# Caching Scalable Videos in the Edge of Wireless Cellular Networks

Xuewei Zhang, Yuan Ren, Tiejun Lv, , and Lajos Hanzo  Manuscript received
August 10, 2021; revised May 7, 2022; accepted July 27, 2022. _(Corresponding
author: Tiejun Lv and Lajos Hanzo)_. X. Zhang and Y. Ren are with the School
of Communications and Information Engineering, Xi’an University of Posts and
Telecommunications (XUPT), Xi’an, China (e-mail:
{zhangxw,renyuan}@xupt.edu.cn). T. Lv is with the School of Information and
Communication Engineering, Beijing University of Posts and Telecommunications
(BUPT), Beijing, China (e-mail: lvtiejun@bupt.edu.cn). L. Hanzo is with
University of Southampton, Southampton, U.K. (email: lh@ecs.soton.ac.uk).

###### Abstract

By pre-fetching popular videos into the local caches of edge nodes, wireless
edge caching provides an effective means of reducing repeated content
deliveries. To meet the various viewing quality requirements of multimedia
users, scalable video coding (SVC) is integrated with edge caching, where the
constituent layers of scalable videos are flexibly cached and transmitted to
users. In this article, we discuss the challenges arising from the different
content popularity and various viewing requirements of scalable videos, and
present the diverse types of cached contents as well as the corresponding
transmission schemes. We provide an overview of the existing caching schemes,
and summarize the criteria of making caching decisions. A case study is then
presented, where the transmission delay is quantified and used as the
performance metric. Simulation results confirm that giving cognizance to the
realistic requirements of end users is capable of significantly reducing the
content transmission delay, compared to the existing caching schemes operating
without SVC. The results also verify that the transmission delay of the
proposed random caching scheme is lower than that of the caching scheme which
only provides local caching gain.

###### Index Terms:

Content delivery, wireless edge caching, scalable videos, transmission delay.

##  I Introduction

The world is witnessing a brand-new information age, where data traffic is
escalating rapidly right across the globe [1]. The pressure on the network
backhaul which delivers data traffic from the core network to wireless edge
nodes is also escalating, resulting in the degradation of the quality of
service (QoS) provided to end users. Multimedia video services account for the
majority of the increased global data traffic, and the repeated deliveries of
popular contents substantially contribute to the surge of multimedia traffic.
For example, there are 8 Billion video requests in Facebook each day, 83$\%$
of which are represented by the top 1$\%$ of the most popular video contents.
To circumvent this retransmission problem, wireless edge caching is advocated
as a powerful technique, since caching the popular contents can satisfy the
majority of video requests [2].

Fig. 1: An example of wireless edge caching and content transmission scenario,
where the edge nodes roaming in the collaborative area can transmit desired
files to users simultaneously. Multicast transmissions are also adopted to
save the system bandwidth and reduce the power consumption.

With the advent of wireless edge caching, multimedia contents can be pre-
fetched and placed in the edge nodes [3], such as small-cell base stations
(SBSs), device-to-device (D2D) helpers and relays, which have finite cache
capacities. In general, content placement for wireless caching takes into
account the long-term system states to optimally exploit the finite caching
resources to provide the best multimedia services for end users. With the aid
of edge caching, backhaul congestion can be significantly alleviated, and the
users are able to enjoy flawless multimedia services at a reduced service
delay.

An wireless edge caching and content transmission scenario is portrayed in
Fig. 1, where each wireless edge node has finite caching capacity and the
contents to be requested can be pre-stored in the local caches of the nodes.
The macro-cell base stations (MBSs) can provide the users with uncached
contents, which are delivered from the content server through their connected
backhauls. In the collaborative area of Fig. 1, the nodes storing the
requested contents can cooperatively transmit the desired files to the users.
Furthermore, multicast transmissions are also allowed, where the serving nodes
can deliver the requested video contents to a group of users who have the same
watching interests.

In contrast to wired caching, wireless edge caching has its unique
characteristics, which are summarized as follows.

  * •

The wireless edge nodes are often power-limited. In contrast to the hosts and
servers in wired networks, the edge nodes, especially the D2D helpers, may not
have reliable power supply. Hence, the power control is a crucial issue for
wireless caching.

  * •

In contrast to wired caching networks, the network topology of wireless
networks is generically time-variant due to the mobility of end users. To this
end, load balancing is required between different parts of the wireless edge
networks.

  * •

The hostile wireless channels tend to have a more grave effect on edge caching
and content transmission, since wireless channels suffer from small-scale and
large-scale fading as well as co-channel interference. As a result, the
network performance quantified in terms of the transmission delay, transmit
power consumption and energy efficiency (EE) will be significantly degraded.
Hence, when designing edge caching schemes, the contents are less likely to be
placed at the local caches of the edge nodes that cause severer fading and
interference.

Taking the above characteristics into account, wired caching schemes cannot be
directly extended to wireless caching. The content placement and delivery
policies should be specifically designed, so that the finite cache sizes of
edge nodes can be efficiently exploited in the face of limited power resource,
dynamic network topology and complex channel conditions.

We also have to pay attention to the different viewing quality requirements of
end users. For supporting diverse requirements, scalable video coding (SVC)
can be employed for efficient edge caching design. The video files encoded by
SVC are called scalable videos. Developed from the advanced video coding (AVC)
[4], SVC can divide a video file into multiple layers. The layer containing
the most fundamental bit stream of the video is termed as the base layer (BL),
because without this layer the so-called enhancement layers (ELs) cannot be
decoded for enhancing the video quality [5]. The users receiving the BL can
only experience standard definition video (SDV), while those acquiring
successive ELs can enjoy high definition video (HDV) [6]. SVC provides
temporal, spatial and quality scalabilities. In this work, we rely on SVC’s
quality scalability to offer different viewing qualities to the end users.
Compared to the non-scalable H.264/AVC, the SVC configured for quality
scalability will however require extra bits to provide the same fidelity. If
an optimized encoder control is introduced, the extra bit rate can be lower
than 10% [4]. As a compressing benefit, SVC can cut down the end-to-end
latency, boost the fault tolerance and much more significantly reduce the
backhaul pressure. If the effect of packet loss is considered, the quality
degradation of H.264/AVC will be much more severe than that of SVC. By
integrating SVC with edge caching, an additional degree-of-freedom can be
obtained. This method does not transmit unnecessary layers, thus decreasing
the power consumption and reducing the content delivery as well as playout
delay experienced by end users. Some research efforts have been devoted to
SVC-based edge caching, and the related caching schemes under different
wireless scenarios are surveyed in Fig. 2.

Although there are benefits in combining SVC with wireless caching, numerous
design challenges also require further consideration. Firstly, by
intrinsically amalgamating SVC with wireless caching, more SVC-based video
layers need to be cached, which makes the caching schemes more complex.
Moreover, there are multiple viewing quality levels, and the viewing quality
preference has to be modeled or predicted. Therefore, it is quite challenging
to design SVC-based edge caching scheme.

Fig. 2: The overview of applying SVC into wireless edge caching.

In this light, we investigate the edge caching of scalable videos in wireless
networks, where users can enjoy multi-quality video services based on their
preference. The main contributions of this paper are summarized in the
following.

  * •

We discuss two key issues in SVC-based wireless edge caching, namely, the
content popularity and quality preference. We also highlight the methods of
dealing with these two important issues, and analyze the differences between
caching conventional videos and scalable ones.

  * •

We reveal the types of cached contents, portray the most attractive
transmission schemes and summarize the caching decisions required for scalable
videos. The new SVC layer placement strategy is also specified in detail, with
the aim to provide scalable videos having different viewing qualities for end
users.

  * •

Relying on the proposed SVC-based random caching scheme, a case study is
conceived for quantifying the transmission delay, which is one of the most
important performance metrics in wireless edge caching. Simulation results
confirm that the SVC-based layer placement scheme is capable of significantly
reducing the transmission delay, as compared to the conventional caching
scheme operating without SVC. We also show that the delay performance of the
proposed random caching design exceeds that of the benchmark scheme, which
only focuses on providing the local caching gain.

##  II Key Issues for Wireless Edge Caching

In SVC-based wireless edge caching, the video popularity and quality
preference are key issues in determining the layer placement and delivery
schemes.

###  II-A Video Popularity Distribution

The video popularity typically has a strong impact on the content placement,
since popular contents are likely to be cached. Most wireless cache placement
researches rely on the Zipf distribution to describe the video popularity [7].
In Zipf distribution, the request concentration is reflected by the skewness
parameter, where a higher skewness represents more concentrated video
requests. The Zipf distribution works well for web services in the context of
wired caching; however, it is less applicable for video requests appearing in
wireless cellular networks. Considering this fact, the Mandelbrot-Zipf (MZ)
law has been developed for capturing cellular video requests [8]. Let us
assume that there are $F$ video files in the content library, and the files
are ranked in descending order of popularity. Following the MZ distribution,
the request probability of the $f$-th video file is given by
$P_{f}=\frac{(f+q)^{-\alpha}}{\sum_{n=1}^{F}(n+q)^{-\alpha}},\ f=1,2,...,F,$
where $\alpha$ is the skewness parameter and $q$ is the plateau factor. With
the increase of $q$, the difference of the request probabilities decreases
among the popular files. Also note that when the plateau factor is zero, the
MZ distribution becomes the conventional Zipf distribution. The parameters of
the distributions are time-varying, hence they should be specified empirically
by applying curve-fitting methods based on real-world datasets.

Given a specific experimental popularity distribution, we can readily
determine the important parameters for accurately characterizing the video
request popularity. In other cases, however, the existing distributions cannot
accurately reflect the video popularity. In this case, with multiple user
requests collected from wireless edge nodes or content server, the video
popularity can be estimated by machine learning algorithms [9]. Based on the
predicted popularity distribution, low-complexity caching placement
strategies, such as the conventional most popular content placement (MPCP),
are available at the edge nodes.

###  II-B Heterogeneous Quality Preference

In the presence of diverse viewing requirements, the quality preferences of
end users deserve careful consideration. The quality preference is typically
predicted either by using curve-fitting methods or machine learning
algorithms. To predict the quality preference, numerous factors have to be
jointly taken into account, such as the types of video files and users as well
as the screen sizes.

The specific type of the video files is one of the most primary factors that
affects the quality preference. A low viewing quality may be adequate for
news, but high perceptual experiences are preferred for entertainment
programs, such as movies, TV series and variety shows. In this case, more
layers of the SVC-based video files are requested to satisfy the viewing
quality requirements, which increases the request probability of the ELs
belonging to these video files.

The quality preference also depends on the type of users. Typically, users
having good channel conditions are likely to request HDVs for superb viewing
experiences, and more ELs are requested and delivered. Often, superior
perceptual qualities are delivered to the VIP users, e.g., those who pay
premiums. Owing to the fact revealed above, the quality preference differs
between different classes of users.

The screen size is the third factor. The devices having large screen sizes
usually require HDVs, since they are sensitive to grainy viewing qualities. By
contrast, devices with smaller screen sizes may tolerate SDVs.

##  III Content Delivery Process for Scalable Videos

Given the layered structure of scalable videos, the content delivery processes
are different from those of their traditional counterparts. Before designing
the content delivery schemes, a discussion about the types of cached contents
is provided.

###  III-A Types of Cached Contents for Scalable Videos

By applying the SVC technique, each video file is encoded into multiple
layers. The SVC-based layers can be locally cached, and the edge nodes can
store the BL and multiple ELs for each scalable video, each of which is
regarded as a caching unit. After end users receive a sufficient number of
layers to satisfy their viewing requirements, the SVC decoding can be
performed to recover the requested videos having the preferred quality levels.

As a design alternative, a set of successive layers can also be viewed as a
caching unit, which we refer to as a super layer (SL) [10]. Each SL consists
of a single BL and a number of successive ELs, where the specific number of
ELs is determined by the particular viewing quality requirements of users.
When the SL is transmitted to the user, it is capable of providing the
requested video file at any of the preferred viewing quality by delivering the
BL and any number of ELs. When SLs are cached and transmitted to users, the
encoding and decoding processes are completed at the content server in the
core network in advance. More specifically, the original video is firstly
partitioned into multiple layers by SVC, and the successive SVC-based layers
are then decoded into the videos having different viewing qualities, i.e.,
SLs, by SVC decoding. As a benefit, if SLs are provided, seamless video
services can be provided, and the decoding process is conveniently avoided at
the receivers’ side.

Naturally, there are a number of trade-offs. Firstly, it is worth pointing out
that caching the individual SVC-based layers can enhance the content
diversity, as compared to storing the SLs, since the size of the SL is
typically larger than that of its constituent BL and ELs. This indicates that,
when the edge nodes are equipped with smaller cache sizes, it is a wise choice
to store individual BLs and ELs. Moreover, when the computational capability
of receivers is limited, the edge nodes prefer to cache SLs instead.

Fig. 3: An example of the ILT scheme, where each video is partitioned into
four layers and the users request the video files colored in purple and
orange. The SBSs caching the requested layers transmit desired layers to users
simultaneously, and the un-cached layers are retrieved and transmitted by the
MBS.

###  III-B Content Delivery Designs

Given the different kinds of cached contents, the transmission policies can be
accordingly designed.

Scenario 1: If multiple individual BLs and ELs are cached in the local caches
of edge nodes, the nodes can individually transmit the requested layers to the
user. This design is referred to as individual layer based transmission (ILT)
in this article. Typically, the serving nodes share the same frequency and
time resources. Hence, they can be tightly coordinated by the MBS covering
these nodes. At the beginning of each time slot, the nodes caching the desired
layers are synchronized, and then the requested layers are transmitted to the
user. When the user receives multiple super-imposed content layers, successive
interference cancellation (SIC) is required for separating the layers [11].
Then the user can decode the desired video file at preferable viewing quality
by performing SVC decoding. An example of this kind of content transmission
design is shown in Fig. 3.

Scenario 2: In another case where the SLs are locally stored, the cached SLs
can be delivered by the serving nodes. This content delivery strategy is
referred to as SL based transmission (SLT). The nodes having the requested SLs
in their caches can serve the end users, and the cooperative transmission and
multicast transmission are also allowed to deliver the requested SLs. The
user, who receives the SL, can directly view the requested multimedia video
without SIC and SVC decoding. As a result, the playback performance can be
substantially improved. This kind of transmission design is portrayed in Fig.
4.

Fig. 4: An illustration of the SLT scheme, in which the cooperative
transmission, multicast transmission and point-to-point transmission are
allowed.

Both Scenarios 1 and 2 have their pros and cons. Although the ILT scheme
requires a complex receiver for SIC and SVC decoding, it allows for the
cooperation of multiple serving nodes, hence avoiding a single-point failure.
Compared to the ILT, the SLT scheme simplifies the receiver design, which
therefore reduces the playout delay. However, when employing the SLT scheme,
if the serving node fails to provide the requested content, the user would
require retransmissions and suffer from longer delays in retrieving the
requested content. This is different from using the ILT scheme, which only has
to deliver the missing BL or ELs. In practical implementations, the specific
choice of the type of cached contents primarily depends on the network
topology, the receiver design and the playout requirements of end users.

##  IV Caching Decisions for Scalable Videos

In this section, we first review the existing wireless caching schemes
conceived for salable videos, and then present a new caching strategy. The
design criteria of the caching decisions are also summarized.

###  IV-A Caching Schemes for Scalable Videos

We first discuss the caching schemes suitable for scalable videos. Typically,
depending on the types of caching decisions, the caching schemes can be
classified into two categories. In the first category, the caching decisions
are continuous variables, ranging from 0 to 1. There are two popular caching
schemes in this category, namely, fractional caching [11] and random caching
[12]. The details of these two schemes are listed as follows.

  * •

Fractional Caching. Under this caching scheme, each single layer or SL is
partly cached in the local caches of edge nodes. The caching fraction for each
layer is determined later. The uncached parts of the requested contents have
to be delivered from the core network via the backhauls. Typically, fractional
caching is suitable for layers with large sizes to improve the layer diversity
and content hit probability.

  * •

Random Caching (also termed as probabilistic caching). Based on this caching
strategy, the edge nodes can probabilistically store the single layers or SLs
based on the caching probability distributions. The layer having a higher
caching probability is more likely to be locally cached. This caching scheme
is capable of operating in the face of user mobility [13].

Furthermore, there is also a new type of layer placement for scalable videos,
where the caching decisions are binary variables, i.e., either 0 or 1.
Specifically, if the caching indicator is 1, the corresponding layer will be
locally cached. Otherwise, the layer cannot be found in the local caches. In
most cases, the optimization problems associated with binary variables are NP-
hard, and cannot be efficiently solved in polynomial time. To this end, it is
important to design efficient layer placement schemes operating at a reduced
computational complexity.

###  IV-B Caching Criteria for Scalable Videos

Under the above caching schemes, the remaining problem is how to determine the
caching decisions for the edge nodes. For different load balancing states,
user requirements and power consumption levels, different caching criteria
have to be considered. In the literature, the following caching criteria have
been studied:

  * •

Average download time [14]. When SVC-based caching schemes are adopted at the
edge nodes, unnecessary ELs do not have to be delivered, thus reducing the
content download delay. If ILT is adopted, the download delay depends on the
maximum time of downloading each requested layer; otherwise, the download
delay is determined by the total time for downloading the requested SLs.

  * •

Transmit power consumption [10]. In contrast to the video files without SVC
processing, caching and transmitting SVC-based layers can decrease the power
consumption of edge nodes. Specifically, multicast transmission can be adopted
for further reducing the total power consumption [10].

  * •

EE [12]. As revealed in [12], there is a trade-off between the data rates of
end users and the total power consumption, and the power savings should not
degrade the users’ QoS. In this sense, the EE has to be carefully considered
when providing scalable video services. It is worth noting that the
cooperation among multiple serving nodes should also be taken into account,
when maximizing the EE in the scenario where individual layers are cached and
transmitted simultaneously. Moreover, when establishing the total power
consumption model, the consumption arising from content caching and backhaul
deliveries should also be accurately modeled.

  * •

Content transmission delay [15]. In modern cellular networks, the finite
capacity of the backhaul may lead to an adverse effect on the content
transmission delay. Severe backhaul congestion usually results in an
unacceptable latency. When scalable videos are cached at the wireless edge,
users can retrieve part of the requested contents without backhaul
involvement, hence the transmission delay caused by backhaul-based retrieval
can be avoided. Moreover, since the number of transmitted layers is tailored
for each user, superfluous layers are not delivered, which further reduces the
transmission delay.

##  V Case Study of SVC-based Wireless Caching

In this section, we present a case study to validate the benefits of caching
and transmitting scalable videos.

###  V-A System Model

Network Model: A three-tier heterogeneous network, consisting of a D2D tier,
an SBS tier and an MBS tier, is observed. The three-tier network model
considered is practical, and it may be readily extended to multi-tier models
or simplified to two-tier models. For the different number of tiers and for
diverse node deployments, the content caching and transmission designs have to
be appropriately adjusted, based on similar steps to those advocated in this
paper. The locations of edge caching nodes follow the classic Poisson point
processes (PPPs) with different distribution densities. The density of D2D
helpers is the highest among the three types of nodes, while the MBSs have the
lowest distribution density. The fading and interference inflicted by wireless
channels are modeled as follows. The small-scale fading follows the Rayleigh
distribution with zero mean and unit variance, i.e., $\mathcal{CN}(0,1)$; the
pathloss is described by $d^{-\alpha}$, where $d$ denotes the distance between
the node and the user and $\alpha$ is the pathloss exponent; finally, the co-
channel interference experiencing both small-scale and large-scale fading is
also included in the denominator of the signal-to-interference-plus-noise
ratio (SINR) experienced by the user.

Transmission and Caching Policies: The D2D helpers and SBSs are equipped with
finite caching storage. They can provide cached layers to nearby users. The
D2D helpers and SBSs that are capable of providing cached layers are
distributed in the serving areas having radii $r_{d}$ and $r_{s}$ around the
requesting user. If the requested layers can be found in the nearby D2D
helpers, the nearest D2D helper in the vicinity of the user and caching the
requested layers is selected as the serving node. If the layers cannot be
found in the caches of the D2D helpers, the user will resort to fetching them
from the SBSs. When edge nodes provide the users with requested layers, the
transmission delay is caused by the downlink transmission from the nodes to
the users. However, if some of the requested layers are not cached locally,
the nearest MBS around the user will provide the missing layers from the
content server via the backhaul. Apart from the delay of downlink transmission
from the MBS to users, the delay contributed by backhaul retrievals is also
non-negligible in this case. Regarding the caching scheme, the random caching
strategy is adopted, where SVC-based layers are probabilistically cached in
the edge nodes.

Problem Formulation and Solutions: Under the proposed transmission and caching
policies, the optimization problem of transmission delay minimization is
formulated, with the constraints of finite cache sizes of each D2D helper and
SBS. The optimization problem is solved by the standard gradient projection
method, which is eminently efficient in solving our optimization problem
having a differentiable objective function over a convex variable set [12].
Then, the random caching probability of each layer is found. In the proposed
algorithm, the partial derivatives of the transmission delay with respect to
caching probabilities have to be calculated for each iteration. The
computational cost of the partial derivative calculations dominates the
complexity of the algorithm, and the number of calculations is proportional to
$F\cdot L$, where $F$ and $L$ are the total numbers of video files and content
layers, respectively. Thus, when employing the ${\cal O}$ function for our
complexity analysis, the overall overhead of the proposed algorithm is ${\cal
O}(F\cdot L)$.

###  V-B Simulation Results

Fig. 5: The transmission delay versus the backhaul delivery rates.

In the simulations, the size of each video without SVC processing is 50 Mbits.
Inspired by [4], we set the size of each SVC-based video file to be 10% larger
than that of the file without SVC. For each scalable video, it is partitioned
into 8 layers. The radii of the serving areas for the D2D helpers and SBSs are
$10$ m and $30$ m, respectively. The path-loss exponent is 4 for D2D helpers,
SBSs and MBSs. The cache sizes of each D2D helper and SBS are 200 Mbits and
500 Mbits. We adopt the most popular layer placement (MPLP) as our benchmark
scheme, where the SVC-based layers from the most popular videos fully occupy
the local caches of D2D helpers and SBSs.

In Fig. 4, we present the relationship between the content transmission delay
and backhaul delivery rate. Clearly, a high backhaul delivery rate will reduce
the transmission delay. In the benchmark schemes, the scheme without caching
yields the worst delay, since all requested layers have to be retrieved by
capacity-limited backhauls. We can also find that by providing SVC-based video
services, the transmission delay can be significantly decreased. This is
because un-requested layers are not delivered to end users. Compared to the
MPLP scheme, random caching scheme can indeed reduce the transmission delay.
The MPLP only focuses on providing the local caching gain that each individual
node can provide, while the random caching can exploit the aggregated cache
sizes of all caching nodes. As a result, random caching significantly reduces
the transmission delay.

Fig. 6: The transmission delay versus the cache sizes of SBSs.

The relationship between the content transmission delay and the cache size of
each SBS is characterized in Fig. 5. Observe that the transmission delay
decreases as the cache size of each SBS grows, since more cached layers are
provided for end users and repeated backhaul deliveries are substantially
reduced. Likewise, the delay of random caching is lower than that of MPLP, and
the SVC-aided caching scheme imposes lower transmission delay than that
without SVC.

##  VI Conclusions and the Future Directions

In this article, wireless edge caching designed for scalable videos was
studied. We discussed the issues of content popularity as well as quality
preference, and collated the types of cached contents and their corresponding
transmission schemes. We elaborated on the caching options in support of
scalable video services and the performance metrics used for making caching
decisions. Finally, a case study was presented to validate the benefits of
caching and transmitting SVC-based layers. From our simulation results, we
concluded that caching and transmitting scalable videos is capable of reducing
the transmission delay. Additionally, it can be observed that the transmission
delay of the proposed random caching scheme is lower than that of the caching
scheme which only focuses on providing the local caching gain.

As a promising technique to reduce repeated content deliveries and relive the
backhaul pressure in future wireless networks, there still remain some crucial
challenges to be solved in the field of edge caching, listed as follows.

  * •

The transmission reliability of the BL is not well documented in the
literature, even though the BL is the most essential part of scalable video
streamings. When offering scalable video services to end users, how to
guarantee the BL is reliably transmitted and decoded remains a future
challenge.

  * •

The trade-off between encoding/decoding complexity and the performance
improvement attained needs a balanced consideration. With more SVC-based
layers, the system performance will be improved, since more layers can be
flexibly cached and transmitted. However, the encoding/decoding complexity
escalates with the number of divided layers. Hence, a compelling trade-off
between the encoding/decoding complexity and the performance improvement is
desired.

  * •

When designing caching schemes, having efficient power control is another
crucial issue that deserves further investigations. Firstly, the edge nodes
providing the scalable video services are often power-limited, especially the
D2D helpers, and hence the power control plays a prominent role. Power control
actions typically have a shorter time scale than content caching, which
imposes a challenge when jointly designing the power control and content
caching schemes. Additionally, for SVC-based edge caching, the power
consumption, video quality and transmission delay are important aspects that
have to be jointly considered. These metrics are often conflicting and coupled
with each other. To strike an attractive trade-off among them, it is of prime
concern to formulate a multi-objective optimization problem, which may be
solved by the sophisticated multi-objective evolutionary algorithm.
Alternatively, it may be transformed into a single-objective optimization
problem associated with $\varepsilon$ constraints. If there are a high number
of parameters, machine learning (ML) may be used for solving large-scale
Pareto optimization problems. With these effective algorithms, all Pareto
optimal points of the Pareto front can be found.

  * •

The security and privacy of end users and D2D helpers may not be guaranteed.
On one hand, when using machine learning approaches for predicting the video
popularity and quality preference, a large amount of information will be
collected by the edge nodes or content server. The collected information is
personal and private, and may be wiretapped by the eavesdroppers. On the other
hand, when D2D helpers assist in caching and transmitting SVC-based layers,
the personal information of the helpers may be exposed to end users, which
also causes rather severe security and privacy problems. Instead of our helper
selection scheme, helpers may also be selected for improving the information
security and minimizing the eavesdropping probability. Furthermore, the
secrecy rate, together with the power consumption, the video quality and the
transmission delay, could also be incorporated into the multi-objective
optimization problem.

These issues deserve research attention and will guide our future directions.

## Acknowledgement

X. Zhang and Y. Ren would like to acknowledge the financial support of the
National Natural Science Foundation of China under Grants 62101442, 61801382,
62071377 and 62001264, the Natural Science Foundation of Shaanxi Province
under Grant 2021JQ-705, and the Young Talent fund of University Association
for Science and Technology in Shaanxi, China under Grant 20210115.

L. Hanzo would like to acknowledge the financial support of the Engineering
and Physical Sciences Research Council projects EP/W016605/1 and EP/P003990/1
(COALESCE) as well as of the European Research Council’s Advanced Fellow Grant
QuantCom (Grant No. 789028).

## References

  * [1] C. Fan, T. Zhang, Y. Liu, and Z. Zeng, “Cache-enabled HetNets with limited backhaul: A stochastic geometry model,” _IEEE Trans. Commun._ , vol. 68, no. 11, pp. 7007–7022, Nov. 2020.
  * [2] K. Wang, J. Li, Y. Yang, W. Chen, and L. Hanzo, “Content-centric heterogeneous fog networks relying on energy efficiency optimization,” _IEEE Trans. Veh. Technol._ , vol. 69, no. 11, pp. 13 579–13 592, Nov. 2020\.
  * [3] X. Wu, Q. Li, V. C. M. Leung, and P. C. Ching, “Joint fronthaul multicast and cooperative beamforming for cache-enabled cloud-based small cell networks: An MDS codes-aided approach,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 10, pp. 4970–4982, Aug. 2019.
  * [4] H. Schwarz, D. Marpe, and T. Wiegand, “Overview of the scalable video coding extension of the H.264/AVC standard,” _IEEE Trans. Circuits Syst. Video Technol._ , vol. 17, no. 9, pp. 1103–1120, Sep. 2007.
  * [5] J. M. Boyce, Y. Ye, J. Chen, and A. K. Ramasubramonian, “Overview of SHVC: Scalable extensions of the high efficiency video coding standard,” _IEEE Trans. Circuits Syst. Video Technol._ , vol. 26, no. 1, pp. 20–34, Jan. 2016.
  * [6] P. Ostovari, J. Wu, A. Khreishah, and N. B. Shroff, “Scalable video streaming with helper nodes using random linear network coding,” _IEEE/ACM Trans. Netw._ , vol. 24, no. 3, pp. 1574–1587, Jun. 2015.
  * [7] L. Breslau, P. Cao, L. Fan, G. Phillips, and S. Shenker, “Web caching and Zipf-like distributions: Evidence and implications,” in _Proc. IEEE Conf. Computer Commun. (INFOCOM)_ , New York, USA, Mar. 1999, pp. 126–134.
  * [8] M. Lee, M. Ji, A. Molisch, and N. Sastry, “Throughput-outage analysis and evaluation of cache-aided D2D networks with measured popularity distributions,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 11, pp. 5316–5332, Nov. 2019.
  * [9] B. Chen and C. Yang, “Caching policy for cache-enabled D2D communications by learning user preference,” _IEEE Trans. Commun._ , vol. 66, no. 12, pp. 6586–6601, Dec. 2018.
  * [10] C. Guo, Y. Cui, D. W. K. Ng, and Z. Liu, “Multi-quality multicast beamforming with scalable video coding,” _IEEE Trans. Commun._ , vol. 66, no. 11, pp. 5662–5677, Nov. 2018.
  * [11] X. Xu and M. Tao, “Modeling, analysis, and optimization of coded caching in small-cell networks,” _IEEE Trans. Commun._ , vol. 65, no. 8, pp. 3415–3428, Aug. 2017.
  * [12] X. Zhang, T. Lv, W. Ni, J. M. Cioffi, N. C. Beaulieu, and Y. J. Guo, “Energy-efficient caching for scalable videos in heterogeneous networks,” _IEEE J. Sel. Areas Commun._ , vol. 36, no. 8, pp. 1802–1815, Aug. 2018.
  * [13] J. Yao, T. Han, and N. Ansari, “On mobile edge caching,” _IEEE Commun. Surveys Tuts._ , vol. 21, no. 3, pp. 2525–2553, 3rd Quart. 2019.
  * [14] C. Zhan and Z. Wen, “Content cache placement for scalable video in heterogeneous wireless network,” _IEEE Commun. Lett._ , vol. 21, no. 12, pp. 2714–2717, Dec. 2017.
  * [15] Y. Li, C. Zhong, M. C. Gursoy, and S. Velipasalar, “Learning-based delay-aware caching in wireless D2D caching networks,” _IEEE Access_ , vol. 6, pp. 77 250–77 264, Nov. 2018.

## Biographies

Xuewei Zhang<EMAIL_ADDRESS>received the Ph.D. degree in information
and communication engineering from the Beijing University of Posts and
Telecommunications (BUPT), Beijing, China, in 2020. She is currently an
Associate Professor with the School of Communications and Information
Engineering, Xi’an University of Posts and Telecommunications (XUPT), Xi’an,
China. Her research interests include wireless edge caching, resource
allocation and heterogeneous networking.

Yuan Ren received the B.Eng. degree in information engineering and the Ph.D.
degree in signal and information processing from the Beijing University of
Posts and Telecommunications (BUPT), Beijing, China, in 2010 and 2017,
respectively. He is currently an Associate Professor with the School of
Communications and Information Engineering, Xi’an University of Posts and
Telecommunications (XUPT), Xi’an, China. His current research interests
include green communications, wireless caching and cooperative communications.

Tiejun Lv received the M.S. and Ph.D. degrees in electronic engineering from
the University of Electronic Science and Technology of China (UESTC), Chengdu,
China, in 1997 and 2000, respectively. He is the author of 3 books, more than
100 published IEEE journal papers and 200 conference papers on the physical
layer of wireless mobile communications. His current research interests
include signal processing, communications theory and networking.

Hanzo Lajos received his doctorate in 1983. He holds an honorary doctorate by
the Technical University of Budapest (2009) and by the University of Edinburgh
(2015). He is a member of the Hungarian Academy of Sciences and a former EIC
of the IEEE Press. He is a Governor of both IEEE ComSoc and VTS.


# Sufficient Loading Conditions for Self-organization in the $n$-dimensional
Version of BML Model with Stochastic Direction Choice

Valery V. Kozlov
Steklov Mathematical Institute
of Russian Academy of Sciences
Moscow, Russial
<EMAIL_ADDRESS>
Alexander G. Tatashev
Department of Higher Mathematics
Moscow Automobile and Road Construction
State Technical University (MADI)
Moscow, Leningradsky avenue, 64, Russia
<EMAIL_ADDRESS>
Marina V. Yashina
Department of Higher Mathematics
Moscow Automobile and Road Construction
State Technical University (MADI)
Moscow, Leningradsky avenue, 64, Russia
<EMAIL_ADDRESS>

###### Abstract

A dynamical system is considered, which comprises an $n$-dimensional lattice
$N_{1}\times N_{2}\times\dots\times N_{n}$ with periodic boundary conditions.
Particles traverse this lattice following a variant of the
Biham–Middleton–Levine (BML) traffic model’s particle movement rules. We have
proved that the BML model, when treated as a dynamical system, constitutes a
specialized class of a Buslaev net. This equivalence allows us to employ
established Buslaev net analysis techniques to investigate the BML model. In
Buslaev nets conception the self-organization property of the system
corresponds to the existance of velocity single point spectrum equal to 1. One
notable aspect of the model under consideration is that particles can change
their type with a certain probability. For simplicity, we assume a constant
probability $q$ that a particle changes type at each step. In the l case where
$q=0$, the system corresponds to the classical version of the BML model. We
define a state of the system where all particles continue to move
indefinitely, both in the present and the future, as a state of free movement.
A sufficient condition for the system to result in a state of free movement
from any initial state (condition for self-organization) has been found. This
condition is that the number of particles be not greater than half the
greatest common divisor of the numbers $N_{1}$ $N_{2},\dots,N_{n}.$

_Keywords_ Dynamical systems $\cdot$ Traffc flow model on networks $\cdot$
Biham–Middleton–Levine (BML) model $\cdot$ Self-organization $\cdot$ Buslaev
nets $\cdot$ Spectrum of Buslaev net

##  1 Introduction

Urban traffic congestion is a critical issue affecting cities worldwide. It
results in significant economic losses, environmental damage, and substantial
disruptions in individuals’ daily lives. This problem is particularly acute in
large megacities, prompting extensive research efforts aimed at improving
transportation infrastructure and minimizing public expenditures. The adoption
of new automotive technologies plays a pivotal role in enhancing traffic
control, while the development of communication networks offers drivers
opportunities to select optimal routes, thereby alleviating congestion in
major cities. Consequently, the development of algorithms for effective
traffic control is of utmost importance. Mathematical models serve as
invaluable tools for deriving optimal solutions to these challenges.
Therefore, predicting and mitigating traffic jams are crucial focal points.
One widely utilized approach for modeling traffic flows in urban networks and
predicting traffic congestion is based on various versions of the renowned
Biham–Middleton-Levine (BML) model, [1]. The BML model, renowned for its
simplicity and efficiency, was introduced to address traffic-related issues.

In O. Biham, A.A. Middleton and D. Levine study [1], a mathematical traffic
model, the BML model, was introduced. In this model, particles navigate a two-
dimensional toroidal lattice comprising $N_{1}$ rows and $N_{2}$ columns,
moving at discrete moments. These particles come in two types: particles of
the first type (referred to as "red particles") move horizontally from left to
right, while particles of the second type (referred to as "blue particles")
move vertically along columns. At any given moment, a cell can either be
vacant or contain a particle of one of these two types. Particle movement
along rows or columns adheres to a rule derived from the elementary cellular
automaton 184 (ECA 184) [2]. According to this rule, a particle advances to an
adjacent vacant cell in the same direction, or it remains stationary if the
cell ahead is occupied. Additionally, a rule is in place to prevent two
particles from entering the same cell simultaneously, one moving along a row
and the other along a column. Specifically, the rule dictates that particles
of the first type move first, followed by particles of the second type.
Alternatively, one can envision a scenario where particles of the first type
move during odd time steps, and those of the second type move during even time
steps.

In a study by Linesh et al [3], results are presented that investigate the
behavior of the BML model across various lattice dimensions and particle
densities. This investigation relies on simulation modeling techniques. One of
the primary challenges in analyzing traffic model systems is determining the
average particle velocity, which quantifies the ratio of particles moving
during a step to the total number of particles. If, starting from a certain
point in time, all particles move without delay, the particle velocity reaches
a value of 1, indicating a state of free movement within the system. This
inherent property of the system to transition into a state of free movement is
called the as self-organization.

In the work by Austin et al [4], it has been provted that if $N_{1}=N_{2}=N$
and the number of particles $m$ is less than $N/2$, the system consistently
evolves into a state of free movement from any initial state. Similarly,
Moradi [5] established that when $m=N/2$, the system also attains a state of
free movement from any initial state.

For a related one-dimensional system, where particles adhere to the rule
ECA184 while traversing a circular lattice with $N$ cells, Blank [6] and
Belitsky and Ferrary [7] provided evidence that if the number of particles
does not exceed $N/2$, the system reliably transitions into a state of free
movement from any starting point. Conversely, if there are more than $N/2$
particles, the average velocity of particles drops below 1. Additionally, they
derived a formula for calculating the average velocity.

In [8], a system is considered such that this system contains two closed one-
dimensional lattices called contours (two-contour system). The lengths of
contours are different. The system belongs to a class of dynamical systems
called Buslaev nets [9], [10]. In [8], it is supposed that there is a common
cell of the contours, and particles move along each contour according to the
rule of ECA 184. After passing a node, a particle passes to another contour
with some probability. It has been proved that, if the number of particles is
not greater than a half of the greatest common divisor of the numbers of cells
in the contours, then the system results in a state of free movement from any
initial state. Under the additional condition that the probability of
transition from the first contour circuit to the second contour and the
probability of transition from the second circuit to the first are not equal
to 0, this sufficient condition for self-organization is also necessary. In
proving these statements, known facts of the number theory relating to the
theory of equations of two variables in integers are used.

Tatashev and Yashina [8] explored a system involving two closed one-
dimensional lattices, referred to as contours, within a two-contour system.
These contours had varying lengths and belonged to the class of dynamical
systems known as Buslaev nets [9, 10]. In this study, it was assumed that
there was a common cell shared by the contours, and particles traversed each
contour following the ECA 184 rule. The research established that when the
number of particles did not exceed half of the greatest common divisor of the
cell numbers in the contours, the system reliably reached a state of free
movement from any initial state. Moreover, when an additional condition
stipulating nonzero transition probabilities from the first contour circuit to
the second, and vice versa, was imposed, this sufficient condition for self-
organization was also proven to be necessary. The proofs of these assertions
relied on well-established principles from number theory, specifically
relating to equations involving two integer variables.

It is worth mentioning that in Wenbin et al’s investigation [11], a version of
the BML model was considered where particles could pass through common cells
of both rows and columns simultaneously. In this scenario, the system was also
represented as a contour network, and certain cells were identified as nodes
while others were not.

Additionally, Ding et al [12] and Yashina et al [13] explored stochastic
versions of the Biham-Middleton-Levine model.

In another study by Chau et al [14], an $n$-dimensional version of the BML
model was introduced within an $N_{1}\times N_{2}\dots\times N_{n}$ lattice,
incorporating periodic boundary conditions. In reference [13], the model was
further examined through simulations under the assumptions that $n=3$ and
$N_{1}=N_{2}=N_{3}$.

In the works of Criado and Wan [15] and Gillman and Martinsson [16], various
mathematical models with a two-dimensional structure are explored, along with
their applications.

In this paper, we extend upon the system discussed in Biham-Middleton-Levine’s
work [1]. This system encompasses an $n$-dimensional lattice with dimensions
$N_{1}\times N_{2}\times\dots\times N_{n}$. At discrete moments in time, each
particle undergoes a type change with a certain probability. For simplicity,
we assume a constant probability denoted as $0\leq q<1$, and the new direction
is chosen with equal probability among available options.

When $n=2$, $N_{1}=N_{2}$, and $q=0$, the system aligns with the one analyzed
in Austin et al’s study [4]. It has been established that a sufficient
condition for the system to evolve into a state of free movement from any
initial state is that the number of particles must not exceed half of the
greatest common divisor of the dimensions $N_{1}$, $N_{2}$, and so on, up to
$N_{n}$. The proof of this condition draws upon methodologies employed in
Austin et al [4], Moradi et al [5], Tatashev and Yashina [8], and incorporates
principles from number theory [17].

Furthermore, we present a description of the system in terms of Buslaev nets,
building upon the established embedding of the BML model into the Buslaev nets
class, as demonstrated in Yashina et al’s work [13]. In this framework, the
rows of lattice cells are referred to as contours, with each cell
corresponding to a node shared by multiple contours. This allows us to apply
methodologies developed for the analysis of contour networks to the study of
the BML model. A crucial concept introduced here is the spectrum of the
Buslaev net, encompassing sets of limit cycles (closed trajectories within the
net’s state space) and the corresponding velocities of particles moving within
the contours.

##  2 Description of model

The system contains an $n$-dimensional lattice $N_{1}\times
N_{2}\times\dots\times N_{n}$ with periodic boundary conditions.

Figure 1:  Classical BML model trajectory with $n=2$

There are $m<N_{1}N_{2}\dots N_{n}$ particles. At each moment, any particle
belongs to one of $n$ types. In general case, the type of a particle may be
changed. Particles move at discrete moments, $t=0,1,2,\dots$ At each moment of
time, first particles of the type 1 move, then particles of the type 2 move,
then particles of the type 3 move, etc, Fig. 1.

After that, type of each particle changes with probability $q$, $0\leq q<1.$
Suppose, at time $t,$ a particle of the $i$th type is in the cell
$(x_{1},\dots,x_{n})$ and, after that the particles of the types $1,\dots,i-1$
could move, the cell $(x_{1},\dots,x_{i-1},x_{i}+1,x_{i+1},\dots,x_{n})$ is
vacant, then, at time $t+1,$ the particle will be in the cell
$(x_{1},\dots,x_{i-1},x_{i}+1,x_{i+1},\dots,x_{n}),$ $i=1,\dots,n,$ $0\leq
x_{j}\leq N_{j},$ $j=1,\dots,N,$ , Fig. 2 with $n=2.$

Note that, if $n=2,$ then we have a BML model in two-dimensional torroidal
lattice.

Figure 2:  Trajectories in BML model with $q$ probability of direction change

##  3 BML model in terms of Buslaev nets

Suppose $q=0.$ Let us describe the model in terms of Buslaev nets.

There are $N_{1}\cdot N_{2}\cdot N_{i-1}\cdot\dots\cdot N_{i+1}\cdot N_{n}$
contours called the contours of the type $i$ such that particles of the $i$th
type move along these contours, $i=1,\dots,N.$ The particles of the type $i$
move in the contours of the type $i,$ $i=1,\dots,n.$ If $n=2,$ then there are
$N_{2}$ contours of the type 1, and there are $N_{1}$ contours of the type 2.
The $i$th contour of the type 1 contains the cells
$(0,i),(1,i),\dots,(N_{1},i),$ $i=1,\dots,N_{2}.$ The $i$th contour of the
type 2 contains the cells $(i,0),(i,1),\dots,(i,N_{2}),$ $i=1,\dots,N_{1}.$

In a Buslaev net, if more than one particle come to a common node
simultaneously, then then a conflict occurs and a particle passes through the
node according to the prescribed conflict resolution rule. In the considered
net, a conflict occurs between a particles different type. In the contour
networks, the conflict may be deterministic or stochastic. In the considered
system a conflict is resolved by that first the particles of the first type
move, and then the particles of the second type move, etc.

##  4 Sufficient condition for self-organization

We say that, at time $t_{0},$ the system is in a state of free movement if, at
any moment $t=t_{0},t_{0}+1,t_{0}+2,\dots,$ all particles move with
probability 1.

Let $d$ be the greatest common divisor of the numbers $N_{1},$
$N_{2},\dots,N_{n}.$

A set of cells $(x_{1},\dots,x_{n})$ such that

$\sum\limits_{i=1}^{n}x_{i}=k\ {\rm(mod}\ d)$

is called the diagonal $k,$ $k=0,1,\dots,d-1.$

By definition, put

$\phi_{s}(t)=k-t\ (mod\ d)$

if, at time $t,$ the $s$th particle is in a cell of the diagonal $k,$
$s=1,\dots,m,$ $k=1,\dots,d.$

Suppose $(x_{1}(t),\dots,x_{d}(t))$ is the vector such that $x_{a}(t)=0$ if,
for any $s=1,\dots,m,$ $\phi_{s}(t)\neq a,$ and $x_{k}(t)=1$ if, there exists
$s\in\\{1,\dots,m\\}$ such that $\phi_{s}(t)=a.$ Lemma 1. Suppose
$\phi_{s}(t)=a.$ If, at time $t,$ the $s$th particle moves, then
$\phi_{s}(t+1)=a,$ and, if the particle does not move, then
$\phi_{s}(t+1)=a-1$ (mod $d)$ Lemma 1 follows from definitions. Lemma 2.
Suppose $x_{a-1}(t)=1,$ $x_{a}(t)=x_{a+1}(t)=\dots=x_{a+l}(t)=0$ (addition
modulo $d),$ $l\geq 1.$ Then $x_{a-1}(t+1)=1,$
$x_{a}(t+1)=x_{a+1}(t+1),\dots,x_{a+l-1}(t+1)=0.$ Proof. At time $t,$ the
$s$th particle such that $\phi_{s}(t)=a-1$ moves. Therefore,
$\phi_{s}(t+1)=a-1.$ Hence,

$x_{a-1}(t+1)=1.$ $None$

Suppose there exists $b\in\\{0,1,\dots,l-1\\}$ such that $x_{a+b}(t+1)=1.$
Then there exists $s\in\\{1,\dots,m\\}$ such that $\phi_{s}(t)=a+b$ or
$\phi_{s}(t)=a+b+1.$ Hence there exists $c\in\\{0,1,\dots,l\\}$ such that
$x_{a+c}(t)=1.$ This contradiction proves that

$x_{a}(t+1)=x_{a+1}(t+1),\dots,x_{a+l-1}(t+1)=0.$ $None$

Combining (1), (2), we get Lemma 2. Lemma 3. The vector $X(t+1)$ contains not
more clusters of zeros of length not less 2 in the vector than the vector
$X(t).$ To any cluster of zeros of length not less than 2 in the vector
$X(t+1)$ corresponds to a cluster of length not less than 2 with the same left
limit in the vector $X(t)$.  Lemma 3 follows from Lemmas 1 and 2. Lemma 4. If
the system never results in the state of free movement, then, for any
realization of the stochastic process, there exists a moment $t_{0}$ such
that, for any $t\geq t_{0},$ the vector $X(t)$ does not contain clusters of
length less than 2. Proof. If the system never results in a free movement,
then exists a particle (let it be a particle $s)$ such that infinite number of
delays occurs. Therefore, as $t$ increases, the function $\phi_{s}(t)$ takes
successively all the values $d-1,d-2,\dots,1,0.$ Suppose, after the end of
this cycle, i. e., at time such that a state is repeated, (suppose this cycle
ends at time $T),$ in the vector $X(T)$ there are clusters of zeros of length
not less 2. Therefore, taking into account Lemma 2, we get that, for any
$t=0,1,\dots,T,$ the vector $X(t)$ contains a cluster of length not less than
2 with the same left limit. Hence the function $\phi_{s}(t)$ can not take all
the values $d-1,d-2,\dots,1,0.$ This concludes the proof of the lemma.

Theorem 1. If $m\leq\frac{d}{2},$ then the system results in a state of free
movement after a time interval with a finite expectation. Proof. If $m<d/2,$
then, for any $t,$ in the vector $X(t),$ there exists, at least one cluster of
zeros such that there exists at least one cluster of length not less than 2.
From this and Lemma 4, taking into account that the number of states is
finite, we get the statement of the theorem under the assumption that $m<d/2.$

Suppose $m=d/2$ and the system does not result in the state of free movement.
This condition may hold only if $d$ is an even number. According to Lemma 3
there exists a moment $t_{0}$ such that, for $t\geq t_{0},$ there exist only
clusters of zeros of length 1 and clusters of zeros of length 1. If a particle
is on the diagonal $k,$ then there are not other particles on the diagonal $k$
and particles on the diagonal $k+1.$ Hence, at any time $t\geq t_{0},$ delays
do not occur. This completes the proof of Theorem 1.

##  5 Conclusion

Versions of BML-models are helpful to obtain optimal solutions for problems of
optimal traffic control.

A system is studied such that, in this system, paticles move along an
$n$-dimensional lattice $N_{1}\times N_{2}\times\dots\times N_{n}$ with
periodic boundary conditions according to rules generalizing the rules of the
BML model. In Buslaev nets conception the self-organization property of the
system corresponds to the existance of velocity single point spectrum equal to
1. A sufficient condition for the self-organization of the system has been
found. According to this condition, if number of particles is less than half
of the greatest common divisor of the numbers $N_{1}$ $N_{2},\dots,N_{n}$,
then the system results in a state of free movement under the assumption that
the number of particles is less than half the greatest common divisor.

## Acknowledgments

The authors would like to thanks our colleague Sergey A. Buruykov for the
calculations of the model realization.

## References

  * [1] O. Biham, A.A. Middleton, D. Levine. Self organization and dynamic transition in traffic models. arXiv preprint cond-mat/9206001 (1992).
  * [2] S. Wolfram. Statistical mechanics of cellular automata. Rev. Mod. Phys., 55(3) (1983) 601–644.
  * [3] N.J. Linesh, R.M. D’Souza. Periodic states, local effects and coexistence in the BML traffic jam model. arXiv:0709.3604v3 [cond-math.stat-mech]21Aug2008
  * [4] T. Austin, I. Benjamini. For what number must self organization occur in the Biham– Middleton–Levine traffic model from any possible starting configuration? arXiv preprint math/0607759 (2006).
  * [5] H.R. Moradi, A. Zardadi, Z. Heydarbeygi. The number of collisions in Biham– Middleton–Levine model on a square lattice with limited number of cars. Appl. Math. E-Notes, 19 (2019) 243–249.
  * [6] M.L. Blank. Exact analysis of dynamical systems arising in models of traffic flow. Russian Mathematical Surveys 55(3) (2000) 562–563.
  * [7] V. Belitsky, P.A. Ferrari. Invariant measures and convergence properties for cellular automaton 184 and related processes. J. Stat. Phys. 118(3) (2005) 589–623.
  * [8] A.Tatashev, M. Yashina. Self-organization of two-contours dynamical system with common node and cross movement. WSEAS Trans. Math. 18(45) (2019) 373.
  * [9] V.V. Kozlov, A.P. Buslaev, A.G. Tatashev. A dynamical communication system on a network. JCAAM, 275 (2015).
  * [10] V.V. Kozlov, A.P. Buslaev, A.G. Tatashev, M.V. Yashina. Dynamical systems on honeycombs, Traffic and granular flow ’13, Springer, 2015, pp. 441–452
  * [11] Hu Wenbin, Yan Liping, Wang Huan, Du Bo, Tao Daucheng. Real-time traffic jams prediction inspired by Biham, Middelton and Levine (BML) model. Information Science, 381 (2017) 209–228.
  * [12] Zhong-Jun Ding, Rui Jiang, Bing-Hong Wang. Traffic flow in the Biham– Middleton-Levine model with random update rule. Physical Review, E83, 047101 (2011).
  * [13] M.V. Yashina, S.A. Biruykov, R.V. Strekalov. Velocity Spectrum of Elementary Bidirectional Traffic Model as Buslaev-Type. In2023 Systems of Signal Synchronization, Generating and Processing in Telecommunications (SYNCHROINFO 2023 Jun 28 (pp. 1-5). IEEE. doi: 10.1109/SYNCHROINFO57872.2023.10178596.
  * [14] Chau H.F., Wan K.Y. Phase diagram of the Biham-Middleton-Levine traffic model in three dimensions. arXiv.cond.-mat/9905014v2 11 Aug 1999.
  * [15] R. Criado, A. García del Amo, B. Hernández-Bermejo, M. Romance. New results on computable efficiency and its stability for complex networks. JCAAM 192 (2006) 59–74
  * [16] A. Gillman, P.G. Martinsson. A fast solver for Poisson problems on infinite regular lattices. JCAAM 258 (2014) 42–56
  * [17] G.E. Andrews. Number Theory. WB Sayders Company (1971–1994).