# Hyperverse, 5-dimensional gravity and multiverses as nested Gogberashvili
shells

Igor Yu. Potemine
Institut de Mathématiques, Université Paul Sabatier, Toulouse, France

###### Abstract

We consider the _Hyperverse_ as a collection of multiverses in 5-dimensional
spacetime with gravitational constant $G$. Each multiverse in our simplified
model is a bouquet of nested spherical Gogberashvili shells. If $g_{k}$ is the
gravitational constant of a thin shell $S_{k}$ and $\varepsilon_{k}$ its
thickness then $G\sim\varepsilon_{k}g_{k}$. The physical universe is supposed
to be one of those shells inside the local nested bouquet called _Local
Multiverse_. We relate this construction to Robinson-Trautman metrics
describing expanding spacetimes with spherical gravitational waves.
Supermassive astronomical black holes, located at cores of elliptic/spiral
galaxies, are also conjecturally described within this theory. Our
constructions are equally consistent with the modern theory of cosmological
coupling.

Keywords: 5-dimensional gravity, black hole, multiverse, spherical shell

###### Contents

  1. 1 Introduction
  2. 2 5-dimensional Einstein equations
  3. 3 Nested Gogberashvili models
  4. 4 Harmony of transcosmic spheres
  5. 5 Sequence of event horizons
  6. 6 Time-amalgamated multiverses
  7. 7 Dark energy and multiversal nurseries
  8. 8 Black holes as multiverses
  9. 9 Multi-solitonic and gyratonic gravitational waves
  10. 10 Elliptic and spiral galaxies
  11. 11 Cosmological coupling
  12. 12 Conclusion

##  1 Introduction

According to the _Newton’s shell theorem_ , the gravitational force, exerted
on any object inside a hollow spherical shell, is zero. Consequently, in the
case of nested hollow shells, one can ignore all spherical shells of greater
radius.

One century ago Kaluza and Klein constructed a 5-dimensional gravitation
theory unified with electromagnetism. However, the extra fifth dimension in
this theory is curled up to an unobservable scale.

An alternative model with a true higher-dimensional Hyperverse where the
matter is trapped on a 4-dimensional domain wall ($D$-brane), appeared, _e.g._
, in the paper by Rubakov and Shaposhnikov [1].

Gogberashvili constructed an exact Schwarzschild-like solution of
5-dimensional Einstein equations, exhibiting an expanding spherical shell [2].
He also solved the _hierarchy problem_ , reducing the particle theory to the
same scale $G$ (5-dimensional gravity constant).

In addition, the trapping of matter on this shell is gravitationally repulsive
and the expansion of Gogberashvili’s spherical shell is accelerating, solving
also the problem of dark energy.

In my recent paper [3], it was argued that our Local Multiverse $\mathcal{M}$
can be considered as a time-amalgamated product of spacetimes. Here, first of
all, we pursue this idea representing the 4-dimensional stratum of
$\mathcal{M}$ as a bouquet of nested Gogberashvili shells.

Cosmological constants around subsequent shells satisfy some recursive
equations reflecting a certain _harmony of the spheres_ (_cf._ sect 4).

In the second part of this article we discuss intriguing relationships between
astronomical black holes, spherical gravitational waves, multi-
solitons/gyratons and Gogberashvili multiverses.

##  2 5-dimensional Einstein equations

Consider the 5-dimensional Einstein equation in the following form:

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\Lambda g_{\mu\nu}+6\pi^{2}GR_{\mu\nu}$
(2.1)

Gogberashvili searched for a Schwarzschild-like solution of the form:

$ds^{2}=A(r)dt^{2}-A(r)^{-1}dr^{2}-r^{2}d\Omega^{2},$ (2.2)

where $r$ is the 4-dimensional radial coordinate and $d\Omega^{2}$ is the
3-dimensional volume element [2, eq. (2)-(4)].

The solution is given in terms of metrics of inner and outer regions:

$\displaystyle ds_{+}^{2}$
$\displaystyle=(1-2MG/r^{2}+\Lambda_{+}r^{2})dt^{2}-$ (2.3)
$\displaystyle-(1-2MG/r^{2}+\Lambda_{+}r^{2})^{-1}dr^{2}-r^{2}d\Omega^{2},$
(2.4) $\displaystyle ds_{-}^{2}$
$\displaystyle=(1+\Lambda_{-}r^{2})dt^{2}-(1+\Lambda_{-}r^{2})^{-1}dr^{2}-r^{2}d\Omega^{2},$
(2.5)

separated by a time-like 4-dimensional spherical shell (brane, bubble) with
FLRW metrics:

$ds^{2}=d\tau^{2}-a^{2}(\tau)d\Omega^{2},$ (2.6)

where $\tau$ is the intrinsic time of this spherical universe [2, eq.
(13)-(14)].

##  3 Nested Gogberashvili models

It is natural to consider the nested variant of considered solution as a
simplified model of _multiverses_ restricted to their $(3+1)$-dimensional
strata. So, we consider a sequence of $n$ nested spherical shells-universes
$S_{k}$ around the same center with masses $m_{k}$, $1\leqslant k\leqslant n$.
Denote by $\displaystyle M_{k}=\sum_{i=1}^{k}m_{k}$ the total mass of first
$k$ shells. Local outer and inner metrics are defined by the following
formulas:

$\displaystyle ds_{+,k}^{2}$
$\displaystyle=(1-2M_{k}G/r^{2}+\Lambda_{+,k}r^{2})dt^{2}-$ (3.1)
$\displaystyle-(1-2M_{k}G/r^{2}+\Lambda_{+,k}r^{2})^{-1}dr^{2}-r^{2}d\Omega^{2},$
(3.2) $\displaystyle ds_{-,k}^{2}$
$\displaystyle=(1-2M_{k-1}G/r^{2}+\Lambda_{+,k-1}r^{2})dt^{2}-$ (3.3)
$\displaystyle-(1-2M_{k-1}G/r^{2}+\Lambda_{+,k-1}r^{2})^{-1}dr^{2}-r^{2}d\Omega^{2}.$
(3.4)

Thus, $ds_{-,k}^{2}=ds_{+,k-1}^{2}$ for $2\leqslant k\leqslant n$ . We put
here $M_{0}=0$, so the innermost Gogberashvili metric has the form:

$ds_{-,1}^{2}=(1+\Lambda_{-}r^{2})dt^{2}-(1+\Lambda_{-}r^{2})^{-1}dr^{2}-r^{2}d\Omega^{2},$
(3.5)

where, by definition, $\Lambda_{-}=\Lambda_{+,0}$.

##  4 Harmony of transcosmic spheres

Let $g_{k}$ denote the gravitational constant on the $k$th shell of thickness
$\varepsilon_{k}$ then $G\sim\varepsilon_{k}g_{k}$. It is natural to suppose
the following relations:

$\Lambda_{+,k-1}-\Lambda_{+,k}=\sigma_{k}g_{k}/G,$ (4.1)

$1\leqslant k\leqslant n$, where $\sigma_{k}$ is the energy density of the
$k$th shell [2, eq. (21)]. Consequently,

$\Lambda_{+,n}=\Lambda_{+,0}-\sum_{k=1}^{n}\sigma_{k}g_{k}/G.$ (4.2)

In this picture, a large 5-dimensional gravitational constant $\Lambda_{+,0}$
could generate a sequence of potential wells (cf. [1, fig. 1]), trapping
matter on a sequence of expanding branes-universes. The expansion rate of the
bubble $S_{k}$ depends on its energy density as well as on $\Lambda_{+,k-1}$
and $\Lambda_{+,k}$. Meanwhile, the 4-dimensional cosmological constant on
$S_{k}$ vanishes.

See pages 1 of nursery.pdf

##  5 Sequence of event horizons

The metric $ds_{+,k}^{2}$ becomes singular when

$1-2M_{k}G/r^{2}+\Lambda_{+,k}r^{2}=0$ (5.1)

So, we have the following growing sequence of event horizons:

$r_{k}=\sqrt{\left(\sqrt{1+8GM_{k}\Lambda_{+,k}}-1\right)/2\Lambda_{+,k}}$
(5.2)

for $1\leqslant k\leqslant n$.

The most meaningful is the _external event horizon_ with radial coordinate
$r_{n}$, located outside of $S_{n}$.

##  6 Time-amalgamated multiverses

Four-dimensional Gogberashvili shells $S_{k}$ with FLRW metrics

$ds_{k}^{2}=d\tau_{k}^{2}-a^{2}(\tau_{k})d\Omega^{2},$ (6.1)

$1\leqslant k\leqslant n$, can be glued into a multiverse $\mathcal{M}$
according to the procedure of time amalgamation, described in my paper [3]. It
means that intrinsic times $\tau_{k}$ become mutually synchronized and there
is a common timeline $\mathbb{T}$ such that

$\mathcal{M}=S_{1}\times_{\mathbb{T}}\times\cdots\times_{\mathbb{T}}S_{n}$
(6.2)

Here $\mathcal{M}$ is considered as a fibered product.

In this way, all shells $S_{k}$ can be studied simultaneously.

##  7 Dark energy and multiversal nurseries

It is possible to endow the Hyperverse with a metric approximately describing
several widely separated Gogberashvili multiverses. For example, one can use
the work of Kashif Alvi on widely separated binary black holes [4].

In our picture, the Hyperverse has numerous powerful sources of dark energy
with large local cosmological constants. Those sources play role of the
_nurseries of multiverses_.

The situation is somewhat similar to the star formation and stellar nurseries
in galaxies. In the latter case, it is related to the collapsing matter in
giant molecular clouds.

Notice that, in our constructions, the dark energy is a higher-dimensional
phenomenon of the Hyperverse.

##  8 Black holes as multiverses

For an external hyperversal observer, any $(3+1)$-dimensional Gogberashvili
multiverse would eventually appear as a supermassive black hole. It suggests
the idea that astronomical black holes in our physical universe are, actually,
nested and spherically embedded $(2+1)$-dimensional branes-universes.

All solutions to $(2+1)$-dimensional Einstein equations were actually
classified [5]. Gogberashvili models correspond in this case to spherical
Robinson-Trautman solutions [6]. In canonical coordinates the Robinson-
Trautman metric is given by the formula:

$ds^{2}=\frac{r^{2}}{P^{2}}dx^{2}+2(er^{2}+f)dudx-2dudr+F(a,e,P,\Lambda;r)du^{2},$
(8.1)

where $a(x,u)$, $e(x,u)$, $f(x,u)$ and $P(x,u)$ are some metric functions
while $F(a,e,P,\Lambda;r)$ is a polynomial of degree 2 with respect to $r$
depending on $a$, $e$, $P$ and the cosmological constant $\Lambda$ [5, eq.
(37)].

Similar solutions in the form

$ds^{2}=g_{pq}dx^{p}dx^{q}+2g_{up}dudx^{p}-2dudr+g_{uu}du^{2},$ (8.2)

exist in all $(d+1)$-dimensional spacetimes with $d\geqslant 2$ (_cf_. [7, eq.
(1)]).

We should also mention innovative works of T.X. Zhang on black hole universes
(_cf._ [8] and references therein).

##  9 Multi-solitonic and gyratonic gravitational waves

So, we are interested in Robinson-Trautman metrics describing expanding
spacetimes with spherical gravitational waves. Multi-soliton solutions of this
type, associated with matter trapping, correspond to our Gogberashvili
multiverses.

If we admit that the null matter field has some internal spin / angular
momentum, there are also more general (multi-)gyratonic solutions. They give
rise to rotating Gogberashvili models/multiverses.

This intriguing topic, including a thorough comparison ot two models, should
be studied in more details. In particular, the harmony of transcosmic spheres
should acquire more complete meaning.

##  10 Elliptic and spiral galaxies

It seems that it partially explains the classification of galaxies, harboring
supermassive black holes at their cores. Indeed, on the one hand, supermassive
black holes of multi-solitonic type could generate elliptic galaxies.

On the other hand, multi-gyratonic black holes could generate spiral galaxies.
The number of spirals and their configuration is related in this case to the
number of multiversal layers and the harmony of the spheres (_cf._ sections 4
and 9).

Relativistic jets, emitting by supermassive black holes at galaxy cores, also
suggest their gyratonic nature.

##  11 Cosmological coupling

Our constructions are consistent with the modern theory of _cosmological
coupling_ [9].

Recent astronomical observations show the abundance of black holes with
unexpectedly high masses. Croker _et al._ proposed a mechanism, explaining the
growth of black holes through anomalous coupling of matter with gravity [10].

It implies that the expansion of the universe and a large (but hidden)
cosmological constant contribute to the growth of black hole masses.

##  12 Conclusion

It turns out that nested Gogberashvili shells represent good models for
multiverses inside the Hyperverse.

In this paper we have restrained the Hyperverse to its local
$(4+1)$-dimensional stratum. However, similar solutions exist in all
dimensions $d+1$ with $d\geqslant 2$. In fact, we can construct an infinite
tower of embedded Gogberashvili multiverses of various dimensions.

An alternative approach is given by Robinson-Trautman metrics describing
expanding spacetimes and spherical gravitational waves.

It provides us with an intriguing idea to consider supermassive astronomical
black holes as expanding $(2+1)$-dimensional multiverses (with possible higher
strata too).

It might give new angles of view on the black hole growth, the cosmological
coupling, relativistic jets and even on the classification of galaxies.

## References

  * [1] V. Rubakov and M. Shaposhnikov.  Extra space-time dimensions: Towards a solution to the cosmological constant problem.  Physics Letters B, 125(2-3):139–143, 1983.
  * [2] M. Gogberashvili.  Our world as an expanding shell.  Europhysics Letters, 7(3):396–399, 2000.
  * [3] I. Potemine.  The amalgamated structure of the Local Multiverse.  Journal of High Energy Physics, Gravitation and Cosmology, 7(4):1213–1218, 2021.
  * [4] K. Alvi.  Approximate binary-black-hole metric.  Physical Review D, 61(12-15), 2000.
  * [5] J. Podolsky, R. Svarc, and H. Maeda.  All solutions of Einstein’s equations in 2+1 dimensions: $\Lambda$-vacuum, pure radiation, or gyratons.  Classical and Quantum Gravity, 36, 2019.
  * [6] I. Robinson and A. Trautman.  Spherical gravitational waves.  Physical Review Letters, 8(4):431–432, 1960.
  * [7] J. Podolsky and R. Svarc.  Gyratons in the Robinson–Trautman and Kundt classes.  Physical Review D, 99, 2019.
  * [8] T.X. Zhang.  The principles and laws of black hole universe.  Journal of Modern Physics, 9(9):1838–1865, 2018.
  * [9] Y. Bisabr.  Gravitational coupling and the cosmological constant.  International Journal of Modern Physics D, 27(8), 2018.
  * [10] K. Croker, M. Zevin, D. Farrah, K. Nishimura, and G. Tarlé.  Cosmologically coupled compact objects: A single-parameter model for LIGO–Virgo mass and redshift distributions.  The Astrophysical Journal Letters, 921(2), 2021.

Igor Potemine

Institut de Mathématiques

Université Paul Sabatier

118, route de Narbonne

31062 Toulouse (France)

e-mail<EMAIL_ADDRESS>


# Stability Conditions for Remote State Estimation of Multiple Systems over
Semi-Markov Fading Channels

Wanchun Liu  Daniel E. Quevedo  Branka Vucetic  Yonghui Li

###### Abstract

††W. Liu, B. Vucetic and Y. Li are with School of Electrical and Information
Engineering, The University of Sydney, Australia. Emails: {wanchun.liu,
branka.vucetic<EMAIL_ADDRESS>D. E. Quevedo is with the School of
Electrical Engineering and Robotics, Queensland University of Technology
(QUT), Brisbane, Australia. Email<EMAIL_ADDRESS>
This work studies remote state estimation of multiple linear time-invariant
systems over shared wireless time-varying communication channels. We model the
channel states by a semi-Markov process which captures both the random holding
period of each channel state and the state transitions. The model is
sufficiently general to be used in both fast and slow fading scenarios. We
derive necessary and sufficient stability conditions of the multi-sensor-
multi-channel system in terms of the system parameters. We further investigate
how the delay of the channel state information availability and the holding
period of channel states affect the stability. In particular, we show that,
from a system stability perspective, fast fading channels may be preferable to
slow fading ones.

###### Index Terms:

Stability of linear systems, control over communications, estimation, Kalman
filtering, Markov processes.

##  I Introduction

The incoming Fourth Industrial Revolution, Industry 4.0, focuses heavily on
interconnectivity, automation, machine learning, and real-time data for
customized and flexible mass production [1]. In particular, with low-cost and
scalable deployment capabilities, wireless remote state estimation from
ubiquitous sensors will be essential in many industrial networked control
applications, such as advanced manufacturing, warehouses automation, mining,
and smart grids [2].

The typical connection density in the Industry 4.0 era is about $10^{6}$/km2;
however, wireless communications have a limited spectrum bandwidth for
transmissions. Therefore, transmission scheduling among sensors is a critical
issue over the shared limited number of frequency channels. Most wireless
scheduling works focus solely on communications performance, including
throughput, latency, and reliability, but are agnostic to upper-layer
applications, such as estimation and control [3]. However, for a multi-sensor-
multi-channel remote estimation system, where each sensor measures an unstable
dynamic plant, the scheduler must guarantee the stability of the remote
estimation of all plant states. Otherwise, some of the plants cannot be
stabilized, leading to catastrophic impacts on real-world systems. The design
of stabilizing multi-sensor-multi-channel remote estimators is a challenging
problem and has drawn significant attention.

Optimal dynamic transmission scheduling policies of multi-plant networked
systems over shared wireless resources were investigated in [4, 5]. However,
the stability conditions of these systems have not been investigated. Sensor
transmission scheduling of remote estimation systems over single and multiple
packet drop channels were investigated in [6] and [7], respectively. Once
sufficient stability conditions were obtained, Markov decision process (MDP)
methods were be adopted for finding the optimal scheduling policies. The work
[8] developed a sufficient stability condition over time-correlated fading
channels, whereas [9] derived a necessary and sufficient stability condition.

Due to shadowing and multi-path propagation, wireless channel states (e.g.,
qualities) are time-varying and time-correlated [10]. Since wireless channel
dynamics have significant impact on the remote estimation quality, accurate
channel modeling is critical for stability analysis. In the multi-system
scheduling works above, time-invariant channels were considered in [6, 7];
time-uncorrelated fading channels were adopted in [4, 5]; recently, more
practical time-correlated fading channels were applied in [8, 9] and modeled
by Markov processes. However, the Markov modeling is suitable for fast fading
channels, i.e., the channel state changes at each time, and is not accurate
for slow fading scenarios. As verified by experiments in industrial
environments [11], semi-Markov processes, which generalize Markov processes,
are suitable for characterizing slow fading channels in factories. In
addition, semi-Markov modeling also captures the time-varying feature of the
channel state holding period in practice, i.e., the channel coherence time.
Note that most of the existing channel models assume fixed coherence time for
tractability [10].

In this work, we focus on the stability analysis of a multi-sensor remote
estimation system over shared semi-Markov fading channels. The novel
contributions include:

$\bullet$ We build up a multi-sensor remote estimation system over practical
multi-level semi-Markov fading channels. To the best of our knowledge, such a
system has never been investigated in the open literature. Note that the
existing works [8, 9] only considered the simpler Markov channel modeling with
binary-level channel states.

$\bullet$ We derive a necessary and sufficient stability condition for remote
estimation and also provide the structure of a stability-guaranteeing
scheduling policy. Our result establishes a fundamental design guideline for
stable remote estimation systems over practical wireless channels.

$\bullet$ We also investigate how the delay of the channel state information
availability and the holding period of channel states affects the stability.
In particular, we show that a fast fading scenario (e.g., with a short average
holding period) is preferable from a stability viewpoint.

##  II System Model

We consider a remote estimation system with $N$ sensors each measuring an
independent physical process, as illustrated in Fig. 1. A local gateway
connected to the sensors collects their measurements and forwards them to a
remote estimator. Connections between sensors and the gateway are reliable and
not scheduled, while the gateway to remote estimator communications are
wireless and scheduled due to bandwidth limitations. There exist infinitely
many dynamic transmission scheduling policies. _It is critical to determine
necessary and sufficient conditions of the remote estimation system under
which there exists a scheduling policy that can stabilize the system._ If such
a condition is not satisfied, then no stabilizing scheduling policy exists and
one should redesign the system. The main focus of the current letter is to
present a necessary and sufficient stability condition, and thereby provide
fundamental design guidelines for stable remote estimation systems.

Figure 1: The multi-sensor-multi-frequency remote estimator. P, S, G, and RE
denote processes, sensors, gateway and remote estimator, respectively.

Each process $n$ is modeled as an LTI system:

$\displaystyle\mathbf{x}_{n}{(t+1)}$
$\displaystyle=\mathbf{A}_{n}\mathbf{x}_{n}(t)+\mathbf{w}_{n}(t),$ (1)
$\displaystyle\mathbf{y}_{n}(t)$
$\displaystyle=\mathbf{C}_{n}\mathbf{x}_{n}(t)+{\mathbf{z}_{n}(t)},$

where $\mathbf{x}_{n}(t)\in\mathbb{R}^{l_{n}}$ and
$\mathbf{y}_{n}(t)\in\mathbb{R}^{r_{n}}$ are the process state and the sensor
measurements at time $t\in\mathbb{N}_{0}$, respectively.
$\mathbf{A}_{n}\in\mathbb{R}^{{l_{n}}\times{l_{n}}}$ and
$\mathbf{C}_{n}\in\mathbb{R}^{{r_{n}}\times{l_{n}}}$ are process $n$’s state
transition matrix and sensor $n$’s measurement matrix, respectively.
$\mathbf{w}_{n}(t)\in\mathbb{R}^{l_{n}}$ and
$\mathbf{z}_{n}(t)\in\mathbb{R}^{r_{n}}$ are the process disturbance and the
measurement noise, and are independent and identically distributed (i.i.d.)
zero-mean Gaussian processes with the covariance matrices $\mathbf{W}_{n}$ and
$\mathbf{Z}_{n}$, respectively.

###  II-A Local Estimation

Each sensor uses a local Kalman filter (KF) to pre-process its measurement
before sending to the gateway [8]. We have

$\displaystyle\mathbf{x}^{s}_{n}({t|t-1})$
$\displaystyle=\mathbf{A}_{n}\mathbf{x}^{s}_{n}({t-1})$ (2)
$\displaystyle\mathbf{P}^{s}_{n}({t|t-1})$
$\displaystyle=\mathbf{A}_{n}\mathbf{P}^{s}_{n}({t-1})\mathbf{A}^{\top}_{n}+\mathbf{W}_{n}$
$\displaystyle\mathbf{K}_{n}(t)$
$\displaystyle=\\!\mathbf{P}^{s}_{n}({t|t-1})\mathbf{C}^{\top}_{n}(\mathbf{C}_{n}\mathbf{P}^{s}_{n}({t|t-1})\mathbf{C}^{\top}_{n}\\!+\\!\mathbf{Z}_{n})^{-1}$
$\displaystyle\mathbf{x}^{s}_{n}(t)$
$\displaystyle=\\!\mathbf{x}^{s}_{n}({t|t-1})\\!+\\!\mathbf{K}_{n}(t)(\mathbf{y}_{n}({t})\\!-\\!\mathbf{C}_{n}\mathbf{x}^{s}_{n}({t|t-1}))$
$\displaystyle\mathbf{P}^{s}_{n}(t)$
$\displaystyle=(\mathbf{I}_{n}-\mathbf{K}_{n}({t})\mathbf{C}_{n})\mathbf{P}^{s}_{n}({t|t-1})$

where $\mathbf{x}^{s}_{n}({t|t-1})$ and $\mathbf{x}^{s}_{n}(t)$ are the
predicted and updated state estimate at time $t$, respectively.
$\mathbf{K}_{n}(t)$ is the Kalman gain. $\mathbf{P}^{s}_{n}({t|t-1})$ and
$\mathbf{P}^{s}_{n}(t)$ are the predicted and updated error covariance,
respectively. $\mathbf{I}_{n}$ is the $l_{n}\times l_{n}$ identity matrix In
particular, $\mathbf{x}^{s}_{n}(t)$ is the optimal estimate of
$\mathbf{x}_{n}(t)$ at time $t$ in terms of the estimation mean-square error,
where the estimation error covariance $\mathbf{P}^{s}_{n}(t)$ is defined as

$\mathbf{P}^{s}_{n}(t)\triangleq\mathsf{E}\left[(\mathbf{x}^{s}_{n}(t)-\mathbf{x}_{n}(t))(\mathbf{x}^{s}_{n}(t)-\mathbf{x}_{n}(t))^{\top}\right].$
(3)

We assume that the _local_ KFs are stable and operate in steady state [8, 12],
i.e., $\mathbf{P}^{s}_{n}(t)=\mathbf{\bar{P}}_{n},\forall
t\in\mathbb{N}_{0},n\in\mathcal{N}\triangleq\\{1,2,\dots,N\\}$, and
investigate the _remote_ estimation stability.

###  II-B Semi-Markov Fading Channel

We assume that there are $M<N$ frequency channels for sensor data
transmission. The channels are not perfect for transmission and can induce
packet dropouts. The channel qualities at the $M$ frequencies are time-
correlated and modeled as a semi-Markov process as below. A better channel
quality leads to a smaller packet drop probability.

Consider an $M$-frequency-multi-level channel quality state
$\mathbf{h}(t)\triangleq[h_{1}(t),h_{2}(t),\dots,h_{M}(t)]$, where the quality
of the $m$th frequency channel $h_{m}(t)$ has $M_{m}$ levels and

$h_{m}\\!(t)\\!\\!\in\\!\mathcal{H}_{m}\\!\\!\triangleq\\!\\{h^{(m)}_{1},h^{(m)}_{2},\dots,h^{(m)}_{M_{m}}\\},\\!\;m\\!\in\\!\mathcal{M}\\!\triangleq\\!\\{1,\dots,M\\}.$

The packet drop probabilities of transmissions at different frequencies and
different channel quality levels can be different.

The channel quality state $\mathbf{h}(t)$ forms a semi-Markov chain with
$\bar{M}=\prod_{m=1}^{M}M_{m}$ irreducible states, where

$\mathbf{h}(t)\in\mathcal{S}=\mathcal{H}_{1}\times\mathcal{H}_{2}\times\dots\times\mathcal{H}_{M}=\\{\mathbf{h}_{1},\mathbf{h}_{2},\dots,\mathbf{h}_{\bar{M}}\\}$

and $\mathbf{h}_{j}=[h_{j,1},h_{j,2},\dots,h_{j,M}]$. The transition instants
between channel quality states are denoted by
$\\{t_{0},t_{1},\dots,t_{l},\dots\\}$, with $t_{0}=0$, and
$t_{0}<t_{1}<t_{2},\dots$ all integers. The $l$th holding period, i.e., the
amounts of time spent in the same channel quality state before the $l$th
channel quality state transitions, is defined as $\varDelta_{l}\triangleq
t_{l+1}-t_{l}$. Assuming that the holding periods are bounded by
$\bar{\varDelta}\in\mathbb{N}$, we have
$\varDelta_{l}\in\Delta\triangleq\\{1,\dots,\bar{\varDelta}\\},\forall l$. See
Fig. 2 for an illustration of the semi-Markov chain of $\\{\mathbf{h}(t)\\}$.
Note that the semi-Markov chain degrades to a Markov chain when
$\bar{\varDelta}=1$.

Figure 2: An illustration of the semi-Markov chain of $\\{\mathbf{h}(t)\\}$.

Let $\mathbf{M}\in\mathbb{R}^{\bar{M}}\times\mathbb{R}^{\bar{M}}$ denote the
channel quality state transition probability matrix of the semi-Markov chain,
where the $i$th-row-$j$th-column element is

$[\mathbf{M}]_{i,j}\triangleq\mathsf{Prob}\left[\mathbf{h}(t_{l+1})=\mathbf{h}_{j}|\mathbf{h}(t_{l})=\mathbf{h}_{i}\right],\forall
l\in\mathbb{N}_{0}.$ (4)

The probability distribution of the holding period given the current channel
quality state is

$\psi_{i}(\delta)=\mathsf{Prob}\left[\varDelta_{l}=\delta|\mathbf{h}(t_{l})=\mathbf{h}_{i}\right],\forall
l\in\mathbb{N}_{0},\mathbf{h}_{i}\in\mathcal{S}.$ (5)

We assume that the channel quality transition and the holding period are
independent, i.e.,

$\mathsf{Prob}\left[\mathbf{h}(t_{l+1})=\mathbf{h}_{j},\varDelta_{l}=\delta|\mathbf{h}(t_{l})=\mathbf{h}_{i}\right]=[\mathbf{M}]_{i,j}\psi_{i}(\delta).$
(6)

Let $\varDelta_{l}(t)\triangleq t-t_{l}+1$ denote the holding time of the
current channel quality state, where $t_{l}\leq t<t_{l+1}$. Similarly, we
define the holding time of the next channel quality state as
$\varDelta_{l^{\prime}}(t+1)\triangleq(t+1)-t_{l^{\prime}}+1$, where $l\leq
l^{\prime}\leq l+1$.

From (5), the probability that the channel quality state transition occurs in
the next time slot is

$\displaystyle\mathsf{Prob}\left[\varDelta_{l^{\prime}}(t+1)=1|\varDelta_{l}(t)=\delta,\mathbf{h}(t)=\mathbf{h}_{i}\right]$
(7)
$\displaystyle=\mathsf{Prob}\left[\varDelta_{l}=\delta|\varDelta_{l}\geq\delta,\mathbf{h}(t)=\mathbf{h}_{i}\right]=\frac{\psi_{i}(\delta)}{\sum_{\delta^{\prime}=\delta}^{\bar{\varDelta}}\psi_{i}(\delta^{\prime})}.$

Now we define the channel state vector $\mathbf{\tilde{h}}(t)$ as the cascaded
state of $\mathbf{h}(t)$ and $\varDelta_{l}(t)$:

$\mathbf{\tilde{h}}(t)\triangleq(\mathbf{h}(t),\varDelta_{l}(t))\in\mathcal{\tilde{S}}\triangleq\mathcal{S}\times\Delta=\\{\mathbf{\tilde{h}}_{1},\mathbf{\tilde{h}}_{2},\cdots,\mathbf{\tilde{h}}_{\tilde{M}}\\},$

where the cardinality
$|\mathcal{\tilde{S}}|=\bar{M}\bar{\varDelta}\triangleq\tilde{M}$. From (4),
(7), and the semi-Markov property of $\\{\mathbf{h}(t)\\}$, it is easy to show
that $\\{\mathbf{\tilde{h}}(t)\\}$ has the Markov property, i.e., given the
current state $\mathbf{\tilde{h}}(t)$, the next state
$\mathbf{\tilde{h}}(t+1)$ is independent of the previous states
$\\{\mathbf{\tilde{h}}(0),\mathbf{\tilde{h}}(1),\dots,\mathbf{\tilde{h}}(t-1)\\}$.
Thus, the original semi-Markov chain $\\{\mathbf{h}(t)\\}$ is converted to the
Markov chain $\\{\mathbf{\tilde{h}}(t)\\}$. In the rest of the paper, we will
use $\mathbf{\tilde{h}}(t)$ in stead of $\mathbf{h}(t)$ for ease of analysis.

Using (4), (5), and (7), the channel state transition probability matrix
$\mathbf{\tilde{M}}\in\mathbb{R}^{\tilde{M}}\times\mathbb{R}^{\tilde{M}}$ of
$\\{\mathbf{\tilde{h}}(t)\\}$ can be obtained, and the $\tilde{i}$th-
row-$\tilde{j}$th-column element is

$\displaystyle[\mathbf{\tilde{M}}]_{\tilde{i},\tilde{j}}\\!\triangleq\\!\mathsf{Prob}\left[\mathbf{\tilde{h}}(t+1)\\!=\\!\mathbf{\tilde{h}}_{\tilde{j}}\\!=\\!(\mathbf{h}_{j},\delta_{j})|\mathbf{\tilde{h}}(t)\\!=\\!\mathbf{\tilde{h}}_{\tilde{i}}\\!=\\!(\mathbf{h}_{i},\delta_{i})\right]$
(8)
$\displaystyle=\begin{cases}[\mathbf{M}]_{i,j}\frac{\psi_{i}(\delta_{i})}{\sum_{\delta=\delta_{i}}^{\bar{\varDelta}}\psi_{i}(\delta)}&\text{
if }\delta_{j}=1,\\\
1-\frac{\psi_{i}(\delta_{i})}{\sum_{\delta=\delta_{i}}^{\bar{\varDelta}}\psi_{i}(\delta)}&\text{
if
}\delta_{j}=\delta_{i}+1\leq\bar{\varDelta},\mathbf{h}_{j}=\mathbf{h}_{i},\\\
0&\text{ otherwise.}\\\ \end{cases}$

For ease of analysis, we assume that $\\{\mathbf{\tilde{h}}(t)\\}$ is an
aperiodic and irreducible Markov chain.

We make the channel state availability assumption as below.

######  Assumption 1 (Known Current Channel State).

At time $t\in\mathbb{N}$, the current channel state $\mathbf{\tilde{h}}(t)$ is
known by the gateway prior to transmission scheduling.

We note that the channel state can be estimated based on standard channel
estimation techniques [10]. The scenario with delayed channel state
information will be investigated at the end of Section III.

We define the transmission failure event at frequency $m$ given the the
channel state $\mathbf{\tilde{h}}_{j}$ as
$\mu\left(\mathbf{\tilde{h}}_{j},m\right)=0$ and the packet drop probability

$d_{j,m}\triangleq\mathsf{Prob}\left[\mu\left(\mathbf{\tilde{h}}_{j},m\right)=0\right]\in[0,1],\forall
j\in\mathcal{\tilde{M}},m\in\mathcal{M},$ (9)

where $\mathcal{\tilde{M}}\triangleq\\{1,2,\dots,\tilde{M}\\}$.

###  II-C Remote Estimation

In each time slot, the gateway collects $N$ packets of the sensor estimates
$\\{\mathbf{\hat{x}}^{s}_{1}(t),\dots,\mathbf{\hat{x}}^{s}_{N}(t)\\}$,
schedules $M$ of them, and sends through $M$ frequency channels to the remote
estimator. Each frequency channel can transmit at most one packet at a time,
and the unscheduled packets are discarded. Each scheduled packet can take at
most one frequency channel for transmission.

Due to the transmission scheduling and packet dropouts, the remote estimator
cannot receive all sensor packets at each time. Let $\gamma_{n}(t)=1$ denote
the event that sensor $n$’s packet is successfully received by the remote
estimator at time $t$. We also define the _age-of-information_ (AoI) for each
sensor, $\phi_{n}(t)\in\mathbb{N}$, which is the time duration between the
previous successful sensor $n$’s packet detection and the current time $t$,
i.e.,
$\phi_{n}(t)\triangleq\min_{\\{t^{\prime}:t^{\prime}<t,\bm{1}(\gamma_{n}(t^{\prime})=1),t^{\prime}\in\mathbb{N}_{0}\\}}(t-t^{\prime}),$
where $\bm{1}(\cdot)$ is the indicator function. Thus, the AoI state has the
updating rule below

$\phi_{n}(t)=\begin{cases}1&\text{ if }\gamma_{n}(t-1)=1\\\
\phi_{n}(t-1)+1&\text{ otherwise.}\\\ \end{cases}$ (10)

The optimal _minimum mean-square error_ (MMSE) remote estimator [12] works as
below, considering the one-step transmission delay [13]

$\hat{\mathbf{x}}_{n}(t)=\begin{cases}\mathbf{A}_{n}{\mathbf{x}}_{n}^{s}(t-1),&\text{
if }\gamma_{n}(t-1)=1,\\\ \mathbf{A}_{n}\hat{\mathbf{x}}_{n}(t-1),&\text{
otherwise.}\end{cases}$ (11)

and can be simplified as

$\hat{\mathbf{x}}_{n}(t)=\mathbf{A}^{\phi_{n}(t)}_{n}{\mathbf{x}}^{s}_{n}(t-\phi_{n}(t)).$
(12)

From (3) and (12), the estimation error covariance of process $n$ is derived
as

$\displaystyle\mathbf{P}_{n}(t)$
$\displaystyle\triangleq\mathsf{E}\left[(\hat{\mathbf{x}}_{n}(t)-\mathbf{x}_{n}(t))(\hat{\mathbf{x}}_{n}(t)-\mathbf{x}_{n}(t))^{\top}\right]=\theta^{\phi_{n}(t)}_{n}(\bar{\mathbf{P}}_{n}),$
(13)

where $\bar{\mathbf{P}}_{n}$ was defined in Section II-A,
$\theta_{n}(\mathbf{X})=\theta_{n}^{1}(\mathbf{X})\triangleq\mathbf{A}_{n}\mathbf{X}\mathbf{A}_{n}^{\top}+\mathbf{W}_{n}$,
and
$\theta_{n}^{m+1}(\cdot)\triangleq\theta_{n}(\theta_{n}^{m}(\cdot)),m\in\mathbb{N}.$
Thus, the remote estimation quality of process $n$ at time $t$ can be
quantified via the sum average estimation error
$\mathsf{E}\left((\hat{\mathbf{x}}_{n}(t)-\mathbf{x}_{n}(t))^{\top}(\hat{\mathbf{x}}_{n}(t)-\mathbf{x}_{n}(t))\right)=\text{Tr}\left(\mathbf{P}_{n}(t)\right)$,
where $\text{Tr}(\cdot)$ is the trace operator. By introducing the following
function

$c_{n}(i)\triangleq\text{Tr}\left(\theta^{i}_{n}(\bar{\mathbf{P}}_{n})\right),\forall
i\in\mathbb{N}$ (14)

and using (13), we have

$\text{Tr}\left(\mathbf{P}_{n}(t)\right)\triangleq c_{n}(\phi_{n}(t)),$ (15)

which is the _estimation cost function_ of process $n$ and is determined by
its AoI state $\phi_{n}(t)\in\mathbb{N}$.

Note that due to the transmission scheduling and error, the AoI state
$\phi_{n}(t)$ can have unbounded support, i.e., the remote estimator may not
receive sensor $n$’s packet for an arbitrarily long time. Thus, the cost
function $c_{n}(\phi_{n}(t))$ in (14) takes values from a countably infinite
set
$\\{\text{Tr}\left(\theta^{1}_{n}(\bar{\mathbf{P}}_{n})\right),\text{Tr}\left(\theta^{2}_{n}(\bar{\mathbf{P}}_{n})\right),\dots\\}.$

As discussed in [12], if $\rho(\mathbf{A}_{n})\geq 1$, the cost grows up
unbounded with the increasing AoI. Our focus is on the remote estimator’s
stochastic stability defined as below.

######  Definition 1 (Average Mean-Square Stability).

The $N$-sensor-$M$-frequency remote estimation system described above is
average mean-square stable, if the long-term average estimation cost $J$ is
bounded, where

$J\triangleq\sum_{n=1}^{N}J_{n},\
J_{n}\triangleq\limsup_{T\to\infty}\frac{1}{T}\sum_{t=1}^{T}c_{n}(\phi_{n}(t)),n\in\mathcal{N}.$
(16)

Using [9, Lemma 1], it is straightforward to establish the following property
of $c_{n}(\cdot)$:

######  Lemma 1.

For any given $\epsilon>0$, there exists positive constants $\kappa$ and
$\eta$ such that

$\displaystyle\text{(Upper bound)
}c_{n}(i)<\kappa\left(\rho^{2}(\mathbf{A}_{n})+\epsilon\right)^{i},$ (17)
$\displaystyle\text{(Lower bound)
}c_{n}(i)\geq\eta(\rho(\mathbf{A}_{n}))^{2i},\forall i\in\mathbb{N}.$ (18)

###  II-D Transmission Scheduling Policy

We solely focus on deterministic stationary scheduling policies. Let
$\bm{\nu}(t)\triangleq[\nu_{1}(t),\dots,\nu_{N}(t)]\in\\{0,1,\dots,M\\}^{N}$
denote the scheduling action for the $N$ sensors at time $t$. In particular,
if $\nu_{n}(t)=0$, sensor $n$ is not scheduled; if $\nu_{n}(t)\in\mathcal{M}$,
sensor $n$ is scheduled at frequency $\nu_{n}(t)$. The scheduling actions are
sent to the sensors via feedback channels. We assume that these transmissions
are error-free due to the small communication overhead.

From (15), the AoI state
$\bm{\phi}(t)\triangleq[\phi_{1}(t),\phi_{2}(t),\dots,\phi_{N}(t)]$ determines
the current estimation cost. From (8) and (11), the current channel state
$\mathbf{\tilde{h}}(t)$ reflects the chances of transmission success in the
current time slot, and will affect the next channel state and hence the
estimation cost in the next step. Then, using the Markov properties (8) and
(10), a scheduling policy $\pi(\cdot)$ should take into account both the
channel and AoI states for decision making, i.e.,

$\bm{\nu}(t)=\pi(\bm{\phi}(t),\mathbf{\tilde{h}}(t)).$ (19)

##  III Stability Conditions

The LTI system model and the semi-Markov channel statistics jointly determine
the stability of the overall remote estimator. Our result is stated in terms
of the channel state transition matrix $\mathbf{\tilde{M}}$, the
length-$\tilde{M}$ channel selection vector
$\mathbf{v}\triangleq[v_{1},v_{2},\dots,v_{\tilde{M}}]\in\mathcal{M}^{\tilde{M}}$,
where the $i$th element $v_{i}\in\mathcal{M}$ denotes the selected frequency
index given the channel state $\mathbf{\tilde{h}}_{i}$, and the
$\tilde{M}\times\tilde{M}$ diagonal packet drop probability matrix
$\mathbf{V}(\mathbf{v})$ given the channel selection vector
$\mathbf{v}\in\mathcal{M}^{\tilde{M}}$:

m,m $\displaystyle\triangleq d_{m,v_{m}},m\in\mathcal{\tilde{M}},$ (20)

where $d_{i,j}$ was defined in (9).

######  Theorem 1.

The $M$-sensor-$N$-frequency remote estimator described in Section II under
Assumption 1 can be stabilized if and only if the following condition holds

$\rho^{2}_{\max}\lambda<1,$ (21)

where $\rho_{\max}\triangleq\max_{n\in\mathcal{N}}\rho(\mathbf{A}_{n})$ is the
largest spectral radius of all processes,

$\lambda\triangleq\rho\left(\mathbf{V}(\mathbf{v}^{\star})\mathbf{\tilde{M}}\right),\
\mathbf{v}^{\star}=[v^{\star}_{1},\dots,v^{\star}_{\tilde{M}}],$ (22)

and

$v^{\star}_{i}=\operatorname*{arg\,min}_{m\in\mathcal{M}}d_{i,m},i\in\mathcal{\tilde{M}}.$
(23)

######  Remark 1.

We see that the stability depends on the system parameter of the most unstable
process, the channel state dynamics, and the packet drop probabilities at
different channel states. Although Theorem 1 does not provide direct insights
on the structure of a stable scheduling policy, we will construct a policy
with stability guarantees in the proof of the sufficient condition. Numerical
examples of the stability condition are provided in Section IV.

We will prove the necessary and sufficiency parts of Theorem 1 in the sequel.
Note that if all processes are stable, i.e., $\rho_{\max}<1$, the remote
estimator is always stable. Thus, in the following, we only focus on the case
with $\rho_{\max}\geq 1$.

###### Proof of Necessity.

The proof has three parts: 1) the construction of a virtual policy that can
always achieve an average cost of the remote estimator lower than any real
scheduling policy; 2) the average cost function analysis of the virtual
policy; 3) the derivation of the necessary condition.

####  III-1 Policy construction

To prove the necessity, we consider a virtual scenario that only the packet of
the sensor corresponding to the most unstable process is scheduled for
transmission in each time slot, while the other sensors’ estimates are
perfectly known by the remote estimator and need not packet transmissions.
Without loss of generality, we assume that process $1$ is the most unstable
one, i.e., $\rho_{\max}=\rho(\mathbf{A}_{1})$. In other words, only sensor
$1$’s is scheduled in each time slot and it can select any of the frequencies
for transmission. For ease of notation, we will drop out the process index $n$
in the following analysis. Furthermore, we replace the cost function $c(i)$
with its lower bound (18).

Thus, the original sensor scheduling policy (19) is reduced to a frequency
selection one

$\nu(t)=\pi(\phi(t),\mathbf{\tilde{h}}(t))\in\mathcal{M},$ (24)

where $\nu(t)$ and $\phi(t)$ are the frequency selection action and the AoI
state of sensor $1$, respectively. Recall that we focus on deterministic
stationary scheduling policies, and hence drop the time index $t$ in the
following.

Let $v_{i}(\phi)\triangleq\pi(\phi,\mathbf{\tilde{h}}_{i})\in\mathcal{M}$
denote the selected channel for transmission given the current AoI state
$\phi$ and the channel state $\mathbf{\tilde{h}}_{i}$. From (24), for given
$\phi$, the frequency selection rule at $\tilde{M}$ different channel states
can be uniformly written as:

$\mathbf{v}(\phi)=[v_{1}(\phi),\dots,v_{\tilde{M}}(\phi)]\in\mathcal{M}^{\tilde{M}}.$
(25)

Given $\phi$ and the frequency selection vector $\mathbf{v}(\phi)$, we obtain
the packet drop probability matrix $\mathbf{V}(\mathbf{v}(\phi))$ based on
(20):

$\displaystyle{}_{m,m}=d_{m,v_{m}(\phi)},\forall m\in\mathcal{\tilde{M}}.$
(26)

From (10), the current frequency selection action $\nu$ will only affect the
next AoI state $\phi^{\prime}$ and cost $c(\phi^{\prime})$, and has no impact
on the next channel state. Given the current AoI $\phi$, the current channel
state $\mathbf{\tilde{h}}_{i}$, and the selected frequency $m$, the
probabilities that $\phi^{\prime}=\phi+1$ and $1$ are $d_{i,m}$ and
$1-d_{i,m}$, respectively. Using the monotonicity of the cost function in
(18), the AoI state $1$ is a better state than $\phi+1$ in terms of the
current and future cost functions. So the frequency selection action that
leads to the lowest chance of state $\phi+1$ is the best action in terms of
the long-term average cost. The optimal frequency selection policy is a greedy
one that always select the frequency with the lowest packet drop probability
at each time, and thus is independent of the AoI state.

######  Lemma 2.

For the remote estimator described above, if there exists deterministic and
stationary frequency selection policies that stabilize the remote estimator,
the optimal policy achieving the minimum average cost is independent of the
AoI state, and is given by

$\nu^{\star}_{i}=\pi^{\star}(\mathbf{\tilde{h}}_{i})\triangleq\operatorname*{arg\,min}_{m\in\mathcal{M}}d_{i,m},\forall
i\in\mathcal{\tilde{M}}.$ (27)

In what follows, we solely need to analyze the average cost of the remote
estimator over the optimal policy above.

####  III-2 Analysis of the average cost and the necessary condition

We adopt an estimation cycle based analysis method that we developed earlier
in [12, 9]. The infinite time horizon are divided into estimation cycles, each
starting after a successful transmission and ending at the next one. Thus, the
AoI state is always equal to $1$ at the beginning of each estimation cycle,
and linearly increases step-by-step. Let $T_{k}$ denote the length of the
$k$th estimation cycle. $C_{k}$ is the sum cost in the $k$th estimation cycle
and is a function of $T_{k}$ as

$C_{k}=g(T_{k})\triangleq\sum_{j=1}^{T_{k}}\eta(\rho(\mathbf{A}_{n}))^{2j}\geq\eta(\rho(\mathbf{A}_{n}))^{2T_{k}}.$
(28)

Define the channel state before the $k$th cycle as
$\mathbf{b}_{k}\in\mathcal{\tilde{S}}$. Without loss of generality, assume
that the first $\tilde{M}_{1}$ channel states of $\mathcal{\tilde{S}}$ can be
pre-cycle states, where $0<\tilde{M}_{1}\leq\tilde{M}$, i.e., not all channel
states in $\mathcal{\tilde{S}}$ have to be a pre-cycle state. The channel
state $\mathbf{\tilde{h}}_{i}$ with a strictly zero chance of transmission
success can never be a pre-cycle state, i.e.,
$\min_{m\in\mathcal{M}}d_{i,m}=1$. Similar to [9, Lemma 1], the Markovian
property of the pre-cycle channel states below can be proved.

######  Lemma 3.

$\\{\mathbf{b}\\}_{\mathbb{N}}$ is a time-homogeneous ergodic Markov chain
with $\tilde{M}_{1}\leq\tilde{M}$ irreducible states of $\mathcal{\tilde{S}}$.
The state transition matrix of $\\{\mathbf{b}\\}_{\mathbb{N}}$ is
$\mathbf{G}^{\prime}$, which is the $\tilde{M}_{1}$-by-$\tilde{M}_{1}$ matrix
taken from the top-left corner of

$\mathbf{G}=\sum_{j=1}^{\infty}\mathbf{\tilde{\Xi}}(j),$ (29)

where
$\mathbf{\tilde{\Xi}}(j)=\mathbf{\mathbf{\Xi}}(j-1)\left(\mathbf{I}-\mathbf{V}^{\star}\right)\mathbf{\tilde{M}},j=1,2,\dots,$
$\mathbf{V}^{\star}\triangleq\mathbf{V}(\mathbf{v}^{\star})$, and

$\mathbf{\mathbf{\Xi}}(j)=\begin{cases}\mathbf{I},&j=0\\\
(\mathbf{V}^{\star}\mathbf{\tilde{M}})^{j},&j>0.\end{cases}$ (30)

We note that the terms $\mathbf{\Xi}(j-1)$ and
$\left(\mathbf{I}-\mathbf{V}^{\star}\right)\mathbf{\tilde{M}}$ in
$\mathbf{\tilde{\Xi}}(j)$ are related to the $(j-1)$ times of consecutive
failed transmissions and the successful transmission right after these of a
length-$j$ estimation cycle. Let
$\bm{\beta}\triangleq[\beta_{1},\dots,\beta_{\tilde{M}_{1}}]^{\top}$ denote
the stationary distribution of the Markov chain
$\\{\mathbf{b}\\}_{\mathbb{N}_{0}}$, which is the unique null-space vector of
$(\mathbf{I-G^{\prime}})^{\top}$ and $\beta_{i}>0,\forall
i\in\mathcal{\tilde{M}}_{1}$, where
$\mathcal{\tilde{M}}_{1}\triangleq\\{1,2,\dots,\tilde{M}_{1}\\}$.

Due to the ergodicity of $\\{\mathbf{b}_{k}\\}$ and the definition in (28), it
directly follows that the random processes $\\{T_{k}\\}$ and $\\{C_{k}\\}$ are
ergodic. Using the property that the time average is equal to the ensemble
average of an ergodic process, we drop the time index of $T_{k}$, $C_{k}$ and
$\mathbf{b}_{k}$, and have
$\mathsf{E}\left[C\right]=\lim\limits_{K\rightarrow\infty}\frac{1}{K}\sum_{k=1}^{K}C_{k}=\sum_{m=1}^{\tilde{M}_{1}}\beta_{m}\mathsf{E}\left[C|\mathbf{b}=\mathbf{\tilde{h}}_{m}\right].$

By following the same steps in the proof of Theorem 1 of [9], we can first
show that the average cost $J<\infty$ if and only if
$\mathsf{E}\left[C\right]<\infty$, and then derive the necessary condition
making $\mathsf{E}\left[C\right]$ bounded as
$\rho^{2}(\mathbf{A})\rho\left(\mathbf{V}^{\star}\mathbf{\tilde{M}}\right)<1$.
∎

###### Proof of Sufficiency.

We construct a _persistent serial scheduling_ policy that persistently
schedules sensor 1’s transmission at a time until it is successful and then
schedules sensor $2$ and so on. For each transmission, the frequency is
selected based on the scheme (27). Using the upper bound of the per-step cost
function (17) and following the similar analytical steps of the average sum
cost per estimation cycle in [9], we can derive the upper bound of each
sensor’s average sum cost per cycle and then obtain the sufficient condition
of Theorem 1. ∎

######  Remark 2.

Although the policy constructed above is a stability-guaranteeing one, it does
not utilize the parallel frequency channels and is strictly not optimal. For
the optimal policy design, once the stability condition is satisfied, we can
first design a suitable MDP problem, and then use classic dynamic programming
or deep reinforcement learning algorithms to solve it, see for example [8,
14].

Extension Scenario: We also investigate the stability condition of the
scenario with delayed channel state information.

######  Assumption 2 (Known Previous Channel State [8, 9]).

At time $t$, only the previous channel state $\mathbf{\tilde{h}}(t-1)$ is
available.

Building on the multi-level Markov channel modeling of
$\\{\mathbf{\tilde{h}}(t)\\}$ in Section II-B and following the same
analytical steps in [9], which focused on the binary-level Markov channel
scenario under Assumption 2, we can derive the stability condition as below.

######  Theorem 2.

The $M$-sensor-$N$-frequency remote estimator described in Section II under
Assumption 2 can be stabilized if and only if
$\rho^{2}_{\max}\lambda_{\infty}<1,$ where
$\lambda_{\infty}\triangleq\min_{L\in\mathbb{N}}\lambda_{L},$

$\lambda_{L}\triangleq\min_{\mathbf{v}_{l}\in\mathcal{M}^{\tilde{M}}}\rho\left(\mathbf{E}(\mathbf{v}_{1})\mathbf{E}(\mathbf{v}_{2})\cdots\mathbf{E}(\mathbf{v}_{L})\right)^{\frac{1}{L}}$
(31)

and $\mathbf{E}(\mathbf{v})$ is an $\tilde{M}\times\tilde{M}$ matrix generated
by the channel selection vector
$\mathbf{v}=[v_{1},\dots,v_{\tilde{M}}]\in\mathcal{M}^{\tilde{M}}$ as

$\displaystyle[\mathbf{E}(\mathbf{v})]_{i,j}\triangleq\mathsf{Prob}\left[\mathbf{\tilde{h}}(t)=\mathbf{\tilde{h}}_{j},\mu\left(\mathbf{\tilde{h}}_{j},v_{i}\right)=0|\mathbf{\tilde{h}}(t-1)=\mathbf{\tilde{h}}_{i}\right]$
(32) $\displaystyle=\mathbf{\tilde{M}}_{i,j}d_{j,{v_{i}}}.$

######  Remark 3.

The stability condition in Theorem 2 is more restrictive than that in Theorem
1. Intuitively, the gap in stability is introduced by the delay of the channel
state information for scheduling. The scheduler with current channel state
information can make better decision for stabilizing the system than the one
with outdated information.

We will show that $\lambda$ in (22) is no larger than
$\min_{L\in\mathbb{N}}\lambda_{L}$ in (31). Using the property that
$\rho(\mathbf{X}^{L})^{1/L}=\rho(\mathbf{X})$, for any square matrix
$\mathbf{X}$ and positive integer $L$, we have
$\lambda=\rho\left(\left(\mathbf{V}(\mathbf{v}^{\star})\mathbf{\tilde{M}}\right)^{L}\right)^{1/L},\forall
L.$ From the definition of $\mathbf{V}(\mathbf{v}^{\star})$ in (23), the
following inequality about non-negative matrices holds element-wise
$\mathbf{V}(\mathbf{v}^{\star})\mathbf{\tilde{M}}\preccurlyeq\mathbf{E}(\mathbf{v}),\forall\mathbf{v}\in\mathcal{M}^{\tilde{M}},$
and hence
$\left(\mathbf{V}(\mathbf{v}^{\star})\mathbf{\tilde{M}}\right)^{L}\preccurlyeq\mathbf{E}(\mathbf{v}_{1})\mathbf{E}(\mathbf{v}_{2})\cdots\mathbf{E}(\mathbf{v}_{L}),\forall\mathbf{v}_{1},\dots,\mathbf{v}_{L}.$
Using the property that the spectral radius of one non-negative matrix is
larger than the other if the former is larger element-wise, it is readily
shown that $\lambda\leq\lambda_{L},\forall L$, and hence
$\lambda\leq\lambda_{\infty}$.

##  IV Numerical Examples and Conclusions

In practice, the maximum frequency number of WiFi can be $48$, and the maximum
holding period can be $100$ calculated by the practical channel coherence time
and packet duration. For the illustration of the stability condition, we
consider a simple 3-sensor-2-frequency remote estimation system with a maximum
holding period $\bar{\varDelta}=2$.

The spectral radii of the three processes are $\rho(\mathbf{A}_{1})=1.5$,
$\rho(\mathbf{A}_{2})=1.2$, and $\rho(\mathbf{A}_{3})=1.1$. Each of the two
frequency channels has two quality states,
$\mathcal{H}_{1}=\\{h^{(1)}_{1},h^{(1)}_{2}\\}$ and
$\mathcal{H}_{2}=\\{h^{(2)}_{1},h^{(2)}_{2}\\}$. The packet drop probability
at frequency-$m$-state-$i$ is denoted as $\tilde{d}_{m,i}$. Thus, the (vector)
channel quality state has $\bar{M}=4$ states in total,
$\mathbf{h}(t)\in\\{\mathbf{h}_{1}=[h^{(1)}_{1},h^{(2)}_{1}],\mathbf{h}_{2}=[h^{(1)}_{1},h^{(2)}_{2}],\mathbf{h}_{3}=[h^{(1)}_{2},h^{(2)}_{1}],\mathbf{h}_{4}=[h^{(1)}_{2},h^{(2)}_{2}]\\}$.
The channel quality transition probability is $\mathbf{M}=[0.1\ 0.2\ 0.3\
0.4;0.2\ 0.1\ 0.4\ 0.3;0.4\ 0.2\ 0.1\ 0.3;0.3\ 0.1\ 0.4\ 0.3]$. The
probability distribution of the holding period is

$\psi_{i}(\delta)=\begin{cases}\tilde{\psi}_{1}&\text{ if }\delta=1\\\
\tilde{\psi}_{2}=1-\tilde{\psi}_{1}&\text{ if }\delta=2\end{cases},\forall
i=1,2,3,4.$ (33)

Thus, the state $\mathbf{\tilde{h}}(t)=(\mathbf{h}(t),\varDelta_{l}(t))$ has
$\tilde{M}=\bar{M}\times\bar{\varDelta}=8$ states:
$(\mathbf{h}_{1},1),(\mathbf{h}_{1},2),\dots,(\mathbf{h}_{4},1),(\mathbf{h}_{4},2)$.
From Theorem 1, the $8$ diagonal elements of the matrix
$\mathbf{V}(\mathbf{v}^{\star})$ are
$\min\\{\tilde{d}_{1,1},\tilde{d}_{2,1}\\}$,
$\min\\{\tilde{d}_{1,1},\tilde{d}_{2,1}\\}$,
$\min\\{\tilde{d}_{1,1},\tilde{d}_{2,2}\\}$,
$\min\\{\tilde{d}_{1,1},\tilde{d}_{2,2}\\}$,
$\min\\{\tilde{d}_{1,2},\tilde{d}_{2,1}\\}$,
$\min\\{\tilde{d}_{1,2},\tilde{d}_{2,1}\\}$,
$\min\\{\tilde{d}_{1,2},\tilde{d}_{2,2}\\}$, and
$\min\\{\tilde{d}_{1,2},\tilde{d}_{2,2}\\}$. From (8), the channel state
transition matrix $\mathbf{\tilde{M}}$ can be obtained directly.

In Fig. 3(a), we plot the stability regions in terms of $\tilde{d}_{11}$ and
$\tilde{d}_{12}$ based on Theorem 1, where $\tilde{d}_{21}=0.2$ and
$\tilde{d}_{22}=0.9$. It is interesting to see that the stability region
increases with $\tilde{\psi}_{1}$, i.e., the probability that the holding
period of the channel condition is $1$. This implies that a fast fading
scenario can lead to a better stability than a slow fading one. Compared to
Fig. 3(a), we increase the packet drop probability at frequency-$2$-state-$1$,
i.e., $\tilde{d}_{2,1}$, from $0.2$ to $0.9$ in Fig. 3(b). We see that the
reduced transmission reliability has lead to diminished stability regions as
expected.

Figure 3: Stability regions (gray colored) with
$\tilde{d}_{21}=0.2,\tilde{d}_{22}=0.9$ in (a) and
$\tilde{d}_{21}=0.9,\tilde{d}_{22}=0.9$ in (b), where the solid, doted, and
dashed lines indicate the regions with $\tilde{\psi}_{1}=0.99,0.5$, and $0.1$,
respectively.

##  V Conclusions

We have investigated the necessary and sufficient stability conditions of the
multi-plant remote estimation system. For future work, in addition to the
local filter-based remote estimation scenario, we will also consider the
extension to transmission of raw measurements and investigate the stability
conditions.

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