=Paper=
{{Paper
|id=Vol-2064/paper07
|storemode=property
|title=
Использование приближения среднего поля для анализа крупномасштабных транспортных сетей с малым параметром
(Mean-field approximation for large-scale transport networks with a small parameter)
|pdfUrl=https://ceur-ws.org/Vol-2064/paper07.pdf
|volume=Vol-2064
|authors=Artem Blinov,Sergey Vasilyev,Leonid Sevasyanov
}}
==
Использование приближения среднего поля для анализа крупномасштабных транспортных сетей с малым параметром
(Mean-field approximation for large-scale transport networks with a small parameter)
==
https://ceur-ws.org/Vol-2064/paper07.pdf
УДК 517.937, 517.928.2, 519.217.2
Блинов А.И., Васильев С.А., Севастьянов Л.А.
Российский университет дружбы народов, г. Москва, Россия
ИСПОЛЬЗОВАНИЕ ПРИБЛИЖЕНИЯ СРЕДНЕГО ПОЛЯ ДЛЯ АНАЛИЗА
КРУПНОМАСШТАБНЫХ ТРАНСПОРТНЫХ СЕТЕЙ С МАЛЫМ ПАРАМЕТРОМ*
Аннотация
Решение задач математического моделирования сложных транспортных сетей на данном
этапе представляет большую сложность по причине большого объема данных, которые
приходится анализировать. Например, огромное количество возможных вариантов
перевозок затрудняет получение достаточно экономного плана эмпирическим или
экспертным путем. Применение математических методов и использование современных
вычислительных алгоритмов в планировании перевозок дает большой экономический
эффект. Проведенный анализ показал, что этот подход является эффективным для
решения широкого круга технических и технологических проблем проектирования,
строительства и функционирования транспортных систем. В рамках этого подхода
удается создать эффективный алгоритм минимизации затрат на проектирование,
строительство и эксплуатацию таких систем. Транспортные задачи могут быть решены
симплексным методом, однако матрица системы ограничений транспортной задачи часто
настолько сложна, что для ее решения разработаны специальные методы. В данной работе
исследуются крупномасштабные транспортные сети с использованием приближения
среднего поля Добрушина. Показано, что анализ эволюции крупномасштабных
транспортных систем можно описать с помощью системы дифференциальных уравнений
бесконечного порядка. Для этой системы можно поставить задачу Коши тихоновского типа
с малым параметром, который вносит сингулярное возмущение. В статье доказана
теорема существования решения этой задачи Коши.
Ключевые слова
Аналитические методы в теории транспортных сетей; системы дифференциальных
уравнений бесконечного порядка; малый параметр; счетные цепи Маркова;
крупномасштабные транспортные сети; приближение среднего поля Добрушина;
транспортная задача; динамика сложных систем.
Blinov A.I., Sevastyanov L.A., Vasilyev S.A.
Peoples' Friendship University of Russia, Moscow, Russia
MEAN-FIELD APPROXIMATION FOR LARGE-SCALE TRANSPORT NETWORKS WITH A
SMALL PARAMETER
Abstract
The solution of mathematical simulation problems of complex transport networks at this stage is
more difficult because of the large amount of data that must be analyzed. For example, a huge number
of possible options of traffic makes it difficult to obtain sufficient economical plan through empirical
or using expert approach. Application of mathematical methods and use of modern computational
algorithms for transport planning gives considerable economic benefit. It is shown this approach is
effective for solving a wide range of technical and technological problems of design, construction and
operation of transport systems. In this approach manages to create an efficient algorithm for
minimizing the cost of design, construction and operation of such systems. The transportation
problem can be solved by simplex method but the matrix of the constraints of the transportation
problem is often so complex that its solution developed special methods. In this paper it is studied
* Труды II Международной научной конференции «Конвергентные когнитивно-
информационные технологии» (Convergent’2017), Москва, 24-26 ноября, 2017
Proceedings of the II International scientific conference "Convergent cognitive information
technologies" (Convergent’2017), Moscow, Russia, November 24-26, 2017
62
large-scale transport network using Dobrushin’s mean-field approximation. It is shown that the
analysis of the evolution of large-scale transport systems can be described using systems of differential
equations of infinite order. For this system, it is formulated the Cauchy problem Tikhon type with a
small parameter , which introduces a singular perturbation. The theorem of existence of the
solution of this Cauchy problem is proved.
Keywords
Analytical methods in transport networks theory; systems of differential equations of infinite order;
small parameter; countable Markov chains; large-scale transport networks; Dobrushin mean-field
approximation; transportation problem; dynamics of complicated systems.
Introduction
In this paper large-scale transport networks are studied using Dobrushin mean-field approach [1,4-5,10,17-
19]. We assume that the transport networks deal with the problem of proving the global convergence of the
solutions of certain infinite systems of ordinary differential equations to a time-independent solution. In work,
[4,5,10] the infinite systems of differential equations modelling large-scale transport systems are studied and the
sufficient conditions of global stability and global asymptotic stability are obtained.
Cauchy problems for the systems of ordinary differential equations of infinite order was investigated
A.N. Tihonov [14], K.P. Persidsky [11], O.A. Zhautykov [20-21], Ju. Korobeinik [6], M.A. Krasnoselsky, P.P. Zabreyko
[8], A.M. Samoilenko, Yu.V. Teplinskii [12] other researchers.
It was studied the singular perturbed systems of ordinary differential equations by A.N. Tihonov [15],
A.B. Vasil'eva [16], S.A. Lomov [9] other researchers.
In papers [3], [7], [13] the authors built various models of large-scale queueing systems and considered their
dynamics.
In paper [2] it was investigated the singular perturbed systems of ordinary differential equations of infinite
order of Tikhonov-type x = F ( x(t , g x ), y (t , g y ), t ) , y = f ( x(t , g x ), y (t , g y ), t ) with the initial conditions x(t0 ) = g x ,
y (t0 ) = g y , where x, g x X , X l1 and y, g y Y , Y R n , t t0 , t1 ( t0 < t1 ), t0 , t1 T , T R , g x and g y are
given vectors, > 0 is a small real parameter.
In this paper we considered large-scale transport network systems that consists of infinite number of network
service nodes with a Poisson input flow of requests. We assume that the queuing system has N nodes and rN
servers. At each node ( N nodes) the arrivals of particles form a Poisson flow of rate . For an empty node a
particle leaves the system. A server at the node takes the particle and moves to a random node. Travelling time is
exponential of mean 1/ . The number of servers at each node (of these N nodes) is bounded by m . We consider
the property of the system for the limiting deterministic process as N . The evolution analysis of large-scale
transport systems can be described using an infinite system of differential equations. It is possible to investigate
Tikhonov type Cauchy problem for this system with small parameter. In this paper we apply Dobrushin mean-field
approaches from [5,10] for analysis of the singular perturbed systems of ordinary differential equations of infinite
order.
It is possible to formulate Tikhonov type Cauchy problem for this system with small parameter and initial
conditions. We study the singular perturbed Tikhonov systems of ordinary differential equations of infinite order
u = f (u (t , gu ), U (t , gU ), t ) , U = F (u (t , gu ),U (t , gU ), t ) with the initial conditions u (0, gu ) = gu , U (0, gU ) = gU ,
where > 0 is a small real parameter. The theorem of existence of solution for this Cauchy problem is proved.
Large-scale transport network model
Let's consider a large-scale transport networks that consist of N nodes, a virtual node and rN servers. At each
node (of these N nodes) the arrivals of particles form a Poisson flow of rate . If a particle arrives at an empty
node then the particle leaves the system. Otherwise, if there is a server at the node then the server takes the particle
and jumps to the virtual node. At this node the server waits for an exponential time of mean t = 1/ . After the
server jumps to a random node with uniform distribution. If the number of servers at the chosen node equals m
then the server waits for the following attempt at the virtual node. The non-negative number of servers at each
node (except virtual) is bounded by m . Consider the fractions f k = nk / N , V = W / N , where nk is the (random)
number of nodes with k servers and W is the number of servers at the virtual node. It is more convenient to
regard the tail probabilities uk = i = k fi . The state space of the corresponding Markov process
m
U N (t ) = (uk (t ),V (t )) is the set X N of all vectors u = (u1 , , um ,V )T in (1/ N ) Z m 1 such that 1 = u0 u1 um ,
63
V 0, u1 um V = r. The generator of U N (t ) is the operator AN (t ) acting on functions and given by
m 1
e e
AN (t ) f (u ) = N (uk uk 1 )[ f u k m 1 f (u )]
k =1 N N
e e
N um [ f u k m 1 f (u )],
N N
m
e e
N V (uk 1 uk )[ f u k m 1 f (u )]
k =1 N N
where ek denotes a vector with the component of number k equal to 1 and others equal to 0 .
The mean-field approximation suggests that the whole process U N (t ) is asymptotically deterministic as
N . More precisely, let X denote the set of all R m 1 vectors defined by (1). Then, if the distribution of the
initial state U N (0) converges to the Dirac delta-measure concentrated at some point g X , the distribution of
U N (t ) is concentrated on the orbit u (t ) X as N where u (t ) is the solution of the following system of
differential equations (mean-field equations)
V (t ) = u1 (t ) V (t ) u0 (t ), u0 (t ) = 1;
uk (t ) = ui 1 (t ) ui (t ) V (t ) ui 1 (t ) ui (t ) ,
k =0
uk (t ) V (t ) = r (t ), r (r ) > 0,
V (0) = V 0, u (0) = g 0, k = 0,1, 2, ,
0 k k
1 = g0 g1 g 2 , , t 0
is a parameter and g = g k is a numerical sequence. The infinite order system (2) is non-linear
where r (t ) > 0
k =1
and its right-hand side depends on time.
Large-scale queueing systems model with a small parameter
We can investigate infinite system of differential equations with small parameter such form
V (t ) = u1 (t ) V (t ) u0 (t ), u0 (t ) = 1;
uk (t ) = uk 1 (t ) uk (t ) V (t ) uk 1 (t ) uk (t ) , k = 1, 2, , n,
sk uk (t ) = uk 1 (t ) uk (t ) V (t ) uk 1 (t ) uk (t ) , k = n 1, n 2, ,
uk (t ) V (t ) = r (t ), r (t ) > 0,
k =0
V (0) = V0 0, uk (0) = g k 0, k = 0,1, 2, ,
1 = g 0 g1 g 2 , , t 0,
where is a small parameter that bring a singular perturbation to the system (2) which allows us to describe the
processes of rapid change of the systems and s = sk k = n 1 ( sk > 0) is a numerical sequence.
Using (3) we can write Tikhonov problems for systems of ordinary differential equations of infinite order with
a small parameter and initial conditions
V (t ) = u1 (t ) V (t ) u0 (t ), u0 (t ) = 1;
u = f (u (t , , , g u ),U (t , , , gU ), t ),
k U = F (U (t , , , gU ), t );
s
V (0) = V 0, u (0, , , g ) = g ,
0 u u
U (0, , , gU ) = gU ,
where u , f X , X R are (n+1)-dimensional functions; U , F Y , Y l1 are infinite-dimensional functions
n 1
and t 0, T0 ( 0 < T0 ), t T , T R ; gu X and gU Y are given vectors
64
( gu = g k k =0 , gU = g k k = n 1 , 1 = g 0 g1 g 2 , ) , > 0
n
is a small real parameter; u (0, gu ) = gu and
U (0, gU ) = gU are the conditions for solutions of (4). Given functions f (u(t , , , gu ),U (t, , , gU ), t ) and
F (U (t , , , gU ), t ) are continuous functions for all variables
f k (u (t , , , gu ), t ) = uk 1 (t ) uk (t ) V (t ) uk 1 (t ) uk (t ) , k = 1, , n,
Fk (U (t , , , gU ), t ) U k 1 (t ) U k (t ) V (t ) U k 1 (t ) U k (t ) , k = n 1, n 2,
Let S is an integral manifold of the system (4) in X Y T . If any point t 0, T0 (u (t ), U (t ), t ) S of
* * * *
trajectory of this system has at least one common point on S this trajectory (u (t , G ),U (t , g ), t ) S belongs the
integral manifold S totally.
If we assume in (4) that = 0 than we have a degenerate system of the ordinary differential equations and a
problem of singular perturbations
V (t ) = u1 (t ) V (t ) u0 (t ), u0 (t ) = 1,
u = f (u (t , , , gu ),U (t ), t ),
0 = F (u (t , , gu ), U (t , , ), t );
u (0, , gu ) = gu ,
where the dimension of this system is less than the dimension of the system (4), since the relations
F (u (t , ),U (t , ), , t ) = 0 in the system (6) are the algebraic equations (not differential equations). Thus for the
system (9) we can use limited number of the initial conditions then for system (4). Most natural for this case we
can use the initial conditions u (0, , gu ) = gu for the system (6) and the initial conditions U (0, , U y ) = gU
disregard otherwise we get the overdefined system. We can solve the system (6) if the equation
F (u (t , ),U (t , ), , t ) = 0 has roots. If it is possible to solve we can find a finite set or countable set of the roots
U q (t , , gu ) = uq (u (t , , gu ), t ) where q N . If the implicit function F (u (t , ),U (t , ), , t ) = 0 has not simple
structure we must investigate the question about the choice of roots. Hence we can use the roots
U q (t , , gu ) = uq (u (t , , gu ), t ) ( q N ) in (10) and solve the degenerate system
ud = f (ud (t , , gu ), uq (ud (t , , gu ), t ), , t );
U d (0, , gu ) = gu .
Since it is not assumed that the roots U q (t , , gu ) = uq (u (t , , gu ), , t ) satisfy the initial conditions of the Cauchy
problem (4) ( U q (0) gu , q N ), the solutions U (t , , gU ) (4) and U q (t , , gu ) do not close to each other at the
initial moments of time t > 0 . Also there is a very interesting question about behaviors of the solutions u (t , , gu )
of the singular perturbed problem (4) and the solutions ud (t , , gu ) of the degenerate problem (6). When t = 0 we
have u (0, , gu ) = ud (0, , gu ) . Do these solutions close to each other when t 0, T0 ? The answer to this question
depends on using roots U q (t , , gu ) = uq (u (t , , gu ), t ) and the initial conditions, which we apply for the systems (7).
Analysis of infinite order system of differential equations
We can rewrite Tikhonov problems (4) for systems of ordinary differential equations of infinite order with a
small parameter and initial conditions in the form
v = FR (v(t , , , , v 0 ), t ),
v(0, , , , v0 ) = v ,
0
where
v = (V , u0 , u1 , , un ,U n1 ,U n2 , ),
FR0 = u1 (t ) V (t ) u0 (t ),
FRk = uk 1 (t ) uk (t ) V (t ) uk 1 (t ) uk (t ) , k = 1, , n, ,
uk 1 (t ) uk (t ) V (t ) uk 1 (t ) uk (t ) , k = n 1, n 2, ,
s s
FRk = k k
0
v = (V0 , gu , gU ),
0 0
where v = V , v = g k , k = 1, 2,
0 k
Using methods from [12], [20-21] we can consider Tikhonov-type problems (8)
65
v = FR (v0 , v1 , , vn , , , , , t ),
v(0, , , , v0 ) = v0 ,
Definition. A function FR v0 , v1 , , vn , , , , , t is called strongly continuous if for any 0 > 0 , there exist N 0
and 0 > 0 such that the inequality | vi vi |< 0 , i = 0,1, 2,
' '
, N 0 , implies the estimate for any 0, 0, > 0
| FR v , v , '
0
'
1 , , FR v , v1' ,
'
0
, , , |< 0 .
Theorem. Assume that the right-hand sides of the system of equations (10)
• are defined for any vi ( , , , t ) R , i = 0,1, 2, , 0, 0, > 0 and all t T0 = 0, t R ;
1 1
• are strongly continuous in v0 , v1 , for fixed t T0 , 0 , 0 , > 0 and measurable in t T0 for fixed
vi ( , , , t ), i = 0,1, 2, ;
• satisfy the inequalities
| FRi t , v0 , v1 , , , , |< M i (t )
for all i = 0,1, 2, , where M i (t ) are functions summable on the segment T0 and for any 0, 0, > 0 .
Then, for any vector
v00 , v10 ,
with real coordinates, there exists at least one solution
v0 ( , , , t ), v1 ( , , , t ), of the system of equations (14) such that vi (0) = vi0 , i = 0,1, 2,
.
Proof. We replace the system of equations (8) by the following system of integral equations:
t
vi (t ) = vi0 FRi t , v0 (t ), v1 (t ), , , , dt , i = 0,1, 2, ,
0
and consider a mapping ( A )
t
zi (t ) = vi0 FRi t , v0 (t ), v1 (t ), , , , dt , i = 0,1, 2, ,
0
which establishes a correspondence between an arbitrary countable system of continuous functions vi (t )i =0 and
another system of this sort zi (t )i =0 . Note that if FR (t , v0 ,
, vn , , , ) is a continuous function of finitely many
variables vi (t )i =0 measurable with respect to t for fixed vi , i = 0, n , then the function
n
(t ) = FR (t , 0 (t ), , n (t ), , , )
is measurable if i (t ), i = 0, n , are measurable.
Thus, the function
n (t ) = FR (t , 0 (t ), , n (t ), 0, 0, , , , )
is measurable and, therefore, the function
FR (t , 0 (t ), 1 (t ), , , , ) = (t , , , )
is also measurable because
(t ) = lim n (t , , , ),
n
which readily follows from the condition of strong continuity. The requirement of summability follows from
condition 3 of Theorem. We consider a system of functions vi (t )i =0 as a point P of an abstract space R . If there
exists a point P invariant under mapping ( A ) (14), then it specifies a solution of the system of equations (13)
and, hence, of system (10).
Consider a set M 0 formed by three points P for which vi (t )i =0 satisfy the conditions
t t
| vk (t ) vk0 | M k (t )dt , | vk (t ) vk (t ) | M k (t ) dt , k = 0,1, 2, .
0 t
It is easy to see that mapping ( A ) (14) maps the set M 0 into itself. We now introduce mapping ( B ) by putting
every point P in correspondence with a set of numbers
a00 an
, , 0 , ,
N0 N0
,
66
a1n ann
, , , ,
nN n nN n
,
t
where N i = vi M i (t ) dt and the numbers an n , r =0 ( an ,
0 0
r
, ann , ) are the coefficients of the Fourier expansion of
0
a function vn (t ) in a certain complete orthogonal system of functions on the segment T0 . By ordering the set of
numbers (16), we obtain a numerical sequence b0 , b1 , , bn , . Moreover, we have
2
t
t t
a = vn (t ) dt vn0 M k (t )dt dt
2 2
k
n
k =0 0 0 0
t
N n2 dt = aN n2 ,
0
whence it follows that
2
ak
1 a 2
i =0
b = n a 2 =
0=1 k =0 nN n n =0 n
i
2
6
.
*
Thus, mapping ( B ) maps the set M 0 into a subset M 0 of the Hilbert space l2 . Therefore, mapping ( A )
induces a mapping ( A* ) of the set M 0 into itself. Further, if mapping ( A* ) has a fixed point P M 0 , then the
* * *
corresponding point P M 0 determines the solution of equation (17) and, hence, (10). To use the Schauder
*
theorem, it suffices to show that the set M 0 is compact and convex. If P = (b0 ,
* * '
, bn' , ) and P * = (b0' , , bn' , )
*
are points from M 0 , then the point
P* P* = ( b0' b0' , b1' b1' , ), = 1, > 0, > 0,
*
belongs to M 0 because it corresponds to the system of functions
v0' (t ) v0' (t ), v1' (t ) v1' (t ), .
specifying a point from the set M 0 . Indeed,
t t
vk' (t ) vk' (t ) vk0 = (vk' (t ) vk0 ) (vk' (t ) vk0 ) ( ) M k (t )dt = M k (t )dt ,
0 0
i.e., condition 1 is satisfied. Similarly, the inequality
t
vk' (t ' ) vk' (t ' ) vk' (t ' ) vk' (t ' ) ( ) M k (t )dt
0
* *
implies condition 2. Hence, the set M 0 is convex. In this set, we choose an arbitrary sequence of points Pi . This
sequence corresponds to the sequence of points Pi v0 (t ), v1 (t ), in the set M . According to conditions 1 and
(i ) (i )
0
(i )
2, the sequence v (t ), i = 0,1, 2,
0 , is uniformly bounded and equicontinuous and, consequently, it contains a
( ) ( ) ( )
subsequence v 0
0
(t ), v
0
1
(t ), s
, v (t ),
0 that converges uniformly in t T0 . However, the sequence
( )
v (t ), h , is also uniformly bounded and equicontinuous and, hence, it also contains a convergent
1
h
subsequence
( ) ( ) ( )
v1 0 (t ), v1 1 (t ), , v1 s (t ), .
This process can be continued infinitely.
We compose the table
( ) ( ) ( )
v0 0 (t )v0 1 (t )v0 2 (t )
( ) ( ) ( )
v1 0 (t )v1 1 (t )v1 2 (t )
( ) ( ) ( )
v2 0 (t )v2 1 (t )v2 2 (t )
and rewrite the set of sequences row by row
( ) ( ) ( )
v0 0 (t )v0 1 (t )v0 2 (t )
67
( ) ( ) ( )
v1 0 (t )v1 1 (t )v1 2 (t )
( ) ( ) ( )
v2 0 (t )v2 1 (t )v2 2 (t )
Each of these sequences converges as a subsequence of a convergent sequence supplemented by finitely many
elements. Thus, the sequence of points
P , P , P , M 0
0 1 2
converges weakly (coordinatewise) to a point P0 M 0 (uniformly in t T0 ). For the sake of convenience, we
rewrite sequence (26) as
P0 , P1 , P2 , , Pn ,
* * * *
Let us show that the sequence of the corresponding points P0 , P1 , P2 , from the set M 0 converges to the point
P0* M 0* in the norm of the Hilbert space l2 . Indeed, the distance between the points P * and P * from M 0* is
given by the formula
t
P* , P* =
1
i =0
(bi' bi' ) 2 = 2 2 (vn' vn' ) 2 dt ,
n =0 n N n 0
whence it follows that
n t 1
P0* , Pk*
0
1
2 2 n
n =0 n N n 0
(v 0 vnk ) 2 dt t 2
n=n n
0
*
is arbitrarily small for sufficiently large n0 and k . This means that the set M 0 is compact. Note that one can easily
*
prove that mapping (B) is a homeomorphism, i.e., the sets M 0 and M 0 are topologically equivalent. Theorem is
proved.
Conclusions
We consider the property of the system for the limiting deterministic process as N . The evolution analysis
of large-scale transport systems can be described using an infinite system of differential equations. It is possible to
formulate Tikhonov type Cauchy problem for this system with small parameter and initial conditions. Tikhonov
type Cauchy problem for this system with small parameter is investigated. The theorems of existence of
solutions for this Cauchy problem is proved with taking into account parameters , , .
Acknowledments
The publication was prepared with the support of the “RUDN University Program 5-100” and partially funded
by RFBF grants № 15-07-08795, № 16-07-00556.
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Об авторах:
Блинов Артем Игоревич, аспирант кафедры прикладной информатики и теории вероятностей,
Российский университет дружбы народов, artemkab@rambler.ru
Васильев Сергей Анатольевич, кандидат физико-математических наук, доцент кафедры прикладной
информатики и теории вероятностей, Российский университет дружбы народов,
svasilyev@sci.pfu.edu.ru
Севастьянов Леонид Антонович, доктор физико-математических наук, профессор кафедры прикладной
информатики и теории вероятностей, Российский университет дружбы народов,
sevast@sci.pfu.edu.ru
Note on the authors:
Blinov Artem I., Postgraduate Student, Department of Applied Probability and Informatics, Peoples' Friendship
University of Russia, artemkab@rambler.ru
Vasilyev Sergey A., Candidate of Physico-mathematical Sciences, Associate Professor, Department of Applied
Probability and Informatics, Peoples' Friendship University of Russia, svasilyev@sci.pfu.edu.ru
Sevasyanov Leonid A., Doctor of Physico-mathematical Sciences, Professor, Department of Applied Probability
and Informatics, Peoples' Friendship University of Russia, sevast@sci.pfu.edu.ru
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