Poisson hypothesis for mean-field
models of generalised Jackson
networks with countable set of
nodes
Alexander Rybko
Joint work with S.Shlosman
Poisson Hypothesis
Infinite Jackson Networks
• Open Jackson network with countable set of nodes J
• A pair (V,P) defines an open Jackson Network
• V   v1,..., vJ  vector of rates of Poisson input flows
in nodes j  1,..., J
• Suppose that service times of all customers are i.i.d
exponentially distributed with mean 1
• P   pij , i  J , j  J  is the routing stochastic matrix;
pij is a probability of the event: a customer comes to
the node j after being serviced in the node i
• Restriction: the matrix P is twice substochastic:
p
iJ
ij
 1,
p
jJ
ij
1

n
• Let   V   P  be the minimal solution of
n 1


(vector) equation

   P  V (1)
• The product of probability measures    j on
jJ
we shall name the multiplicative phases, where
  j  z j   1   j  
J

zj
j
• Let X n be a countable (breaking) Markov chain with
T
the phase space J and the substochastic matrix P
• Let Pj ( J ) be a probability of the event that the
trajectory of X n X 0  j starting from an initial
state X 0  j will never break
Lemma 1.
a) If P j ( J )  0 , there is no more then one
multiplicative phase for Jackson network.
b) If sup  j  1 where  is the minimal solution
jJ
of (1), then the multiplicative phase of Jackson
network is unique iff Pj ( J )  0
Theorem 2. Let Pj ( J )  0 , then
a) the minimal multiplicative phase   is a unique
invariant measure of the Markov process (t )
describing the evolution of the infinite Jackson
network;
b) for any initial state  (0)    J the measure  (t )
weakly converges to   when t  : for any
finite subset J  J and any vector nJ  n j , j  J



lim Pr  J (t )  n J  (0)     1   j 
t 
jJ
where J (t ) is a projection of the process  (t )
on the subset J  J

nj
j
Self-averaging property of nonhomogenous systems M (t ) G 1, 
Let’s consider the system M (t ) G 1,  with
Poisson input flow of variable intensity  (t )
and with stationary ergodic sequence of
service times with mean value 1.
Let the function  (t ) satisfy the non-overload
0
condition I
1
limsup   ( x)dx  1
T  T T
Let 0  (t ) also satisfy the conditions

  ( x)dx  ,

  ( x)dx  
0
Theorem 3.
Let b(t ) be the rate of an output flow of the system M (t ) G 1,  .
Then b(t ) satisfies the equation:

b(t ) 
  (t  x)q
,t
( x)dx,    t  

and the kernels q ,t ( ) depend on  ( ) by restriction  ( )   ,t 
only. And more over, the kernels q ,t ( ) are stochastic:
for any t

 q
,t
( x)dx  1
0
where q ,t ( x)  0 when x  0
Poisson hypothesis for symmetrical Jackson networks
vj
vj
pij
pij
p ji
k
vk
vj
vj
j
j
i
vj
p ji
r
r
i
k
vk
vk
vk
r
vk
A sequence of finite generalized
Jackson networks X (t )
J ,R
• Set of nodes J r1J ,...,rJJ : for each j  J the
network contains r j identical nodes.
J
J
J
J
So the total number of nodes is J r1 ,...,rJ   rj
j 1
Let’s denote by R  r1J ,..., rjJ ,..., rJJ
• In each node of class j the service time
distribution is


1   xdFj ( x),
0
 j ( y) 

F ' j ( y)
1  Fj ( y)
,
 j  C1 (

)
• In each node of class j the input rate of Poisson
flow is equal to v j : v  v , k  1,..., r J
jk
j
j
• For a routing matrix P we have
pik ' jk ''
J ,R
1
 pij J
rj
• Let the increasing sequence J n converges to J
and the ratios ri
Jn
r
Jn
j
converge to the limiting
uniformly boundered ratios
ri
rj
Non-linear Markov Processes
(Linear) Markov chain X n  S.
Configurations = points in S. State = probability measure on S.
Transition matrix P  P( s, t ),
P(s, t )  1

t
State  is transformed to  by   P
Non-linear Markov chain:
Transition probability to go from s to t depends also on the state
The Non-linear Markov chain is defined by the collection of 
transition matrices P  P (s, t ),
P (s, t )  1,

t
and state  is transformed to  by 
Evolution:
 P
Limiting dynamical system as the

nonlinear Markov process  (t )
Dynamical system x(t , 0 ) : evolution of probability measure
of nonlinear Markov process: countable set J of systems
Mi (t ) GIi 1, i  J

bi (t )    i ( y)d  ( y )dy
k
(k )
t
k 1
bi (t )
i (t )
Additional equations:
where
ri
lim
ri J
i (t )  vi   b j (t ) p ji
J
r
j
jJ
j
rj
ri
Theorem 3.
For any t  0 and any function F weakly continuous on
P the equation lim sup TJ , R (t )F ( 0 ) F  x(t , 0 )   0
holds.
J  J P
J ,RJ
RJ RJ
J
The convergence is uniform on any finite interval  0,T 
Theorem 4.
t 
X J ,R (t )
Stationary measure
 J ,R ( )
J  J,
J  J,
R R
RJ  R
J
t 
x(t , 0 )
Where
Where

  is a stationary measure of

M
i
GI i 1 , 
J independent systems
is the unique solution of countable set of equations:
i  vi    j p ji
jJ
rj
ri
,
i, j  J
Closed networks, J-finite
p
 1, vi  0, ( P  ergodic)
j
rj
i (t )   b j (t ) pij
ri
j
b j (t )  i ( )  qi ,i ( ),t ( )  (t )
ij
i,n :  i (t ) 1,..., m, t  0,..., n
 i (0)   i (n)  i,  i (t )  i, t  1,..., n 1
P  i (t ) 
 
n
i ,n
n
 P
k 1
( k 1), ( k )
P ( )  1
(2)
(3)
Proposition
i (t )  
n

P( )  ...   q


i ,n

n

( n 1),  ( n1),t

( x1 ) 

 q ( n  2), ( n2),t x ( x2 )  ...  q (0), ( 0),t x ...x ( xn ) 
1

1
n1

i  t  x1  ...  ...  xn  dx1...dxn  i ( )  q  ,t ( ) (t )
Theorem 4
i (t )  i where   1,..., i ,..., J 
is the solution of a system of equations
rj
i    j pij
ri
j
Open networks
i (t )  vi   b j (t ) p ji
rj
(4)
ri
b j (t )   j ( )  q j ( ),t ( ) (t )
(5)
Stationary solution:
  V  P where
(6)

j

  i , i  J , i
V  Vi , i  J ,Vi
i ri 
vi ri 
  l
minimal solution (6)
(0)  V  VP  VP2  ...
Proposition:  is unique 
  P
0
has only trivial solution
Theorem 5
Suppose that i  J ,

1
ri
0
i
Then the Poisson Hypothesis holds.
Proof: i (t )  Vi  B j (t ) p ji

B j (t )    j ( )  q j ,n ( )  t
j
Let X
and
then
X

limsup X (t )
t 

lim inf X (t )
t 

i

i

i

i
 B B 
L     , j  J  ,   B  B , j  J 

j
L

j

j
L  BP
L LP 
B
L
LP  0
n

j
References
1. Kel’bert M.Ya., Kontsevich M.L., Rybko A.N.
Infinite Jackson Networks, Theor.Probab. And
Appl. 1988 v.33
2. Stolyar P.I.T. 1989 v.25#4
3. Rybko, Shlosman Moscow Mathematical Journal
2005 v.5#3, v.5#4, 2008 v.8#1
4. Dobrushin, Karpelevich, Vvedenskaya P.I.T.
1996 v.32#1
5. Karpelevich, Rybko P.I.T. 2000 v.36#2
6. Rybko, Shlosman, Vladimirov P.I.T. 2006 v.42#4
7. Rybko, Shlosman P.I.T. 2005 #3
8. Rybko, Shlosman, Vladimirov J.of Stat.Physics
2009 v.134#1
Open Questions
• Is Poisson Hypothesis true for generalized
Jackson networks with several types of
customers? For example in the case when their
service times are exponentially distributed with
mean values depending on their types.
We can not prove Poisson Hypothesis in this
situation even in the case of a complete graph
with an increasing number of nodes.
• Is Poisson Hypothesis true for non-FIFO service
discipline?
• What kind of self-averaging properties between
inputs and outputs are true in these situations?