Poisson hypothesis for mean-field models of generalised Jackson networks with countable set of nodes

Alexander RybkoJoint work with S.Shlosman

Poisson Hypothesis

• Open Jackson network with countable set of nodes J

• A pair (V,P) defines an open Jackson Network

• vector of rates of Poisson input flows in nodes

• Suppose that service times of all customers are i.i.d exponentially distributed with mean 1

• is the routing stochastic matrix; 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:

Infinite Jackson Networks

1,..., JV v v1,...,j J

, ,ijP p i J j J ijp

1, 1ij iji J j J

p p

• Let be the minimal solution of

(vector) equation

(1)

• The product of probability measures on

we shall name the multiplicative phases, where

• Let be a countable (breaking) Markov chain with the phase space J and the substochastic matrix

• Let be a probability of the event that the trajectory of starting from an initial state will never break

1

n

n

V P

P V j

j J

J

1 jz

j j j jz

nXTP

( )jP J

0nX X j0X j

Lemma 1. a) If , there is no more then one multiplicative phase for Jackson network.

b) If where is the minimal solution of (1), then the multiplicative phase of Jackson

network is unique iff

( ) 0jP J

sup 1jj J

( ) 0jP J

Theorem 2. Let , then a)the minimal multiplicative phase is a unique

invariant measure of the Markov process describing the evolution of the infinite Jackson network;

b)for any initial state the measure weakly converges to when : for any finite subset and any vector

where is a projection of the processon the subset

( ) 0jP J

( )t

(0) J ( )t t

J J ,jJn n j J

lim Pr ( ) (0) 1 jn

j jJ Jtj J

t n

( )

Jt ( )t

J J

Self-averaging property of non-homogenous systems

Let’s consider the system with Poisson input flow of variable intensity and with stationary ergodic sequence of service times with mean value 1.

Let the function satisfy the non-overload

condition

Let also satisfy the conditions

( ) 1,M t G ( ) 1,M t G

( )t

( )t01

limsup ( ) 1T T

I x dxT

( )t0

0

( ) , ( )x dx x dx

Theorem 3.

Let be the rate of an output flow of the system .

Then satisfies the equation:

and the kernels depend on by restriction

only. And more over, the kernels are stochastic:

for any t

where when

,( ) ( ) ( ) ,tb t t x q x dx t

( )b t ( ) 1,M t G ( )b t

, ( )tq ( ) ,( )

t

, ( )tq

,

0

( ) 1tq x dx

, ( ) 0tq x 0x

Poisson hypothesis for symmetrical Jackson networks

r

ijp

jip

jvjv

kv

ijpr jip

r

jvjv jv

kv kvkv

kv

kk

jj

i i

A sequence of finite generalized Jackson networks

• Set of nodes : for each the network contains identical nodes.

So the total number of nodes is

Let’s denote by

• In each node of class j the service time distribution is

1 ,...,J JJ

r rJ j J

jr

1 ,...,

1

J JJ

JJ

r r jj

J r

1

0

' ( )1 ( ), ( ) , ( )

1 ( )j

j j jj

F yxdF x y C

F y

1 ,..., ,...,J J Jj J

R r r r

,( )

J RX t

• In each node of class j the input rate of Poisson flow is equal to :

• For a routing matrix we have

• Let the increasing sequence converges to

and the ratios converge to the limiting

uniformly boundered ratios

jv , 1,...,k

Jj j jv v k r

,J RP

' ''

1k ki j ij J

j

p pr

nJ J

n

n

Ji

Jj

rr

i

j

rr

Non-linear Markov Processes(Linear) Markov chain

Configurations = points in S. State = probability measure on S.

Transition matrix

State is transformed to by

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

and state is transformed to by

Evolution:

.nX S

( , ), ( , ) 1t

P s t P s t P

P

P

( , ), ( , ) 1,t

P s t P s t P

Limiting dynamical system as the nonlinear Markov process

( )

1

( ) ( ) ( )k

i i tk

b t y d y dy

Additional equations: ( ) ( ) ji i j ji

j i

rt v b t p

r

Dynamical system : evolution of probability measure of nonlinear Markov process: countable set J of systems

( )t

( )i t

( )ib tk

0( , )x t

limJ

ii J

jj J

rr

r

where

( ) 1,i iM t GI i J

Theorem 3.For any and any function weakly continuous on

the equation

holds.

The convergence is uniform on any finite interval

0t F

P ,

0 0,lim sup ( ) ( ) ( , ) 0

JJ RJJJ

J RJ JR R

T t x t

P

F F

0,T

,( )

J R

Stationary measure

0( , )x t

t

t

,J

J J

R R

Where is a stationary measure of J independent systems

Where is the unique solution of countable set of equations:

, ,ji i j ji

j J i

rv p i j J

r

,( )

J RX t

,J

J J

R R

1 ,i iM GI

Theorem 4.

Closed networks, J-finite

1, 0, ( )ij ij

p v P ergodic ( ) ( ) j

i j ijj i

rt b t p

r (2)

, ( ),( ) ( ) ( ) ( )ij i i tb t q t (3)

, : ( ) 1,..., , 0,...,i n i t m t n

(0) ( ) , ( ) , 1,..., 1i i in i t i t n

( 1), ( )1

( )n

i k kk

P t P

,

( ) 1i nn

P

( 1),

,

( 2), (0), ...1 1 1

( 1), 1

( 2), 2 (0),

1 1 ,

( ) ( ) ... ( )

( ) ... ( )

... ... ... ( ) ( ) ( )

n t

i n

n t x t x xn

i nn n

n n

i n n i t

t P q x

q x q x

t x x dx dx q t

Proposition

( )i it Theorem 4

where 1,..., ,...,i J is the solution of a system of equations

ji j ij

j i

rp

r

Open networks( ) ( ) j

i i j jij i

rt v b t p

r (4)

(5) ( ),( ) ( ) ( ) ( )jj j tb t q t

Stationary solution:V P where

, ,

, ,

i i i i

i i i i

i J r

V V i J V v r

l

minimal solution (6)

(6)

(0) 2 ...V VP VP

Proposition: is unique

P

has only trivial solution

0

0

, 1i

i

i Jr

Theorem 5

Suppose that

Then the Poisson Hypothesis holds.Proof: ( ) ( )i i j ji

j

t V B t p

and

,( ) ( ) ( )j j j nB t q t

Let limsup ( )t

X X t

liminf ( )t

X X t

theni i i iB B

, , ,j j j jL j J B B j J

L B L BP

L LP

0nL LP

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#43. Rybko, Shlosman Moscow Mathematical Journal

2005 v.5#3, v.5#4, 2008 v.8#14. Dobrushin, Karpelevich, Vvedenskaya P.I.T.

1996 v.32#15. Karpelevich, Rybko P.I.T. 2000 v.36#26. Rybko, Shlosman, Vladimirov P.I.T. 2006

v.42#47. Rybko, Shlosman P.I.T. 2005 #38. Rybko, Shlosman, Vladimirov J.of Stat.Physics

2009 v.134#1

References

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?