The asymptotic analysis of the income in closed HM-structure with priority messages

THE ASYMPTOTIC ANALYSIS OF THE INCOME IN CLOSED HM-STRUCTURE WITH PRIORITY MESSAGES

Mikhail Matalytski¹, Olga Kiturko²

1 Institute of Mathematics, Czestochowa University of Technology Czestochowa, Poland

2 Faculty of Mathematics and Computer Science, Grodno State University Grodno, Belarus

1 m.matalytski@gmail.com, ² sytaya_om@mail.ru

Abstract. In the article the asymptotic analysis of closed exponential queueing HM-structure with priority messages is carried out with a large total number of messages, depending on time. The number of service lines in systems, the intensity of service messages in them, the probabilities of message transitions between systems also depends on time. It is proved that the density of the income distribution in the network systems in asymptotic satisfies differential equations in partial derivatives. This provided the inhomogeneous differential equation for the expected incomes system structure. An example of transport logistics shows how to solve such equations.

Introduction

The concept of the closed queueing structure (CQS) with a central system of service has been introduced in monography [1]. In article [2] an asymptotic analy- sis CQS of arbitrary architecture with one-type messages and parametres, depend- ing on the time, is carried out. In this article, such an analysis is carried out for CQS with priority messages.

Let's consider the closed exponential queueing structure with priority messages consisting of n+1 queueing systems S₀, S₁, …, S_n, where S₀ - environment, as which condition we will understand a vector k(t)=(k,t)=(k₁₁(t),k₁₂(t);k₂₁(t),

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

( _{1 2}

22 t k t k t

k

n

n , where k₁(t)

i , k₂(t)

i - accordingly the number of priority and priority-free messages in system S_i in an instant t . Priority messages have an absolute priority in relation to priority-free messages [3]. Let K₁(t) and K₂(t) - respectively be the number of the priority and priority-free messages served in CQS, K₁(t)+K₂(t)=K(t) - total number of served messages in an instant t. It is apparent that in system S₀ the number of priority and usual messages in an instant

t equal respectively

∑

=

−

=

n

i i t k t K t k

1 1 1

01( ) ( ) ( ) and

∑

=

−

=

n

i

i t

k t K t k

1 2 2

02( ) ( ) ( ). The number of service lines in queuing system (QS) m(t)

i , i=1,n, m₀(t)=K(t), prob-

abilities of messages transition between them p_ij(t), i,j=0,n, depend on time.

Disciplines of service of messages of both types in each QS are FIFO. Let's desig- nate through µ_js(t)- service intensity of messages of type sin j-th QS in an instant

t. Let's also enter the following designations:

)}, ( ), ( min{

)) ( ), (

( _{1 1}

1 k t m t k t m t

i j j

j

j =

ε









=

≥

≥ +

<

−

<

+

=

. , 0 ), ( ) ( , 0

), ( ) ( ) ( ), ( ) ( ), ( ) (

), ( ) ( ) ( ), ( ))

( ), ( ), ( (

1

2 1

1 1

2 1

2 2

1 2

n j t m t k

t m t k t k t m t k t k t m

t m t k t k t k t m t k t k

j j

j j

j j j

j j

j j

j j j

j j j

ε (1)

1. Equation conclusion for density of distribution of the expected income of separate system

Let's designate through v^*(k,t)

с the full expected income which will receive QS

с

S of the closed structure with priority messages in time t , if in an initial timepoint of CQS is in a condition k . During a small period t∆ of CQS can remain in condi- tion ( tk, ) or make transition to condition (k I₁ I₁,t t)

j

i − +∆

+ , (k I₂ I ₂,t t)

j

i − +∆

+ ,

thus for simplicity we will consider that p_ii(t)=0, i=0,n. Here I0_s - vector con- sisting of zero, I_is- 2n-vector with zero components, except for a component with the number 2(i− )1 +s which is equal to 1, i=1,n, s=1 ,2.

Theorem. Income distribution density p_vc^*(x,t) under a condition that it is differentiated on t and is twice sectionally continuous differentiated on x_is, x_js,

n j

i, =1, , s=1,2, satisfies the terms of order of smallness ε²(t) to the following differential equation in partial derivatives:

− +

−

=

∑ ∑ ∑ ∑

= =

= =

n

j

i s is js

vc ijs

n

i s is

vc is

vc

дx дx

t x p t д x t B

дx t x t дp x дt A

t x дp

1 ,

2

1

* 2

1 2

1

*

* ( , )

) , 2 (

) ( ) , ) ( , ) (

,

( ε

) , ( ) , ( ) ( ) (

2nK t ′ t p^*_vc x t +r_c^* x t

− ε , (2)

where











 −

+

=

∑ ∑

=

=

n

j j i

n

j

j j j ji j

i x

t K

t t K p t t

l x t p t t

x A

1 1 1

0 01 1

1 1

* 1

1 ()

) ) ( ( ) ( )) ( , ( ) ( ) ( )

,

( µ ε µ ,











 −

+

=

∑ ∑

=

=

n

j j i

n

j

j j j j ji j

i x

t K

t t K p t t

l x x t p t t

x A

1 2 2

0 02 1

2 1 2

* 2

2 ()

) ) ( ( ) ( )) ( , , ( ) ( ) ( )

,

( µ ε µ , (3)

)) ( , ( ) ( ) ( )

,

( _{1 1 1}

1 x t t p t x l t

B_ij =−µ_{j ji} ε_{j j j} ^,

∑

=

= ⁿ

j

j j j ji j

ii x t t q t x l t

B

0

1 1

* 1

1( , ) µ () ()ε ( , ()), (4) ))

( , , ( ) ( ) ( )

,

( _{2 2 1 2}

2 xt t p t x x l t

B_ij =−µ_{j ji} ε_{j j j j} ^,

∑

=

=

n

j

j j j j ji j

ii x t t q t x x l t

B

0

2 1 2

* 2

2( , ) µ ( ) ( )ε ( , , ( )), (5)

)

*( t p_ji





≠

=

= −

, ), (

, , 1 ) (

i j t p

i j t p

ji

ji q^*_ji(t)





≠

=

= +

, ), (

, ), ( 1

i j t p

i j t p

ji ji

+

=

∑

= n

j i

jic j j j j

c x t K t t x l t r t

r

0 ,

) 1 ( 1

1 1

*( , ) () [µ ()ε ( , ()) ()

) ( ) ( )]

( )) ( , , ( )

( _{2 1 2} ⁽²⁾

2 t _j x_j x_j l_j t r_jic t p_ji t r_c t

j +

+µ ε ,

) ( ) ( )

( ^{( )}

)

( t K t R t

r ^s

jic n s

jic = , r (t) K (t)R (t)

c n

c = , i,j,c=1,n, s=1 ,2, income from transitions between conditions of CQS R^(s)(t)

jic , R (t)

c is defined in the proof.

Proof.

Let's say that if on an interval of time [t,t+∆t] of CQS makes transition from condition ( tk, ) to condition (k I₁ I₁,t t)

j

i − +∆

+ (it can happen to probability )

( ) ( )) ( ) ( ( )) ( ( )) ( ), ( ( )

( _{1 1 1 1 1}

1 t _j k_j t m_j t u k_j t u K t k_i t p_ji t t o t

j ε − ∆ + ∆

µ ), where u(x)- func-

tion of Heaviside, the income of system S_с will make R⁽¹₁⁾(t)

ji , therefore the income of this QS in an instant t+∆t will be equal to this size plus the expected income

) ,

( _{1 1}

* k I I t

vc + i − j which it receives for the remaining time t if the condition was initial (k I₁ I₁,t)

j i −

+ . Similarly, if on interval [t,t+∆t] of CQS makes transition from condition ( tk, ) to condition (k I₂ I ₂,t t)

j

i − +∆

+ with probability

)) ( ( )) ( ), ( ), ( ( )

( _{2 1 2 2}

2 t k t k t m t u k t

j j

j j j

j ε

µ u(K₂(t)−k_i2(t))p_ji(t)∆t+o(∆t), the income of system S_с will make R⁽²₁⁾(t)

ji plus the expected income v^*(k I₂ I ₂,t)

j i

c + − which

it receives for theremaining timetif theconditionwas initial(k+I_i2−I_j2,t). Let’s consider besides, that the system S_сreceives income R (t)

с for a unit of time during the stay of CQS in condition ( tk, ). CQS remains in condition ( tk, ) during time

∆t with probability −

∑

+ ×

= n

j

j j j j j j

j j

j t k t m t t k t k t m t

0

2 1 2 2 1

1

1( ) ( ( ), ( )) ( ) ( ( ), ( ), ( ))]

[

1 µ ε µ ε

) ( t o t+ ∆

∆

× , thus the income of system S_с will make R (t) t v^*(k,t)

c

c ∆ + .

Owing to the aforesaid, the full expected income v^*(k,t t)

c +∆ of system S_с in an instant t+∆t satisfies the system of the difference equations:

×









 − + ∆

=

∆

+

∑

=

t t m t k t k t t

m t k t t

t k v

n

j

j j j j j j

j j j c

0

2 1 2 2 1

1 1

*( , ) 1 [µ ()ε ( ( ), ()) µ ()ε ( (), (), ())]

(

∆ +

)

+

× R (t) t v^*(k,t)

c

c

∑ [

=

×

∆

−

n

j i

ji i j

j j j

j t k t m t u k t u K t k t p t t

0 ,

1 1 1

1 1

1( ε) ( (), ()) ( ( )) ( () ()) ()

µ

(

+ + −

)

+ ×

× R⁽¹⁾(t) v^*(k I₁ I ₁,t) ₂(t) ₂(k ₁(t),k ₂(t),m (t))u(k ₂(t))

j j

j j j j j

i c

jic µ ε

(

^{( ) ( , )}

) ]

^{( )}

) ( )) ( ) (

(K₂ t k₂ t p t t R⁽²⁾ t v^* k I₂ I ₂ t o t

u − _{i ji} ∆ _jic + _c + _i − _j + ∆

× . (6)

Owing to the expression definition ε_j1,

2

εj , according to (1) and to definition of function of Heaviside, in the right part of a relation (6) we can lower functions

)) ( (k ₁ t

u j , u(k ₂(t))

j . Besides, hereinafter we will carry out the asymptotic analysis at K t ≤ N→∞

s( ) ,s=1 ,2, therefore itis possible toconsiderthatu(K₁(t)−k₁(t))=

i

1 )) ( ) (

( ₂ − ₂ =

=u K t k t

i . Considering ∆t→0 from (6) we receive a system of differ- ence-differential equations (DDE) for expected incomes of system :

Sс

( )

[ {

∑

=

+

−

− +

=

n

j i

c j

i c j j j j

c t k t m t v k I I t v k t

dt t k dv

0 ,

* 1 1

* 1

1 1

*

) , ( ) , (

)) ( ), ( ( ) ) (

,

( µ ε

(

+ − −

) ] }

+

+µ_j2(t)ε_j2(k_j1(t),k_j2(t),m_j(t))v_c^*(k I_i2 I_j2,t) v_c^*(k,t) p_ji(t)

[

∑

=

+ +

n

j i

jic j j j

j t k t m t R t

0 ,

) 1 ( 1

1

1( ε) ( ( ), ( )) ( )

µ

]

^{( ) ( )}

) ( )) ( ), ( ), ( ( )

( _{2 1 2} ⁽²⁾

2 t _j k_j t k_j t m_j t R_jic t p_ji t R_c t

j +

+µ ε . (7)

Let's pass to density of distribution of the income of QS S_с p_vc^* (x,t). Consider- ing a case of a large number of messages 1<<K(t)<N and passing to a vector of relative variables







=

) (

) , ( ) (

)

;...; ( ) (

) , ( ) (

)

; ( ) (

) , ( ) (

) ) (

( ^{11 12 21 22 1 2}

*

t K

t k t K

t k t K

t k t K

t k t K

t k t K

t

t k ^{n n}

ξ ,

whose possible values belong to the limited closed set







 = ≥ = + ≤

=

∑

= n

i

i i is

n

n x x s x x

x x x x x x G

1

2 1 2

1 22 21 12 11

* ( , ; , ;...; , ): 0, 1,2; ( ) 1 ,

in which they place in knots of 2n-dimensional lattice on distance ε(t) from each other, we can use function approximation v^*(k,t)

c : K² (t)v^*(k,t)=

c n

) , ( ) ), ( ( )

( ^{* *}

2 t v xK t t p x t

K ⁿ _c = _vc

= , where p_vc^* (x,t) - density of distribution of proba- bilities of a vector ξ^*(t).

Let e_i1 =ε(t)I_i1, e_i2 =ε(t)I_i2, i=1,n, and let at K(t)→∞: )

( ) ( )

( ^{(1) (1)}

2 t R t r t

K jic jic

n = , K² (t)R⁽²⁾(t) r⁽²⁾(t)

jic jic

n = , K² (t)R (t) r (t)

с с

n = . (8)

Let's increase both parts (7) on K²ⁿ(t) and having added to both parts

=

′

−() () ( , )

2nK^{2 1} t K t v^* k t

c

n 2nK⁻¹(t)K′(t)p_vc^* (x,t)=2nε(t)K′(t)p^*_vc(x,t)=−2nK(t)× )

, ( ) (t p_vc^* x t ε′

× , we receive

{ [

∑

=

× +

′

−

=

n

j i

j j j j vc

vc nK t t p x t K t t x t l t

дt t x дp

0 ,

1 1 1

*

*

)) ( ), ( ( ) ( )

( ) , ( ) ( ) ( ) 2

,

( ε µ ε

(

+ − −

)

+ ×

× p^*_vc(x e_i1 e_j1,t) p_vc^* (x,t) µ_j2(t)ε_j2(x_j1(t),x_j2(t),l_j(t))

(

p_vc* ⁽x+e_i2−e_j2^,t⁾−p^*_vc⁽x^,t⁾

) ]

p_ji⁽t⁾

}

+r_c^*(x^,t⁾

× . (9)

Let’s spread out functions p^*_vc(x+e_is−e_js,t), p^*_vc(x+e_0s−e_js,t)= p^*_vc(x−e_js,t), )

, ( ) ,

( ₀ ^*

* x e e t p x e t

p_vc + _is− _s = _vc + _is abreast Taylor in a point neighbourhood ( tx, ):

+













∂

−∂

∂ + ∂

=

− +

js vc is

vc vc

js is

vc x

t x p x

t x t p

t x p t e e x

p ( , ) ( , )

) ( ) , ( ) , (

*

* *

* ε

)) ( ) (

, ( )

, 2 (

) , ( 2

)

( ₂

2

* 2

* 2 2

* 2 2

t о x

t x p x

x t x p x

t x p t

js vc js

is vc is

vc ε

ε +











∂ +∂

∂

∂

− ∂

∂

+ ∂ ,

+

∂

− ∂

=

−

js vc vc

js

vc x

t x t p t x p t e x

p ( , )

) ( ) , ( ) , (

* *

* ε ( , ) ( ())

2 )

( ₂

2

* 2 2

t о x

t x p t

js

vc ε

ε +

∂

∂ ,

+

∂ + ∂

= +

is vc vc

is

vc x

t x t p t x p t e x

p ( , )

) ( ) , ( ) , (

*

*

* ε ( , ) ( ())

2 )

( ₂

2

* 2 2

t о x

t x p t

is

vc ε

ε +

∂

∂ ,

n c j

i, , =0, , s=1 ,2. Having substituted this decomposition in the equation (9):

 +

































∂

−∂

∂

=

∑ ∑

∂

= =

n

i n

j j

vc i

vc j

j j ji j vc

x t x p x

t x t p

l t x t p дt t

t x дp

1 1 1

*

1

* 1

1 1

* ( , ) ( , )

)) ( ), ( ( ) ( ) ) (

,

( µ ε

+

























∂ +∂

∂

∂

− ∂

∂ + ∂

2 1

* 2

1 1

* 2 2

1

*

2 ( , ) ( , )

) 2 , ( 2

) (

j vc j

i vc i

vc

x t x p x

x t x p x

t x p t ε