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 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 guarantees:
+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.
         \item A performance guarantee: a \at that wants to start or stop running is not prevented by other threads wanting to do the same.
+        \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}
+It is important to note that these guarantees are expected only up to a point.
+\Glspl{at} that are ready to run should not be prevented from doing so, but they still share the limited hardware resources.
+Therefore, the guarantee is considered respected if a \at gets access to a \emph{fair share} of the hardware resources, even if that share is very small.
+Similar to 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.
+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 guarantee is considered achieved.
+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 that are less fair.
         \item Faster than other schedulers that have equal or better fairness.
+        \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: true starvation freedom and ``fast'' load balancing.
+\paragraph{True 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.
+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 in this case.
 Now, this requirement begs the question, what about preemption?
+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 load balancing should happen faster than preemption would normally allow.
+For interactive applications that need to run at 60, 90 or 120 frames per second, \ats having to wait for several milliseconds to run are effectively starved.
+Therefore load-balancing should be done at a faster pace, one that can detect starvation at the microsecond scale.
+With that said, this is a much fuzzier requirement since it depends on the number of \procs, the number of \ats and the general \gls{load} of the system.
+\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}
 …
 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 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.
+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 goals do not necessarily overlap temporally.
+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.
+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}
 …
 \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 the central ones here before looking at the design.
+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}
+The most basic performance challenge of a scheduler is scalability.
+Given a large number of \procs and an even larger number of \ats, scalability measures how fast \procs can enqueue and dequeue \ats.
+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}
 …
 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 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 and the sub-queues are accessed by multiple \glspl{hthrd} without interfering.
+Before going into the design of \CFA's scheduler, it is relevant to discuss two sharding solutions that served as the inspiration scheduler in this thesis.
+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 happens 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: 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.
+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 indefinite starvation.
 Based on these observations, the conclusion is that a \emph{perfect} scheduler should behave similarly to work-stealing in the steady-state case, but load balance proactively when the need arises.
+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 to create a ``relaxed-FIFO'' queue, as in \cite{alistarh2018relaxed}.
+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:
 …
 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.
+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.
 …
 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 suffers.
+It requires wide sharding, \eg at least 4 queues per \gls{hthrd}, and finding non-empty queues is difficult when there are few ready \ats.
+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 good performance with many \ats make the relaxed-FIFO queue a good candidate to form the basis of a new scheduler.
+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.
 …
 As this is the most obvious challenge, it is worth addressing first.
+The obvious solution is to supplement each sharded sub-queue with data that indicates if the queue is empty/nonempty to simplify finding nonempty queues, \ie ready \glspl{at}.
+This sharded data can be organized in different forms, \eg a bitmask or a binary tree that tracks the nonempty sub-queues.
+Specifically, many modern architectures have powerful bitmask manipulation instructions or searching a binary tree has good Big-O complexity.
+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 denser than the sharded sub-queues.
+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.
 …
 \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}.
+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.
+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}
 …
 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 $B$ backwards.
 \item
 When a \proc attempts to enqueue a \at, it runs $F$ forward picking a sub-queue to enqueue to.
 If the enqueue is successful, state $B$ is overwritten with the content of $F$.
+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.
 Tests showed this approach performs better than relaxed-FIFO 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 constant randomness limits how much locality the scheduler offers.
+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.
 …
 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 needs timestamps for each \at to make this possible.
+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 A pool of sub-queues offers the sharding, two per \proc.
+        \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.}
 …
 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 width can be adjusted based on contention.
 Note, as an optimization, the TS 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.
+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 \at is dequeued to provide global fairness.
+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 have some damping effect on helping.
 Random effects like cache misses and preemption can add spurious but short bursts of latency negating the attempt to help.
 These bursts can cause increased migrations and make this work-stealing approach slow down to the level of relaxed-FIFO.
+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}
 …
 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 because the \at at the head of a sub-queue is still waiting, so its wait time has not ended.
+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 choice of the weight for the moving average or the bias is not important, \ie weights and biases of similar \emph{magnitudes} have similar effects.
 With these additions to work stealing, scheduling can be made as fair as the relaxed-FIFO approach, avoiding the majority of unnecessary migrations.
+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, \ie, writes.
 To make things worst, 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.
+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 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 to have a consensus on which \at is the oldest.
+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.
 …
         \centering
         \input{base_ts2.pstex_t}
         \caption[\CFA design with Redundant Timestamps]{\CFA design with Redundant Timestamps \smallskip\newline An array is added containing a copy of the timestamps.
+        \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}
 …
 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.
+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.
 …
 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 a CPU containing multiple hyper threads and cores and multiple CPU sockets, cache misses can be satisfied from the caches on the same (local) CPU, or by a CPU on a different (remote) socket.
+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.
 …
 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 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.
+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 a wide L3 cache, similar to Figure~\ref{fig:cache-share}, and less well on architectures with many narrower L3 cache instances, similar to Figure~\ref{fig:cache-noshare}.
+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.
 …
 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.
+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.}
+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.
 …
 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 worst, the small number of \procs means that few helping attempts are made.
+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} with narrow cache sharing, and a 100:1 bias, it can take multiple seconds for a \at to get dequeued from a remote queue.
+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.
 …
 \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.
+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, each \proc queries the CPU id and the corresponding cache instance.
 Since sub-queues are tied to \procs, each \proc can then update the cache instance mapped to the local sub-queue(s).
+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.

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