Analysis of HM-networks with stochastic incomes from transitions between states

(1)

ANALYSIS OF HM-NETWORKS WITH STOCHASTIC INCOMES FROM TRANSITIONS BETWEEN STATES

Michal Matalycki¹, Katarzyna Koluzaewa²

1 Institute of Mathematics, Czestochowa University of Technology, Poland

2 Faculty of Mathematics and Computer Science, Grodno State University koluzaeva@gmail.com

Abstract. In article method of finding of expected incomes in systems of HM-network of arbitrary topology when incomes from transitions between network states are stochastic variables with given mean values is proposed. For expected incomes the system of linear non-homogeneous ordinary differential equations was obtained, to solve it we can find incomes in network systems.

Introduction

Markov queueing networks (QN) with incomes were examined in works [1-3]

at the first time. They are described with help of Markov chains with continuous time and incomes which were introduced by R. Howard [4]. So at the recent time they are called HM(Howard-Matalytski)-networks [5, 6]. Before investigation of closed networks with account number of states was carry out. Herewith the next cases were examined: a) incomes from transitions between network states depend on states and time or b) incomes are stochastic variables (SV) with known finite moments of the first and the second orders.

For expected incomes of network systems in case a) the system of difference- -differential equations which are reduced to the system of linear non-homogeneous ordinary differential equations (ODE) can be obtained. For their solution different methods - method of multidimensional z-transformations and known methods: method of Laplace transformation, matrix method, numerical methods were proposed [7-9].

In [5, 10] approximate relations for expected incomes and income variations in systems of exponential HM-networks in case b) were obtained. Technique of re- ceiving of these relations is based on interval partition of the network functioning by big number m of small intervals of size ∆t, income estimation on every interval and summing of these incomes by means of passage to the limit m→∞, ∆t→0.

Herewith mean value of messages in network system in unsteady condition was

found with help of developed recurrence by time moments method. Notice that

different methods of analysis and optimization of Markov HM-networks and their

(2)

application were described in [11]. In the present work method of finding of expected incomes of network systems in case b) which is based on solution of the linear non-homogeneous ODE, which were received for expected incomes and mean values of messages, is proposed.

1. Finding of expected incomes in systems

Let us examine open exponential QN of arbitrary topology with one type messages which consists of n queueing systems (QS) S

1

, S

2

,…,S

n

with m

i

service channels in system S ,

_i

i = 1 , n . Denote via k ( t ) = ( k

₁

( t ), k

₂

( t ), ..., k

_n

( t ) ) - vector of network states, where k

_i

(t ) - message number in system S

i

(in queue and service) at the moment t. Poisson flow of messages of the rate λ enters the network. Service rate of messages at the moment t µ

i

(k

i

(t)) in system S

i

depends on message number in this system, i = 1 , n . Message when transiting from one QS to another brings to the last system some stochastic income and income of the first system reduces by this value correspondingly.

Let us consider dynamics of income changes of some network system S

i

. De- note it’s income at the moment t as V

i

(t). Let at the initial moment income of system equals V

i

(0) = v

i0

. Income of this QS at the moment t + ∆t can be presented as

) , ( ) ( )

( t t V t V t t

V

_i

+ ∆ =

_i

+ ∆

_i

∆ (1)

where ∆V

ⁱ

(t,∆t) - income change of system S

ⁱ

on time interval [t, t+∆t). For finding of this value we write probabilities of events which can appear during time

∆t

and changes of incomes of system S

i

which are connected with these events.

1. Message from the outside with probability λ

p₀_i∆t +o(∆t)

will enter to system S

i

and will bring to it income of size r

₀_i

, where r

₀_i

- RV with mathematical expectation (m.e.) M { } r

₀i

= a

₀i

, p

₀_i

- probability of message enter from outside to the system S

i

, i = 1 , n .

2. Message from the system S

i

with probability µ

_i

( k

_i

( t )) u ( k

_i

( t )) p

_i₀

∆ t + o ( ∆ t ) will pass to the outside and income of the system S

_i

will decrease by value R

_i₀

, where R

_i₀

- RV with m.e. M { } R

i0

= b

i0

, p

_i₀

- probability of message leaving from system S

_i

to the outside, i = 1 , n ,

 



≤

= >

, 0 , 0

, 0 , ) 1

( x

x x

u - Heavyside function.

3. Message from the system S

j

with probability µ

_j(k_j(t))u(k_j(t))p_ji∆t+o(∆t)

will

pass to the system S

i

and income of the system S

i

will increase by value r

_ji

and

income of the system S

j

will decrease by this value, where r

_ji

- RV with m.e.

(3)

{ } r

ji

a

ji

M = , p

_ji

- probability of message transition from the system S

j

to the system S

i

, i , j = 1 , n , i ≠ j .

4. Message from the system S

i

with probability µ

_i

( k

_i

( t )) u ( k

_i

( t )) p

_ij

∆ t + o ( ∆ t ) will pass to the system S

j

and income of the QS S

i

will decrease by value R

_ij

and income of the system S

j

increase by this value, where R

_ij

- RV with m.e.

{ } R

ij

b

ij

M = , i , j = 1 , n , i ≠ . j

5. State changes of system S

i

on the time interval [t,t+∆t) with probability

 







∆ +

 ∆

 



 +

+

− ∑

≠

=

) ( ))

( ( )) ( ( ))

( ( )) ( ( 1

1

0

k t u k t k t u k t p t o t

p

n

i j j

ji j j

j i

i i

i

µ µ

λ will not appear,

n i = 1 , .

Besides that during each small time interval t ∆ the system S

i

increases its income by value r

_i

∆ t , where r

_i

- RV with m.e. M { r

_i

} = c

_i

, i = 1 , n . Let also suppose that RV r

_ji

, R

_ij

, r

₀_i

, R

_i₀

are independent with respect to RV r

_i

, i , j = 1 , n .

Evidently that r

_ji

= R

_ji

with probability 1, i.e.

n j i b

a

_ji

=

_ji

, , = 1 , (2)

Then from said follows

 



 





∆ +

 ∆

 



 +

 

 + +

∆ −

∆ +

∆

∆ +

−

∆ +

∆

∆ +

∆

∆ +

−

∆ +

∆

∆ +

=

∆

∑

≠

=

) ( ))

( ( )) ( (

)) ( ( )) ( ( 1

y probabilit with

) ( ))

( ( )) ( ( y

probabilit with

) ( ))

( ( )) ( ( y

probabilit with

) ( ))

( ( )) ( ( y

probabilit with

) ( y

probabilit with

) , (

1 0

0 0

t o t p t k u t k

t k u t k t p

r

t o t p t k u t k t

r R

t o t p t k u t k t

r r

t o t p t k u t k t

r R

t o t p t

r r

t t V

n

i j j

ji j j

j

i i i i i

ij i i i i

ij

ji j j

j i

ji

i i i i i

i

i i

i

µ

µ λ µ µ

µ λ

, , , ,

(3)

Under fixed realization of the process k (t ) and taking to account (3) it can be

written:

(4)

{ }

) ( ))

( ( )) ( (

)) ( ( )) ( ( )

( / ) , (

1

1 0 0 0

0

t o t a p t k u t k

b p b

p t k u t k c a p t

k t t V M

n

i j j

ji ji j j

j

n

i j j

ij ij i

i i i i i i i i

∆ +

∆

 





 +

+

 







 







 







+

− +

=

∆

∑

≠

=

≠

=

µ

µ λ

Average the last relation by

k(t)

with taking to account normalization condition

( ( ) = ) = 1

∑

k

k t k

P for income change of system S

_i

we will obtain

{ } ( ) { }

( )

{ }

( )

( ( ) ) ( ( )) ( ( )) ( ) )) ( ( )) ( ( ) (

)) ( ..., ), ( ), ( ( ) ( / ) , (

)) ( ..., ), ( ), ( ( ) ( ...

) ( / ) , ( )

( )

, (

1

1 0 0 0

0

2 1

0 0 0

2 1

1 2

t o t t k u t k k t k P a p

t k u t k k t k P b p b

p c a p

t k t k t k t k t t V M

t k t k t k t k P

t k t t V M k t k P t

t V M

n

i j j

j j

j k

ji ji

i i i k

n

i j j

ij ij i

i i i i

n i

k k k

n k

i i

n

∆ +

∆

 







= +

 







+

 =

 





 







+

− +

=

∆

×

=

∆

=

∆

∑ ∑

∑

∑ ∑ ∑

∑

≠

=

≠

=

∞

=

∞

=

∞

=

µ

µ λ

Let the system S

_i

contains m

_i

identical service channels, time of message service in every channel is distributed under exponential law with parameter µ

_i

, i = 1 n , . In this case

 



>

= ≤

, ) ( ,

, ) ( ), )) (

( (

i i

i i i i i

i

m k t m

m t k t t k

k µ

µ µ µ

_i

( k

_i

( t )) u ( k

_i

( t )) = µ

_i

min( k

_i

( t ), m

_i

) , i = 1 , n

Let us suppose that averaging of expression µ

_i

( k

_i

( t )) u ( k

_i

( t )) brings )

), (

min(

_i _i

i

N t m

µ , i.e.

) ), ( min(

) ), (

min( k

_i

t m

_i

N

_i

t m

_i

M = (4)

(5)

where N

_i

(t ) - average number of messages (waiting and serving) in the system S

_i

at the moment , t i = 1 , n . Taking in account this assumption we obtain the next approximate relation

{ }

) ( )

), ( min(

) ), ( min(

) , (

1

1 0 0 0

0

t o t a p m t N

b p b

p m t N c

a p t

t V M

n

i j j

ji ji j j j

n

i j j

ij ij i

i i i i

∆ +

∆

 





 +

 







 +

 





 







+

− +

=

∆

∑

≠

=

≠

=

µ

µ λ

(5)

Since Poisson flow of messages with rate λ enters the network, i.e. probability of entering of l messages in the system S

_i

during time ∆ has the appearance t

( )

p t

l i

l

e

ⁱ

l t t p

P ∆

⁻ ^∆

=

∆

⁰

) !

( λ

⁰ ^λ

, l = 0 , 1 , 2 , ... , so average number of messages which entered to the system S

_i

from the outside during time t ∆ equals λ p

₀_i

∆ t . Denote the average number of busy service channels in system S

_i

at the moment t as ρ

_i

(t ) ,

n

i = 1 , . Then µ

_i

ρ

_i

( t ) ∆ t - average number of messages that left the system S

_i

during time ∆ t and ∑

≠

=

∆

n

i j j

ji j

j

t p t

1

) ρ (

µ - average number of messages that entered in the

system S

_i

from another QS during time ∆ . So t

t t t

p t t

p t N t t

N

_i _i

n

i j j

ji j j i

i

+ ∆ − = ∆ + ∑ ∆ − ∆

≠

=

) ( )

( )

( ) (

1

0

µ ρ µ ρ

λ , i = 1 , n

whence with ∆t → 0 system of ODE for N

_i

(t ) follows:

i i

i n

i j j

ji j j

i

t p t p

dt t dN

0 1

) ( )

) (

( = ∑ µ ρ − µ ρ + λ

≠

=

, i = 1 , n (6)

It is impossible to find variable ρ

_i

(t ) exactly so as we did earlier we approximate it by expression

 

 =

>

= ≤ min( ( ), )

, ) ( ,

, ) ( ), ) (

(

_i _i

i i i

i

N t m

m t N m

m t N t t N

ρ

(6)

Then system of equations (6) will take on form

i i i i

n

i j j

j j ji

j

i

p N t m N t m p

dt t dN

0 1

) ), ( min(

) ), ( ) min(

( = ∑ µ − µ + λ

≠

=

, i = 1 , n (7)

This system is system of linear ODE with discontinuous right parts. It’s neces- sary to solve it by means of segmentation of the phase space by set of the areas and finding solution in each of them. System (7) can be solved for example by using means of system of computer mathematics Maple 8.

Let us introduce denotation v

_i

( t ) = M { V

_i

( t )} , i = 1 , n . From (1), (5) we obtain

) ( )

), ( min(

) ), ( min(

) (

)}

, ( { ) ( ) (

1

1 0 0 0

0

t o t a p m t N

b p b

p m t N c

a p t v

t t V M t v t t v

n

j

ji ji j j j

n

i j j

ij ij i

i i i i

i i

i

∆ +

 ∆





 +

 







 +

 





 







+

− + +

=

∆

∆ +

=

∆ +

∑

=

≠

=

µ

µ λ

Then pass to limit with ∆t → 0 we receive non-homogeneous linear ODE of the first order

n i c a p a p m t N

b p b

p m t dt N

t dv

i n

i j j

i i ji ji j j j

n

i j j

ij ij i

i i i i

i

, 1 , )

), ( min(

) ), ( ) min(

(

1

0 0

1 0 0

= + +

+

 +

 





 







+

−

=

∑

≠

=

≠

=

λ µ

µ

(8)

Specify initial conditions v

_i

( 0 ) = v

_i₀

, i = 1 , n , it is possible to find expected incomes of network systems.

If the network is functioning so that queues are not observed in it upon the average, i.e. min( N

_i

( t ), m

_i

) = N

_i

( t ) , i = 1 , n , then systems (7), (8) will take the appearance:

i i

i n

i j j

j ji j

i

p N t N t p

dt t dN

0 1

) ( )

) (

( = ∑ µ − µ + λ

≠

=

, i = 1 , n (9)

(7)

 



 



=

+ +

 +







 



 +

−

= ∑ ∑

≠

=

n i v v

c a p t N a p t

N b p b

dt p t dv

i i

i n

i j j

i i j

ji ji j i

n

j ij ij i

i i i

, 1 , ) 0 (

, )

( )

) ( (

0

1

0 0 1

0

µ λ

µ

(10)

System (9) can be rewrote in matrix form f t dt QN

t

dN ( ) = ( ) +

(11) where N

^T

( t ) = ( N

₁

( t ), N

₂

( t ),..., N

_n

( t )) , Q - quadratic matrix which consists of elements q

_ij

= µ

_j

p

_ji

, if suppose p

_ii

= − 1 , i , j = 1 , n , f - column vector with elements

p

₀i

λ , i = 1 n , . Solution of the system (11) is

∫

⁻

+

=

t t Q

Qt

f e d

e N t N

0 )

)

(

0 ( )

(

^τ

τ

where N ( 0 ) - some given initial conditions, but finding of elements of matrix e

^Qt

is difficult problem even for rather small values of n.

Fig. 1. Structure of the network with central QS

Let us consider closed network with central QS that consists of n systems (Fig. 1).

Let suppose that queues are not observed in peripheral systems of the network under the average, i.e. min( N

_i

( t ), m

_i

) = N

_i

( t ) , i = n 1 , − 1 , and central QS functions in the condition of heavy traffic, i.e. min( N

_n

( t ), m

_n

) = m

_n

. System (7) in this case will re- write as

 



 



−

=

−

= +

−

=

∑

⁻

=

n n n

i i i n

ni n n i i i

m t dt N

t dN

n i p m t dt N

t dN

µ µ

1

) ) (

(

, 1 , 1 , )

) ( (

(12)

S₁ S₂ S_n

…

S_n

(8)

General solution of the system (12) with initial conditions N

_i

( 0 ), i = 1 , n , equals

∑

⁻

=

−

 



 

 +

−

=

−

= +

=

1

) (

, 1 , 1 , )

(

n

i i

ni n t n i n

i ni n t n i i

p e m

g K

t N

n p i

e m g t N

i i

µ µ µ

µ

µ µ

where

i ni n n i

i

p N m

g µ

− µ

= ( 0 ) , ∑

=

n

i i

t N K

1

)

( - number of messages in the network.

For such N

_i

( t ), i = 1 , n , system (10) for expected incomes of the network systems will take the appearance

n j

nj n t n j n

j jn j

n

i i

ni n t n i n

j nj nj n n

i n

j j

nj n t n j ni

ni n

i ni n t n i in i i

p c e m

g a

p K e m

g b

dt p t dv

n i p c

e m g K

a p

p e m

g dt b

t dv

j i j

i

 +







 



 +

+

 +

 



 



  −





 

 +

=

−

=

 +







 





 





 



 +

− +

 +





 

 +

−

=

− −

=

−

=

−

=

−

=

−

∑

µ µ µ

µ

1

1 1

1

) (

1 , 1 , )

(

Integrate given ODE with initial conditions v

_i

( 0 ) = v

_i₀

, i = 1 , n , we will obtain

1 , 1 , )

(

0 1

1

1 1

1

−

= +

−

 +







 





 +







 





−

− +

+ +

=

∑

−

=

−

=

− −

=

−

n i v g g b a p

t c b p m a m Ka

p

e g b g e

a p t v

i i in n

j j

j ni ni n

i in n n

j j

nj ni n n ni ni n

t i in n

j

t j j ni ni n

i ⁱ

j

µ µ

µ µ µ

µ µ

^µ ^µ

(13)

(9)

0 1

1 1

1

1 1

1

1 1

1

) (

n n

j j jn n

j j

j n

i ni ni n

n n

j

jn nj n n

j nj nj n

i i

ni n

j nj nj n n n

n t j

j jn t

n

i i

i n

j nj nj n n

v g g a

b p

t c a p m b p p K

b p m

e g a g e

b p t

v

ⁱ ^j

+ +

+

 +







 





 +







 





+

− +

+

−

=

∑

−

=

−

=

−

=

−

=

−

=

−

=

−

=

− −

=

− −

=

−

=

µ µ µ µ µ

µ µ

^µ ^µ

(14)

2. Numerical example

Let us examine the network which was described in the previous section with

= 30

n , K = 61 , where K - the number of messages in the network. Service rates of messages in channels of network systems equal: µ

₁

= µ

₁₃

= µ

₁₉

= µ

₂₉

= 4 , µ

₆

= 6 ,

,

16

5

10 9

3

= µ = µ = µ =

µ µ

₄

= µ

₈

= µ

₁₂

= µ

₁₄

= µ

₂₁

= µ

₂₄

= µ

₂₆

= µ

₂₈

= 2 , µ

₂₅

= 1 , ,

27

= 7

µ µ

₂

= µ

₅

= µ

₇

= µ

₁₁

= µ

₁₅

= µ

₁₇

= µ

₁₈

= µ

₂₀

= µ

₂₂

= µ

₂₃

= 3 , µ

₃₀

= 40 , channel number in the central system − m

₃₀

= 2 , probabilities of message transitions between network QS – p

₃₀_i

= 1 29 , p

_i₃₀

= 1 , i = 1 , 29 , define also p

_ii

= − 1 , i = 1 , 30 , the rest of probabilities p

_ij

= 0 , i , j = 1 , 30 . Let also N

_i

( 0 ) = 2 , i = 1 , 29 , N

₃₀

( 0 ) = 3 . Charts of average number of messages in the QS are shown in Figures 2-5.

Fig. 2. Average number of messages in network systems S_i,i=1,10

(10)

Fig. 3. Average number of messages in network systems S_i, i=11,20

Fig. 4. Average number of messages in network systemsS_i, i=21,29

(11)

Fig. 5. Average number of messages in central QS

Specify values of m.e. of incomes from transitions between network states:

29 , 1 , exp 100 ,

sin 50

30 , 1 2 , cos

20

30

=

 



 

= 

=

 =



 



 −

=

n i a i

n a i

n i c i

i i

i

π

π π

Then using relation (13), (14) with initial condition v

_i

( 0 ) = 100 , i = 1 , 29 , 150

) 0

30

( =

v , the expressions for expected incomes of network systems were obtained. For example expression for expected income of central system is

1 . 15870 2

. 28654 1

. 93 7 . 823

7 . 149 9

. 2354 1

. 720 5

. 2970 4

. 795 ) (

7

6 2

5 3

4 30

+

−

− +

+ +

+

=

−

t e

e

e e

e t

v

t t

t

Charts of expected incomes of the network systems are shown in Figures 6-9.

(12)

Fig. 6. Expected incomes of the systemsS_i, i=1,10

Fig. 7. Expected incomes of the systems S_i,i=11,20

(13)

Fig. 8. Expected incomes of the systemsS_i,i=20,29

Fig. 9. Expected incomes of the central QS

(14)

References

[1] Matalytski M.A., Pankov A.V. Application of operation calculus for the investigation of banking models, Proc. of 17 Int. Conf, Modern Mathematical Methods of Analysis and Optimization of Telecommunication Networks, BSU, Minsk 2003, 172-177.

[2] Matalytski M., Pankov A., Incomes probabilistic models of the banking network, Scientific Re- search of the Institute of Mathematics and Computer Science, Czestochowa, University of Tech- nology 2003, 1(2), 99-104.

[3] Matalytski M., Pankov A., Probability analysis of incomes in banking networks, Westnik BSU, 2004, 2, 86-91.

[4] Howard R., Dynamic programming and Markov processes, Moscow, 1964 (in Russian).

[5] Matalytski M., Koluzaeva E. Analysis and optimization of Markov HM-networks with stochastic incomes from transition between their states, Scientific Research of the Institute of Mathemat- ics and Computer Science, Czestochowa University of Technology 2008, 1(7), 51-62.

[6] Matalytski M.A., Koluzaeva E.V., About methods of analysis and applications of queueing HM-networks, Review of Applied and Industrial Mathematics 2008, 3(15), 564-566 (in Russian).

[7] Matalytski M., Pankov A., Analysis of the stochastic model of the changing of incomes in the open banking network, Computer Science 2003, 5, 19-29.

[8] Matalytski M., Pankov A., Numerical analysis of the probabilistic model of incomes changing in banking network, Computer Science 2004, 6, 7-11.

[9] Matalytski M., Koluzaeva K., Finding of expected incomes in open exponential networks of arbitrary architecture, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology 2007, 6, 179-190.

[10] Koluzaeva E.V., Matalytski M.A., Analysis of incomes in open HM-networks of arbitrary architecture, Westnik GrSU, 2008, 1, 22-29 (in Russian).

[11] Matalytski M.A., About some results of analysis and optimization of Markov networks with incomes and their application, Automatics and Remote Control 2009, 10 (in Russian).