UDC 517.937, 517.928.2, 519.217.2
Dobrushin mean-eld approach with application in
queueing large-scale networks analysis
∗
∗
S. A. Vasilyev , G. O. Tzareva
∗
Department of Applied Probability and Informatics,
RUDN University,
Miklukho-Maklaya str. 6, Moscow, 117198, Russia
In this paper it is considered a system that consists of innite number
of servers with a Poisson input ow of requests of intensity N λ. Each requests
arriving to the system randomly selects two servers and is instantly sent to the
one with the shorter queue. In this case a share uk (t) of the servers that have the
queues lengths with not less than k can be described using an innite system of
dierential equations. It is possible to investigate Tikhonov type Cauchy problem
for this system with small parameter µ for building the solutions uk (t). The
evolution analysis of uk (t) (k = 1, 2, . . .) be applied to application in queueing
large-scale networks analysis.
Abstract.
analytical methods in probability theory, systems of dierential
equations of innite order, singular perturbated systems of dierential equations,
small parameter, countable Markov chains, large network modeling.
Keywords:
1.
Introduction
The recent research of service networks with complex routing discipline
in [8], [9], [10] transport networks faced with the problem of proving the
global convergence of the solutions of certain innite systems of ordinary
dierential equations to a time-independent solution. Scattered results of
these studies, however, allow a common approach to their justication.
This approach will be expounded here. In work [2] the countable systems
of dierential equations with bounded Jacobi operators are studied and
the sucient conditions of global stability and global asymptotic stability are obtained. In this paper we apply Dobrushin mean-eld approachs
from [8] for the singular perturbated systems of ordinary dierential equations of innite order of Tikhonov type. Cauchy problems for the systems of ordinary dierential equations of innite order was investigated
A.N.Tihonov [4], R. Bellman [1], K.P.Persidsky [3], O.A.Zhautykov [11]
and other researchers. It was studied the singular perturbated systems of
ordinary dierential equations by A.N. Tihonov [5], A.B.Vasil'eva [7] and
other researchers.
In the papers [6] we investigated same questions which relate to systems
of ordinary dierential equations of innite order with a small parameter
and initial conditions. It was proved same essential theorems such as the
local existence of solutions theorem.
In this paper we considered a system that consists of innite number
of servers with a Poisson input ow of requests of intensity N λ. Each
requests arriving to the system randomly selects two servers and is instantly sent to the one with the shorter queue. In this case a share
uk (t)
of the servers that have the queues lengths with not less than k can be
described using an innite system of dierential equations. It is possible
to investigate Tikhonov type Cauchy problem for this system with small
parameter µ and initial conditions using the singular perturbation methods [5, 7].
Using the truncation method it was studied a nite system
of dierential equations with small parameter
analysis of
uk (t) (k = 1, 2, . . . , N )
µ
order
N.
The evolution
be applied to application in queueing
large-scale networks analysis
2.
Large-scale network model
Let's consider a system that consists of N servers with a Poisson input
ow of requests of intensity N λ. Each request arriving to the system
randomly selects two servers and is instantly sent to the one with the
shorter queue. The service time is distributed exponentially with mean
t̄ = 1.
uk (t) be a share servers that have the queues lengths with
k . It is possible to investigate the asymptotic distribution
queue lengths as N → ∞ and λ < 1 [8]. The considered system of
Let
not less than
of the
the servers is described by ergodic Markov chain. There is a stationary
probability distribution for the states of the system and if N → ∞ the
evolution of the values
uk (t)
becomes deterministic and the Markov chain
asymptotically converges to a dynamic system the evolution of which is
described by innite system of dierential-dierence equations


u̇k (t) = uk+1 (t) − uk (t) + λ (uk−1 (t))2 − (uk (t))2 ,
(1)
u (0) = g ≥ 0, k = 0, 1, 2, . . . ,
k
k
where
∞
g = {gk }k=1
is a numerical sequence (1
= g0 ≥ g1 ≥ g2 , . . .)
[8].
We can investigate innite system of dierential-dierence equations
with small parameter such form



u̇k (t) = uk+1 (t) − uk (t) + λ (uk−1 (t))2 − (uk (t))2 ,






k = 0, 1, . . . , n,
µu̇k (t) = uk+1 (t) − uk (t) + λ (uk−1 (t))2 − (uk (t))2 ,




k = n + 1, . . . ,




uk (0) = gk ≥ 0, k = 0, 1, 2, . . . ,
where
µ
(2)
is a small parameter that bring a singular perturbation to the
system (1) which allows us to describe the processes of rapid change of the
systems.
1.14
u0
u1
1.12
u2
u4
1.1
u5
1.08
1.06
1.04
1.02
1
0
2
4
6
8
10
Figure 1. Evolution analysis of uk (µ = 0.1).
3.
Truncation large-scale network model and numerical
analysis
Using (2) we can write the truncation system of dierential-dierence
equations



u̇k (t) = uk+1 (t) − uk (t) + λ (uk−1 (t))2 − (uk (t))2 ,






k = 0, 1, . . . , n,




µu̇k (t) = uk+1 (t) − uk (t) + λ (uk−1 (t))2 − (uk (t))2 ,
k = n + 1, . . . , N,







uk (0) = gk ≥ 0, k = 0, 1, 2, . . . , N,



uN +1 = gN +1 ≥ 0.
The numerical example is presented in the gure (see Fig. 1,
2) where
n = 2, N = 4, l = 0.2, gk = 1, k = 0,¯5 and a small parameter µ = 0.1
(Fig. 1) , µ = 0.01 (Fig. 2). In this numerical example it was shown
the existence of stady state conditions for evolutions ui (t), i = 0, 1, 2 and
quasi-periodic conditions with boundary layers for evolutions ui (t), i =
3, 4.
1.07
u0
u1
1.06
u2
u3
1.05
u4
1.04
1.03
1.02
1.01
1
0
0.5
1
1.5
2
Figure 2. Evolution analysis of uk (µ = 0.01).
4.
Conclusions
We investigated the large-scale network model that consists of innite
number of servers with a Poisson input ow of requests of intensity N λ.
Each requests arriving to the system randomly selects two servers and is
instantly sent to the one with the shorter queue. In this case a share
uk (t)
of the servers that have the queues lengths with not less than k can be
described using an innite system of dierential equations. For Tikhonov
problem for innite system of dierential equations with small parameter
µ and initial conditions we applied the truncation method and studied a
nite system of dierential equations with small parameter µ order N .
The numerical analysis of
uk (t) (k = 1, 2, . . . , 5)
be applied for queueing
large-scale networks evolution conditions. It was shown the existence of
stady state conditions for evolutions
ui (t), i = 0, 1, 2 and quasi-periodic
ui (t), i = 3, 4.
conditions with boundary layers for evolutions
Acknowledgments
1
The work is partially supported by RFBR grant No 12-34-56789.
1 This
section may be omitted if not relevant.
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