guarantees so that, as long as they respect the mental model, the system will also respect this model.
     8As 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.
    99
    10 For threading, a simple and common mental model is the ``Ideal multi-tasking CPU'' :
     10For threading, a simple and common execution mental-model is the ``Ideal multi-tasking CPU'' :
    1111
    1212\begin{displayquote}[Linux CFS\cit{https://www.kernel.org/doc/Documentation/scheduler/sched-design-CFS.txt}]
    1313        {[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.
     14        \label{q:LinuxCFS}
    1415\end{displayquote}
    1516
    1617Applied 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.
    1718
    18 In general, the expectation at the center of this model is that ready \glspl{thrd} do not interfere with eachother but simply share the hardware. This makes it easier to reason about threading because ready \glspl{thrd} can be taken 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:
     19In 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:
    1920\begin{enumerate}
    20         \item A fairness guarantee: a \gls{thrd} that is ready to run will not be prevented to do so by another thread.
    21         \item A performance guarantee: a \gls{thrd} that wants to start or stop running will not be slowed down by other threads wanting to do the same.
     21        \item A fairness guarantee: a \gls{thrd} that is ready to run is not prevented by another thread.
     22        \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.
    2223\end{enumerate}
    2324
    24 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 need to share a limited amount of hardware. Therefore, the guarantee is considered respected if a \gls{thrd} gets access to a \emph{fair share} of the hardware, even if that share is very small.
     25It 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.
    2526
    26 Similarly the performance guarantee, the lack of interferance between threads, is only relevant up to a point. Ideally the cost of running and blocking would 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 will attempt to show that the cost of scheduling is at worst equivalent to existing algorithms used in popular languages. This demonstration can be made by comparing application built in \CFA to applications built with other languages or other models. Recall from a few paragraphs ago that the expectation of programmers is that the impact of the scheduler can be ignored. Therefore, if the cost of scheduling is equivalent or lower to other popular languages, I will consider the guarantee achieved.
     27Similarly 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.
    2728
    2829More precisely the scheduler should be:
    2930\begin{itemize}
    3031        \item As fast as other schedulers that are less fair.
    31         \item Faster than other scheduler that have equal or better fairness.
     32        \item Faster than other schedulers that have equal or better fairness.
    3233\end{itemize}
    3334
    3435\subsection{Fairness vs Scheduler Locality}
    35 An important performance factor in modern architectures is cache locality. Waiting for data not present in the cache can have a major impact on performance, and having multiple \glspl{hthrd} writing to the same cache lines can lead to cache lines that need to be waited on again. It is therefore preferable to divide the data among each \gls{hthrd}\footnote{This can be an explicit division up front or using data structures where different \glspl{hthrd} are naturally routed to different cache lines.}.
     36An 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.}.
    3637
    37 For a scheduler, having good locality\footnote{This section discusses \emph{internal} locality, \ie, the locality of the data used by the scheduler. \emph{External locality}, \ie, how the data used by the application is affected by scheduling, is a much more complicated subject and will be discussed in the chapters on evaluation.}, \ie, having the data be 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 \gls{thrd}, and as a consequence cache lines, to \glspl{hthrd} that are currently more appropriate.
     38For 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.
    3839
    39 However, I claim that in practice it is possible to strike a balance between fairness and performance because the need for these do not necessarily overlap temporaly. Figure~\ref{fig:fair} shows an visual representation of this behaviour. As mentionned, a little bit of unfairness can be acceptable, therefore it can be desirable to have an algorithm that prioritizes cache locality as long as no threads is left behind for too long.
     40However, 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.
    4041
    4142\begin{figure}
    42         \begin{center}
    43                 \input{fairness.pstex_t}
    44         \end{center}
    45         \caption{Fairness vs Locality}
     43        \centering
     44        \input{fairness.pstex_t}
     45        \vspace*{-10pt}
     46        \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.}
    4647        \label{fig:fair}
    47         Rule of thumb graph: Importance of Fairness and Locality while a ready \gls{thrd} waits run.
    48         As the time a ready \gls{thrd} waits increases, ``Ready Time'', the chances that its data is still in cache 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 mentionned above. Since the actual values and curves of this graph can be highly variable, the graph is left intentionally fuzzy and innacurate.
    4948\end{figure}
    5049
    5150\section{Design}
    52 A naive strictly \glsxtrshort{fifo} ready-queue does not offer sufficient performance. As shown in the evaluation sections, most production schedulers scale when adding multiple \glspl{hthrd} and that is not possible with a single point of contention. Therefore it is vital to shard the ready-queue so that multiple \glspl{hthrd} can access the ready-queue without performance degradation.
     51In 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 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 though lack of contention.
    5352
    5453\subsection{Sharding} \label{sec:sharding}
    55 An interesting approach to sharding a queue is presented in \cit{Trevors paper}. This algorithm represents a queue with relaxed \glsxtrshort{fifo} guarantee using an array of strictly \glsxtrshort{fifo} sublists as shown in Figure~\ref{fig:base}. Each cell of the array contains a linked-list with a lock and each node in these list is marked with a timestamp indicating when they were added to the list. Push operations are done by picking a random cell and attempting to push to its list. If the cell is already locked, the operation is simply retried on a new cell until a lock is acquired. Pop operations are done in a similar fashion except two random cells are picked. If both cells are not already locked and both cells contain non-empty lists, the operation pops the node with the oldest timestamp. If only one of the cell is unlocked and non-empty, the operation pops from that cell. If both cells are either locked or empty, the operation picks two new cells and tries again.
     54An 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 and 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 cell 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.
    5655
    5756\begin{figure}
    58         \begin{center}
    59                 \input{base.pstex_t}
    60         \end{center}
    61         \caption{Relaxed FIFO list}
     57        \centering
     58        \input{base.pstex_t}
     59        \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.}
    6260        \label{fig:base}
    63         List at the base of the scheduler: an array of strictly FIFO lists.
    64         The timestamp is in all nodes and cell arrays.
    6561\end{figure}
    6662
    6763\subsection{Finding threads}
    68 Once threads have been distributed onto multiple queues, indentifying which queues are empty and which are not can become a problem. Indeed, if the number of \glspl{thrd} does not far exceed the number of queues, it is probable that several of these 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.
    69 
     64Once 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.
    7065
    7166\begin{figure}
    72         \begin{center}
    73                 \input{empty.pstex_t}
    74         \end{center}
    75         \caption{``More empty'' Relaxed FIFO list}
     67        \centering
     68        \input{empty.pstex_t}
     69        \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.}
    7670        \label{fig:empty}
    77         Emptier state of the queue: the array contains many empty cells, that is strictly FIFO lists containing no elements.
    7871\end{figure}
    7972
    80 This can lead to performance problems since picks that do not yield a \gls{thrd} are not useful and do not necessarily help make more informed guesses.
     73There 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:
    8174
    82 Solutions to this problem can take many forms, but they ultimately all have to encode where the threads are in some form. My results show that 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:
     75\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.
    8376
    8477\begin{figure}
    85         \begin{center}
    86                 {\resizebox{0.73\textwidth}{!}{\input{emptybit.pstex_t}}}
    87         \end{center}
     78        \centering
    8879        \vspace*{-5pt}
    89         \caption{Underloaded queue with added bitmask to indicate which array cells have items.}
     80        {\resizebox{0.75\textwidth}{!}{\input{emptybit.pstex_t}}}
     81        \vspace*{-5pt}
     82        \caption[Underloaded queue with bitmask]{Underloaded queue with bitmask indicating array cells with items.}
    9083        \label{fig:emptybit}
    91         \begin{center}
    92                 {\resizebox{0.73\textwidth}{!}{\input{emptytree.pstex_t}}}
    93         \end{center}
     84
     85        \vspace*{10pt}
     86        {\resizebox{0.75\textwidth}{!}{\input{emptytree.pstex_t}}}
    9487        \vspace*{-5pt}
    95         \caption{Underloaded queue with added binary search tree indicate which array cells have items.}
     88        \caption[Underloaded queue with binary search-tree]{Underloaded queue with binary search-tree indicating array cells with items.}
    9689        \label{fig:emptytree}
    97         \begin{center}
    98                 {\resizebox{0.9\textwidth}{!}{\input{emptytls.pstex_t}}}
    99         \end{center}
     90
     91        \vspace*{10pt}
     92        {\resizebox{0.95\textwidth}{!}{\input{emptytls.pstex_t}}}
    10093        \vspace*{-5pt}
    101         \caption{Underloaded queue with added per processor bitmask to indicate which array cells have items.}
     94        \caption[Underloaded queue with per processor bitmask]{Underloaded queue with per processor bitmask indicating array cells with items.}
    10295        \label{fig:emptytls}
    10396\end{figure}
    10497
    105 \paragraph{Dense Information} Figure~\ref{fig:emptybit} shows a dense bitmask to identify which inner queues are 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 using 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 number of underlying queues in the ready queue 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 will nolonger be constant time. Finally, a dense bitmap, either single or multi-word, causes additional contention problems which 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.
     98\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.
    10699
    107 \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 more contention on the nodes of the tree if the tree is not deep enough.
     100\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.
    108101
    109 \paragraph{Local Information} Figure~\ref{fig:emptytls} shows an approach using dense information, similar to the bitmap, but have each thread keep its own independent copy of it. 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 few tries.
    110 
    111 I built a prototype of these approach and none of these techniques offer satisfying performance when few threads are present. All of these approach hit the same 2 problems. First, blindly picking two sub-queues is very fast which means that any improvement to the hit rate can easily be countered by a slow-down in look-up speed. Second, the array is already as sharded as is needed to avoid contention bottlenecks, so any denser data structure will tend to become a bottleneck. In all cases, these factors meant that the best cases scenerio, many threads, would get worst throughput and the worst case scenario, few threads, would get a better hit rate, but an equivalent throughput. As a result I tried an entirely different approach.
     102I 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 but means any improvement to the hit rate can easily be countered by a slow-down in look-up speed when there are empty lists. Second, the array is already as 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.
    112103
    113104\subsection{Dynamic Entropy}\cit{https://xkcd.com/2318/}
    114 In the worst case scenario there are few \glspl{thrd} ready to run, or more accuratly given $P$ \glspl{proc}, $T$ \glspl{thrd} and $\epsilon$, as usual, a very small number, in this case $\epsilon \ll P$, we have $T = P + \epsilon$. An important thing to note is that in this case, fairness is effectively irrelevant. Indeed, this case is close to \emph{actually matching} the model of the ``Ideal multi-tasking CPU'' presented in this chapter\footnote{For simplicity, this assumes there is a one-to-one match between \glspl{proc} and \glspl{hthrd}.}. Therefore, in this context it is possible to use a purely internal locality based approach and still meet the fairness requirements. This approach would simply have each \gls{proc} running a single \gls{thrd} repeatedly. Or from the shared ready-queue viewpoint, each \gls{proc} would push to a given sub-queue and then pop from the \emph{same} subqueue. Ideally, the scheduler would achieve this without affecting the fairness guarantees in cases where $T \gg P$.
     105In 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 $\epsilon \ll P$, than $T = P + \epsilon$. 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 popes from the \emph{same} subqueue. In cases where $T \gg P$, the scheduler should also achieves similar performance without affecting the fairness guarantees.
    115106
    116 To achieve this, I use a communication channel I have not mentionned yet and which I believe I use in a novel way : the \glsxtrshort{prng}. If the scheduler has a \glsxtrshort{prng} instance per \gls{proc} exclusively used for scheduling, its seed effectively encodes a list of all the accessed subqueues, from the latest to the oldest. The only requirement to achieve this is to be able to ``replay'' the \glsxtrshort{prng} backwards. As it turns out, this is an entirely reasonnable requirement and there already exist \glsxtrshort{prng}s that are fast, compact \emph{and} can be run forward and backwards. Linear congruential generators\cite{wiki:lcg} are an example of \glsxtrshort{prng}s that match these requirements.
     107To handle this case, I use a pseudo random-number generator, \glsxtrshort{prng} in a novel way. When the scheduler uses a \glsxtrshort{prng} instance per \gls{proc} exclusively, the random-number seed effectively starts an encoding that produces a list of all accessed subqueues, from latest to oldest. The novel approach is to be able to ``replay'' the \glsxtrshort{prng} backwards and there exist \glsxtrshort{prng}s that are fast, compact \emph{and} can be run forward and backwards. Linear congruential generators~\cite{wiki:lcg} are an example of \glsxtrshort{prng}s that match these requirements.
    117108
    118 The algorithm works as follows :
     109The algorithm works as follows:
    119110\begin{itemize}
    120111        \item Each \gls{proc} has two \glsxtrshort{prng} instances, $F$ and $B$.
    121         \item Push and Pop operations happen as mentionned in Section~\ref{sec:sharding} with the following exceptions:
     112        \item Push and Pop operations occur as discussed in Section~\ref{sec:sharding} with the following exceptions:
    122113        \begin{itemize}
    123114                \item Push operations use $F$ going forward on each try and on success $F$ is copied into $B$.
    … …  
    126117\end{itemize}
    127118
    128 The main benefit of this technique is that it basically repects the desired properties of Figure~\ref{fig:fair}. When looking for work, \glspl{proc} will look first at the last cells they pushed to, if any, and then move backwards through the cells. As the \glspl{proc} continue looking for work, $F$ moves back 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.
     119The 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.
    129120
    130121\section{Details}
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