# Towards a Cost vs. Quality Sweet Spot for Monitoring Networks

Nofel Yaseen‡⋄, Behnaz Arzani‡, Krishna Chintalapudi‡, Vaishnavi Ranganathan‡,
Felipe Frujeri‡, Kevin Hsieh‡, Daniel Berger‡, Vincent Liu⋄, Srikanth Kandula‡
Microsoft‡ and University of Pennsylvania⋄

(2021)

###### Abstract.

Continuously monitoring a wide variety of performance and fault metrics has
become a crucial part of operating large-scale datacenter networks. In this
work, we ask whether we can reduce the costs to monitor – in terms of
collection, storage and analysis – by judiciously controlling how much and
which measurements we collect. By positing that we can treat almost all
measured signals as sampled time-series, we show that we can use signal
processing techniques such as the Nyquist-Shannon theorem to avoid wasteful
data collection. We show that large savings appear possible by analyzing tens
of popular measurements from a production datacenter network. We also discuss
the technical challenges that must be solved when applying these techniques in
practice.

††copyright: rightsretained††doi: 10.475/123_4††isbn:
978-1-4503-7020-2††conference: The 20th ACM Workshop on Hot Topics in
Networks; November 10-12, 2021; Virtual Event, UK††journalyear: 2021

##  1\. Introduction

Figure 1. The fraction of devices (collection points) at which our production
data center currently measures various metrics above the Nyquist rate; each
bar coalesces information from O($10^{3}$) devices.

High availability guarantees are etched into the service-level agreements of
data centers, and any failure in their adherence results in significant
monetary impact. As a result, data centers deploy large-scale monitoring
systems that continuously monitor various performance metrics to help quickly
identify/predict service disruptions and alleviate them. For the most common
class of monitoring systems, typical operation entails periodically sampling
operational parameters across various components of a data center (e.g., CPU
temperatures, packet drop counts, and path latencies); operators collect and
analyze these measurements in real-time or store them for later analysis.

At the scale of modern data centers, monitoring systems can use a sizable
portion of valuable storage, network bandwidth, and CPU resources. Thus,
efficiency is crucial for production monitoring systems. Furthermore, there is
a fundamental trade-off between monitoring cost and quality– for a large class
of underlying measurements be they interface counters, latency of pings and
traceroutes, or results from sketches, better quality generally requires
greater cost but the actual trade-off remains nebulous.

Today, the typical reaction to this dilemma is ad-hoc and rather conservative.
In our survey of measurements collected from many production monitoring
systems of a large cloud provider, we observe admins typically choose to
collect as much information as possible subject to some (arbitrarily set)
resource constraints. They often cannot answer whether their ad-hoc’ly chosen
measurement rate is adequate for a given metric or whether measuring more (or
less) frequently will lead to better (or no worse) insights? In fact, admins
often express concern that collecting less information could lead to missing
out on important insights.

Information theory provides techniques that help quantify this trade-off. For
instance, the Nyquist rate of a signal is the minimum sampling rate that fully
captures the information in the signal. Some technical conditions apply such
as the signal being bounded in the frequency domain which holds by definition
for many practical metrics; for instance, when measuring the temperature of
CPUs, the underlying thermodynamics limit the maximum rate at which
temperatures change. We discuss other complications in Section 4.

A key contribution of this paper is the somewhat surprising result that we can
treat a wide range of metrics in production data centers as digital signals
and find appropriate measurement frequencies (using nyquist rate) which are
several orders of magnitude smaller than the actual rates at which these
metrics are measured today in production. This indicates we can achieve
similar measurement quality at much smaller monitoring costs. Figure 1 shows
operators are over-sampling a diverse set of metrics at a vast majority of
measurement points (methodology in Section 3.2).

We offer a method that given a metric trace can compute the Nyquist rate; key
issues here are how to cope with measurement noise and quantization effects.
Using this method we quantify the possible gains on many real-world metrics.

We note that computing the appropriate measurement rate aposteriori does not
suffice: to reduce monitoring costs, we must use an appropriate measurement
rate while measuring the metric of interest; also, the choice of measurement
rate must be robust to variations in the metric value such as sudden changes
and phase shifts. To this end we propose a dynamic sampling method which can
adapt to changes in the underlying signal’s Nyquist rate in near real-time and
outline the research questions that arise in using such a system in practice.
We further show preliminary results demonstrating the effectiveness of this
approach by comparing its output with that of existing monitoring systems
(which operate at ad-hoc sampling rates chosen by network operators) in our
technical report (arzani2021DSP).

While signal processing techniques such as compressive sensing and sparse FFT
have been applied before, to our knowledge, we are unaware of any prior work
that applies the Nyquist principle to find appropriate measurement rates for
datacenter metrics. Our initial results are promising and point to large
untapped gains. Finally, we believe that treating datacenter measurements as
signals can lead to fruitful future work since there is a vast literature on
signal processing beyond the Nyquist principle.

Figure 2. Showing the result, in the frequency domain, when sampling above and
below the Nyquist rate. Sampling a signal at frequency $f_{1}$ and
reconstructing it can be thought off, in the frequency domain, as adding
copies of the signal which are $f_{1}$ apart.

##  2\. A Primer on Nyquist–Shannon

Signals are functions of one or more independent variables, with the primary
independent variable in most signals being time (though the theory applies
equally to other variables). We can further divide these signals into two
categories: continuous- and discrete-time signals where the difference is in
the domain of the function. More concretely, continuous-time signals are
functions, $f(t)$, where $t:\mathbb{R}\rightarrow\mathbb{R}$, and discrete-
time signals are functions, $f(T)$, where $T:\mathbb{N}\rightarrow\mathbb{R}$.
The outputs of today’s measurement systems are discrete-time signals. Sampling
a signal converts continuous-time signals into discrete-time signals or down-
samples discrete-time signals to reduce the costs to monitor and store
telemetry.

(a) Original signal

(b) Sampled above Nyquist

(c) Sampled below Nyquist

(d) Sampled far below Nyquist

(e) Original signal

(f) Reconstructed: 3(b)

(g) Reconstructed: 3(c)

(h) Reconstructed: 3(d)

Figure 3.  A signal and its sampled versions in the frequency domain (top
row). The original signal (bottom) is the superposition of two $\sin$ waves at
$400$ and $440$ Hz. The time-domain representations of the sampled versions
are reconstructed and upsampled. Variants shown are: (a) the original signal,
(b) the signal sampled at above the Nyquist rate ($890$ Hz), (c) the signal
sampled at slightly below the Nyquist rate ($800$ Hz), and (d) the signal
sampled at significantly below the Nyquist rate ($600$ Hz). Aliasing is
observable in the frequency domain of (c) and (d). Reconstructing a signal (f)
from the DFT of (d) results in a distorted result.

The Fourier transform. One important property of signals is that they can be
converted from functions over time into the frequency-domain (i.e., as
mathematical functions of frequency). These conversions from the time to
frequency domain are lossless: we can recover the time-domain representation
of the signal from its frequency domain representation and vice-versa. The
procedure to accomplish this translation is the Fourier transform, which
produces a function whose magnitude at frequency $f$ is the amount of that
frequency present in the original function. An Inverse Fourier Transform
converts the signal back to the time-domain.

The Fast Fourier Transform (FFT) is an efficient algorithm that applies over
discrete sampled signals. Given an input $N$ which can either be specified by
the user or determined by the length of the signal, the FFT divides the
frequency space from 0 to the maximum frequency in the signal into $N$
discrete bins and computes the signal power in each bin. Bin $j$ is computed
as $j*(\text{sampling frequency}/N)$. The square of these per-bin magnitudes
is called the the Power Spectral Density (PSD); we will use this later in
Section 3.2 to analyze the Nyquist rate for deployed monitoring systems.

Nyquist rates and Fourier transforms in practice. The Nyquist–Shannon theorem
states: if a function $x(t)$ contains no frequencies higher than $f_{0}$,
then, sampling it at a rate at or above $2f_{0}$ ensures the original signal
can be recovered completely; we call $2f_{0}$ the Nyquist rate of $x(t)$
(Oppenheim and Schafer, 1975).

Figure 2 shows the effects of sampling at a rate above or below the Nyquist
rate. When the sampling frequency is below the nyquist rate, as we see in the
bottom half of Figure 2, _aliasing_ occurs which distorts the PSD and prevents
recovering the original signal. Figure 3 demonstrates these effects on an
example signal.

The implication of this theorem in practice is that when a monitored signal is
sampled at or above the Nyquist rate of that signal, then operators can rest
assured no information is being lost due to sampling. On the other hand, when
aliasing occurs, the extent of the information loss depends on the difference
between the PSD of the aliased signal and that of the original. The impact of
information loss depends on the application using the measurements and
operators must determine what level of aliasing (if any) is acceptable.

(a) 5-pct CPU util

(b) FCS errors

(c) In-bound dis

(d) Link util

(e) Lossy paths

(f) Memory usage

(g) Multicast bytes

(h) Multicast drops

(i) Peak egress BW

(j) Peak ingress BW

(k) Temperature

(l) Unicast bytes

Figure 4. CDFs of the ratio between the actual sampling rate and the computed
Nyquist rate. Note x axes is in log scale and $x=10$ indicates $10\times$
over-sampling. Each datapoint is one day’s worth of data from a distinct
device. We do not show the cases where we cannot reliably detect the Nyquist
rate.

##  3\. Quantifying the Opportunity

Through this new lens of viewing existing network monitoring systems as
collecting (sensing) signals, we can optimize many existing monitoring
systems. While we may need to address many challenges before realizing these
techniques in practice (see Section 4), as a first step, we answer the
question: what do we stand to gain if we use them?

To answer this question, we examine the network of a major cloud provider and
the monitoring systems it has deployed. We describe why many of the existing
monitoring systems this provider (and others similar to it) have deployed are
sub-optimal. For this discussion, we will restrict our focus to monitoring
systems that periodically poll a numeric metric. Later, we introduce
opportunities for taking these observations further and applying them to other
types of monitoring systems our community has developed.

###  3.1. Today’s Monitoring Systems

Every aspect of the task of monitoring—collection, transmission, analysis, and
storage—all consume resources that, when considering the scale of modern data
centers, represent a non-negligible overhead. In an effort to reduce the costs
of monitoring, for non-event-based data, today’s systems generally sample
their target metrics periodically/randomly so as to capture the gist of the
metric without sacrificing too many resources. Examples include systems that
periodically poll switch counters (Hare, 2011; Zhang et al., 2017; Yaseen et
al., 2018), sample packets to construct flow records (Phaal et al., 2001;
Claise et al., 2004), or send packets through the network to extract the
instantaneous latency of the network (Guo et al., 2015; Tan et al., 2021).

The degree of sampling for the majority of these systems is entirely arbitrary
(Zhang et al., 2017; Arzani et al., 2020). Often, the sampling rates are not
governed by signal processing principles but chosen based on defaults and
vague ‘gut feelings’ about the desired granularity of the data and the system
and network-level overheads (e.g., in PrivateEye (Arzani et al., 2020) this is
the cost of locking the virtual switch’s flow-table and the volume of data
that would have to be aggregated at the central collector, or in (Guo et al.,
2015) it’s the network overhead of sending too many pings). For such
monitoring systems aliasing effects are never evaluated and the chosen
granularities are never re-considered.

The end result is that most of these systems are either (a) over-sampling
(increasing overheads and wasting resources) or (b) under-sampling without any
idea of how much information is lost. This has significant implications on the
resource usage of existing monitoring systems.

###  3.2. Case Study: A Large Cloud Provider

We demonstrate empirically the opportunity afforded by applying the Nyquist-
Shannon theorem. The monitoring systems we study encompass a wide range of
monitoring systems in a large cloud provider including device temperature,
packet drops, FCS errors (Zhuo et al., 2017), and link utilization.

Our goal is to identify the Nyquist rate of each of the signals these systems
monitor. To do so: (a) for a given trace (where each trace is the data for a
metric being measured on a single device), we compute the FFT and compute the
total energy in the signal—the sum of the PSD across all FFT bins; (b) we add
the PSD components in each FFT bin until we reach $99\%$ of the total energy
in the signal computed in (a). If we need all bins of the FFT to achieve
$99\%$ of the total energy we conclude the signal is probably already aliased
and record -1 as the Nyquist rate; (c) otherwise, we report twice the
frequency at which we capture $99\%$ of the total energy of the signal as the
Nyquist rate.

The approach above can uncover the Nyquist rate of the underlying signal if
the measured trace is sampled above its Nyquist rate. We find this to be the
common case in our traces. The converse case is challenging because when
aliasing has already happened in a measured signal (e.g., the bottom right
portion of Figure 2), the method above does not reliably produce the Nyquist
rate. We discuss how to detect aliasing (under-sampling) in Section 4. Our
choice of the $99\%$ cut-off on total energy is a workaround to compensate for
measurement noise. Using a higher parameter value such as $99.99\%$ would
increase our estimate of the Nyquist rate and reduce performance gains but, in
our experience, does not necessarily lead to a lower reconstruction error
since the delta that is being captured is often just the noise.

In practice, monitoring systems do not produce perfectly sampled
signals—samples are not always spaced at equi-distant points in time. In such
situations, we pre-clean the signal using nearest neighbor re-sampling
(Thévenaz et al., 2000, 2000); that is, we add values for missing samples
based on nearby samples.

Figure 5. A box plot of the Nyquist rate of each monitoring system.

Figure 4 reports for each measured statistic the CDF of the ratio between the
current sampling and the Nyquist rate we identified through the above
approach. The ratio indicates the degree to which we are currently
oversampling the underlying signal. We observe that in $20\%$ of the examples
the sampling rate can be reduced by a factor of $1000\times$.

In total, we studied $1613$ metric and device pairs ($14$ distinct metrics).
Of these, $89\%$ were sampling at higher than their Nyquist rate. Within a
metric, the Nyquist rate varies widely across devices (Figure 5). For example,
for the temperature signal, the Nyquist rate ranges from $7.99\times 10^{-7}$
Hz to $0.003$ Hz across the monitored devices. We also notice different
Nyquist rate at different time periods on the same device. This indicates that
the properties of the underlying metric vary over time and across devices and
so dynamically adapting the measurement rate would be useful. Furthermore,
adaptation must be quick because under-sampling would lead to aliasing and
information loss. We discuss some relevant prior work (Penny et al., 2003a)
and propose a dynamic sampling method in Section 4; Figure 6 shows an example
of how the proposed approach would work in practice.

Finally, in our experiments the existing sampling rate is below the Nyquist
rate of the underlying signal in about $11\%$ of the metric-device pairs.
These cases require more careful inspection; it is possible that existing
monitoring systems must increase their sampling rates in order to fully
capture these metrics but we want to also rule out issues such as measurement
noise, data loss or data corruption that may have lead to an incorrect
assessment of the Nyquist rate. We defer doing so to our future work.

##  4\. Toward Dynamic Monitoring

In the previous section, we showed that there is a significant opportunity to
save resources in deployed monitoring solutions by analyzing the signals
produced by those systems. In some cases, the actual measurement may be
inexpensive relative to the cost to store the metric or the cost of downstream
analysis; in such cases, we can use the above techniques a posteriori, i.e.,
measure at a high rate, compute the nyquist rate over the measurements and
store or present for later analysis only the measurements that are re-sampled
at the lower nyquist rate. More generally, acquiring the measurements may
itself be a large expense and we discuss a new dynamic sampling method below
to address this case. The dynamic sampling method also applies when the
Nyquist rate of a metric varies across devices or across time (recall the
cases that we saw in Section 3); we do not fully understand the reasons for
these changes but nevertheless use dynamic sampling to ensure robustness to
changes to the nyquist rate of the underlying signal.

A complete solution to these challenges necessitates further exploration;
however, the signal processing field again provides some initial avenues of
exploration. We outline potential components to a solution based on those
principles.

###  4.1. Detecting aliasing

One critical component to a solution is a method to detect when a chosen
sampling rate has dipped below the Nyquist rate of the underlying signal. For
this, prior work has previously proposed possible solutions, e.g., Penny et
al. (Penny et al., 2003b). In that work, the authors propose to sample at two
distinct frequencies, $f_{1}$ and $f_{2}$, where $f_{1}>f_{2}$ and
$\frac{f_{1}}{f_{2}}$ is not an integer. If aliasing occurs, i.e., the
underlying signal has frequency terms that are larger than $\frac{f_{2}}{2}$,
then comparing the discrete fourier transforms of the two sampled signals
would show discrepancies; for example, frequencies below $\frac{f_{2}}{2}$
will match in both spectra but the higher frequencies will not match.111this
requires $f_{2}$ to not be a factor of $f_{1}$ as conditioned above. A
complicating aspect here is noise at higher frequencies, but noise especially
of a small amplitude can be filtered using standard techniques (TODO).

Collecting samples at two frequencies roughly doubles measurement cost but we
still expect sizable net benefit since, as we saw in Section 3 current
systems, over-sample by well over $2\times$. Furthermore, after checking for
aliasing, we can discard excess measurements by resampling at the identified
nyquist rate. We believe that further improvements are possible for example by
using an aliasing detector that is specific to the actual frequencies and
changes that appear in datacenter measurements.

###  4.2. Adapting the sampling rate

Upon detecting aliasing, we increase the sampling rate. There are several
possible approaches to manage the rate adaptation process and the choice among
these depends on the properties of the signal (e.g., whether the high-
frequency changes are one-off occurrences or sustained shifts) and the
requirements for the measurement (e.g., how important is it to capture every
spike in Nyquist frequency and how costly is it to oversample).

Consider the problem of quantifying link failures by sampling frame checksum
(Zhuo et al., 2017) errors. Initially, we do not know the Nyquist rate of the
underlying signal and so we must probe, i.e., multiplicatively increase the
measurement rate along witht he method in Section 4.1, i.e., measuring at two
different frequencies to detect aliasing. While aliasing persists, we remain
in probe mode. Once we no longer detect aliasing, we use the method in in
Section 3.2 which will successfully identify the Nyquist rate of the signal.

We can add memory into the system and/or leverage temporal stability to make
adaptation faster and less expensive. If the frequency increases exhibit
temporal locality (as it does in, e.g., fail-stop or link-flap scenarios), we
can optimize the system by also adaptively decreasing the sampling rate if we
observe the Nyquist rate returning to a lower value. We can even ‘remember’
previous maximum Nyquist rates to ramp up more quickly in the future.
Similarly, we may be able to learn information about applications’ Nyquist
shift distributions from other (oversampled) datasets from the same
application. Again, the optimal strategy will vary depending on the properties
of the signal.

Perhaps the most challenging scenario for such a system is how the system
should handle a first-of-its-kind event. Maintaining ample headroom may be
helpful in these cases (many of the deployed systems we examine in Section 3
are already sampling at rates well above the Nyquist rate). Additional
mitigations are a subject of future work. We note, however, the most critical
issues in modern systems are those that recur—after all, ongoing fires are
typically a higher priority for operators to debug than events that occur only
once and never again. An adaptive strategy will eventually be able to detect
this more severe category of issues.

###  4.3. Quantization Noise and Reconstruction

In practice, measurement readings are quantized. For example, a temperature
sensor may emit readings that are rounded to the nearest integer. Such
quantization adds noise which in the frequency domain appears at higher
frequencies; the larger the quanta relative to the range of values that a
signal can take, the higher the noise level. Quantization noise impacts our
techniques in a few different ways: (a) identifying the true Nyquist rate of a
signal becomes more challenging; and (b) upsampling and recovering the signal
after it is downsampled. For (a) we use the thresholding approach proposed in
Section 3.2 so as to discard higher-order frequencies introduced by
quantization. For (b) we can add the same quantization in order to recover the
signal more accurately. However, in such cases the signal is no longer
“perfectly recoverable” and the recovered signal may be slightly different
from the original. Figure 6 shows the effectiveness of this approach in the
context of the temperature signal.

Using the Nyquist principle and the adaptive sampling approach described
above, we are able to reduce the overhead on the monitoring system. However,
in order to reconstruct the signal, operators would have to pass the signal
through a low-pass filter (for example, by taking an FFT of the sampled
signal, setting all frequency components above $f_{0}$ to $0$ and then taking
the IFFT). This reconstruction takes time and may not be acceptable to
applications that expect low-latency. However, in many cases this
reconstruction cost is acceptable. For instance, in machine learning models
where some delay in recovering the data in return for higher-fidelity is
desired.

Figure 6. Comparing an actual temperature signal in blue (sampled every $5$
minutes) with the signal in red that was downsampled to the nyquist rate and
then upsampled back again just for the purpose of comparison. The L2 distance
between these signals is $0$. Here, we used the method in Section 4.2 to
dynamically adapt the sampling rate; the inferred nyquist rates are shown as
well in Figure 7. Figure 7. The inferred Nyquist rates over time for the
signal depicted in Figure 6. The timestamps mark the beginning of the moving
window. We use a step of 5 minutes for the moving window and a window size of
6 hours.

##  5\. Related Work

In this paper, we proposed a different perspective on how we approach network
monitoring. The research we propose builds on top of the vast array of
knowledge in the field of signal processing and relates to the following
categories of work in signal processing and network monitoring:

Signal processing theory and its applications to networking. The
Nyquist–Shannon sampling theorem is ubiquitous and has been applied in several
other domains (Farrow et al., 2011; Romo-Cárdenas et al., 2018; Zamaruiev,
2017) and been expanded in its applicability (Song et al., 2012). Prior work
has also focused on how to detect aliasing in practice (Penny et al., 2003a).
We build on all of this work.

In network monitoring, the most closely related work are the existing, but
isolated examples of signal processing techniques in systems and networks.
These include techniques like compressive sensing, which shares both
motivation and intuition with our work. In (Huang et al., 2021), for example,
authors regard flow statistics as signals to design new sketch algorithms that
bound the information loss of the system. Similarly, in (Zhang et al., 2005)
the authors use FFTs to identify network faults. Others, characterize the
underlying properties of network traffic efficiently and accurately (Thompson
et al., 1997; Owezarski and Larrieu, 2004; Demaine et al., 2002; Benson et
al., 2010; Stoev et al., 2005). Stoev et al. (Stoev et al., 2005) developed
analyses to better characterize the properties of Internet traffic in the
presence of high-frequency oscillations and non-stationarities. Finally,
(Demaine et al., 2002) presents an algorithm that can track categories of
traffic with frequencies above a certain value with high probability.

These examples are complementary to our work, and they serve as examples of
the potential of viewing the data center as a collection of digital signals.

Network monitoring systems. More generally, many others have also tried to
address the problem of scalability/overhead in both the collection and storage
of network measurement. Sketches like the one cited above (Huang et al., 2021)
and (Ivkin et al., 2019; Moshref et al., 2015) are one approach that
researchers have used to do so. Sketches reduce the space required for storing
data plane statistics, but those statistics must still be sampled when
summarized for users or stored for later analysis.

There is also a vast body of various network monitoring systems, from simple,
periodic ones like (Guo et al., 2015; Zhuo et al., 2017), to those that are
further optimized (Moshref et al., 2014; Tammana et al., 2016; Holterbach et
al., 2019; Moshref et al., 2016). It may be possible to improve the efficiency
of these systems as well by re-framing the monitored information as time-
varying signal and bringing to bear the vast theory of signal processing as
already demonstrated by (Huang et al., 2021).

Finally, many network protocols also measure (most prominently TCP) and
continuously monitor the network. Here too we may be able to improve
efficiency through our new lens of viewing the information they monitor (in
the case of TCP this would be the network round trip time, and the bottleneck
bandwidth) as a continuous or discrete-time signals.

In summary, this paper advocates for a broader re-examination of modern data
center systems with an eye toward directly improving the efficiency of those
systems by bringing to bear the vast theory present in signal processing. More
specifically, we pose the significant gains possible by bringing to bear the
rather simply stated (but more nuanced in practice) Nyquist-Shannon sampling
theorem.

##  6\. Conclusion and Future Work

Monitoring systems are a crucial part of ensuring high availability for data
centers. Yet, efficient measurements based on information theoretic concepts
are not widely studied. We argue that the next stage of network monitoring
should go beyond vague ‘gut feelings’ about the granularity of data and
instead leverage concepts from signal processing to help avoid wasteful
collection.

We show how Nyquist-Shannon might provide guidance to optimize existing
monitoring systems and propose a straw-man approach for adaptive sampling.

Beyond numbers. In our experiments and discussions above we focused on
monitoring systems that already fit nicely into the Nyquist model of sampling:
they periodically sample a numeric metric at a fixed frequency. Further
research is needed to apply the same techniques to other more complex systems
such as (Moshref et al., 2014, 2015, 2016; Huang et al., 2021; Yaseen et al.,
2020), which may measure events, sets of metrics, or text representations. We
defer solving this problem to future work.

Multivariate signals. Many applications may monitor and use multiple different
signals. The correlation and joint distribution of these signals may be
important to such applications. As long as we sample each individual signal at
a rate higher than its Nyquist rate, we can recover the original signal and
preserve any correlations. However, if the Nyquist rate changes frequently and
the system attempts to use the dynamic increase/decrease approach of Section
4, the algorithm we presented may encounter pathological cases. For example,
even the sampling of every individual signal converges to the Nyquist rate,
the global optimal is not guaranteed.

Luckily, the signal processing community has worked on extending the Nyquist
theorem to multivariate signals (Nathan, 1978; Freeden and Nashed, 2018).
Incorporating these extensions into a practical system is left to future work.

Beyond Nyquist. Although we focused on the Nyquist–Shannon theorem in this
paper and how it can help improve the efficiency of deployed monitoring
systems, it is only one example of a technique from the realm of digital
signal processing that can be applied to data centers.

Another is ergodicity. In statistical mechanics, an ergotic process is one in
which the statistical properties of a sufficiently long random sample of the
process are equivalent to the properties of a random sample of a statistical
ensemble of the process. As an example, consider a system that monitors the
CPU utilization of all servers in a data center. Samples from the system are
ergotic if the statistical properties of a set of samples derived from a
single CPU over a sufficiently long sequence of time are equivalent to those
of a set of samples derived from measuring the entire fleet at once.

Operators often assume ergodicity implicitly. One example is the practice of
canarying, where an update is rolled out to a handful of
servers/racks/switches to evaluate its effects before deploying the update
more broadly. Extrapolating canary results to other devices relies on
ergodicity. Does this assumption hold in practice? How long of an observation
period is required for the assumption to hold true? Is there a way to leverage
ergodicity to reduce the number of devices that we need to sample or the time
for which we need to sample them? The literature here is equally rich and
should be explored.

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# Pursuit of $CP$ violation in hyperon decays at $e^{+}e^{-}$ colliders

Xiao-Gang He<EMAIL_ADDRESS>Tsung-Dao Lee Institute, KLPAC and SKLPPC
laboratories, School of Physics and Astronomy, Shanghai Jiao Tong University,
Shanghai 200240, China
Department of Physics, National Taiwan University, Taipei 10617, Taiwan  Jusak
Tandean<EMAIL_ADDRESS>Gayungsari III/20, Surabaya 60235, Indonesia
German Valencia<EMAIL_ADDRESS>School of Physics and Astronomy,
Monash University, Melbourne VIC-3800, Australia

The phenomenon of $CP$ violation—the breaking of the combination of charge-
conjugation symmetry ($C$) and parity symmetry ($P$)—is one of the least
understood aspects of high-energy physics. The first evidence for $CP$
violation was reported in 1964 by a group led by Cronin and Fitch [1] after
studying neutral-kaon decays. It was later recognized in 1967 by Sakharov [2]
that $CP$ violation is an important ingredient for resolving the mystery of
why our Universe is dominated by matter over antimatter if they were created
in equal quantities by the Big Bang. The quest for understanding $CP$
violation has since drawn a great deal of attention from both the particle
physics and cosmology communities. In 1973 Kobayashi and Maskawa [3] showed
that $CP$ violation would emerge naturally in the presence of three families
of quarks, establishing its mechanism in the standard model (SM) of particle
physics.

After the aforesaid historical discovery of $CP$ violation, numerous different
searches for it were carried out. Some of them yielded positive results,
empirically confirming its existence not only in the kaon sector but also in
the beauty-meson and charmed-meson systems, and all of the observations to
date are compatible with the SM expectations [4]. Despite its successes,
however, it is widely believed that the SM falls short of being the complete
theory of fundamental interactions. One of the reasons is that the amount of
$CP$ violation that the SM could generate is thought to be inadequate if it is
to be solely responsible for the matter-antimatter imbalance of the Universe
[5]. This suggests the necessity of continuing to hunt for signals of $CP$
violation in many other processes, the outcomes of which could further check
its SM explanation and, moreover, might help pin down potential sources of
$CP$ violation from beyond the SM.

Outside the meson sectors mentioned above, baryon systems have not yet
supplied firm indications of $CP$ violation, but there are ongoing and
forthcoming efforts to change the situation, a few of which will be discussed
shortly. Having odd half-integer spins, baryons can offer environments where
additional tests of $CP$-violation may be available besides the traditional
particle-antiparticle decay-rate differences and mixing-related observables.

Hereafter we focus on the lightest spin-1/2 hyperons (strange-flavored
baryons), specifically $\Lambda$, $\Sigma$, and $\Xi$. Their nonleptonic two-
body decays could provide several observables potentially sensitive to
$CP$-violating effects, as pointed out decades ago [6, 7, 8, 9, 10, 11, 12,
13, 14]. The $\Omega^{-}$ hyperon, which has spin 3/2, could play an analogous
role [15, 16].

Early experimental endeavors to realize this were performed from the 1980s
through 2010 by a number of groups [4, 17, 18, 19, 20, 21] investigating the
decays of $\Lambda$ and $\Xi^{-}$ as well as $\Omega^{-}$. Most recently there
has been a renewal of similar attempts at the Beijing Electron-Positron
Collider II by the BESIII Collaboration [22, 23, 24, 25, 26, 27]. In its clean
high-intensity environment, a charmonium resonance can form in $e^{+}e^{-}$
annihilation and subsequently convert into a quantum-entangled pair of hyperon
and antihyperon which decay nonleptonically, allowing for the simultaneous
determination of various parameters pertaining to $CP$ violation in the
decays. Its predecessors, DM2 [18] and BES [21], utilized $e^{+}e^{-}$
collisions as well but had accumulated datasets which were much smaller than
that of BESIII. Although all of the foregoing searches have so far come up
empty, there remains open a large window into possible new physics beyond the
SM to be explored by BESIII in its current program. Further in the future, the
quests are anticipated to be taken up by the proposed super tau-charm
factories [28, 29], with total luminosities increased by two orders of
magnitude, which might enable them to attain SM-level sensitivity.

It is worth remarking that one could also seek $CP$-violation signs in the
concurrent decays of a hyperon and antihyperon produced pairwise in proton-
antiproton scattering [12, 14]. This was tried in the past [4, 17], but with
limited statistics, and will be undertaken again by the upcoming PANDA
experiment in Germany [30].

In the main processes of interest here, a spin-1/2 hyperon, ${\cal B}_{i}$,
turns into another spin-1/2 baryon, ${\cal B}_{f}$, and a pion, $\pi$, causing
the strangeness quantum number to change by one unit. This is described by an
amplitude expressible as ${\cal
M}=\chi_{f}^{\dagger}\big{(}S+\bm{\sigma}\cdot\hat{\textbf{{p}}}_{\\!f}P\big{)}\chi_{i}$,
where $S$ and $P$ are generally complex constants corresponding to parity-
violating S-wave and parity-conserving P-wave transitions, respectively,
$\bm{\sigma}$ and $\chi_{f},\chi_{i}$ represent Pauli matrices and spinors,
and $\hat{\textbf{{p}}}_{\\!f}$ is the momentum unit-vector of ${\cal B}_{f}$.
In the rest frame of ${\cal B}_{i}$, if it has a polarization
$\textbf{{P}}_{\\!i}$, the angular distribution of ${\cal B}_{f}$ is given by

$\displaystyle\frac{d\Gamma_{{\cal B}_{i}\to{\cal B}_{f}\pi}}{d\Omega_{f}}$
$\displaystyle=\frac{\Gamma_{{\cal B}_{i}\to{\cal
B}_{f}\pi}}{4\pi}\big{(}1+\alpha\,\textbf{{P}}\\!_{i}\cdot\hat{\textbf{{p}}}_{\\!f}\big{)}$
(1)

and its polarization

$\textbf{{P}}\\!_{f}=\frac{\big{(}\alpha+\textbf{{P}}_{\\!i}\\!\cdot\\!\hat{\textbf{{p}}}_{\\!f}\big{)}\hat{\textbf{{p}}}_{\\!f}+\beta\textbf{{P}}\\!_{i}\\!\times\\!\hat{\textbf{{p}}}_{\\!f}+\gamma\,\hat{\textbf{{p}}}_{\\!f}\\!\times\\!\big{(}\textbf{{P}}\\!_{i}\\!\times\\!\hat{\textbf{{p}}}_{\\!f}\big{)}}{1+\alpha\,\textbf{{P}}\\!_{i}\cdot\hat{\textbf{{p}}}_{\\!f}},$
(2)

where $\Gamma_{{\cal B}_{i}\to{\cal B}_{f}\pi}$ denotes the partial rate,
$\Omega_{f}$ is the solid angle of ${\cal B}_{f}$, and the parameters
$\alpha$, $\beta$, and $\gamma$ depend on $S$ and $P$ according to

$\alpha=\frac{2\,{\rm Re}(S^{*}P)}{|S|^{2}+|P|^{2}}\,,~{}\beta=\frac{2\,{\rm
Im}(S^{*}P)}{|S|^{2}+|P|^{2}}\,,~{}\gamma=\frac{|S|^{2}-|P|^{2}}{|S|^{2}+|P|^{2}}$
(3)

and satisfy $\alpha^{2}+\beta^{2}+\gamma^{2}=1$. Another parameter $\phi$,
which is linked to the second and third by $\beta=\sqrt{1-\alpha^{2}}\sin\phi$
and $\gamma=\sqrt{1-\alpha^{2}}\cos\phi$, and $\alpha$ are the most closely
connected to experiment and essentially uncorrelated [4]. For the antiparticle
process $\overline{{\cal B}_{i}}\to\overline{{\cal B}_{f}}\overline{\pi}$, the
amplitude has the form $\overline{\cal
M}=\chi_{f}^{\dagger}\big{(}\overline{S}+\bm{\sigma}\cdot\hat{\textbf{{p}}}_{\\!f}\overline{P}\big{)}\chi_{i}$
and the partial rate, $\overline{\alpha}$, $\overline{\beta}$, etc. are
related to $\overline{S}$ and $\overline{P}$ in the same way as $\Gamma_{{\cal
B}_{i}\to{\cal B}_{f}\pi}$, $\alpha$, $\beta$, etc. to $S$ and $P$.

Before addressing the $CP$ tests of interest, we write in the customary
parameterization

$\displaystyle S$
$\displaystyle=\raisebox{2.0pt}{\footnotesize$\displaystyle\sum_{r}$}\,S_{r}\,e^{i(\delta_{S}^{r}+\xi_{S}^{r})}\,,$
$\displaystyle\overline{S}$
$\displaystyle=-\raisebox{2.0pt}{\footnotesize$\displaystyle\sum_{r}$}\,S_{r}\,e^{i(\delta_{S}^{r}-\xi_{S}^{r})}\,,$
$\displaystyle P$
$\displaystyle=\raisebox{2.0pt}{\footnotesize$\displaystyle\sum_{r}$}\,P_{r}\,e^{i(\delta_{P}^{r}+\xi_{P}^{r})}\,,$
$\displaystyle\overline{P}$
$\displaystyle=\raisebox{2.0pt}{\footnotesize$\displaystyle\sum_{r}$}\,P_{r}\,e^{i(\delta_{P}^{r}-\xi_{P}^{r})}\,,$
(4)

where the index $r$ in the sums runs over all possible amplitudes for
different final-isospin states and changes $\Delta I$ in isospin, $S_{r}$ and
$P_{r}$ are real numbers, $\delta_{S}^{r}$ and $\delta_{P}^{r}$ designate the
$CP$-conserving phase-shifts arising from strong rescattering of the final
states, and $\xi_{S}^{r}$ and $\xi_{P}^{r}$ stand for weak phases encoding the
$CP$ violation. In eq. (Pursuit of $CP$ violation in hyperon decays at
$e^{+}e^{-}$ colliders) the extra minus sign of $\overline{S}$ relative to
$\overline{P}$ and the flipped signs of $\xi_{S}^{r}$ and $\xi_{P}^{r}$ in
them compared to $S$ and $P$ are attributable to the assumed invariance of the
interaction Hamiltonian under the combined $CP$ and time-reversal
transformation and to the application of unitarity. If $CP$ symmetry holds,
$\xi_{S}^{r}=\xi_{P}^{r}=0$ in eq. (Pursuit of $CP$ violation in hyperon
decays at $e^{+}e^{-}$ colliders), which implies that
$\alpha=-\overline{\alpha}$ and $\beta=-\overline{\beta}$. It follows that

$\displaystyle A_{CP}$
$\displaystyle=\frac{\alpha+\overline{\alpha}}{\alpha-\overline{\alpha}}\,,$
$\displaystyle B_{CP}$
$\displaystyle=\frac{\beta+\overline{\beta}}{\alpha-\overline{\alpha}}$ (5)

are tests of $CP$ violation. Each of them would probe, at the same time, the
parity-even (P-wave) and parity-odd (S-wave) parts of the responsible
interactions. Therefore, its measurements could supply valuable information
which is complementary to that gained from experimental studies on the so-
called indirect and direct types of $CP$ violation in the kaon system, which
originate from the parity-even and -odd components, respectively, of the
underlying physics.

For ${\cal B}_{i}=\Lambda$ or $\Xi$, it is known from data that the term
corresponding to $\Delta I=1/2$ dominates each of $S$, $P$, $\overline{S}$,
and $\overline{P}$ in eq. (Pursuit of $CP$ violation in hyperon decays at
$e^{+}e^{-}$ colliders). Accordingly, the leading expression for $A_{CP}$ is
[11]

$\displaystyle A_{CP}$
$\displaystyle=-\tan(\delta_{P}-\delta_{S})\,\tan(\xi_{P}-\xi_{S})\,,$ (6)

where the $r$ superscript has been dropped for visual simplicity. The majority
of past searches for hyperon $CP$-violation targeted this asymmetry [4].

It is clear from eq. (6) that to make $A_{CP}$ nonzero requires both
$\delta_{P}\neq\delta_{S}$ and $\xi_{P}\neq\xi_{S}$ and that a small phase-
shift difference, $\delta_{P}-\delta_{S}$, will bring about a suppressive
factor in $A_{CP}$ which reduces its sensitivity to $CP$-violating effects.
Moreover, extracting a precise value of $\xi_{P}-\xi_{S}$ from it will be
unattainable if the phase shifts are poorly known. An observable that is not
subject to these issues is $B_{CP}$ in eq. (5) which at leading order becomes
[10, 14]

$\displaystyle B_{CP}$ $\displaystyle=\tan(\xi_{P}-\xi_{S})\,,$ (7)

independent of $\delta_{P,S}$. As a consequence, $B_{CP}$ would exceed
$A_{CP}$ in size by up to an order of magnitude [10]. There is, however, a
limitation to pursuing $B_{CP}$, namely that $\beta$ is more challenging to
measure than $\alpha$ if one relies on a single-step transition ${\cal
B}_{i}\to{\cal B}_{f}\pi$, as this would have to involve a dedicated detector
for the polarization of ${\cal B}_{f}$. We note that the usual partial-rate
asymmetry is even harder to reach than $A_{CP}$ due to additional suppression
by small $\Delta I=3/2$ contributions.

As it turns out, it is possible to derive simultaneously not only $\alpha$ and
$\phi$ of ${\cal B}_{i}\to{\cal B}_{f}\pi$ but also $\overline{\alpha}$ and
$\overline{\phi}$ of $\overline{{\cal B}_{i}}\to\overline{{\cal
B}_{f}}\,\overline{\pi}$ if ${\cal B}_{i}$ and $\overline{{\cal B}_{i}}$ are
pair-produced in the decay of a charmonium (e.g., the $J/\psi$ meson) created
by $e^{+}e^{-}$ annihilation and each undergo a nonleptonic two-step weak
decay. After translating the findings into $\beta$ and $\overline{\beta}$, the
value of $B_{CP}$ can be obtained. This has recently been achieved by BESIII
[25], upon implementing a recently proposed method [31, 32] to the reaction
$e^{-}e^{+}\to J/\psi\to\Xi^{-}\overline{\Xi}{}^{+}$ and the ensuing
sequential transitions $\Xi^{-}\to\Lambda\pi^{-}\to p\pi^{-}\pi^{-}$ and
$\overline{\Xi}{}^{+}\to\overline{\Lambda}\pi^{+}\to\overline{p}\pi^{+}\pi^{+}$.
The acquired data led to the determination of $\alpha_{\Xi}$ and $\phi_{\Xi}$
for $\Xi^{-}\to\Lambda\pi^{-}$ and $\overline{\alpha}{}_{\Xi}$ and
$\overline{\phi}{}_{\Xi}$ for
$\overline{\Xi}{}^{+}\to\overline{\Lambda}\pi^{+}$, as well as
$\alpha_{\Lambda}$ for $\Lambda\to p\pi^{-}$ and
$\overline{\alpha}{}_{\Lambda}$ for
$\overline{\Lambda}\to\overline{p}\pi^{+}$. This allowed BESIII to report the
$CP$ asymmetries [25]

$\displaystyle A_{CP}^{\Xi}$ $\displaystyle=(6\pm 13\pm 6)\times 10^{-3}\,,$
$\displaystyle B_{CP}^{\Xi}$
$\displaystyle\simeq\,\xi_{P}^{\Xi}-\xi_{S}^{\Xi}=(1.2\pm 3.4\pm 0.8)\times
10^{-2}\,,~{}~{}~{}$ $\displaystyle A_{CP}^{\Lambda}$ $\displaystyle=(-4\pm
12\pm 9)\times 10^{-3}$ (8)

where the first and second uncertainties are statistical and systematic,
respectively, and the superscript $\Xi$ ($\Lambda$) refers to the
$\Xi\to\Lambda\pi$ ($\Lambda\to p\pi$) case. The $CP$-conserving phase-shifts
$(\delta_{P}-\delta_{S})$ in the $\Xi$ modes were also derived, but again with
sizable uncertainties [25]. It is worth mentioning that the $A_{CP}^{\Xi}$ and
$B_{CP}^{\Xi}$ measurements had never been done before and that the
$A_{CP}^{\Lambda}$ result agrees, and is compatible in precision, with the
best earlier measurement [23]. Very recently BESIII [26] has announced the
improved value $A_{CP}^{\Lambda}=(-2.5\pm 4.6\pm 1.1)\times 10^{-3}$, which is
the most precise to date, after directly examining $e^{-}e^{+}\to
J/\psi\to\Lambda\overline{\Lambda}\to p\pi^{-}\overline{p}\pi^{+}$.

It is of interest to compare these BESIII findings to the expectations within
and beyond the SM. Currently the theoretical estimates suffer from
considerable uncertainties and therefore can yield only ranges of values [33,
34]. For our quantities of concern, the SM predicts

$\displaystyle 0.5\times 10^{-5}$
$\displaystyle\leq\big{(}A_{CP}^{\Xi}\big{)}{}_{\rm SM}\leq 6\times
10^{-5}\,,$ $\displaystyle-3.8\times 10^{-4}$
$\displaystyle\leq\big{(}\xi_{P}^{\Xi}-\xi_{S}^{\Xi}\big{)}{}_{\rm
SM}\leq-0.3\times 10^{-4}\,,~{}~{}~{}$ $\displaystyle-3\times 10^{-5}$
$\displaystyle\leq\big{(}A_{CP}^{\Lambda}\big{)}{}_{\rm SM}\leq 3\times
10^{-5}\,,$ (9)

as updated in ref. [35] from the analysis of ref. [33]. These are below their
empirical counterparts given in the previous paragraph by two orders of
magnitude or more and might be out of BESIII scope ultimately. Nevertheless,
the predictions in eq. (Pursuit of $CP$ violation in hyperon decays at
$e^{+}e^{-}$ colliders) could be within reach of the proposed super tau-charm
factories [28, 29] because their total luminosities are planned to surpass
that of BESIII by two orders of magnitude, implying better sensitivity by at
least ten times, and further improvement can be gained if polarized-electron
beams are utilized [35].

Figure 1: Correlation between theoretical $A_{CP}^{\Xi}$ and
$\xi_{P}^{\Xi}-\xi_{S}^{\Xi}$. The (darkly shaded) red patch comes from the SM
alone and lies within the (lightly shaded) cyan region which includes the BSM
contribution described in the text.

Figure 2: The latest BESIII results for $A_{CP}^{\Xi}$ and
$\xi_{P}^{\Xi}-\xi_{S}^{\Xi}$ from ref. [25] and $A_{CP}^{\Lambda}$ from ref.
[26] (the statistical and systematic uncertainties combined in quadrature)
compared to their respective theoretical predictions.

In the presence of physics beyond the SM (BSM), there could be new sources of
$CP$ violation which enhance $A_{CP}$ and $\xi_{P}-\xi_{S}$ in size with
respect to the SM expectations. This possibility can be realized with, for
instance, the so-called chromomagnetic-penguin interactions [36, 37]. Since
their parity-even and -odd components also affect, respectively, the indirect
and direct $CP$-violation in the kaon system, restrictions from the relevant
kaon data apply. Taking the latter into account and employing the work of ref.
[34], with parameter inputs from ref. [35], we arrive at $|A_{CP}^{\Xi}|_{\rm
BSM}<5.9\times 10^{-4}$, $|\xi_{P}^{\Xi}-\xi_{S}^{\Xi}|_{\rm BSM}<3.7\times
10^{-3}$, and $|A_{CP}^{\Lambda}|_{\rm BSM}<7.0\times 10^{-4}$ for the
chromomagnetic-penguin contributions. These numbers are higher than the
corresponding ones in eq. (Pursuit of $CP$ violation in hyperon decays at
$e^{+}e^{-}$ colliders) from the SM by about an order of magnitude. The
correlation between $\xi_{P}^{\Xi}-\xi_{S}^{\Xi}$ and $A_{CP}^{\Xi}$ in the
two scenarios is displayed in fig. 2. On the other hand, the preceding BSM
results are still lower than their experimental counterparts quoted earlier by
roughly a factor of ten. The situation is depicted in fig. 2. Notwithstanding,
forthcoming findings from BESIII may yet be able to test these predictions and
in the future the super tau-charm factories would certainly probe them more
stringently.

In conclusion, hyperon nonleptonic decays potentially serve as a promising
hunting ground for $CP$ violation in the strangeness-changing sector caused by
new physics beyond the SM. Given that they involve fermions, these processes
could supply information about the underlying interactions which is different
from that furnished by kaon transitions. It is therefore exciting that there
are now quests for hyperon $CP$-violation that are ongoing intensively by
BESIII and also planned or proposed ones at other facilities. It is hoped that
fresh hyperon data will appear soon which reveal evidence for new physics or,
if not, meaningfully constrain BSM contributions.

As final remarks, since charmed baryons are copiously produced at running
$e^{+}e^{-}$ machines and the LHC and some of them can undergo a Cabibbo-
favored nonleptonic or semileptonic decay into a charmless final-state
containing a hyperon with a branching fraction at the percent level or bigger
[4], it might additionally be feasible to look for hyperon $CP$-violation by
means of such processes [35, 38]. The first attempt in this direction has very
recently been conducted by the Belle Collaboration [39] studying
$\Lambda_{c}^{+}\to\Lambda\pi^{+}$, although the search came up negative.

Acknowledgments This work was supported in part by the Australian Government
through the Australian Research Council. XGH was supported in part by the NSFC
(Grant Nos. 11735010, 11975149, and 12090064), and also supported in part by
the MOST (Grant No. MOST 109-2112-M-002-017-MY3).

## References

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