# Multi-Stream TSN Gate Control Scheduling in the Presence of Clock
Synchronization

Aviroop Ghosh  Carleton UniversityOttawaCanada<EMAIL_ADDRESS>,
Saleh Yousefi  Urmia UniversityUrmiaIran<EMAIL_ADDRESS>and  Thomas
Kunz  Carleton UniversityOttawaCanada<EMAIL_ADDRESS>
(2025)

###### Abstract.

With the advancement of technologies like Industry 4.0, communication networks
must meet stringent requirements of applications demanding deterministic and
bounded latencies. The problem is further compounded by the need to
periodically synchronize network devices to a common time reference to address
clock drifts. Existing solutions often simplify the problem by assuming either
perfect synchronization or a worst-case error. Additionally, these approaches
delay the scheduling process in network devices until the scheduled frame is
guaranteed to have arrived in the device queue, inducing additional delays to
the stream. A novel approach that completely avoids queuing delays is
proposed, enabling it to meet even the strictest deadline requirement.
Furthermore, both approaches can be enhanced by incorporating network-derived
time-synchronization information. This is not only convenient for meeting
deadline requirements but also improves bandwidth efficiency.

Ethernet, IEEE 802.1Qbv, IEEE 802.1AS, time-sensitive network, time aware
shaper, time synchronization, multi-stream scheduling, Integer Linear
Programming.

††copyright: acmlicensed††journalyear: 2025††doi: XXXXXXX.XXXXXXX††conference:
Under Review; July 2024; ††ccs: Networks††ccs: Networks Network types

##  1\. Introduction

Modern cyber-physical systems aim to provide efficient, flexible and reliable
systems by leveraging recent advances in artificial intelligence, cloud
computing and networking technologies. These systems require deterministic and
low-latencies communication. To achieve this, it is essential to guarantee
bounded end-to-end latencies and minimal jitter in data delivery times.
Ethernet is such a technology that can meet these requirements in a cost-
effective manner. The IEEE Time-Sensitive Networking (TSN) Task Group (TG) is
standardizing these enhancements to achieve the necessary performance
improvements (Cavalcanti et al., 2019; IEEE 802.1, 2021).

The IEEE 802.1Qbv standard (IEEE Std 802.1Qbv, 2016), also known as Time Aware
Shaper (TAS), is used for guaranteeing deterministic and bounded latencies for
periodic time-sensitive traffic streams. The TAS mechanism achieves this is by
enabling specific queue/queues on a egress port of a switch for transmissions
while disabling others. The process is managed using Gate Control Lists (GCLs)
which are periodic schedules that control the queue opening and closing given
the periodicity of time-sensitive streams. This mechanism allows for the time-
domain isolation of time-sensitive streams from non-time-sensitive streams.
Determinism is ensured by setting these GCLs such that each switch on a given
time-sensitive stream route opens its queues for transmission at a
predetermined time interval, synchronized with the expected arrival time of
the frames. This coordinated timing across the network ensures that time-
sensitive data is transmitted with deterministic delay and jitter. However,
for the TAS mechanism to work effectively, accurate time-synchronization is
required such that all network devices maintain a common clock reference.

Each device maintains its own internal clock, which, even if initially
synchronized with other devices, will begin to drift due to physical
properties and external influences affecting its local clock oscillator. This
phenomenon is known as clock drift. To mitigate the impacts of clock drift,
IEEE has a standard specifying a time-synchronization mechanism, detailed in
the IEEE 802.1AS standard (IEEE Std 802.1AS-2020 (2020), Revision of IEEE Std
802.1AS-2011). The standard defines a protocol known as generalized Precision
Time Protocol (gPTP). A common device in a network, known as the Grandmaster
(GM), sends out periodic messages to synchronize all network devices to its
interval clock. These messages are passed on hop by hop, where each node uses
the information received by the GM to synchronize with its internal clock,
accounting for delays on the path such as propagation, processing and
transmission delays.

Deriving GCL schedules for multiple time-sensitive (TS) streams is inherently
complex due to varying stream features such as periodicity and frame size, all
while meeting specific deadlines. The impact of clock drift adds further
complications. To simplify this, existing scheduling approaches either assume
perfect synchronization, rendering clock drift impacts inconsequential, or
adopt a worst-case error which is a constant factor signifying the maximum
possible time difference between any two network devices. The former approach
is impractical in realistic networks where clock drift impacts are
significant. The latter, as will be demonstrated in this study, introduces
unnecessary delays to TS frames. This approach is ineffective for streams
having tight jitter requirements such as zero jitter (IIC, 2019), unless
specialised adjustments are made to the GCLs. Furthermore, this study also
hypothesises that deriving clock drift information from network measurements
related to the gPTP mechanism, and using it for formulating GCLs leads to more
efficient schedules.

Considering these, the contributions of this paper are as follows:

  * •

Alternative Approaches: Proposing a new scheduling approach for multi-stream
GCL scheduling that ensures the minimum possible end-to-end delays while
guaranteeing deadline requirements for TS streams.

  * •

Augmentation with Network Measurements: Both the existing approach and the
proposed approach are augmented by considering network derived measurements in
the form of clock drift.

  * •

Quantifiable Comparison Method: A quantifiable comparison method is introduced
to compare the different scheduling methods devised.

  * •

Elaborated Network Scenarios: Detailed network scenarios are examined to
illustrate how incorporating clock drift measurements optimize bandwidth
efficiency.

The rest of the paper is organized as follows: in Section 2, a survey of the
relevant literature is provided. Section 3 covers the relevant background and
system model. Building on these, the scheduling approaches for these studies
are developed in Section 4. Finally, in Section 5 results and analysis are
discussed, and a conclusion is provided in Section 6.

##  2\. Literature Survey

This section reviews the influential works in the field to provide relevant
context. Note that the emphasis is on offline scheduling algorithms, where all
streams are known in advance and a single schedule is generated. Online
scheduling algorithms, which handle streams added dynamically have a different
evaluation methodology and formulation (Stüber et al., 2024), are outside the
scope of this study.

Many algorithms aim to reduce end-to-end latency for TS streams and prevent
frames from different streams to overlap over a given egress port. This is
complicated due to non-determinism introduced by clock drift. For this,
isolation techniques, such as assigning streams to different egress queues,
offsetting transmission start times, and allowing only a single queue to
transmit, are used to prevent frame interference.

Isolation techniques are used by Craciunas et al. in (Craciunas et al., 2016).
They use Satisfiability Modulo Theories (SMT) to create GCL schedules.
Building on this, Raagaard et al. in (Raagaard and Pop, 2017) use an Integer
Linear Programming (ILP) method to develop GCL schedules as presented in their
technical report. Barazegaran et al. in (Barzegaran et al., 2021) use a
constraint programming technique to formulate GCLs. Other studies, such as
(Vlk et al., 2020), propose switch hardware modifications to ensure detection
of frame arrival based on a pre-defined schedule. The authors propose an ILP
scheduling approach to determine GCLs and maintain determinism by idling the
queue if a frame is missing or delayed.

Some studies focus on jitter control as well. The authors in (Dürr and Nayak,
2016) employ a no-wait approach by avoiding queuing delays entirely. They use
an ILP model and tabu search method to generate GCL schedules that ensure the
minimum possible latency (no queuing delays) for TS streams, hence no jitter.
Other works such as (Serna Oliver et al., 2018; Chaine et al., 2022), use
SMT/ILP models for developing GCL schedules while also providing specific
provisions for ensuring zero jitter for streams.

Most studies focus on improving algorithms by striking a balance between
finding the optimal solution versus solving times. There are studies which
involve other aspects such as deriving GCL schedules after a brief period of
loss of synchronization (Craciunas and Oliver, 2021). However, all of the
studies either consider perfect synchronization or a worst-case error thereby
failing to integrate the time synchronization process with TAS. Furthermore,
studies considering a worst-case error address the problem by incorporating
queuing delays. Specifically, at each switch, the derived GCL will only open a
queue to forward time-sensitive traffic once it is confirmed that the frame
has been received by considering the worst-case clock drifts between the
adjacent devices. Previous research on single TS streams has shown that
incorporating network derived measurement, such as clock drift from time
synchronization can lead to more bandwidth-efficient schedules (Ghosh et al.,
2023). However, no studies have addressed scheduling multiple TS streams in
this context. Therefore, this study aims to address this gap.

##  3\. Background and System Model

This section provides the necessary background beginning with a brief
description of the how TSN switches are configured and the topology considered
followed by detailing the system model.

###  3.1. Background on TSN Standard

The TSN standards specify three architectural topologies for configuring TSN
switches, namely, fully distributed, hybrid and centralized configuration
model. Out of the three, this study considers the centralized configuration
model since it is best suited for offline scheduling methods.

The centralized configuration comprises of the Centralized User Configuration
(CUC), which manages user information, and the Centralized Network Controller
(CNC), which configures the End Stations (ES) and TSN Switches. The CNC, with
a global view of all network devices, communicates with the CUC through the
User/Network Interface (UNI). This comprehensive information enables the CNC
to devise Gate Control List (GCL) schedules. GCLs, defined for the egress port
of a device, operate on periodic cycles, allowing frame transmission when
queue gates are open. Frames are assigned to queues based on their Priority
Code Point (PCP) markings, and only frames in open gates transmit. If multiple
gates are open, the frame in the highest priority queue is transmitted.

Gate open durations for TS traffic are limited, requiring precise scheduling
to ensure frames are transmitted within their allocated time slots (scheduling
durations). Accurate time synchronization is crucial, as misaligned frame
arrivals can cause queuing delays and unnecessary interference. Figure 1
illustrates the GCL for the egress port of a switch. The schedule repeats
every $T$ duration and consisting of sub-periods $T1,T2,T3$ and $T4$. Time-
slots $T1$ and $T3$ are allocated for TS traffic, with Queue 7 open, while the
other queues are closed (value associated with Queue 7 is set to 1 i.e., ON
while others are 0 i.e., OFF). If there is frame misalignment from the
previous device during $T1$ and the frame is unable to transmit, then it
queues for a duration of at least $T2$ and transmits during $T3$, potentially
interfering with the frame scheduled for $T3$.

Figure 1. Egress port GCL

As explained previously, the worst-case error parameter ($\delta$) used in the
literature mitigates these timing inaccuracies. The gPTP synchronization
mechanism provides measurements, including the relative clock difference
between devices and the GM. These measurements allow for calculating the
constant clock drift of a device and establish the relative drift between
devices, which should be less than or equal to $\delta$. As an example,
consider the worst-case constant clock drift range (measured in parts per
million or ppm) as specified by the IEEE 802.1AS standard which is between
[$-100$ ppm, $100$ ppm] (IEEE Std 802.1AS-2020 (2020), Revision of IEEE Std
802.1AS-2011), i.e., a relative difference of $200$ ppm. With a default
synchronization period of $125$ ms, the value of $\delta$ $=$ $25$ $\mu{s}$.
However, if the relative difference between any two devices is only $50$ ppm,
then the maximum time-difference for the devices will only be $12.5$ $\mu{s}$.
Previous studies have shown how the gPTP measurement parameters can be exposed
to the TSN control plane (Thi et al., 2020), while the study in (Ghosh et al.,
2023) showed how to derive the clock drift value from these measurements.

###  3.2. System Model

To model the network, a directed graph $\mathcal{G}=$ {$\mathcal{E,V}$} is
considered where $\mathcal{E}$ represents the set of edges that represents the
data links and $\mathcal{V}$ represents the set of vertices or the set of
devices, namely the End-Stations ($ES$) and Switches ($SW$). A bi-directional,
full-duplex link between two consecutive Devices $v_{a}$ and $v_{b}$ is
denoted by [$v_{a}$,$v_{b}$], indicating the direction of flow of traffic for
the stream.

A Stream $s_{i}$ comprises of periodic data transmission from $ES$ to another
$ES$ over a given Route $r_{i}$. Each Stream $s_{i}$ is associated with its
Payload size $s_{i}.P$, Periodicity $s_{i}.T$ and Deadline $s_{i}.D$. Given
that periodic streams can have different periods, they can be aligned into a
common periodic network cycle known as a hyperperiod ($hp$). The value is
calculated based on the least common multiple (lcm) of the stream periods.
Therefore, there can be multiple repetitions of frames from a stream within a
$hp$. Consider an example, there are three streams of periodicity
$s_{i}.T=[100,150,300]$ $\mu{s}$. The $hp$ $=$ $lcm(100,150,300)$ $\mu{s}$ $=$
$300$ $\mu{s}$. Therefore, the first stream has 3 repetitions, the second
stream has 2 and the third stream has 1 repeat over the $hp$.

Each stream comprises of a frame transmitted over the route. A frame for the
stream is represented as $f_{i}^{[v_{a},v_{b}]}$. For example, in Fig. 2,
Frame $f_{i}^{[v_{x},v_{a}]}$ on Link [$v_{x}$,$v_{a}$] travels from the
egress of Device $v_{x}$ to the ingress of Device $v_{a}$, while for Stream
$j$, Frame $f_{j}^{[v_{a},v_{b}]}$ on Link [$v_{a}$, $v_{b}$] travels from the
egress of Device $v_{a}$ towards Device $v_{b}$. The link speed for the link
between $v_{a}$ and $v_{b}$ is denoted by $[v_{a},v_{b}].c$. Denote the set of
routes, streams and frames in the network by $\mathcal{R}$, $\mathcal{S}$ and
$\mathcal{F}$ respectively.

Figure 2. Network model

Each frame is associated with a specific set of delays given as:

  * •

$[v_{a}].p$: Processing delay in Device $v_{a}$.

  * •

$[v_{a},v_{b}].d$: Propagation delay on Link [$v_{a}$,$v_{b}$].

  * •

$f_{i}^{[v_{a},v_{b}]}.t$: Frame transmission duration of Frame
$f_{i}^{[v_{a},v_{b}]}$ on egress port of Device $v_{a}$.

  * •

$f_{i}^{[v_{a},v_{b}]}.q$: Queuing delay of Frame $f_{i}^{[v_{a},v_{b}]}$ in
Device $v_{a}$.

  * •

$f_{i}^{[v_{a},v_{b}]}.L$: Denotes the sum of $f_{i}^{[v_{a},v_{b}]}.t$,
$[v_{a},v_{b}].d$ and $[v_{b}].p$.

Each frame is transmitted at a specified time based on its GCL schedule. The
timing is determined by an offset integer variable $\phi$, where
$f_{i}^{[v_{a},v_{b}]}.\phi$ specifies when the transmission of Frame
$f_{i}^{[v_{a},v_{b}]}$ can start on the egress port of Device $v_{a}$ (i.e.,
start of the scheduling duration). These offsets are crucial to control the
timing of the gate control operations in the networks and are expressed as
integer multiples of macroticks ($mt$). In this study, the value of $mt$ $=$
$0.1$ $\mu{s}$. From the previous example, the $hp$ when expressed in $mt$ is
3000. Now if the offset value of Frame $f_{1}^{[v_{a},v_{b}]}.\phi$ $=$ $120$
$mt$, this means that the frame is scheduled (i.e., the gate is opened) at
$12$ $\mu{s}$ within the $hp$ of 300 $\mu{s}$.

Finally, it is important to discuss the parameters associated with the gPTP
mechanism. Each Device $v_{a}$ has a clock subject to a constant clock drift
$v_{a}.\rho$ and a time-varying clock drift component
$v_{a}.\rho^{\prime}(t)$. In this study, the time-varying component is
neglected ($v_{a}.\rho^{\prime}(t)$ $=$ $0$), since it is not considered in
the IEEE 802.1AS standard. Given the clock drift and the periodic
synchronization (denoted by $T_{s}$), the maximum time difference between two
Devices $v_{a}$ and $v_{b}$ occurs just before the $T_{sync}$ as the clocks
are re-synchronized after this interval. After synchronization, the time
difference gradually increases until the next synchronization cycle. For
example, assume that $v_{a}.\rho$ $=$ $10$ ppm and $v_{b}.\rho$ $=$ $-5$ ppm,
with the synchronization interval of $T_{s}$ $=$ $125$ ms, after the initial
alignment, the internal clock of $v_{a}$ will be 1.875 $\mu{s}$ faster than
that of $v_{b}$ after $125$ ms. While formulating GCLs on the basis of gPTP
measurements, other factors such as synchronization errors and estimation
errors for clock drift could be considered (Ghosh et al., 2023). However, for
simplicity in formulation and analysis, these have not been included in this
study.

###  3.3. GCL Formation

In this sub-section, the formation of GCLs based on the offset value is
explained briefly. Consider the previous example of three Streams $s_{1}$,
$s_{2}$ and $s_{3}$ with periodicity of $[100,150,300]$ $\mu{s}$ respectively
and $hp$ is $300$ $\mu{s}$.

An offset integer for a frame in an egress queue $[v_{a},v_{b}]$ specifies the
offset within its time period applied consistently across the $hp$. For
example, as shown in Fig. 3, if $f_{1}^{[v_{a},v_{b}]}.\phi$ $=$ $0$, the
scheduling duration of Frame $f_{1}^{[v_{a},v_{b}]}$ starts with the start of
the $hp$ and repeats twice with a periodicity of $s_{1}.T$. Scheduling
duration of Frames $f_{2}^{[v_{a},v_{b}]}$ and $f_{3}^{[v_{a},v_{b}]}$ are
delayed by $f_{2}^{[v_{a},v_{b}]}.\phi$ and $f_{3}^{[v_{a},v_{b}]}.\phi$
respectively with Frame $f_{2}^{[v_{a},v_{b}]}$ occurring once more in the
$hp$.

Figure 3. GCL formation example

Thus, based on the offset values and scheduling durations, the GCLs for all
egress ports can be calculated, indicating when the queue should be active for
transmitting TS frames and when it should be inactive.

##  4\. Scheduling Approach with Worst-Case Error and Clock Drift

In the previous sections the primary motivations of this study were elaborated
along with the relevant background and system model description. In this
section, this is taken further to define the different scheduling approaches.
Before proceeding, it is essential to define the scheduling duration in the
context of this study. The scheduling duration for a single TS frame is
defined as the transmission duration plus any added slack to mitigate the
impacts of uncertainties, such as clock drift. Stating this, there are four
approaches considered in this study:

  * •

Worst-Case Delay (WCD) Method: Based on delaying the scheduling duration using
a worst-case clock drift estimation ($\delta$).

  * •

Worst-Case Adjustment (WCA) Method: Based on adjusting the scheduling
durations to mitigate the impacts of a worst-case clock drift.

  * •

Network-Derived Clock Drift Delay (NCD) Method: Enhances the WCD method by
considering network-derived measurements of clock drift.

  * •

Network-Derived Clock Drift Adjustment (NCA) Method: Enhances the WCA method
by considering network-derived measurements of clock drift.

###  4.1. Scheduling Duration and Clock Drift

Scheduling durations are crucial for forming the GCLs. Inadequate scheduling
durations can result in queuing delays, detrimental to TS applications.
Conversely, overly generous scheduling durations lead to inefficient bandwidth
usage and under-utilization of network resources for other traffic types.
Additionally, these challenges are exacerbated by clock drift. To address
these issues, two main approaches are used in approaches with respect to
scheduling durations: delaying it or adjusting it. This subsection explains
the principles behind each approach and the influence of clock drift
measurements.

The WCD scheduling approach, based on works such as (Raagaard and Pop, 2017;
Vlk et al., 2020; Barzegaran et al., 2021), uses the scheduling duration delay
method. As a TS frame is forwarded through each consecutive switch, the
scheduling duration is delayed by a factor of $\delta$. This ensures that the
frame is completely received before transmission. Here, the scheduling
duration is equal to the transmission duration, resulting in a queuing delay
between 0 and $\delta$ for each switch. This is shown in Fig. 4(a). However,
with clock drift considerations, the relative drift between switches can be
calculated and adjusted accordingly. For example, if the relative drift
between two switches is zero, there is no need for any scheduling duration
delay, thereby eliminating any queuing delay. This mechanism is addressed in
the NCD approach.

The objective of the WCA method is to match the minimum end-to-end (e2e)
latency (therefore, zero jitter). To achieve this, TS frames must be
transmitted promptly upon arrival to the queue. Therefore, the scheduling
durations are extended to ensure the frames can be transmitted regardless of
their arrival times. Therefore, the scheduling duration is adjusted such that
both positive and negative clock drifts are considered. This is shown in Fig.
4(b). With clock drift measurements, these adjustment durations can be
shortened and be based on the actual relative drift between devices instead of
the worst-case value. This mechanism is captured in the NCA approach.

(a) Delaying scheduling duration

(b) Adjusting scheduling duration

Figure 4. Illustration of scheduling duration delay and adjustment mechanisms

###  4.2. Scheduling Approaches

In this sub-section, the four scheduling approaches are further elaborated
using an Integer Linear Programming (ILP) framework. As detailed in Section 2,
ILP has been frequently used to determine the offsets that govern the GCLs.
Furthermore, ILP mechanisms are guaranteed to find an optimal solution as
opposed to heuristic solutions (Stüber et al., 2024). Solving the ILP
optimization problem yields the offsets ($\phi$) for each stream on each
switch, which are then used to create GCL schedules in a post-processing
phase. The optimization objective is defined in (1) $\forall
s_{i}\in\mathcal{S}$.

(1) $\min[\sum{(\lambda_{i}-\lambda_{i}^{min})}]$

The objective minimizes the total excess latency as the difference between the
measured e2e latency $\lambda_{i}$ and the minimum possible e2e latency
($\lambda_{i}^{min}$) i.e., latency without queuing delays, summed across all
streams in the network. While other studies also focus on minimizing the
excess number of queues assigned to TS traffic as a part of the optimization
problem, this study assumes that only a single queue is dedicated to TS
traffic.

Following the definition of the objective function, the constraints applicable
to all four approaches are defined. The unit of expression of all the terms
are in macroticks.

####  4.2.1. Boundary Conditions:

All offset values must be greater than or equal to zero. Thus, the lower bound
of the offsets for an Stream $i$ for transmission on Link $[v_{a},v_{b}]$ is
defined accordingly, as shown in the (2).

(2) $f_{i}^{[v_{a},v_{b}]}.\phi\geq 0$

The first offset i.e., the offset from the source device of Stream $i$
(denoted by $v_{src\\_i}$) is upper-bounded by the hyperperiod as shown in (3)

(3) $f_{i}^{[v_{src\\_i},v_{x}]}.\phi\leq hp$

Both the relations (2) and (3) are valid $\forall s_{i}\in\mathcal{S}$ and
$\forall\\{[v_{a},v_{b}],\newline [v_{src\\_i},v_{x}]\\}\in\mathcal{E}$.

####  4.2.2. Non-Overlapping Condition:

A non-overlapping condition is applied to all the scheduling approaches
developed as a part of this study. This condition states that the absolute
difference of transmission offsets of any two Streams $i$ and $j$ on Egress
Port $[v_{a},v_{b}]$ represented as $f_{i}^{[v_{a},v_{b}]}.\phi$ and
$f_{j}^{[v_{a},v_{b}]}.\phi$ respectively, must be less than the hyperperiod.
This is shown in (4).

(4) $\displaystyle\forall
f_{i}^{[v_{a},v_{b}]},f_{j}^{[v_{a},v_{b}]}\in\mathcal{F},\forall\\{[v_{a},v_{b}]\\}\in\mathcal{E},\forall
s_{i},s_{j}\in\mathcal{S},\;i\neq j:$
$\displaystyle|f_{i}^{[v_{a},v_{b}]}.\phi-f_{j}^{[v_{a},v_{b}]}.\phi|\leq
hp-1$

This prevents any overlaps while deriving GCL schedules. For example, consider
$hp$ $=$ $3000$ and $f_{i}^{[v_{a},v_{b}]}.\phi$ $=$ $0$, if
$f_{j}^{[v_{a},v_{b}]}.\phi$ $=$ $3000$, then this indicates that the offset
for Stream $j$ wraps around the GCL period and overlaps with Stream $i$,
thereby resulting in an invalid GCL schedule.

Table 1. Set of Expressions Variable |  Expression | Approach | Constraint/Condition
---|---|---|---
$\kappa_{11}$ |  $\lceil f_{i}^{[v_{x},v_{a}]}.L+\delta\rceil$ | WCD | Consecutive Offsets
$\kappa_{12}$ |  $\lfloor f_{i}^{[v_{x},v_{a}]}.L+\gamma\times\delta\rfloor$ | WCA | Consecutive Offsets
$\kappa_{13}$ |  $\lceil f_{i}^{[v_{x},v_{a}]}.L+\max\left(|(v_{a}.\rho-v_{x}.\rho)T_{s}|,\psi\times|(v^{\prime}_{a}.\rho-v^{\prime}_{x}.\rho)T_{s}|\right)\rceil$ | NCD | Consecutive Offsets
$\kappa_{14}$ |  $\begin{aligned} &\lfloor f_{i}^{[v_{x},v_{a}]}.L+\min\left[0,(v_{a}.\rho-v_{src\\_i}.\rho)T_{s},\psi\times(v^{\prime}_{a}.\rho-v^{\prime}_{src\\_i}.\rho)T_{s}\right]\\\ &-\min\left[0,(v_{x}.\rho-v_{src\\_i}.\rho)T_{s},\psi\times(v^{\prime}_{x}.\rho-v^{\prime}_{src\\_i}.\rho)T_{s}\right]\rfloor\end{aligned}$ | NCA | Consecutive Offsets
$\kappa_{21}$ |  $(\alpha_{i}\times s_{i}.T-\beta_{j}\times s_{j}.T)-\lceil f_{j}^{[v_{a},v_{b}]}.t+1\rceil$ | WCD, NCD | Scheduling Duration
$\kappa_{22}$ |  $(\alpha_{i}\times s_{i}.T-\beta_{j}\times s_{j}.T)-\lceil f_{j}^{[v_{a},v_{b}]}.t+2\delta+1\rceil$ | WCA | Scheduling Duration
$\kappa_{23}$ |  $\begin{aligned} &\left(\alpha_{i}\times s_{i}.T-\beta_{j}\times s_{j}.T\right)\\\ &-\lceil f_{j}^{[v_{a},v_{b}]}.t+\max\left[0,(v_{a}.\rho-v_{src\\_j}.\rho)T_{s},\psi\times(v^{\prime}_{a}.\rho-v^{\prime}_{src\\_j}.\rho)T_{s}\right]\\\ &-\min\left[0,(v_{a}.\rho-v_{src\\_j}.\rho)T_{s},\psi\times(v^{\prime}_{a}.\rho-v^{\prime}_{src\\_j}.\rho)T_{s}\right]+2\rceil\end{aligned}$ | NCA | Scheduling Duration
$\kappa_{24}$ |  $(\beta_{j}\times s_{j}.T-\alpha_{i}\times s_{i}.T)-\lceil f_{i}^{[v_{a},v_{b}]}.t+1\rceil$ | WCD, NCD | Scheduling Duration
$\kappa_{25}$ |  $(\beta_{j}\times s_{j}.T-\alpha_{i}\times s_{i}.T)-\lceil f_{i}^{[v_{a},v_{b}]}.t+2\delta+1\rceil$ | WCA | Scheduling Duration
$\kappa_{26}$ |  $\begin{aligned} &\left(\beta_{j}\times s_{j}.T-\alpha_{i}\times s_{i}.T\right)\\\ &-\lceil f_{i}^{[v_{a},v_{b}]}.t+\max\left[0,(v_{a}.\rho-v_{src\\_i}.\rho)T_{s},\psi\times(v^{\prime}_{a}.\rho-v^{\prime}_{src\\_i}.\rho)T_{s}\right]\\\ &-\min\left[0,(v_{a}.\rho-v_{src\\_i}.\rho)T_{s},\psi\times(v^{\prime}_{a}.\rho-v^{\prime}_{src\\_i}.\rho)T_{s}\right]+2\rceil\end{aligned}$ | NCA | Scheduling Duration
$\kappa_{31}$ |  $(\beta_{j}\times s_{j}.T-\alpha_{i}\times s_{i}.T)+\lceil f_{j}^{[v_{y},v_{a}]}.L+\delta\rceil$ | WCD, WCA | Frame Arrival
$\kappa_{32}$ |  $(\beta_{j}\times s_{j}.T-\alpha_{i}\times s_{i}.T)+\lceil f_{j}^{[v_{y},v_{a}]}.L+\max[0,(v_{y}.\rho-v_{a}.\rho)T_{s},\psi\times(v^{\prime}_{y}.\rho-v^{\prime}_{a}.\rho)T_{s}]\rceil$ | NCD, NCA | Frame Arrival
$\kappa_{33}$ |  $(\alpha_{i}\times s_{i}.T-\beta_{j}\times s_{j}.T)+\lceil f_{i}^{[v_{x},v_{a}]}.L+\delta\rceil$ | WCD, WCA | Frame Arrival
$\kappa_{34}$ |  $(\alpha_{i}\times s_{i}.T-\beta_{j}\times s_{j}.T)+\lceil f_{i}^{[v_{x},v_{a}]}.L+\max[0,(v_{x}.\rho-v_{a}.\rho)T_{s},\psi\times(v^{\prime}_{x}.\rho-v^{\prime}_{a}.\rho)T_{s}]\rceil$ | NCD, NCA | Frame Arrival
$\kappa_{41}$ |  $f_{i}^{[v_{n\\_i},v_{dst\\_i}]}.\phi-f_{i}^{[v_{src\\_i},v_{1\\_i}]}.\phi+f_{i}^{[v_{n\\_i},v_{dst\\_i}]}.t+f_{i}^{[v_{n\\_i},v_{dst\\_i}]}.d$ | WCD, NCD | End-to-End Latency
$\kappa_{42}$ |  $f_{i}^{[v_{n\\_i},v_{dst\\_i}]}.\phi-f_{i}^{[v_{src\\_i},v_{1\\_i}]}.\phi+f_{i}^{[v_{n\\_i},v_{dst\\_i}]}.t+f_{i}^{[v_{n\\_i},v_{dst\\_i}]}.d+\delta$ | WCA | End-to-End Latency
$\kappa_{43}$ |  $\begin{aligned} &f_{i}^{[v_{n\\_i},v_{dst\\_i}]}.\phi-f_{i}^{[v_{src\\_i},v_{1\\_i}]}.\phi+f_{i}^{[v_{n\\_i},v_{dst\\_i}]}.t+f_{i}^{[v_{n\\_i},v_{dst\\_i}]}.d\\\ &-\min[0,(v_{n\\_i}.\rho-v_{src\\_i}.\rho)T_{s},\psi\times(v^{\prime}_{n\\_i}.\rho-v^{\prime}_{src\\_i}.\rho)T_{s}]\end{aligned}$ | NCA | End-to-End Latency

####  4.2.3. Consecutive Offsets:

Consecutive offsets ensure that frame transmission on a link starts only after
the same frame has been fully transmitted and received by the current device
from the previous link (adjusted to clock drift considerations). In the
reference topology shown in Fig. 2, Frame $f_{i}^{[v_{a},v_{b}]}$ for Stream
$i$ can only be transmitted by $v_{a}$ on Link $[v_{a},v_{b}]$ after Frame
$f_{i}^{[v_{x},v_{a}]}$ is received by Switch $v_{a}$. The constraint is
captured in (5). Due to spatial limitations, the equations are expanded in
Table 1 with the $\kappa$ variables capturing the remaining part of the
expression.

(5) $\displaystyle\forall
f_{i}^{[v_{x},v_{a}]},f_{i}^{[v_{a},v_{b}]}\in\mathcal{F},\;\\{[v_{x},v_{a}],[v_{a},v_{b}]\\}\in\mathcal{E},\forall
s_{i}\in\mathcal{S}:$ $\displaystyle f_{i}^{[v_{a},v_{b}]}.\phi\geq
f_{i}^{[v_{x},v_{a}]}.\phi+\begin{cases}\kappa_{11}&\text{for WCD}\\\
\kappa_{12}&\text{for WCA}\\\ \kappa_{13}&\text{for NCD}\\\
\kappa_{14}&\text{for NCA}\\\ \end{cases}$

For WCD, the expression dictates that Link $[v_{a},v_{b}]$ can begin
transmission only after receiving Frame $f_{i}^{[v_{x},v_{a}]}$, adjusted for
transport delays and the worst-case clock difference between $v_{x}$ and
$v_{a}$. Similarly, WCA is adjusted with $\delta$, and a term $\gamma$ is
included. Here, $\gamma$ $=$ 1 only if the previous Device $v_{x}$ is an ES;
otherwise, $\gamma$ $=$ 0\. This adjustment is necessary because the
scheduling duration of a switch can be modified to accommodate frame
transmission based on the arrival of the frame, while ES transmission time
remains fixed, requiring the switch to adjust accordingly.

In the NCD approach, scheduling can begin only after the frame is fully
received, accounting for the absolute relative difference in clock drift
between devices (considering the polarities of the clock drift values). The
consecutive offset for the NCA approach requires that the start of
transmission from a device must be at least equal to the start of transmission
from the previous device, adjusted for the relative clock drift between the
devices and the source ES device ($v_{src\\_i}$). The source clock drift
serves as a reference point for comparing the relative drifts of $v_{x}$ and
$v_{a}$.

Synchronization pathways must also be considered in the NCD and NCA approaches
since they are based on network derived clock drift measurements. The term
$(v^{\prime}_{x}.\rho-v^{\prime}_{src\\_i}.\rho)$ in NCD is introduced to
denote whether one device is synchronized with the GM before the other. For
example, in Fig. 2, if $v_{y}$ is the GM, hop-by-hop synchronization ensures
$v_{a}$ is synchronized before $v_{x}$. Thus, Frame $f_{i}^{[v_{x},v_{a}]}$
could be transmitted by $v_{x}$ while $v_{a}$ is already synchronized and
$v_{x}$ is not. Here, the clock drift of $v_{a}$ will be equivalent to the GM
($v_{GM}.\rho$) while $v_{x}$ remains $v_{x}.\rho$. Therefore, the relative
clock difference is ($v_{GM}.\rho$ $-$ $v_{x}.\rho$) instead of ($v_{a}.\rho$
$-$ $v_{x}.\rho$). In this case $v_{a}^{\prime}.\rho$ $=$ $v_{GM}.\rho$ while
$v_{x}^{\prime}.\rho$ $=$ $v_{x}.\rho$. The variable $\psi=1$, only if there
is a possibility one device is synchronized before another otherwise,
$\psi=0$.

Note that the ceiling function is employed in WCD and NCD methods, whereas the
floor function is utilized in WCA and NCA methods. This distinction arises
from the nature of the scheduling approaches: the former initiate scheduling
duration after full frame reception, while the latter requires it to start
just before the frame reaches the queue, reflecting differing principles in
the approaches.

####  4.2.4. Scheduling Duration Constraint:

Each link can transmit at most one frame at a time. If there are two frames
from different streams, then at any given time, the scheduling duration of
frames cannot overlap in time domain.

In (6), the offset of frames from Streams $i$ and $j$ are set such that they
do not interfere when transmitted via Link $[v_{a},v_{b}]$ (with reference to
Fig. 2). Here, $\alpha_{i}$ and $\beta_{j}$ represent the number of frame
repetitions within the $hp$, such that,
$\alpha_{i}=\\{0,...,\nicefrac{{lcm(s_{i}.T,s_{j}.T)}}{{s_{i}.T}}-1\\}$ and
$\beta_{j}=\\{0,...,\nicefrac{{lcm(s_{i}.T,s_{j}.T)}}{{s_{j}.T}}-1\\}$. A
logical disjunction is considered such that the scheduling duration of Frame
$f_{i}^{[v_{a},v_{b}]}$ is before that of Frame $f_{j}^{[v_{a},v_{b}]}$ or
after. The source ES device for Streams $i$ and $j$ are denoted by
$v_{src\\_i}$ and $v_{src\\_j}$ respectively.

(6) $\displaystyle\forall
f_{i}^{[v_{a},v_{b}]},f_{j}^{[v_{a},v_{b}]}\in\mathcal{F},\;\forall\\{[v_{a},v_{b}]\\}\in\mathcal{L},$
$\displaystyle\forall s_{i},s_{j}\in\mathcal{S},\;i\neq j:$ $\displaystyle
f_{j}^{[v_{a},v_{b}]}.\phi-
f_{i}^{[v_{a},v_{b}]}.\phi\leq\begin{cases}\kappa_{21}&\text{for WCD, NCD}\\\
\kappa_{22}&\text{for WCA}\\\ \kappa_{23}&\text{for NCA}\\\ \end{cases}$
$\displaystyle\vee$ $\displaystyle f_{i}^{[v_{a},v_{b}]}.\phi-
f_{j}^{[v_{a},v_{b}]}.\phi\leq\begin{cases}\kappa_{24}&\text{for WCD, NCD}\\\
\kappa_{25}&\text{for WCA}\\\ \kappa_{26}&\text{for NCA}\\\ \end{cases}$

For the WCD and NCD approaches, the scheduling duration is equal to the
transmission duration. For the WCA method the scheduling duration adds a
factor of $2\delta$ (Section 4.1). In these approaches, a slack of 1 $mt$ is
added to ensure that the transmission end of one frame does not overlap with
the beginning of another.

For the NCA method, the start of the scheduling durations of TS frames are
adjusted based on the relative drift between the switch and the source,
similar to the end of the scheduling duration. Furthermore, this is also
adjusted considering the value of $\psi$ where the source or the switch
assumes the value of the GM. Therefore, total scheduling duration is the
transmission duration plus the maximum adjustment required subtracted with the
minimum adjustment required. Furthermore, an adjustment factor of 2 $mts$ is
considered. The second $mt$ is introduced to ensure that the scheduling
duration encompasses the frame transmission. For example, consider Frame
$f_{i}^{[v_{a},v_{b}]}$, if the frame arrives at $12.09$ $\mu{s}$, then with
at $mt$ $=$ $0.1$ $\mu{s}$, the scheduling start
$f_{i}^{[v_{a},v_{b}]}.\phi=120$ $mts$. If the frame requires a transmission
time of $10$ $\mu{s}$ (or $100$ $mts$) i.e., transmits at $22.09$ $\mu{s}$,
then the scheduling duration of $101$ $mts$ ensures complete transmission.

All approaches apply a ceiling function to the scheduling durations to ensure
complete frame transmission and total isolation of the frame during its
scheduled time.

####  4.2.5. Frame Arrival Constraint:

The non-deterministic effects of clock drift and synchronization errors may
cause two frames from different ingress ports to arrive simultaneously at the
same egress port. To prevent this, the frame arrival constraint is required.
The constraint is shown in (7).

(7) $\displaystyle\forall s_{i},s_{j}\in\mathcal{S},\;\forall
f_{i}^{[v_{a},v_{b}]},f_{i}^{[v_{x},v_{a}]},f_{j}^{[v_{y},v_{a}]}\in\mathcal{F},\;i\neq
j$
$\displaystyle\forall\\{[v_{x},v_{a}],[v_{y},v_{a}],[v_{a},v_{b}]\\}\in\mathcal{E}:$
$\displaystyle f_{i}^{[v_{a},v_{b}]}.\phi-
f_{j}^{[v_{y},v_{a}]}.\phi\leq\begin{cases}\kappa_{31}&\text{for WCD, WCA}\\\
\kappa_{32}&\text{for NCD, NCA}\\\ \end{cases}$ $\displaystyle\vee$
$\displaystyle f_{j}^{[v_{a},v_{b}]}.\phi-
f_{i}^{[v_{x},v_{a}]}.\phi\leq\begin{cases}\kappa_{33}&\text{for WCD, WCA}\\\
\kappa_{34}&\text{for NCD, NCA}\\\ \end{cases}$

The constraint ensures that, the arrival of Frame $f_{j}^{[v_{y},v_{a}]}$ to
$v_{a}$ is after the start of scheduling $f_{i}^{[v_{a},v_{b}]}.\phi$ or that
Frame $f_{i}^{[v_{x},v_{a}]}$ arrives after $f_{j}^{[v_{a},v_{b}]}.\phi$. The
scheduling duration constraint prevents any overlap between the frames.

Timing mismatches are considered as frames arrive from different devices. For
WCD and WCA approaches, a worst-case timing error ($\delta$) is used,
representing the maximum possible relative drift between Device
$v_{x}$/$v_{y}$ and $v_{a}$. For NCD and NCA approaches, the adjustment is
based on the maximum possible relative drift between the devices.

####  4.2.6. End-to-End Latency:

The end-to-end latency equality constraint for Stream $i$ relates the start of
the transmission from the source to the first switch (denoted by $v_{1\\_i}$)
on Route $r_{i}$, to the transmission onto the final link between the last
switch ($v_{n\\_i}$) and the destination ES ($v_{dst\\_i}$). The e2e latency
constraints are listed in (8).

(8) $\displaystyle\forall s_{i}\in\mathcal{S},\;\forall
f_{i}^{[v_{src\\_i},v_{1\\_i}]},f_{i}^{[v_{n\\_i},v_{dst\\_i}]}\in\mathcal{F},$
$\displaystyle\forall\\{[v_{src\\_i},v_{1\\_i}],[v_{n\\_i},v_{dst\\_i}]\\}\in\mathcal{E}:$
$\displaystyle\lambda_{i}=\begin{cases}\kappa_{41}&\text{for WCD, NCD}\\\
\kappa_{42}&\text{for WCA}\\\ \kappa_{43}&\text{for NCA}\\\ \end{cases}$

Note that in case of the WCA approach, $\delta$ is used to establish the
differences in clock between $v_{n\\_i}$ and $v_{src\\_i}$. While, in the NCA
approach if the clock of $v_{n\\_i}$ is slower than $v_{src\\_i}$, an
adjustment is necessary because the scheduling start duration will be earlier
than when all devices are synchronized.

The boundary conditions for the e2e latency ($\lambda_{i}$) are shown in (9)
for the WCD and NCD approaches, and in (10) for the WCA and NCA approaches.
The latter uses equality between $\lambda_{i}$ and $\lambda_{i}^{min}$ since
these approaches ensure minimum e2e latency for all frames. The relations are
valid $\forall s_{i}\in\mathcal{S}$.

(9) $\lambda_{i}^{min}\leq\lambda_{i}\leq s_{i}.D$ (10)
$\lambda_{i}^{min}=\lambda_{i}\leq s_{i}.D$

##  5\. Results and Analysis

Before presenting the results and analysis, it is crucial to establish a
metric for comparing the different scheduling methods, especially given the
two distinct approaches: delay-based and adjustment-based. This is performed
in a multi-staged, criteria-based approach:

  * •

Criteria 1: Deadlines must be kept
For the GCL schedule generated for each egress port in all the switches in the
network to be valid, the deadlines for all frames from every stream across the
network must be met. If any frame misses its deadline, the entire schedule is
rendered invalid.

  * •

Criteria 2: Infeasible schedule
A schedule is deemed infeasible if the scheduling durations of all the TS
streams egressing through an egress port cannot be accommodated within the
$hp$. Consider $f_{i}^{[v_{a},v_{b}]}.SD$ to be the scheduling duration of
Frame $f_{i}^{[v_{a},v_{b}]}$ transmitting on Link $[v_{a},v_{b}]$ from Stream
$i$. The relation in (11) shows the schedule is infeasible for $v_{a}$ when
the cumulative scheduling durations for all streams (along with their
repetitions) is greater than $hp$. Here, $[v_{a},v_{b}].N_{s}$ represents the
number of streams transmitting on Link $[v_{a},v_{b}]$, $\forall
s_{i}\in\mathcal{S}$ and $\forall\\{[v_{a},v_{b}]\\}\in\mathcal{E}$.

(11)
$hp<\sum_{i=1}^{[v_{a},v_{b}].N_{s}}(\frac{hp}{s_{i}.T})f_{i}^{[v_{a},v_{b}]}.SD$

  * •

Criteria 3: Schedulability Cost
The Schedulability Cost ($SC$) qualifies as the cumulative scheduling
durations for all the streams across all the switches encountered over an
individual route within the specified $hp$. As shown in (12), for each Stream
$i$, the total scheduling duration of a frame across Route $r_{i}$ (denoted by
$r_{i}.SD$) is multiplied by the number of frame occurrences within the $hp$.
Summing this parameter across the total number of TS streams in the network
(denoted by $N_{s}$) gives the value of $SC$, $\forall s_{i}\in\mathcal{S}$
and $\forall r_{i}\in\mathcal{R}$.

(12)
$sc=\sum_{i=1}^{N_{s}}\frac{hp}{s_{i}.T}\times\frac{r_{i}.SD}{hp}=\sum_{i=1}^{N_{s}}\frac{r_{i}.SD}{s_{i}.T}$

A schedule is stated to be more bandwidth efficient if it has a lower $SC$
than the schedule it is being compared with.

###  5.1. Case Studies

In this section, a series of case studies are conducted to validate the
scheduling approaches and compare the schedules. The approaches are executed
using the CPLEX optimization tool to generate the transmission offsets.
Subsequently, MATLAB is utilized in a post-processing phase to create the GCL
schedules. Finally, the INET framework (version 4.4) for OMNeT++ is employed
for network simulation. The simulation period runs for 1 second, allowing
sufficient time to analyze the results.

First, the network topology is defined as shown in Fig. 5. In this setup,
$ES1$ and $ES2$ are the sources, while $ES3$ is the destination. All the links
in the network are 1 Gbps Ethernet (GE). There are three traffic streams:
$s_{1}$, $s_{2}$ and $s_{3}$. Both $s_{1}$ and $s_{3}$ originate from $ES1$
and terminate at $ES3$. Stream $s_{2}$ originates from $ES2$ and terminates at
$ES3$. The routes for the streams are: $r_{1}=([ES1,SW1],[SW1,SW2],[SW2,ES3])$
for $s_{1}$, $r_{2}=([ES2,SW1],\newline [SW1,SW2],[SW2,ES3])$ for $s_{2}$ and
$r_{3}=([ES1,SW1],\newline [SW1,SW2],[SW2,ES3])$ for $s_{3}$. Also note, in
this topology, $ES2$ is considered the GM.

Figure 5. Case study network topology

Table 2 shows the network traffic profiles. The periodicity, deadlines and
frame sizes specified for TS traffic are consistent with the requirements
specified in (IIC, 2019). A further assumption is that all the associated
delays such as propagation, processing in the network and devices are fixed
across the network and the devices. The minimum and maximum clock drift
values, represented by $[v_{a}^{min}].\rho$ and $[v_{a}^{max}].\rho$
respectively, are chosen specifically for demonstration in this case study.

The case studies (CS) include three scenarios based on different clock drift
values for the devices:

  1. (1)

$[ES1,ES2,SW1,SW2]$ $=$ $[0,0,+10,-10]$ $ppm$.

  2. (2)

$[ES1,ES2,SW1,SW2]$ $=$ $[+10,-10,-10,+10]$ $ppm$.

  3. (3)

$[ES1,ES2,SW1,SW2]$ $=$ $[-5,+5,+5,+5]$ $ppm$.

CS1 tests the scheduling approach with a perfect sources (and GM) and extreme
clock drift in switches. CS2 explores extreme clock drift in both sources and
switches, with interleaving clock drift polarities for devices in routes
$r_{1}$ and $r_{3}$. CS3 uses lower and closer clock drift values to
demonstrate the improvements of NCD and NCA approaches over WCD and WCA
respectively, with a polarity change in $ES1$ to prevent uniform clock drift
across all devices.

The results presented in Table 3 displays the end-to-end latency ranges for
each stream across various scenarios and approaches. Furthermore, the $SC$
value is calculated for each scenario based on the specific scheduling
approach used.

Table 2. System Model Details and Requirements Parameter |  Symbol |  Value
---|---|---
Deadline |  $s_{i}.D$; $\forall s_{i}\in\mathcal{S}$ |  45.00 $\mu$s
Stream
periodicity |  $[s_{1}.T,s_{2}.T,s_{3}.T]$ |  $[100,150,300]$
$\mu$s
Macrotick |  $mt$ |  0.1 $\mu$s
Hyperperiod |  $hp$ |  3000
Propagation
delay |  $[v_{a},v_{b}].d$;
$\forall[v_{a},v_{b}]\in\mathcal{E}$ |  50 ns
Processing delay |  $[v_{a}].p$; $\forall[v_{a}]\in\mathcal{V}$ |  5 $\mu$s
Transmission
delay |  $f_{i}^{[v_{a},v_{b}]}.t$;
$\forall s_{i}\in\mathcal{S}$, $\forall[v_{a},v_{b}]\in\mathcal{E}$ |  12.144 $\mu{s}$
Frame Size |  $f_{i}$; $\forall s_{i}\in\mathcal{S}$, $\forall f_{i}\in\mathcal{F}$ |  1464 Bytes
Min e2e latency |  $\lambda_{i}^{min}$; $\forall s_{i}\in\mathcal{S}$ |  39.682 $\mu$s
Clock drift range |  $[v_{a}^{min}.\rho,\;v_{a}^{max}.\rho]$; $\forall v_{a}\in\mathcal{V}$ |  $[-10,10]$ ppm
Synchronization
periodicity |  $T_{sync}$ |  $125$ ms
Worst-case error |  $\delta$ |  2.5 $\mu{s}$
Table 3. Case Study Results Method | e2e Latency Range (unit $\bm{\mu}$s) | Schedulability Cost
---|---|---
Scenario 1 |  Scenario 2 |  Scenario 3 |  Scenario 1 |  Scenario 2 |  Scenario 3
WCD |  Stream 1: [44.674, 46.043]
Stream 2: [44.734, 46.044]
Stream 3: [44.734, 46.043] |  Stream 1: [44.673, 44.793]
Stream 2: [42.235, 44.792]
Stream 3: [44.793, 47.293] |  Stream 1: [43.424, 44.793]
Stream 2: [44.733, 44.793]
Stream 3: [43.543, 44.793] |  0.4290 |  0.4290 |  0.4290
WCA |  Stream 1: [39.682, 39.682]
Stream 2: [39.682, 39.682]
Stream 3: [39.682, 39.682] |  Stream 1: [39.682, 39.682]
Stream 2: [39.682, 39.682]
Stream 3: [39.682, 39.682] |  Stream 1: [39.682, 39.682]
Stream 2: [39.682, 39.682]
Stream 3: [39.682, 39.682] |  0.6920 |  0.6920 |  0.6920
NCD |  Stream 1: [43.374, 44.743]
Stream 2: [43.434, 44.743]
Stream 3: [43.394, 44.743] |  Stream 1: [44.673, 44.793]
Stream 2: [39.735, 42.292]
Stream 3: [44.793, 44.793] |  Stream 1: [39.682, 40.993]
Stream 2: [39.733, 39.793]
Stream 3: [39.743, 40.993] |  0.4290 |  0.4290 |  0.4290
NCA |  Stream 1: [39.682, 39.682]
Stream 2: [39.682, 39.682]
Stream 3: [39.682, 39.682] |  Stream 1: [39.682, 39.682]
Stream 2: [39.682, 39.682]
Stream 3: [39.682, 39.682] |  Stream 1: [39.682, 39.682]
Stream 2: [39.682, 39.682]
Stream 3: [39.682, 39.682] |  0.5440 |  0.5763 |  0.5280

The results show that both the WCA and NCA approaches ensure the minimum end-
to-end (e2e) latency for all frames, as evident by the fact that both the
minimum and maximum latency values are equal to the minimum possible e2e
latency ($\lambda_{i}^{min}$). The $SC$ for the WCA approach is constant (due
to factoring a worst-case error), while the $SC$ parameters for NCA are 16% to
24% lower than that of WCA, indicating greater bandwidth-efficiency. The NCA
has shorter scheduling durations due to considering relative clock drifts.
Even within the NCA results, Scenario 3 provides the lowest $SC$ due to having
the smallest relative drift between the devices, while Scenario 2 has the
largest $SC$.

The scheduling durations of the WCD and NCD methods are the same, therefore,
the $SC$ for them are equal across all the scenarios. However, factoring clock
drift measurements are crucial. Across all the scenarios, the NCD method not
only achieves lower relative difference in e2e latency ranges (thereby, lower
jitter) but also reduces the maximum e2e latency when compared to the WCD
method. Furthermore, the WCD method, due to the delays and considering a
worst-case error, violates the deadline requirement (Criteria 1) as observed
in Scenarios 1 and 2. Although, the ILP method provides a feasible solution,
the deadline constraint for some frames is violated due to clock drift
interactions.

Overall, the WCA and the NCA methods guarantee meeting deadlines with their
minimum possible e2e latency (and zero jitter) approach. Therefore, they do
not face the issue of violating Criteria 1, unlike the WCD and NCD methods.
However, the $SC$ is higher because the scheduling durations are longer.
Therefore, there could be an increased propensity of violating Criteria 2.

###  5.2. Comparing the Scheduling Approaches

In this section, the system model and topology from Section 5.1 is extended to
compare the WCD and NCD, and WCA and NCA methods. When comparing the WCD and
NCD methods, these yield the same $SC$, however, the WCD method introduces
greater e2e delays. The difference in the upper-bound of the e2e latencies
between the WCD and NCD method will always be equal to $\delta$. This is
because the upper-bound considers the extremities of clock drift interleaving,
with the source and last switch having the same polarity and intermediate
device having the opposite polarity. Consequently, if the maximum queuing time
in $SW1$ is just before synchronization, then this correspond to the minimum
at $SW2$ and vice-versa, thereby reducing the overall queuing delay. This
assumption cannot be extended to the WCD method. Hence there will always be a
difference of $\delta$ between the two scheduling approaches. Note, this was
observed for Scenario 2, Stream 3, in Tab. 3, where the difference in upper-
bound e2e latency is exactly $2.5$ $\mu{s}$.

The value of $\delta$ depends on the relative clock drift and synchronization
periodicity. As either the clock drift or synchronization periodicity
increases, so does $\delta$. Consequently, the upper bound of end-to-end
latencies also increases. This is illustrated in Fig. 6, where the latency
difference ranges from $0$ to $50$ $\mu{s}$, with bluer shades representing a
lower difference and the yellow shades a higher difference. The significant
extent of this difference is noteworthy. Given that this calculation is for a
3-hop network, any further increase in number of hops further increases the
difference.

Figure 6. Upper-bound e2e latency difference: WCD and NCD methods

Conversely, the WCA and NCA methods are designed to achieve the minimum
possible e2e latency. Consequently, any improvements by considering clock
drift values focus on reducing the $SC$. Using the same topology and system
model as in Section 5.1, Fig. 7 illustrates how the $SC$ varies with changes
in synchronization periodicity and the maximum possible relative clock drift
range between two devices.

(a) Schedulability cost for the WCA method

(b) Maximum schedulability cost for the NCA method

Figure 7. Schedulability cost variation for the WCA and NCA methods

The $SC$ range for the WCA and NCA methods are shown in Fig. 7(a) and Fig.
7(b) respectively. The range of values is from $[0.492,2]$, with bluer shades
representing a lower $SC$ and yellow shades representing higher $SC$ values. A
point to point comparison shows, the the NCA approach gives lower $SC$ values
than the WCA approach. Furthermore, beyond a certain threshold, Criteria 2 is
violated for both the methods, as the schedule cannot fit within the
hyperperiod. This is depicted by the unshaded (white) region. From the
figures, it is clear that the NCA approach has a lower unshaded area than the
WCA method due to tighter scheduling durations. This demonstrates that
considering clock drift measurements results in more bandwidth-efficient
schedules.

##  6\. Conclusion and Future Work

In this study, different scheduling approaches for developing GCLs were
derived and compared. Furthermore, a criteria for evaluating GCL schedules
including measuring the bandwidth effectiveness was proposed. The scheduling
approach primarily used in literature, the WCD approach was augmented by
incorporating network derived clock drift measurements, resulting in the NCD
approach. As discussed in Section 5.2, the e2e latency increases as the clocks
are synchronized less frequently and/or there is an increase in the number of
hops. However, the NCD approach was shown to be more effective at reducing
unnecessary delays to the e2e latency and the delay bounds were tighter for TS
streams.

Irrespective of the scheduling approaches, these delay bounds can quickly
exceed application deadline requirements based on standardized specifications,
necessitating a different approach. This violation of deadlines was especially
true of the literature-based WCD approach. Therefore, an alternative and novel
scheduling approach was proposed in the form of the WCA which guarantees the
minimum e2e latency. However, as discussed in Sections 5, this results in
higher bandwidth usage as indicated by the $SC$ values. By incorporating clock
drift measurements and augmenting the approach in the form the NCA method
which was demonstrably more bandwidth effective as shown by the $SC$
comparison.

The feasibility and validity of the approaches, along with the effectiveness
of using clock drift measurements, were shown using a series of case studies.
Consequently, it can be concluded that the TAS mechanism must be considered in
conjunction with time synchronization. Factoring in clock drift measurements
leads to more efficient schedules by reducing excessive end-to-end delays or
improving bandwidth allocations for TS streams.

As a future work, the approaches will be extended to consider other realistic
factors such as measurement errors during the synchronization process and GCL
schedule development in complex non-linear topologies. Also, as indicated in
Section 5.2, the synchronization periodicity has an impact on the efficiency
of GCL schedules irrespective of the scheduling approaches. Therefore, the
schedules can be further optimized by varying the synchronization periodicity.

## References

  * (1)
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11institutetext: School of Physics and Astronomy, Sir William Henry Bragg
Building, Woodhouse Ln., University of Leeds, Leeds LS2 9JT, UK
11email<EMAIL_ADDRESS>22institutetext: Konkoly Observatory, Research
Centre for Astronomy and Earth Sciences, Eötvös Loránd Research Network
(ELKH), Konkoly-Thege Miklós út 15-17, 1121 Budapest, Hungary 33institutetext:
CSFK, MTA Centre of Excellence, Budapest, Konkoly Thege Miklós út 15-17, 1121,
Hungary 44institutetext: School of Physics and Astronomy, University of
Exeter, Stocker Road, Exeter EX4 4QL, UK 55institutetext: Institute of
Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK
66institutetext: School of Physics and Astronomy, University of Leicester,
Leicester LE1 7RH, UK

# A trio of month long flares in the nova-like variable V704 And

Gabriella Zsidi 112233 C. J. Nixon 11 T. Naylor 44 J. E. Pringle 55 K. L. Page
66

(Received )

We present the discovery of an unusual set of flares in the nova-like variable
V704 And. Using data from AAVSO, ASAS-SN, and ZTF, of the nova-like variable
V704 And, we have discovered a trio of brightening events that occurred during
the high state. These events elevate the optical brightness of the source from
$\sim 13.5$ magnitude to $\sim 12.5$ magnitude. The events last for roughly a
month, and exhibit the unusual shape of a slow rise and faster decay. Just
after the third event we obtained data from regular monitoring with Swift,
although by this time the flares had ceased and the source returned to its
pre-flare level of activity in the high-state. The Swift observations confirm
that during the high-state the source is detectable in the X-rays, and provide
simultaneous UV and optical fluxes. As the source is already in the high-state
prior to the flares, and thus the disc is expected to already be in the high-
viscosity state, we conclude that the driver of the variations must be changes
in the mass transfer rate from the companion star and we discuss possible
mechanisms for such short-timescale mass transfer variations to occur.

###### Key Words.:

stars: individual: V704 And – accretion, accretion disks – stars: novae,
cataclysmic variables

##  1 Introduction

Cataclysmic variables (CVs) are close interacting binary systems, in which a
white dwarf is accreting material from a companion via Roche-lobe overflow. In
the case of weakly or non-magnetic CVs, the material transferred from the
secondary component forms an accretion disk around primary. These systems may
be classified based on their photometric and spectroscopic characteristics
(Warner 1995). For example, dwarf novae (DNe) show recurring accretion related
outbursts, and sometimes superoutbutsts, lasting from a few days to $\sim 20$
days. The nova-like variables (NLs), defined as those that do not display nova
or dwarf-nova outbursts, remain in the high accretion, therefore bright
photometric state, for a prolonged time.

The VY Sculptoris stars (also called “antidwarf novae”) are an interesting
subclass of NLs, whose bright photometric states are occasionally interrupted
by rapid drops of brightness (low states). The low states are believed to be
caused by decreased mass transfer from the companion, which can eventually
lead to the disappearance of the disc (Hameury & Lasota 2002). However, it is
argued that sometimes the accretion disc does not disappear completely during
these periods (Schmidtobreick et al. 2018). The exact process causing the
reduced mass transfer is still unknown, but one mechanism that was suggested
by previous works is starspots on the secondary blocking the L1 point and
hence suppressing the mass transfer (Livio & Pringle 1994).

V704 Andromedae, hereafter V704 And, is a poorly studied NL; its variable
nature was discovered only roughly two decades ago by Dahlmark (1999), who
reported brightness changes in the $V$ band between 14.8 and 12.6 mag. The
source was further studied in the early 2000s by Papadaki et al. (2006) within
a photometric survey of not well known CVs, mainly focusing on nova-like
variables. Based on optical photometric measurements between 2002 and 2005,
they found that V704 And showed fading episodes in 2003 and 2005. Long-term
historical light curves confirm the occasional fading of the system, which
suggests that V704 And belongs to the VY Scl subclass of the nova-like
variables, and the optical study by Weil et al. (2018) supports this argument.
Bruch (2022) further investigated the system and measured an orbital period of
$P_{\textrm{orb}}=0.15424(3)$ days (3.7 h). This is within the typical range
of orbital periods for VY Scl nova-like variables. Furthermore, they estimated
the spectral type of the secondary to be M3–M4 using optical spectra obtained
during a low state. Bruch (2022) revealed a strong negative superhump in the
V704 And system by analysing the TESS satellite light curves with a period of
$P_{\textrm{nSH}}=0.14772(3)$ days. The system was also included in the Gaia
survey, which placed it at a distance of 410.4${}^{+6.3}_{-5.3}$ pc (Bailer-
Jones et al. 2021).

##  2 Observations

Figure 1: Light curves of V704 And. Top panel: Optical light curves of V704
And. The yellow and dark green x-symbols show the ASAS-SN $V$ and $g$ band
measurements, the orange and the light green filled circles indicate the ZTF
$r$ and $g$ band observations; and yellow, blue and grey circles mark the
AAVSO B, V band and unfiltered CCD (CV) measurements, respectively. The cyan
and dark orange star symbols are measurements from the ATLAS survey with the
$c$ and $o$ filters, respectively. We indicated, with the grey shaded area,
the period which is shown in the bottom panel. We note that the ATLAS
measurements from 2016 suggest that the system was as faint as $\sim 18$ mag
during this period, however, the ASAS-SN data indicated a brightness around 16
mag. Since the sensitivity of ATLAS observations is higher, and the ASAS-SN
measurements from this period include several upper limit measurements, we
excluded from the figure those ASAS-SN data which were fainter than 16 mag.
Bottom panel: Optical and UV measurements obtained since 2022 April (grey
shaded area in the top panel). The empty circles show the Swift optical and UV
data. Yellow, blue and black filled circles are the AAVSO $V$, $B$ and visual
band observations, respectively.

In order to analyse the long-term behaviour of V704 And, we inspected
photometric data from public surveys. We studied the data obtained by ZTF
(Bellm et al. 2019), which aims to observe the northern sky with a $\sim$2-day
cadence. We downloaded the measurements for V704 And from their
archive111https://irsa.ipac.caltech.edu/Missions/ztf.html, which provided
photometric measurements between 2018 March and 2023 January in the
Palomar/ZTF $g$ and $r$ bands for the latest data release (DR16). In order to
remove bad-quality observations, we applied the “catflags=0” filter of the
catalogue. Johnson $V$ and Sloan $g$ band photometric observations are also
available from the ASAS-SN survey. The ASAS-SN team performs aperture
photometry using a 2-pixel aperture radius, and the reduced data are available
from their online catalogue222https://asas-sn.osu.edu/. $V$ band measurements
were obtained between 2013 December and 2018 August, whereas the $g$ band
observations are available since 2017 December (Kochanek et al. 2017; Shappee
et al. 2014). Despite both surveys providing $g$ band data, we analysed them
separately as the sensitivity of the two surveys differ: the ASAS-SN is
sensitive up to $\sim$17 mag, whereas the ZTF is accurate up to about 20.8
mag. We also complemented our dataset with observations from the ATLAS survey
(Heinze et al. 2018), which provided measurements in “cyan” and “orange”
filters333The “cyan” ($c$) filter covers the 420–650 nm wavelength range, and
the “orange” ($o$) filter covers the 560–820 nm band. between 2015 and 2017,
which have a sensitivity up to $\sim 19$ mag. Furthermore, we examined the
AAVSO optical light curves of V704 And, which date back to 1966 and allow to
study the historic behaviour of the system. However, we must note that the
early AAVSO observations for V704 And are rather incomplete, and they either
did not provide information on the filter used, or applied a “visual” band.
The more recent measurements often provide Johnson $B$ or $V$ band
observations besides the visual ones444https://www.aavso.org/filters.

We also have been monitoring V704 And with the Ultraviolet/Optical Telescope
(UVOT) instrument of the Swift satellite between 2022 November 29 and December
26 with a three-day cadence (ToO ID 18163, PI: G. Zsidi). We used the “0x30ed”
mode of UVOT, which provided photometric measurements in the $V$, $B$, and $U$
optical and the $UVW1$, $UVM2$, and $UVW2$ UV filters555Central wavelengths of
the UVOT filters ($\AA$): $V$ – 5468, $B$ – 4392, $U$ – 3465, $UVW1$ – 2600,
$UVM2$ – 2246, $UVW2$ – 1928.. However, due to scheduling conflict, one
observation was not executed, and the one on 2022 December 14 was carried out
only in the $UVW1$ filter. We reduced the UVOT data using the version 6.31.1
of the heasoft software. All UV and optical light curves are displayed in Fig.
1.

##  3 Results

The brightness of V704 And varies around 13 mag in the high state but this is
occasionally interrupted by low state periods, during which the brightness of
the system decreases by $3-5$ mag. According to the most recent optical ASAS-
SN and the ZTF light curves (Fig. 1, top panel), V704 And underwent three low
state periods during the last six years (i.e. since 2017), each of which
lasted about six months. Before 2017, the system exhibited an inverse
behaviour to some extent, as it spent a longer period of time in the low
brightness state. This behaviour was characteristic of the system since the
mid-2000s, as revealed by the AAVSO light curves (Fig. 3). Despite the fact
that the AAVSO observations date back to 1966, the early measurements were
rather incomplete, therefore, we cannot make far reaching conclusions about
the historical behaviour of the system. The available data suggest that V704
And spent significant time in the high state between 1966 and 1995, but
intervening drops into the low state during this time cannot be ruled out.

###  3.1 The trio of flares in the latest high state

The most recent low state of V704 And finished around March/April in 2022. The
source initially returned to a regular high state for a few months, albeit at
a slightly lower flux level than before the latest minimum ($g\approx 13.5$
mag rather than $g\approx 12.8$ mag prior). However, the system then began to
show peculiar outburst-like events, which we call flares to distinguish from
ordinary CV outbursts, with amplitudes of $\sim$1 mag and lasting for about a
month each. The bottom panel of Fig. 1 shows an enlarged excerpt of the light
curve highlighting these events. The outbursts seem to have an asymmetric
evolution: it takes $18-20$ days to reach the maximum, and $10-12$ days to
return to the initial brightness level666We also note that around
$\textrm{HJD}-2450000=8500$, V704 And exhibited a small amplitude brightening
event. However, due to the sparseness of the data during that period, it is
not possible to tell with certainty whether this event was similar to the trio
of flares in 2022..

The light curves of the flares obtained with different filters are remarkably
similar, we did not detect any noticeable time delay in reaching the maximum
brightness at different wavelengths. This might be caused by two factors. One
is that, due to the cadence of the light curves, the peaks of the flares are
not well defined in all filters. The other cause could be that the available
optical filters indeed probe the same region of the system; therefore, no time
delay is expected in these observations.

It is instructive to compare the outbursts of V704 And with the dwarf novae
outbursts. The two types of outbursts might look similar in amplitude and in
length, but the dwarf nova outbursts normally exhibit an asymmetry with an
opposite trend, i.e. they show faster rise than decay (Cannizzo et al. 1986).
Furthermore, the increase of the far-UV flux can be delayed compared to the
optical. This second aspect, unfortunately, cannot be probed for V704 And, as
we have only optical light curves during the flares.

###  3.2 Colour changes

The multi-filter photometric observations allow us to examine the colour
variations of V704 And. The ZTF survey has long enough coverage to study the
changes in the system on yearly timescale. We constructed the colour-magnitude
diagram using the simultaneous $g$ and $r$ band measurements (Fig. 2). Our
results show that when the object is in the low and fainter state, the data
points cover a wider range in the colour space. However, as the system
brightens, the overall colour is more concentrated on the blue side.

It is interesting to separate those observations when the system underwent the
three flares in order to investigate their nature. For this reason, we
highlighted the data points that cover the period between
$\textrm{HJD}-2\,450\,000=9740$ and $9890$ with orange symbols. Furthermore,
we indicated the earlier measurements with darker shades and the later ones
with lighter in order to investigate the temporal evolution. We found that no
significant colour change was observed when V704 And brightened by $\sim$1
mag. However, we must note that the simultaneous $g$ and $r$ band data do not
cover the flares entirely.

The Swift measurements provide information on the colour of the high state
after the trio of flares. During this time, the brightness of the system did
not change substantially. Fig. 9 shows, that no significant colour change was
detected in the optical filters. The most apparent evolution is observed at
the shortest wavelengths ($UVM2$ and $UVW2$ filters), in which the system
becomes redder as it brightens.

Figure 2: Optical colour-magnitude diagram based on the ZTF measurements. The
blue filled circles show the entire dataset, whereas the orange ones cover
only the period between $\textrm{JD}-2450000=9740\ \textrm{and}\ 9890$, i.e.
when the three flares occurred. The measured errors are smaller than or equal
to the symbol size.

##  4 Discussion

V704 And is a nova-like variable star that typically displays a steady high-
state brightness corresponding to $g\approx 13-13.5$ mag, with aperiodic low
states lasting several months. In the high-state the accretion disc shows no
evidence for DNe type outbursts, and thus the whole disc is expected to be
fully ionised with well-established hydro-magnetic viscosity. In similar
systems, primarily the DNe in outburst, the disc viscosity parameter $\alpha$
(Shakura & Sunyaev 1973) is found to be high with $\alpha\approx 0.2-0.4$
(Martin et al. 2019). As the disc is already in the high-state for the months
preceding the trio of flares it is unlikely that these flares are caused by
variations in $\alpha$. Instead, they are likely to be caused by variations in
the accretion rate through the disc, which is controlled by the mass supply
rate from the companion star. To determine whether the duration of the flares
is consistent with varying mass supply to the outer disc regions, we compute
the viscous timescale in V704 And and then we discuss possible mechanisms to
induce a change to the mass transfer rate in this system.

###  4.1 Viscous timescale in V704 And

The timescale on which the central accretion rate responds to changes in the
mass transfer rate from the companion star is the viscous timescale given by
$t_{\nu}=R^{2}/\nu$. Here $\nu$ is the Shakura & Sunyaev (1973) kinematic
viscosity and $R$ is the radius in the disc at which the mass transferred from
the companion star is added. For our estimate, we take this to be the outer
disc edge where viscous spreading of the disc is impeded by tides from the
companion star. To evaluate $R$ and $\nu$ we require estimates for the system
parameters, e.g. $M_{1}$, $M_{2}$, $P_{\rm orb}$, and $\dot{M}$.

The estimates of the system parameters can be uncertain, therefore, we only
summarise here the main steps of the calculation. Our detailed calculation and
the estimates of the system parameters can be found in Appendix C. By using
$P_{\rm orb}=0.15424(3)$ days, $q=0.34$, $M_{2}=0.3\,M_{\odot}$, and
$M_{1}=0.9\,M_{\odot}$, we have that the binary separation is $a=8.9\times
10^{10}$ cm, and therefore the outer disc radius is $R\approx 2.3\times
10^{10}$ cm. We also assume a typical white dwarf radius of $R=0.01$
$R_{\odot}$, and take $\alpha=0.3$. Then using equation 5.50 in Frank et al.
(2002), we estimate the viscous timescale through the disc to be 9.6 days for
an accretion rate of $5\times 10^{-9}M_{\odot}$/yr, and 7.7 days for an
accretion rate of $10^{-8}M_{\odot}$/yr. We therefore conclude that the
viscous timescale is approximately a week, and maybe up to 10 days. This
timescale is consistent with the breadth of the flares, and also consistent
with the viscous timescale inferred from measurements of the decay of DNe
light curves (e.g. Cannizzo 1994).

###  4.2 Origin of the flares

The origin of the flares is puzzling. At the optical wavelength range, the
accretion disc is the most dominant component of a NL system in high state
(Pringle & Wade 1985), however, disc instability typical of DNe is an unlikely
cause of the observed flares of V704 And, as their characteristics differ from
DN outbursts. We propose that variable mass transfer from the secondary is
responsible for inducing the flares. Nonetheless, VY Scl type variables are
expected to have the highest mass transfer rate during the high state (Warner
1987). For this reason, there has to be an additional mechanism that induces
the increase of the mass transfer rate. Here, we discuss a few mechanisms to
achieve this.

Variable mass transfer rate can be achieved by the variations of the radius of
the secondary (Warner 1988; King et al. 1995). King et al. (1995) proposes
that variable irradiation by the accreting component may induce changes in the
radius of the Roche lobe-filling star, which causes a limit-cycle variation.
However, the timescale of this is significantly longer ($10^{5}-10^{6}$ yr)
than the variations we see here. Another mechanism that could contribute to
the variation of the radius of the secondary component is solar-type magnetic
cycles. Warner (1988) report such quasiperiodic variations in CVs but these
also appear on several-years timescale.

Magnetic activity, at the same time, can have short-term effects as well
(Baliunas & Vaughan 1985). The light curves of main sequence stars often
reveal the presence of star spots and flares, and as the secondaries of CVs
are main sequence stars, stellar activity is expected in these systems as
well. Indeed, e.g. Webb et al. (2002) have found spectroscopic evidence for
starspots on the secondary star of SS Cyg, and Dunford et al. (2012) found a
high-latitude starspot on RU Peg with Roche tomography. Flares or coronal mass
ejections of the secondaries of a CV may induce additional large amounts of
matter, however, we cannot directly test this theory for V704 And, as the
optical light curves of CVs are dominated by the emission from the disc, which
veils the variability of any stellar activity.

###  4.3 Comparison with other VY Scl type systems

We have compiled a list of VY Scl type NLs and examined their short- and long-
term light curves based on AAVSO, ZTF and ASAS-SN observations with the aim of
looking for any similar behaviour to the one observed in V704 And. The
majority of these objects are taken from Table 4.1 of Warner (1995), which we
complemented with a few additional systems which were identified as VY Scl
type NLs since that table was compiled. See our complete list of the inspected
objects in Table 1. We found that most of these systems behave as typical VY
Scl type objects with high states occasionally interrupted by low states. We
identified only a few systems, which show any kind of outburst-like events
(see Fig. 5). For example, the light curves of HV And, BZ Cam, LQ Peg, and HS
0506+7725 each reveal a small amplitude brightening, but these events show
more complex structure than those of V704 And. In contrast, V1082 Sgr and VZ
Scl show more stochastic brightness variations.

When individual outbursts are examined, some VY Scl type systems show
superficial similarities to the flares of V704 And. Certain brightening events
in the light curves of FY Per, CM Del, or LN UMa seem to have similar duration
and/or amplitude to the flares seen in the light curves of V704 And. However,
there are significant differences between V704 And and these other NLs.
Firstly, the shape, amplitude, and duration of the brightening events of the
latter vary from outburst to outburst, whereas the three flares of V704 And
are remarkably similar. This is also supported by the high cadence TESS light
curve of FY Per (Fig. 8) which shows a faster rise than decay, i.e. the
opposite of the flares in V704 And. Secondly, the long-term light curves
reveal that the outbursts can be continuously present in the light curves of
FY Per and CM Del over timescales of decades (see Fig. 6), which is best
revealed by the higher cadence ASAS-SN and ZTF data; in contrast V704 And
displayed three identical flares over a period of three months.

Among the sources that show features which are superficially similar to the
V704 And flares, LN UMa is the closest match of any system to V704 And. LN UMa
shows alternating low states and high states typical of a VY Scl type object
(Hillwig et al. 1998). However, during the last decade the lightcurve of LN
UMa appears significantly more variable than the regular high states of V704
And. In particular, the high states of LN UMa are often interrupted by a
series of outburst-like events, which can remain present for an extended
period of time. This behaviour is not characteristic of VY Scl type objects,
and to our knowledge has not been reported for this source in the literature
before. It is tempting to assign to the outbursts seen in the LN UMa
lightcurve the same physical mechanism as for the flares seen in V704 And.
However, we note that the accretion rate of LN UMa is likely to be very close
to the critical accretion rate for disc instability; using the
974${}^{+22}_{-21}$ pc distance measured by Gaia (Bailer-Jones et al. 2021)
and the correlation between the absolute magnitude of the disc and the
accretion rate (see eq. 7 in Puebla et al. 2007) yields an estimate for the
accretion rate that (depending on the system inclination) can be above or
below the critical accretion rate. As such it is not currently possible to
decide with any certainty whether LN UMa is above, on, or below the critical
accretion rate, and thus we cannot be confident which physical effect is
responsible for the variability in the LN UMa lightcurve.

Some of the above-mentioned NLs (e.g. FY Per, V794 Aql) are known to exhibit
quasi-periodic stunted outbursts777The cadence of the lightcurve for V794 Aql
is insufficient to accurately characterise the shape of the outbursts (see
Fig. 5), which could be caused by a variety of mechanisms, which include mass
transfer modulations, DN type outburst, DN type outbursts in a truncated disc,
or Z Cam-like outbursts (Honeycutt et al. 1998). Honeycutt (2001) argues that
among these, the accretion disc instability is the favoured driving mechanism.
More recent works show strong indications that some of the stunted outbursts
are caused by mass transfer variations (see e.g. Robertson et al. 2018).
However, these latter authors remain uncertain of the nature of the stunted
outbursts due to discrepancies between their findings, as some of the results
could be explained by both mass transfer variations and accretion disc
instability related to the Z Cam phenomenon. The contradiction regarding the
origin of the stunted outburst is also discussed in the review by Hameury
(2020).

As some of the outbursts in the repeating systems (particularly FY Per, LN UMa
and CM Del) show some similarity to the three isolated flares in V704 And it
is possible that the same mechanism is the driver of both phenomena, including
the stunted outbursts. If these phenomena are all caused by the same mechanism
then, for the reasons argued in Sect. 4.2, this leads to the conclusion that
the stunted outbursts and the variability in e.g. CM Del are driven by mass
transfer variations rather than disc instability. However, we emphasise that
there are significant differences between the three flares seen in V704 And
and the variability we have found in other systems; notably the typical shape
and number of the events. For this reason, we conclude that the flares
reported here in V704 And are more likely to be physically distinct and
therefore could in principle be driven by a different mechanism.

All in all, the flare nature observed in V704 And is extraordinary; it not
only differs from the dwarf nova outbursts, but no similar behaviour has
previously been reported in other VY Scl type NLs. The light curves of V704
And show typical VY Scl type behaviour with a constant brightness in the high
state occasionally interrupted by short periods of low states. As the system
recovered from the latest low state in 2022 it re-established its regular high
state. However, instead of remaining at the usual high state brightness level,
it exhibited three flares and only then it returned to the usual plateau of
the high state (see lower panel of Fig. 1). This makes V704 And the first
clear example of a system which exhibits flares on the plateau of the high
state.

##  5 Conclusions

We report here the discovery of a trio of flares occurring during the high
state of the VY Scl type variable V704 And, behaviour which has never been
reported for a nova-like system before. The observed phenomenon also differs
from the dwarf nova outbursts, therefore, it is unlikely to be caused by disc
instability. Here, we propose that the flares are caused by increased mass
transfer from the secondary and suggest a few mechanisms, such as stellar
activity of the secondary, that could induce it even when the system is
already in the high state. Future observations of V704 And during a prolonged
minimum would be desirable, and would allow for a better understanding of the
nature and activity of the secondary star.

###### Acknowledgements.

We thank the referee for useful comments which helped to improve the
manuscript. We thank Phil Evans for providing technical support with the
installation of the Swift data analysis software. This work made use of data
supplied by the UK Swift Science Data Centre at the University of Leicester.
GZs and CJN acknowledge support from the Leverhulme Trust (grant number
RPG-2021-380). CJN acknowledges support from the Science and Technology
Facilities Council (grant number ST/Y000544/1). KLP acknowledges the support
from the UK Space Agency. We acknowledge with thanks the variable star
observations from the AAVSO International Database contributed by observers
worldwide and used in this research.

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##  Appendix A AAVSO light curve of V704 And

AAVSO has been carrying out observations of V704 And since 1966. We show the
entire available AAVSO light curve for V704 And in Fig. 3.

Figure 3: AAVSO optical light curves of V704 And. The yellow, blue, black, and
grey circles show the B, V, visual, and unfiltered CCD measurements,
respectively. We indicated with the grey shaded area the period, which is
shown in the top panel of Fig. 1.

##  Appendix B Swift X-ray observations

Figure 4: Top: XRT light curve. Bottom: XRT spectra. The different colours
correspond to the spectra taken at the different epochs, as in the top panel.

The Swift satellite also provided X-ray measurements obtained by the X-Ray
Telescope (XRT; operates in the $0.3-10$ keV range) contemporaneously with the
UVOT observations. The X-ray data were obtained in photon counting mode for
about 1.5 ks each. The spectra were created using the XRT online product
generator (Evans et al. 2009), which provided the X-ray count rate and
spectrum for each epoch. We show the XRT light curve in Fig. 4, which does not
reveal high variability during the time of the observations. The brightness of
the source fluctuates around 0.05 counts s-1. We show the X-ray spectra in the
bottom panel of Fig. 4. The different colours correspond to the spectra taken
at the different epochs.

##  Appendix C Confirmation of the viscous timescale in V704 And

In order to confirm the viscous timescale of V704 And we require estimates for
the system parameters, e.g. $M_{1}$, $M_{2}$, $P_{\rm orb}$, and $\dot{M}$.
The orbital period has been measured by Bruch (2022) as $P_{\rm
orb}=0.15424(3)$ days. To estimate the primary and secondary masses we use the
superhump period to determine the mass ratio, and the spectral type of the
secondary to determine $M_{2}$. Wood et al. (2009) presents a relationship
between the mass ratio $q=M_{2}/M_{1}$, the orbital period, and the negative
superhump period deficit
$\varepsilon_{-}=(P_{\textrm{nSH}}-P_{\textrm{orb}})/P_{\textrm{orb}}$. Using
the $P_{\textrm{orb}}$ and $P_{\textrm{nSH}}$ values determined by Bruch
(2022), the mass ratio for the system is $q=0.34$. Weil et al. (2018) estimate
the spectral type of the secondary to be M3$-$M4, and based on this we can
estimate the mass of the secondary to be around
0.3$M_{\odot}$888http://www.pas.rochester.edu/~emamajek/EEM_dwarf_UBVIJHK_colors_Teff.txt
(Pecaut & Mamajek 2013).999This is consistent with the mass estimated from the
mass-period relation for CVs which yields $M_{2}\approx 0.33\,M_{\odot}$ (e.g.
eq. 2.100 of Warner 1995). Combined with the mass ratio of $q=0.34$, this
means that the white dwarf mass is $M_{1}=0.9\,M_{\odot}$.

To estimate the accretion rate through the disc we use the weak correlation
between $P_{\textrm{orb}}$ and $\dot{M}$ (Puebla et al. 2007, see also Warner
1987), which suggests ${\dot{M}}\approx 5\times 10^{-9}$
$M_{\odot}/\textrm{yr}$. As this is a weak correlation, we expect that values
of up to ${\dot{M}}\approx 10^{-8}\,M_{\odot}$/yr are plausible, so we provide
a range of timescales based on a range of accretion rates below. We note that
these accretion rates result in a surface temperature of $\approx(2-3)\times
10^{4}$ K in the outer disc regions, which is sufficient to ensure that the
source remains in the high-state without exhibiting dwarf nova like outbursts.
Furthermore, the absolute magnitude of V704 And is comparable with other
members of the VY Scl type variables, which also supports the argument that
the high state of V704 And is similar to that of other prolonged high states
of this class.

Frank et al. (2002) provide the outer radius of the disc due to tides as
$R\approx 0.9R_{\rm RL}$ (their equation 5.122), where $R_{\rm RL}$ is the
Roche lobe radius given by (Eggleton 1983)

$\frac{R_{\rm RL}}{a}=\frac{0.49q^{2/3}}{0.6q^{2/3}+\ln(1+q^{1/3})}\,$ (1)

where $a$ is the binary separation. From the period and masses given above, we
have that the binary separation is $a=8.9\times 10^{10}$ cm, and therefore the
outer disc radius is $R\approx 2.3\times 10^{10}$ cm.

Finally, we assume a typical white dwarf radius of $R=0.01$ $R_{\odot}$, and
take $\alpha=0.3$. Then using equation 5.50 in Frank et al. (2002) we estimate
the viscous timescale through the disc to be 9.6 days for an accretion rate of
$5\times 10^{-9}M_{\odot}$/yr, and 7.7 days for an accretion rate of
$10^{-8}M_{\odot}$/yr. We therefore conclude that the viscous timescale is
approximately a week, and maybe up to 10 days. The system parameter estimates
also allowed us to determine $\dot{M}_{crit}=1.26\times
10^{-9}M_{\odot}/\textrm{yr}$ for V704 And using equation 13 in Hameury (2020,
see ), which is lower than $\dot{M}$, i.e. the system is expected to be in the
stable regime.

##  Appendix D Other VY Scl type nova-like variables

We have examined a list of VY Scl type nova-like variables in order to find
similar outbursting behaviour to the one observed in V704 And. In general,
none of these objects show outbursts with similar nature, i.e. slower rise and
faster decay. Here, we give comments on a few individual objects, which
exhibit some kind of outburst-like event, and show the corresponding excerpts
of their light curves in Fig. 5

#### HV And

The source exhibits one brightening event which lasts for about 30 days but
has small amplitude and asymmetric shape.

#### V794 Aql

The system shows $\sim$2 mag asymmetric oscillation with $\sim$25 day period.
Honeycutt et al. (2014) studied these stunted outbursts in more detail.

#### BZ Cam

The measurements of BZ Cam showed a brightening event lasting for about 30
days with 0.7 mag amplitude and less symmetry.

#### VZ Scl

The object seem to exhibit a $\sim$1 mag oscillation.

#### LX Ser

The system showed an outburst-like event in 2021 lasting for about 7 months.

#### CM Del

The light curves of CM Del reveal a continuous outbursting behaviour with
varying amplitude. We also note that some works classified this object as a
dwarf nova (La Dous 1991; Lyons et al. 2001).

#### LQ Peg

A 0.4 mag outburst appeared on a $\sim$25-day timescale in the photometric
observations of LQ Peg.

#### HS 0506+7725

The light curves of HS 0506+7725 shows a brightening event with $\approx$1 mag
amplitude and lasting for about 40 days.

#### LN UMa

Several brightening events occur in the light curves of LN UMa with various
shapes and amplitudes. Their duration is typically $20-30$ days.

#### V1082 Sgr

Numerous small amplitude brightenings are revealed by the optical light curves
of V1082 Sgr.

#### TW Pic

The brightness of the system is increased by $\approx$1 mag for the period of
a few days.

#### FY Per

This system shows several stunted outbursts, similarly to V794 Aql. The
stunted outbursts of FY Per are discussed in Honeycutt (2001).

Table 1: List of VY Scl type nova-like variables Object | Reference
---|---
HV And | Henden & Honeycutt (1995)
| Schwope & Reinsch (1992)
PX And | Warner (1995)
MV Lyr | Warner (1995)
V425 Cas | Warner (1995)
V751 Cyg | Warner (1995)
KR Aur | Warner (1995)
V794 Aql | Warner (1995)
DW UMa | Warner (1995)
LY Hya | Warner (1995)
TT Ari | Warner (1995)
BZ Cam | Warner (1995)
WX Ari | Warner (1995)
V442 Oph | Warner (1995)
VZ Scl | Warner (1995)
BH Lyn | Warner (1995)
LX Ser | Warner (1995)
CM Del | Warner (1995)
VY Scl | Warner (1995)
HS 0506+7725 | Downes et al. (2001)
BH Lyn | Downes et al. (2001)
LN UMa | Downes et al. (2001)
V504 Cen | Downes et al. (2001)
V1082 Sgr | Downes et al. (2001)
RX J2337+4308 | Downes et al. (2001)
TW Pic | Puebla et al. (2007)
V533 Her | Honeycutt & Kafka (2004)
FY Per | Honeycutt & Kafka (2004)
LQ Peg | Honeycutt & Kafka (2004)

We have inspected the long-term light curves of a few above-mentioned VY Scl
type sources using observations from AAVSO, ASAS-SN and ZTF. We show the light
curves in Figs. 6 and 7. For their description, see Sect. 4.3.

The Transiting Exoplanet Survey Satellite (TESS, Ricker et al. 2015) observed
FY Per in 2019 December and in 2022 December with 2-min cadence, providing a
$\sim$26-day-long observing sequence for both Sectors. These unprecedentedly
high cadence light curves cover one stunted outburst entirely and the decline
of presumably another outburst (Fig. 8) The 2019 measurements reveal the
detailed shape of a stunted outburst with fast rise and slower decay.

Figure 5: Excerpts of the optical light curves of a few VY Scl type nova-like
variables. The yellow and the dark green symbols show the ASAS-SN $V$ and $g$
band observations, respectively, and the orange and the light green filled
circles indicate the ZTF $r$ and $g$ band observations. The names of the
systems are displayed in the top left corner of each panel.

Figure 6: Long-term lightcurves of FY Per and LN UMa. The top four panels show
the available AAVSO (grey), ASAS-SN $V$ (yellow), ASAS-SN $g$ (dark green),
ZTF $r$ (orange), ZTF $g$ (light green) light curves for FY Per. The bottom
two panel show the long-term light curves for LN UMa based on the same
catalogues.

Figure 7: Long-term lightcurves of CM Del and V794 Aql. The top three panels
show the available AAVSO (grey), ASAS-SN $V$ (yellow), ASAS-SN $g$ (dark
green), ZTF $r$ (orange), ZTF $g$ (light green) light curves for CM Del. The
bottom two panels show the long-term light curves for V794 Aql based on the
same catalogues. We note the lack of available data for the latter source
between $\sim$2005 and $\sim$2010. Figure 8: High cadence TESS light curve of
FY Per from 2019 and 2022. Both TESS Sectors provided an $\sim$26-day-long
light curve with 2-min cadence, which also reveal the
$P_{orb}=6.20\,\textrm{hr}$ (Ritter & Kolb 2003) of FY Per.

##  Appendix E Swift UVOT colour-magnitude diagrams

Figure 9: Optical and UV colour-magnitude diagrams of the Swift/UVOT
measurements.

The multifilter Swift/UVOT measurements allowed us to obtain colour-magnitude
diagrams as well, which are displayed in Fig. 9. These represent the high
state colour variations of the system.