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Large transportation networks with
nite-dimensional state space. Asimptotic
approach.
L.G. Afanassieva D.V. Khmelev
August 1998
Department of Mathematics and Mechanics Moscow Lomonosov State Uni-
versity e-mail dima@vvv.srcc.msu.su
Consider an open network consisting of N nodes (stations) and V (N) cars,
which circulate among the stations. The network is divided into n clusters.
As N increases the number of clusters does not change. Cluster j contains
d N
j N stations,
P n
j=1 d N
j = 1. There exists d j > 0 so that
p
N(d N
j d j ) N!1
! 0
for j = 1; n. Later j and v denote cluster numbers and they take values from
1 to n. The arrival process to a station of cluster j is a Poisson one with the
rate  j .
If arrived customer nds a car at the node he takes it to reach his destina-
tion. Cluster v is chosen via routing matrix P = fp jv g j;v=1;n . The destination
station in the cluster v is chosen uniformly. After having reached their des-
tinations, customers leave the network.
The fully symmetric network was considered in [1]. Our model represents
an asymmetric generalization of [1]. The travel time from a station in cluster
j to a station in cluster v is exponentionally distributed with a parameter
 jv . The travel time between two stations in cluster j is also exponentionally
distributed with a parameter  jj . A customer which upon arrival does not
nd an available car and nds free waiting place joins the queue. Otherwise
he leaves the network. Capacities of waiting rooms for customers in cluster
v are supposed limited by k v for each station. There are m v parking lots
for servers at each station. If a car nds a station in cluster v empty and
number of cars at the station is less then m v , it stops and waits for the next
1

customer at this node. Otherwise, it is directed to cluster l via routing matrix
e
P = f~p vl g v;l=1;n . The car chooses a station inside cluster l uniformly.
Initially, each station in cluster j has r j servers where r j is integer from 0
to m j , V (N) = (r 1 d N
1 +


+ r n d N
n )N . Matrix P e
P is assumed to be ergodic.
Let x j;i (t) be a fraction of nodes in state i at cluster j among all N
stations,
P m j
i= k j
x j;i (t) = d N
j . Let f
M jv (t) denotes the number of cars driving
from cluster j to cluster v at the moment t. One can describe the state of the
network as a vector x 2 R  with  = n 2 +
P n
v=1 (k v +m v + 1) components:
x = (M 11 , M 12 , : : :, M 1n , M 21 , : : :, M nn , x 1; k1 , x 1; k1+1 , : : :, x 1;m1 , x 2; k2 ,
: : :, x n;mn ), where M jv (t) = f
M jv (t)=N . It is clear that the stochastic process
X N
t = x(t) is the ergodic Markov chain.
Let x t (x) be a solution of the following system of ordinary dierential
equations with the initial point x 0 (x) = x:
_
x j; k j
= ( j d j x j; k j +1 M j x j; k j
)=d j ;
_
x j;i = ( j d j x j;i+1 ( j d j +M j )x j;i +M j x j;i 1 )=d j ;
_
x j;m j
= (  j d j x j;m j
+M j x j;m j 1 )=d j ;
_
M jv = p jv  j d j S +
j +M j S j


 jv M jv + d j M j x j;m j
~
p jv
for j, v = 1; n, i = k j + 1; m j 1. Here S +
j  d j
P m j
i=1 x j;i , S j  d j
P 1
i= k j
x j;i ,
M j 
P n
l=1  lj M lj . Let " x denote a distribution which is concentrated in a
point x 2 R  .
Theorem. If X N
0 ! " x weakly then
(i) sup
0st
jX N
s x s (x)j P
! 0 for all t  0 as N !1,
(ii) proceses
p
N(X N
t x t (x)) weakly converges to a continuous process
Y with independent increments and covariation function ^
C(x). Here ^
C(x) =
t
R
0
c(x s (x))ds and c is a matrix function which can be written in explicit form.
The results obtained allow to study some parameters of X N
t with the help
of non-linear dynamical system x t (x).
[1] L.G. Afanassieva, G. Fayolle, S.Yu. Popov. Models for transportation
networks.
2