Macrosystem models of flows in
communication-computing networks
(GRID-technology)
Yuri S. Popkov
Institute for Systems Analysis
of the Russian Academy of Sciences
[email protected]
GRID — distributed computer
Real-time operation mode
•
•
•
•
network as a computer
response time is a random value
which depends on the flows in
network
random delay
random delay depends on flows in
network
A
x 1  f1 ( x1 , x2 )
x 2  f 2 ( x1 , x2 )
x1[( n  1)h]  x1[nh]  hf1 ( x1[nh], x2[nh   1 ])
x2[( n  1)h]  x2[nh]  hf2 ( x1[nh   2 ], x2[nh])
B
Transportation flows in Moscow traffic system
(middle of the day)
T = 25 min
Change of transportation flows in Moscow traffic system
(morning)
T = 32 min
Change of transportation flows in Moscow traffic system
(evening)
T = 29 min
GRID — Stochastic network — Dynamic system
History
Transportation networks (passanger, cargo)
Pipe-line networks (oil, gas)
Computer networks (Internet, Intranet)
Energy networks
State
GRID
Distribution of
Information
flows
Dynamic stochastic network
Macrosystem theory
Stochastic factors
Inertia
GRID states
• Spatial distribution of information and computing
resources
relaxation time
r
• Distribution of correspondence flows
relaxation time
X (t )
Y (t )
f
 r   f
Problems for study
A.
B.
Formation of quasi-stationary states of corresponding flows
Spatial-temporary evolution of network: interaction between “slow” and “fast”
processes in network
GRID phenomenology
Network
Flows
Correspondences
I (t )
Assignment
Macrostate
Y ( t )  [ yij ( t )]
- correspondence flows
Model of quasi-stationary states
Probabilistic characteristics
Time interval
t
t
t  t
Information and computing resources  Number of information portions
X (t )
Correspondence flows  Number of information portions per time unit
Y (t )
Prior probabilities
B( X , t )  A( X , t ) t
Flows
Y (t )
Volumes G( t )  Y ( t ) t
Generalized Boltzmann information entropy
H B (G , t , t )    gij ( t ) ln
gij
e bij ( X , t )
Model of quasi-stationary states
Probabilistic characteristics
Throughputs
Eij1 , Eij2 ,, Eijs
Feasible correspondence flows
 Eijr ( t )
C ij ( t )  
m
max
E
ij
 1 m  s
Volume of correspondences
0  gij (t )  Cij
Generalized Fermi-Dirac information entropy
H F (G , t , t )    gij ln
gij
 (C ij ( t )  gij ( t )) ln( C ij ( t )  gij ( t )),
~
e bij ( X , t )
~
bij ( X , t ) 
bij ( X , t )
1  bij ( X , t )
Model of quasi-stationary states
Feasible sets
Cost constraints
 ij
i
— transmission cost of an information portion for ( i j ) – correspondence
— transmission cost of an information portion per time unit for i–th resource
 gij (t ) ij  X i (t )i t ,
j  1, n
j
Balance constraints
- demands
 gij ( t ) ij  q j ( t ) t  Q j ( t ),
j  1, m
i
- throughput constraints
1, route of ( i j )  correspondence belongs to arc k
0, otherwise
kij  
 gij (t ) kij  W k t ,
где W
– throughput of k-th arc
i , jM k
General model
H [G , t , t , X ( t )]  max
G  D( X , t , t )
Classification of the model of quasi-stationary states
(MQSS)
I.
MQSS for constant capacity of correspondences
H F (Y , X ( t ), C )  max, Y  D( X , t )
D: y 
ij
ij
  i X i ( t ) i  1, n
j
 yij  q j
j  1, m
i
 yij kij  W k
k  1, r
ij
II.
MQSS for variable capacity of correspondences
H F (Y , t , X ( t ), C ( t ))  max, Y  D( X , t )
D: y 
ij
ij
  i X i ( t ) i  1, n
j
 yij  q j
j  1, m
i
III.
MQSS for small network loading
H B (Y , X ( t ))  max, Y  D( X , t )
D: y 
ij
ij
  i X i ( t ) i  1, n
j
 yij  q j
i
j  1, m
Illustration of adequacy of the MQSS
(transport network)
Dynamic models of stochastic network
Regional structure of network
X i (t )
— volume of computing resources in i-th region
(slow variables)
yij (t )
— information flows between regions i and j (fast
variables)
0  X (t )  M (t )
0  Y (t )  C (t )
Change factors of information and computing resources
• ageing (depends on X(t))
• renewal (external influence U(t))
• information flows (Y(t))
Change factors of information flows
• information and computing resources (X(t))
• demand (Q(t))
• information flows (Y(t))
or
0  Y ( t )  C ( t )
Dynamic model
dX ~
dY
 F [ X ( t ), U ( t ),Y ( t )];
 Ф[Y ( t ), X ( t ), Q( t )]
dt
dt
А. Resource dynamic
- positiveness
~
Fi ( X 1 ,  , X i 1 ,0, X i 1 ,  , X n | U , Y )  0 i  1, n
~
Fi ( X | U , Y )   ( X i ) Fi ( X , U , Y ), where  ( X )  0, F ( )  
dX
 X  F ( X , U ,Y )
dt
- boundedness
Fi ( X | U , Y )  0 для X   i
 i  { X : 0  X j ( t )  M j ( t ); X i ( t )  M i ( t ),
j  1, n; j  i }
Example:
Fi ( X | U ,Y )  bi (U ,Y )  X i si (U ,Y ) i  1, n
Model types
1. Ageing with constant rate
Fi ( X ,Y )  bi  X i P iY i
bi  const
2. Ageing and renewal with constant rate
~
Fi ( X ,Y )  bi  biU  X i P iY i
~
bi , bi  const
3. Renewal with constant rate
~
Fi ( X ,U )  biU  X i P iY i
~
bi  const
P – (m x n) matrix;
Pi – i –th row of matrix P;
Yi – i –th column of matrix Y;
B. Quasi-stationary states of the information flows distribution
max H (Y , X , t )
Y  D( x )
General dynamic model of stochastic network
dX
 X  ( b(U ,Y * ( X , t ))  X  s(U ,Y * ( X , t )))
dt
Y * ( X , t )  arg max( H (Y , X , t ) | Y  D( x ))
Positive dynamic system with entropy operator
Conclusion
GRID-technology
GRID as a system
Hardware, software, technical tools and etc.
Information and computing resources, information
flows, distributed on-line computing
Interestingly: new class of
dynamic systems
Why it is necessary to study
System properties of GRID?
Usefully: active and strategic
control, prediction
Tools
Macrosystem modelling
Quasi-stationary states
Resources evolution
Entropy maximization models
Models of dynamic systems with
entropy operator
Numerical methods, sensitivity,
smothness
Existing, boundedness, stability