Finding of expected incomes in open exponential networks of arbitrary architecture

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FINDING OF EXPECTED INCOMES IN OPEN EXPONENTIAL NETWORKS OF ARBITRARY ARCHITECTURE

Michal Matalycki¹, Katarzyna Koluzaewa²

1Institute of Mathematics and Computer Science, Czestochowa University of Technology, Poland e-mail: m.matalytski@gmail.com

2Faculty of Mathematics and Computer Science, Grodno State University, Belarus e-mail: e.koluzaeva@grsu.by

Abstract. Investigation of the open exponential queueing network of arbitrary architecture with incomes is carry out in paper. The incomes of transitions between network’s states are random variables with specified moments of two first orders. The expressions for expected incomes and variances of incomes are received.

Introduction

Exponential queueing networks with incomes in case, when incomes of transitions between stations of network were values that depended from network’s state and time were examined in [1, 2]. The expressions for expected incomes and variances of incomes in systems of such networks were obtained in [3, 4] in the case, when incomes of transitions between network’s stations were random variables (RV) with specified moments of the first and the second degrees.

However, only networks with central system were examined in these works. This work is devoted to finding of expected incomes in exponential networks of arbitrary architecture.

Let’s consider open exponential queueing network with one-type messages that consists of n queueing systems S₁,S₂,...,S_n. State of such network could be dis- cribed by vector ^k(^t)=

(

^k₁,^k₂,...,^kn,^t

)

, where k - number of messages in system _i

S in the moment t, i i=1,n. The incoming flow arrives in the network with rate λ. Let’s denote the service rate in system S (when there are _i k messages in it) as _i

( )

i

i k

µ ^; p₀j - probability that the message comes in system S_j from the outside, 1

1

0 =

∑

= n

j

p j ; p - probability that the message after service in system _ij S comes to _i

S , j 1

1

∑

=

= n

j

pij , i=1,n. Message that comes from one system to another brings

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the last one some random income, so the income of the first system descened on this random value.

1. Expressions for expected incomes of network’s systems

Let’s consider variation dynamics of incomes in system S_i. Let’s denote its income in the moment t as V_n(t). Let at any moment t₀ income of this system equals to v_i₀. We are interesting in incomes V_i(t₀ +t) of system S_i at the moment

t

t0+ . We will broke segment [t₀,t₀ +t] by m equal parts with length ∆t=t/m, where m is rather large. Let’s give out all events which can occur at l-th range.

Following situations are possible.

1. Message arrives to system S from the outside with the probability _i )

0 t o( t

p _i∆ + ∆

λ and increases its income by r₀_i, where r₀_i – RV with distribution function (DF). F₀_j(x).

2. Message goes from system S to the outside with probability _i

( )

k_i u⁽k_i⁾p_i0 t o⁽ t⁾

i ∆ + ∆

µ , where





≤

= >

0 , 0

, 0 , ) 1

( x

x x

u - Heaviside’s function, and decreases an income byR , where _i₀ R - RV with DF _i₀ F_i₀(x).

3. Message arrives from S to _j S with probability _i µj

( )

^kj û⁽^kj⁾^pji∆^t+ô⁽∆^t⁾ând income of system S increses by _i r , and income of system _ji S decreses by the _j same value, j=1,n,j≠i, where r – RV with DF _ji F₁_ji(x).

4. Message arrives from S to _i S with probability _j µi

( )

^ki û⁽^ki⁾^pij∆^t+ô⁽∆^t⁾ând income of system S decreses by _i R , and income of system _ij S increases by _j the same value, where R – RV with DF _ij F₂_ij(x), i,j=1,n.

5. No changes of network’s state are happened during time period t∆ with probability ¹−

(

λp0_i +µ_i⁽k_i⁾u⁽k_i⁾

)

∆t+o⁽∆t⁾. It is evident that r_ji =R_ji with probability 1, i, j=1,n, that is

) ( )

( ₂

1 x F x

F_ij = _ij , i,j=1,n (1)

Besides, system S increases its income by _i r_i∆t during any short time period

∆t at the cost of money percents it stores. Let r is RV with DF _i F_i(x),i=1,n. Let ∆V_il(t)=V_il(t+∆t)−V_il(t) is a change of income of i-th system at l-th time segment. It follows from above that

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( )











∆ +

−

∆ = ≠

∆ +

∆ ∆ +

−

≠

=

∆ +

∆ ∆ +

∆ +

∆

∆ +

−

∆ +

∆

∆ +

=

∆

).

( ) ( ) ( 1

y probabilit with

, , , 1

), ( )

( ) y (

probabilit with

, , , 1

), ( )

( ) y (

probabilit with

), ( )

( ) ( y probabilit with

), ( y

probabilit with

) (

0

0 0

t o t k u k p

t r

i j n j

t o t p k u t k

r R

i j n j

t o t p k u t k

r r

t o t p k u k t

r R

t o t p t

r r

t V

i i i i i

ij i i i i

ij

ji j j j i

ji

i i i i i

i

j i

i

il

µ λ µ

µµ λ

Income of S equals _i

∑

=

∆ +

= ^m

l il i

i t v V t

V

1

0 ( )

)

( (2)

Let’s introduce denotation for corresponding expectation values:

{ }

^r^ji ^xdF^ji ^x ^a^ji

M =^∞

∫

=

0

1 ( ) ,

{ }

^,

0

2_ji( ) _ij

ij xdF x b

R

M =

∫

^∞ = ^,^M

^{{ }}

^rⁱ ⁼^∞

∫

^xdFⁱ ^x ⁼^сⁱ

0

) (

{ }

ri xdFi x a i

M ₀

0 0

0 =

∫

^∞ ( )= ^,

^{{ }}

⁰

0 0

0 _i ( ) _i

i xdF x b

R

M =^∞

∫

= ^,ⁱ^, ^j⁼¹^,ⁿ

in view of (1)

ji

ji b

a = , i, j=1,n (3)

Let’s get a relationship for expectation value of income of system S at time _i moment t. With fixed member function k(t) we get:

{ } ∑

=

−

∆ +

∆

−

∆

= ⁿ

j

ji j j j ji i

i i i i i i

il t k t a p t b k u k p t a k u k p t

V M

1 0

0 0

0 ( ) ( ) ( ) ( )

) ( / )

( λ µ µ

(

¹ ⁽ ⁾ ⁽ ⁾

)

⁽ ⁾

) ( )

( ₀

1

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

b _i _i _i _i _i

n

j

ij i i i

ij ∆ + ∆ − ∆ − ∆ + ∆

−

∑

=

µ λ

µ

that is

{ }

= ∆ +

∑

∆ −

= n

j

ji j j j ji i

i

il t k t a p t a k u k p t

V M

1 0

0 ( ) ( )

) ( / )

( λ µ

) ( )

( ) (

0

t o t c t p k u k

b _i

n

j

ij i i i

ij ∆ + ∆ + ∆

−

∑

= µ ^,ⁱ=¹^,ⁿ (4)

(4)

Therefore, taking into account that m t= t

∆ , we get from (2), (4)

{ }

= +

∑ {

∆

}

=

= m

l

il i

i t k t v M V t k t

V M

1

0 ( )/ ( )

) ( / ) (

=

∆











 + − +

+

=

∑ ∑

=

t c p k u k b p

k u k a p

a

v _i

n

j

ij i i i ij n

j

ji j j j ji i

i i

0 1

0 0

0 λ µ ( ) ( ) µ ( ) ( )

) ( )

( ) ( )

( ) (

0 1

0 0

0 a p a k u k p b k u k p c t o t

v _i

n

j

ij i i i ij n

j

ji j j j ji i

i

i + ∆











 + − +

+

=

∑ ∑

=

µ µ

λ

So, when ∆t→0, we will have

{

^V ^k ^k ^t

}

^v ^c â ^p â ^k û ^k ^p ^b ^k û ^k ^p ^t

M

n

j

ij i i i ij n

j

ji j j j ji i

i i i

i 









 + + −

+

=

∑ ∑

=

=1 0

0 0

0 ( ) ( ) ( ) ( )

) ( / )

( λ µ µ

Considering

∑ (

⁽ ⁾=

)

=¹

k

k t k

P we get average by k(t)

=

{ }

= +

∑ (

=

) { }

=

k

i i

i

i t M V t v P k t k M V t k t

v ( ) ( ) ₀ ( ) ( )/ ( )

(

=

) {

=

}

=

∑ ∑ ∑

^∞

=

∞

=

∞

=

0 0 0

2 1 2

1

1 2

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

k k k

n i

n

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

( )

−





 + + =

+

=

∑ ∑

= n

j k

j j j ji

ji i

i i

i c a p a p P k t k k u k

v

1 0 0

0 λ ( ) µ ( ) ( )

(

^k ^t ^k

)

^k ^u ^k ^t

P p b

n

j k

i i i ij

ij 



= 

−

∑ ∑

=0

) ( ) ( )

( µ

Then we suppose that all network systems are multi-line, any system S has _i m _i identical service lines, service time of messages have exponential distribution with

µi at any line, i=1,n. In this case

) , , min(

,

, ) ,

( _i _i _i

i i i i

i i i i i

i k m

m k m

m k

k k µ

µ

µ µ =





>

= ≤ , i=1,n

(5)

and it’s normally to suppose that average of µ_i(k_i)u(k_i) will be )

), (

min( _i _i

i N t m

µ ^{, where} ^Ni^(t⁾ - mean number of messages (being waiting and servicing) in system S at time interval _i

[

^t0,^t0+^t

]

, i=1,n. So finally we will have

( )





 + + −

+

=

∑

= n

j

ji j j j

ji i

i i i

i t v c a p a N t m p

v

1 0 0

0 min ( ),

)

( λ µ

(

^N ^t ^m

)

ⁿ ^b ^p ^t

j ij ij i i

i 



−

∑



=0

), (

µ min ⁽⁵⁾

For expected income of the whole queueing network we have

∑

=

= ⁿ

i i t V t

W

1

) ( )

(

{ } ∑ ∑ ∑ ( )

= =

= 



 + + −

+

= ⁿ

i

n

j

ji j j j

ji i

i i n

i

i c a p a N t m p

v t

W M

1 1

0 0 1

0 min (),

)

( λ µ

(

^N ^t ^m

)

ⁿ ^b ^p ^t

j ij ij i i

i 



−

∑



=1

), (

µ min ⁽⁶⁾

but, taking into acount (3),

( ) ∑ ( ) ∑

∑∑

= =

= ⁿ

i

n

j ij ij i i i

n

i n

j

ji j j j

ji N t m p N t m b p

a

1 1

), ( min ),

(

min µ

µ

so

{ }

^W ^t ^v

[

^c ^a ^p ^b ^p

]

^t

M

n

i

i i i i i i n

i

∑

= =

− +

+

=

1

0 0 0

0 1

) 0

( λ µ ⁽⁷⁾

Note, that general expected income of network depends on r_ij,R_ij,i,j=1,n, as these incomes extinguish each other (if message from one system of network arrivals to another then income of the last one increases by some value and the first system’s income decreases by the same value).

Example. Let’s consider close queueing network with central system, that consists of central system S and peripheral systems _n S₁,S₂,...,S_n₋₁. Let

0 ,

1 ,

0 = ≠

= pin pni

λ , other p_ij =0,p₀_i = p_i₀ =0,i=1,n. In this case, from (5), (6) we recieve expressions for total expected income of the network

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{ }

^W ^t ^v ^c^t

M

n

i i n

i

∑

= =

+

=

1 1

) 0

( (8)

expected income of central system is

(

^N ^t ^m

)

^b ^p

(

^N ^t ^m

)

^t

a c

v t

v _nj _n _nj _n _n

n

j

j j j

jn n

n

n 









 + −

+

=

∑

⁻

=

), ( min ),

( min )

(

1

0 µ µ ⁽⁹⁾

and expected incomes of peripheral systems taking into (3) are

( ) ( )

[

⁺ ⁻

]

⁼

+

=v c a N t m p b N t m t

t

v_i() _i₀ _i _niµ_nmin _n(), _n _ni _inµ_imin _i(), _i

=^vi₀+

[

^ci−^ainµimin

(

^Ni(^t),^mi

)

+^bniµnmin

(

^Nn(^t),^mn

)

^pni

]

^t (10) Note that exprassions (8)-(10) concur with expressions for corresponding expected incomes in network with central system [5].

2. Recurrent mean-value analysis method with respect to the moments of time for open queueing network

From relations (5), (6) follows that we have know expressions of average number of messages N_i(t), i=1,n, for finding of expected system’s incomes and the whole network. Finding of these values in analytic form for network that functions in transient behaviour is rather complicated problem. Before recurrent mean-value analysis method with respect to the moments of time was developed for finding of series of average characteristics for arbitrary closed queueing networks with one-type messages and multi-line systems [6]. In this paper we suggest substantiation of such method for open queueing network with arbitrary distributions of service times of messages in network’s system lines.

Let’s consider open queueing network with one-type messages and multi-line systems. Let M_i(t)is average rate of massage’s flow from the i-th system, ρ_i(t)^,

)

i(t

τ - average number of busy lines and average sojourn time (include wating) at i-th system at time period [t₀,t₀+t], i=1,n, correspondently. Then expression is correct

) ( ) ( )

(t t N t

M_i τ_i = _i ^,i=1,n (11)

For proving it we can use technique [7]. Let Z_i(t)- total number of messages wich have left i -th system during time period [t₀,t₀+t], Θ_i(t) - total sojourn time in i-th system during time period [t₀,t₀ +t]. So

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t t t MZ t M

t t M

t N MZ

t

t M _i ⁱ _i ⁱ

i i i

) ) (

( ), ) (

( ), (

) ) (

( = Θ = Θ =

τ , i=1,n

therefore, formula (11) is correct.

As ρ_i(t) we can approximately take value min(N_i(t),m_i) because at any time moment t

) ), ( min(

)

( _i _i

i t = k t m

ρ

where ρ_i(t) - number of busy lines in i-th system at the moment t.

Let λ_i - arrival rate of messages in i-th system, i=1,n. It is known [8] that

∑

=

+

= ⁿ

j ji j i

i p e p

1

0 λ

λ

λ ^,ⁱ=¹^,ⁿ, (12)

where λ - arrival rate of messages in network, p - probability of message’s tran-_ji sition between j-th and i-th systems, i,j=1,n, p₀_i - probability that message comes from outside to the i-th system, values e , _j j=1,n, meet the system of linear equations

∑

=

+

= ⁿ

j ij i j

j p e p

e

1

0 , j=1,n

It follows from (11)

) (

) ) (

( t

t t N

i i

i i i

i µ ρ

τ λ

λ = (13)

but

) ( ) (t N_i t

i

iτ ≈

λ ⁽¹⁴⁾

what follows from formula that is similar to Littl’s formula [3].

From the account above we recieve recurrent mean-value analysis method with respect to the moments of time for open queueing networks:

(

^N ^t ^m

)

ⁱ ⁿ

t _i _i

i()=min (), , =1,

ρ ⁽¹⁵⁾

n t i

t t N

i i

i

i , 1,

) (

) ) (

( = =

ρ

τ µ ⁽¹⁶⁾

n i t t

t

N_i( +∆)=λ_iτ_i( ), =1, ⁽¹⁷⁾

(8)

where λ_i^, ⁱ=¹^,ⁿ, meet the system of equation (12). Initial conditions can be choosen in following way: N_i(t₀)≠0,i=1,n.

3. Variation of incomes of network’s systems

Taking into account (3) we will rewrite expression of expected income of system S as _i

( )





 + + −

+

=

∑

= n

j

ji j j j

ji i

i i i

i t v c a p a N t m p

v

1 0 0

0 min (),

)

( λ µ

( )

⁼

















 +

−

∑

=

t p a p

b m t N

n

j ij ij i

i i i i

1 0

), 0

( µ min

( )

[

⁺ ⁻ ⁺

+

=v_i0 c_i a0_iλp0_i b_i0µ_imin N_i(t),m_i p_i0

( ) ( )

(

^a ^N ^t ^m ^p ^a ^N ^t ^m ^p

)

^t

n

j

ij i i i

ij ji j j j

ji 



−  +

∑

=1

), ( min ),

(

min µ

µ (18)

V_i(t)−v_i0

)

²:

( )

_ ⁼







 ∆

 =







 + ∆ −

=

−

∑ ∑

=

2

1 2

0 1

0 2

0 ( ) ( )

) (

m

l il i

m

l il i

i

i t v v V t v V t

V

n i t V t V t

V

m

l m

j l

j

ij il m

l

il ( ) ( ) ( ), 1,

1 1

1

2 + ∆ ∆ =

∆

=

∑ ∑∑

= ≠=

=

(19)

Let’s find an expectation value of summands in the right part of the last equation.

⁼^c ^∆^t

[

⁻

^∆ ⁺

+

∑

≠= n

i j j

ji j j j i

i ji

ji a c t c t k u k p t

a

1

2 2

2 2 ( ) µ ( ) ( )

[

⁻ ^∆ ⁺ ^∆

]

^∆ ⁺ ^∆ ⁼

+

∑

≠=

) ( )

( ) ( ) ( 2

1

2 2

2 b c t c t k u k p t o t

b

n

i j j

ij i i i i

i ij

ij µ

[

⁺ ⁺

= λa20_ip0_i b2_i0µ_i(k_i)u(k_i)p_i0

(

⁽ ⁾ ⁽ ⁾ ⁽ ⁾ ⁽ ⁾

)

⁽ ⁾

1

2

2 k u k p b k u k p t o t

a

n

i j j

ij i i i ij ji j j j

ji ∆ + ∆







 +

+

∑

≠= µ µ ⁽²⁴⁾

Taking into account that with fixed member function k(t) values ∆V_il(t) and )

(t V_ij

∆ independent when l≠ j, using (3), (4) it can be fined

{

^∆Vil t ^∆Vij t k t

}

⁼

[{

a i p i ⁻ai i ki u ki pi ⁺ci⁺

M ( ) ( )/ ( ) ₀λ ₀ ₀µ ( ) ( ) ₀

(

⁽ ⁾ ⁽ ⁾ ⁽ ⁾ ⁽ ⁾

)

⁽ ⁾ ² ⁽ ⁾

1

t o t c t p k u k a p k u k a

n

j

ji = ∆





∆  +

∆





−  +

∑

=

µ

µ (25)

Then, making extreme transition by ∆t→0, it follows from (19), (24), (25)

(10)

( )

{

⁻

}

⁼

^∑ {

^∆

}

⁺

= m

l

il i

i t v k t M V t k t

V M

1

2 2

0 ( ) ( )/ ( )

) (

{

^∆ ^∆

}

⁼

[

⁺ ⁺

+

∑∑

= ≠= 20 0 20 0

1 1

) ( ) ( )

( / ) ( )

( _i _i _i _i _i _i _i

m

l m

j l

j

ij

il t V t k t a p b k u k p

V

M λ µ

(

â ^k û ^k ^p ^b ^k û ^k ^p

)

^t

n

i j j

ij i i i ij ji j j j ji







 +

+

∑

≠=1

2

2 µ ( ) ( ) µ ( ) ( ) ^,ⁱ=¹^,ⁿ

Get average by k(t) we will have

( )

{

⁻

}

⁼

[

20 0 ⁺ 20

( )

0 ⁺ 2

0 ( ) min ( ),

)

( _i _i _i _i _i _i _i

i t v k t a p b N t m p

V

M λ

( ) ( )

( )

⁼







 +

+

∑

≠=

t p m t N b

p m t N a

n

i j j

ij i i ij

ji i i ji

1

2

2 min ( ), min ( ),

(

^N ^t ^m

)

^p

(

^N ^t ^m

)

^b ^p ^t

a p a

n

i j j

ij ij i i n

i j j

ji i i ji

i i

















+

=

∑ ∑

≠=

≠= 0

2 1

2 0

20 min ( ), min ( ),

λ ⁽²⁶⁾

Let’s find

{ (

0

) }

2 V_i(t) v_i

M − with help of (3), (4) for calculation of the variation using standard formula:

( )

{ } { }

^ ⁼



 



 ∆

=





 ∆

=

−

∑ ∑

=

2

1 1

2 0

2 V (t) v M V (t)k(t) M V (t)k(t)

M

m

l

il m

l il i

i

[

ji 



−  +

∑

=

µ

µ ^,^for ∆t→⁰

(11)

that is

( )

{

⁻ 0

}

⁼

{ [

0 0 ⁻ 0 0⁺

2 V_i(t) v_i k(t) a _i p _i b_i _i(k_i)u(k_i)p_i

M λ µ

( )

⁺

[

⁻ ⁺





−  +

∑

= 0 0 0 0

2

1

) ( ) ( 2

) ( ) ( )

( )

( _i _i _i _i _i _i _i _i

n

j

ji k u k p a k u k p c a p b k u k p

a µ µ λ µ

( )

² ²

1

) ( ) ( )

( )

(k u k p a k u k p c t

a _n

n

j

ji 

 + 





−  +

∑

= µ µ

Average by k(t) with taking in account (18) give

( )

{ } ( )[



 = − +

=

− 0

∑

0 0 0 0

2 ( ) ( ) _i _i _i _i( _i) ( _i) _i

k i

i t v Pk t k a p b k u k p

V

M λ µ

( )

⁼





 +



 



 − −

 +





−  +

∑

=

2 0

2

1

) 2 (

) ( ) ( )

( )

( c c t

t v t c V p

k u k a p k u k

a _i ⁱ ⁱ _i _i

n

j

jiµ µ

( ) [



 = − +

=

∑

() 0_i 0_i _i0 _i( _i) ( _i) _i0 k

p k u k b p a k t k

P λ µ

(

â ^k û ^k ^p â ^k û ^k ^p

)

^ci ^t ^ci

⁽

^vi ^t ^vi

⁾

^t n

j

ji 0

2 2 1

) ( 2 )

( ) ( )

( )

( + −





− 





−  +

∑

=

µ

µ ⁽²⁷⁾

where v_i(t), i=1,n, is finding by formula (18).

⁼

= ( ) ₀ ( ) ₀ ² ² ( ) ₀ )

( _i _i _i _i _i _i

i t DV t v M V t v M V t v

DV

( ) ( )









− +

+

=

∑ ∑

≠=

n

i j j

ij ij i i n

i j j

ji i i ji

i

ip a N t m p N t m b p

a

0 2 1

2 0

20 min ( ), min ( ),

λ

( )] ( )[



 − = − +

+

−

∑

0 0 0 0

2

0 () ( ) ( )

) (

2 _i _i _i _i _i _i _i

k i i

i

i v t v t c P k t k a p b k u k p

c λ µ

( )

² ²

1

) ( ) ( )

( )

(k u k p a k u k p t

a

n

j

ji 









−  +

∑

=

µ

µ , i=1,n

(12)

References

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

[2] Matalytski M., Pankov A., Analysis of stochastic model of the changing of incomes in the open banking network, Computer Science, Czestochowa University of Technology 2003, 3(5), 19-29.

[3] Matalytski M., Pankov A., Finding time-probabilistic characteristics of close Markov network with central system and incomes, Wiestnik GrUP 2006, 3(46), 22-28 (in Russian).

[4] Matalytski M., Pankov A., Finding of expected incomes in closed stochastic network with central system, Computer Science, Czestochowa University of Technology 2007, 7(11).

[5] Pankov A., Analysis of expected incomes in closed queueing network with central system, Wiestnik GrUP 2006, 2(45), 48-56 (in Russian).

[6] Matalytski M., Recurrent mean-value analysis method with respect to the moments of time, Automation and Remote Control 2000, 60, 11, 1, 1552-1557.

[7] Eilon S., A simpler proof of L = λW, Operations Research 1969, 17, 915-918.

[8] Matalytski M., Rusilko T., Mathematical analysis of stochastic models for claim processing in insurance companies, GrSU, Grodno 2007 (in Russian).