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On the Trio Unification of Information Geometry, Einsteinian Relativity, and Transient Queues the Parthasarathian Case

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Submitted:

31 January 2024

Posted:

02 February 2024

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Abstract
The current study characterizes the transient M/M/1 queue manifold info-geometrically, through devising Fisher Information matrix(FIM) and its inverse(IFIM). Additionally, the impact of stability on the existence of IFIM and explore the geodesic equations of motion has been revealed. More potentially, the paper discusses the relationship between stability and the Gaussian curvature, as well as the connections between queueing theory, information geometry, , Riemannian geometry, and the Theory of Relativity.
Keywords: 
Subject: Computer Science and Mathematics  -   Probability and Statistics

1. Introduction

Numerous study areas, including statistical sciences, have extensively used IG [1]. In other words, the goal of IG is to use statistics to apply the methods of differential geometry (DG), which indicates the major goal of IG is to use stochastic processes and probability theoretic in the applications of methodologies of non-Euclidean geometry.
A manifold is a topological finite-dimensional Cartesian space, R n , in which an infinite-dimensional manifold exists[2]. Additionally, IG supports SMs’ descriptions that are based on intuitive reasoning. One might have a greater understanding of the crucial significance of IG[3,4]. Parametrization of a Statistical Manifold (SM) is visualized by figure 1[4].
Figure 1. SM’s parametrization(Nielsen 2020).
Figure 1. SM’s parametrization(Nielsen 2020).
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According to the literature, a paper by[5] examined info-geometrically the stable M/D/1 queue on the basis of queue length routes ‘characteristics, was the real motivator for the current study.
The main deliverables of this paper are described below.
Revealing the new discovery of the geodesic equations of motion of the coordinates of the transient M/M/ .
In the context of the research paper, a novel 𝛼-connection [6] is introduced, which maps each coordinate to a value. The paper also discusses the concept of incompressibility or solenoidality in the transient M/M/1 queue, where closed surfaces have no net flux [7], and includes definitions and physical interpretations of Gaussian and Ricci curvatures.
Revolutionary relativistic and transient queueing theoretic connections are established.
The remainder of the paper is divided into the following sections: Preliminary definitions of Information Geometry (IG) are given in Section 2. The transient M/M/1 QM’s FIM and it IFIM are introduced in Section 3. While Section 5 gives the Information Geometric Equations of Motion (IMEs) for the coordinates of the transient M/M/1 QM, Section 4 addresses the -connection of the queue manifold. The potential function, divergence measurements, stability, and curvature of the M/M/1 QM are all discussed in detail in Sections 6 to 13. Section 14 wraps up the report and provides a plan for further research.

2. Main Definitions

2.1. Main Definition on IG

Definition 1. Statistical Manifold (SM)
M = { p ( x , θ ) | θ ϵ Θ }  is called an SM[8] if x is a random variable in sample space  X  and  p ( x , θ )  is the probability density function, which satisfies certain regular conditions. Here,  θ = ( θ 1 , θ 2 , . . , θ n ) ϵ Θ  is an n-dimensional vector in some open subset  Θ R n , and   θ  can be viewed as the coordinates on manifold M.
Definition 2. Potential Function
The potential function Ψ θ (c.f., (2.1)) [8] is the distinguished negative function of the coordinates alone of ( L x ; θ = l n p x ; θ ).
Definition 3. Fisher’s Information Matrix(FIM)
The FIM (or, Fisher’s metric) [ g i j ][6] is given by the Hessian (the nxn matrix of the partial derivatives of the potential function Ψ ( θ ) with respect to the coordinates) i.e.,
g i j = 2 θ i θ j Ψ θ , i , j = 1,2 , . . , n
with respect to natural coordinates.
Definition 4. IFIM
Given the FIM, the inverse matrix of [ g i j ] is defined by[8]
[ g i j ] = ( [ g   i j ] ) ) 1 = a d j g i j
= det g i j
Definition 5.
α -Connection For each  α ϵ R ,  the  α (or  ( α ) )-connection [6] is the torsion-free affine connection with components:
Γ i j , k ( α ) = ( 1 α 2 ) ( i j k ( Ψ θ ) )
where Ψ ( θ ) is the potential function and i = θ i .
Definition 6.
(1)The geodesic equations of manifold M with coordinate system  θ = θ 1 , θ 2 , . . , θ n  read as [8]
d 2 θ k d t 2 + Γ i j k 0 d θ i d t ( d θ j d t ) = 0 , i , j = 1,2 , , n , Γ i j k α = Γ i j , s α g s k
By the above definition, it is clear that the geodesic equations are interpreted physically as the information geometric equations of motion , shortly (IGEMs), or the relativitstic equations of motion (REMs) , or the Riemannian equations of motion.
At this stage, the current study provides a ground- breaking discovery of the IG analysis of transient queueuing systems in comparison to that of non-time dependent queueing systems, namely IG analysis of stable queueues[9,10].
Definition 7.
1. The  α c u r v a t u r e   R i e m a n n i a n Tensors,    R i j k l ( α ) [6] read as:
  R i j k l ( α ) = j Γ i k s α i Γ j k s α g s l + Γ j β , l α Γ i k β α Γ i β , l α Γ j k β α ,   i , j , k , l , s , β = 1,2 , 3 , . , n
Where
Γ i j k α = Γ i j , s α g s k
2.  K 1212 α  =  K α  is  α G a u s s i a n   c u r v a t u r e   and reads as [6]
K α = R 1212 ( α ) d e t g i j
Definition 8.
(i) Developable surfaces are a specific class of ruled surfaces that may be mapped onto a flat surface without the curves being altered in any way; each curve is generated from such a surface and keeps its original shape [11].
Figure 2. Three kinds of developable surfaces: Tangential on Figure. 5a (on the left), Conical on Fig. 5b (on the centre) and Figure. 5c (on the right), Cylindrical. Note that curves in bold are directrix or base curves and straight lines in bold are directors or generating lines (curves) [11].
Figure 2. Three kinds of developable surfaces: Tangential on Figure. 5a (on the left), Conical on Fig. 5b (on the centre) and Figure. 5c (on the right), Cylindrical. Note that curves in bold are directrix or base curves and straight lines in bold are directors or generating lines (curves) [11].
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Definition 9.
In mathematics, a function is considered well-defined if it consistently produces the same output regardless of how the input is represented. This means that even if the input is expressed differently, if the value remains the same, the function will yield the same result. This concept ensures the reliability and consistency of mathematical operations and calculations.
Definition 10 [13].
1. function f is said to be one-to-one, or injective, if and only if  f x = f y implies  x = y   for all x, y in the domain of   f . A function is said to be an injection if it is one-to-one. Alternative: A function is one-to-one if and only if  f x f y , whenever x ≠ y. This is the contrapositive of the definition.
2.A function f from A to B is called onto, or surjective, if and only if for every  b B  there is an element  a A  such that  f ( a ) = b . Alternative: all co-domain elements are covered.
3. A function f is called a bijection if it is both one-to-one (injection) and onto (surjection).
Theorem 1
[14] Let f be a function that is defined and differentiable on an open interval (c,d).
If   f x > 0 ( < 0   )   for   all   x ( c , d )   , then   f is   increasing ( decreasing )   on   c , d .
Taylor expansions [15]) are widely used to approximate functions by expansions. We have for all x around zero,
ln 1 + x = n = 1 ( 1 ) n + 1 x n n
1 1 + x = n = 0 ( 1 ) n x n

2.2. Important Inequalities [16]

1. Chebyshev inequality
x 1 + x ln 1 + x x   f o r   x 1
(12) can be rewritten as
  1 1 x ln x x 1   f o r   x 1
2. Lehmer inequality
  x n n ! < x n n ! + 1 e x   for   all   x , n > 0   

3. Fim and Ifim of the Transient M/M/1QM

A single-channel model with exponential inter-arrival times, service times, and FIFO queue discipline is the simplest probabilistic queueing model that can be addressed analytically. In queueing theory, this is referred to as the M/M/1 queue [17].
Based on the introduction of the function, Parthasarathy[18] suggested a straightforward method for the transient solution of the M/M/1 system to read as:
p n t = ρ p 0 t e λ + μ t [ m = 1 n r m ( t ) + e λ + μ t ]   ,    n = 0,1 , 2 , .  
where λ   stands for the mean rate per unit time at which arrival instants occur, μ is the mean rate of service time, ρ = λ ( t ) μ ( t ) defines the traffic intensity or utilisation factor,
r n t = μ β 2 n a 1 1 δ 0 a 1 I n a 1 β 1 t I n + a 1 β 1 t + λ β 2 n 1 a 1 ( I n + 1 a 1 β 1 t I n 1 a 1 β 1 t )
w i t h   β 1 = 2 λ μ , β 2 = 2 λ μ , I n ( x ) [19] is the modified Bessel function.
And
p 0 t = 0 t r 1 e ( λ + μ ) y d y + δ 0 a 1
Theorem 2.
The underlying queue of (15) has:
(i) FIM reads as
[ g i j ] = a b c b 0 h c h l
where
a = 1 ρ 2 1 ( 1 ρ ) 2
b = t
c =   ρ . ρ 2 + ρ . ( 1 ρ ) 2 + t μ . + μ ρ .
   h = 1 + ρ + t ρ .
l = ρ . 2 ρ ρ . . ρ 2 + ρ . 2 + ( 1 ρ ) ρ . . ( 1 ρ ) 2 + ( ( 1 + ρ ) t μ . . + μ + 2 ρ . μ . + 2 μ ρ . + 2 ρ μ . + μ t ρ . .
Δ = d e t ( g i j )
= a h 2 b b l c h + c ( b h )
Provided that, . refers to the temporal derivative d d t .
(iii) [ g i j ] reads as
[ g i j ] = a d j g i j =         A B L B E F L F H
where
A = ( h 2 )
B = ( c h b l )
L = ( b h )
E = ( a l c 2 )
F = ( b c a h )
H = ( b 2 )
Proof. 
(i)Following (15), we have:
L x ; θ = l n p n x ; θ                            =   l n ρ p 0 t e λ + μ t + l n ρ p 0 t e λ + μ t [ m = 1 n r m ( t ) + e λ + μ t ]  
    θ = ( θ 1 ,   θ 2 , θ 3 ) = ( ρ , μ , t )
We have
Ψ θ = l n ρ l n 1 ρ + μ 1 + ρ t ,    p 0 t = 1 ρ
Thus,