Analysis and optimization of markov Hm-networks with stochastic incomes from transitions between their states

ANALYSIS AND OPTIMIZATION OF MARKOV HM-NETWORKS WITH STOCHASTIC INCOMES FROM TRANSITIONS BETWEEN THEIR STATES

Michal Matalycki¹, Katarzyna Koluzaewa²

1Institute of Mathematics, Czestochowa University of Technology, Poland

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

Abstract. Expressions for expected incomes and variations of incomes in systems of Mar- kov HM-networks, when service rates of messages are depending or not depending of network’s states are obtained. The case when incomes of transitions between network’s states are random variables with the set moments of first two orders is considered. Solution of some optimization problems for HM-networks are resulted.

Introduction

It is known that functioning of any Markov queueing network (QN) is described with a help of Markovian chain with continuous time. In Howard’s work [1] the conception of Markovian chains with incomes that were constants was introduced and there were proposed to use method of Laplace transformation and method of z-transformation for analysis of such chains with small number of states. This con- cept has laid down in a basis of definition of Markov QN with incomes that were examined in works [2-4] at first. Later open and closed networks with central queue- ing system (QS) were investigated in cases when a) incomes from transitions be- tween network’s states depend on states and time [5-7] or b) incomes are random variables (RV) with the set moments of the first and the second orders [8, 9]. Re- cently QN with incomes refer to as HM (Howard-Matalytski)-networks. In the present article expressions in case b) were obtained for expected incomes and in- come’s variations in systems of open exponential HM-networks and Jackson HM- networks of arbitrary architecture.

Let us examine open exponential QN with one type messages that consist of n QS S₁,S₂,...,S_n. Vector ^k(^t) (^k,^t)



^k₁,^k₂,...,^kn,^t



is state of network where k_i is number of messages in system S at the moment_i t,t t[ ₀,f), i 1 n, . For un- ification of designation let us introduce system S (outside medium) from which the ₀ Poisson flow of messages with arrival rate O comes into network. At first we will

examine the case when service rate of messages P_i ^{in system} S doesn’t depend on i

number of messages in it, i 1,n. Let p_0j is probability of message entry from system S to system₀ S_j, 1;

1

¦

ⁿ 0 j

p j p is probability of message transition after _ij

its service in system S to system _i S_j, 1,

¦

ⁿ0 j

pij i 1 n, .Message during its transi- tion from system S to system_i S_j brings to system S_j some random income and income of system S descend on this value correspondently,_i i,j 0,n. It is needed to find expected (average) incomes of network’s systems during time t considering that network’s state in the starting time t₀ is known. At first we will suppose that network is functioning in the term of high capacity, i.e. k_i(t)!0, t[t₀,f),

. , 1 n i

1. Expected incomes of network systems

Let us examine dymamics of income’s change of some network’s system S_i. Let us denote its income at the moment t as V_i(t). The income of this system at some moment t is equal to ₀ v_i0. We will be interested in system’s incomeV_i(t₀ t) at the moment t₀ To a finding of the income we shall apply a following technique. Let t. us divide interval

>

^t0,t0t

@

into m equal portions with length ǻ ,

m

t t and consid- er that m is big. For finding of system’s income Let us write probability of all events that can occur at the l -th time interval, l 1 m, . The next situations are poss- ible: a) message from outside will arrive to system S_i with probability

ǻ ) ǻ (

0 t o t

p _i 

O and will bring income of size r_0i to it, where r_0i – RV with distribu- tion function (d.f.) F_0i(x); b) message from system S will transmit with probability _i

)

0 t o( t p_i

i '  '

P into outside medium and system’s income will decrease on value

0,

Ri where R_i0 – RV with d.f. F_i0(x); c) message from system S_j will transmit with probability P_jp_ji'to( t' ) into system S and income of system _i S will in-_i crease on value r_ji and income of system S_j will decrease on this value,

, , , 1n j i

j z where r_ji – RV with d.f. F_1ji(x); d) message from system S will_i transmit with probability P_ip_ij'to( t' ) into system S and income of system _j S_j

will increase on value R and income of system _ij S will decrease on this, where _i R_ij – RV with d.f. F_2ij(x), i, j 1,n; e) change of system S state won’t occur with _i probability ¹



Op0_i P_i



'to⁽'t⁾ at the time interval 't.

It’s evident that r_ji R_ji with probability 1, i,j 1,n, i.e.

) ( )

( ₂

1 x F x

F_{ij ij} , i,j 1,n (1)

Besides system S increases its income on value_i r_i' during every small time inter-t val 't due to interest on money that are in it; let us suppose that r is RV with d.f. _i

. , 1 ), (x i n

F_i We will suppose also that RV r_ji, R_ij, r_0i, R_i0, r are independent in _i pairs, i,j 1,n.

Let d_il( t' is income change of) i-th QS at the l-th time interval of length ǻt. Then from all that was saying before it follows

 

°°

°

¯

°°

°

®

­

'

 '



 '

z '

 ' '





z '

 ' '



'

 ' '





'

 ' '

 '

).

( 1

y probabilit with

, , , 1 ), ( y

probabilit with

, , , 1 ), ( y

probabilit with

), ( y

probabilit with

), ( y

probabilit with

) (

0 0 0

0 0

t o t p

t r

i j n j t o t p t

r R

i j n j t o t p t

r r

t o t p t

r R

t o t p t

r r

t d

i i i

ij i i

ij

ji j i

ji

i i i

i

i i

i

il

P O

P P

P O

(2)

Income of system S_i equals

¦

^'

 ^m

l il i

i t v d t

V

1

0 ( )

)

( (3)

Let us introduce definition for expectation values (e.v.) correspondently:

^ `

^{rji xdFji x aji}

M ^f

³

0

1 ( ) ,

^ `

^,

0

2_ij( ) _ij

ij xdF x b

R

M

³

^{f M}

^{^ `}

^{ri f}

³

^{xdFi x ɫi}

0

) (

^ `

ri xdFi x a i

M ₀

0 0

0 ^f

³

( ) ^,

^{^ `}

⁰

0 0

0 _i ( ) _i

i xdF x b

R

M

³

^{f , i,j 1,n (4)}

in view of (1)

ji

ji b

a , i,j 1,n (5)

Let us derive expression for e.v. of income of systemS_i at the moment t.

According to formula for e.v. of discrete RQ

^

^{( )}

`

^{( )}

0 1

0

0 p a p b p c t o t

a t

d

M _i

n

j ij ij i n

j

ji ji j i

i

il ¸¸'  '

¹

·

¨¨

©

§   

' ^O

¦

^{P P}

¦

^{, i 1,n (6)}

As it follows from (3) taking into account mǻt whent, 'to0

^ `

^{V t v c a p a p b p t}

M t v

n

j ij ij i n

j

ji ji j i

i i i i

i ««

¬ ª

»»

¼

 º







¦ ¦

0 1

0 0

) 0

( )

( O P P ⁽⁷⁾

For expected income for whole network considering (5), we have:

^ ` ¦ ¦

ⁿ

>

^



^{ }

 @

i

i i i i i i i n

i

i t v c a p b p t

v t

W M

1

0 0 0

0 0

1

) ( )

( O P ⁽⁸⁾

Let us notice that common expected network’s income doesn’t depend on ,

, 1 , ,

,R i j n

r_{ij ij} as that incomes absorb each other (if message transmits from one network’s system to another then income of the last one increases on some value and income of the first QS decreases on the same value).

2. Variations of network systems incomes

From expressions (2), (3) it follows that

^ `  

^{2 2}

1 0 0 0

0 0

2 V(t) v c a p b p a p b p t

M

n

j

ij ij i ji ji j i

i i i i i i

i ««

¬ ª

»»

¼

 º







 ^{O P}

¦

^{P P}

Let us introduce denotations:

^ `

^{ri a i}

M ₀² ₂₀, ^M

^ `

^{Ri02 b2i0, M}

^ `

^{rji2 a2ji, M}

^ `

^{Rij2 b2ij, i, j 1,n (9)}

Let us consider expression



^



_¨¨©^{§ }

¦

^{' } _¸¸¹^· _¨¨©^§

¦

^' _¸¸¹^·2

1 2

0 1

0 2

0 ( ) ( )

) (

m

l il i

m

l il i

i

i t v M v d t v M d t

V M



^{d t d t}



^{i n}

M t

Md

m

l m

l j j

ij il m

l

il( ) ( ) ( ), 1,

1 1

1

2 ' 

¦¦

' '

¦

z

(10)

Taking into account (2)-(9), we have:

^

^{d 't}

`

^M

> 

^{r  r r 'tr 't}



^{p 't}

@

^

M _il²( ) ₀²_i 2 _{0i i i}²( )² O _0i

 

> @ _¦ > 

z

 '



 ' '

 '



 ⁿ

i j j

i ji ji ij

i i

i

i R r t r t p t M r r r t

R M

1 2 2

2 0 0 2

0 2 ( ) P 2



^'

@

^

> 

^{ '  '}



^'

@

^

'



¦

z n

i j j

ij i i

i ij ij ji

j

i t p t M R R r t r t p t

r

1

2 2 2

2

2( ) P 2 ( ) P

 

>

^{  '}

@

^{ '}

'

Mr_i²( t)²1 Op_0i P_i t o( t)

 

^{( )}

1

2 2

0 0 2 0

20 p b p a p b p t o t

a

n

i j j

ij ij i ji ji j i

i i i

i '  '

»»

»

¼ º

««

«

¬ ª







¦

z

P P

P

O ^{, i 1,n (11)}

Besides from independence of d_il( t' and) d_ij( tǻ), jz considering (2) it fol-i, lows that ^M



^{dil(ǻt)diq(ǻt)}



^o(ǻt)2. So as it’s fallowed from (10), (11) and

ǻt t,

m for variation of income of i-th QS when 'to0 we obtain the following expression



^{( ) 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

 

^

««

«

¬ ª

»»

»

¼ º







¦

z

t p b p a p

b p

a

n

i j j

ij ij i ji ji j i

i i i i

1

2 2

0 0 2 0

20 P P P

O

 

²

1 0

0 0

0p a p c a p a p t

b

n

j

ji ji j ij ij i i

i i i i

« i

«¬ ª

»»

¼

 º







 ^{P O}

¦

^{P P , i 1,n (12)}

3. Expected incomes of Jackson HM-network systems

Let us consider now the case when service rate of messages P_i(k_i) ^{in system} Si

depends on number of messages that are present in it, i 1 n, . So we take off limita- tions that network must functioning in the term of high capacity. In our case expres- sion (2) will look as

 

°°

°

¯

°°

°

®

­

'

 '



 '

'

 ' '





'

 ' '



'

 ' '





'

 ' '



'

), ( ))

( ( )) ( ( 1

y probabilit with

), ( ))

( ( )) ( ( y probabilit with

), ( ))

( ( )) ( ( y probabilit with

), ( ))

( ( )) ( ( y probabilit with

), ( y

probabilit with

) (

0

0 0

0 0

t o t l k u l k p

t r

t o t p l k u l k t

r R

t o t p l k u l k t

r r

t o t p l k u l k t

r R

t o t p t

r r

t d

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

il

P O

P P

P O

i j n

j 1, , z (13)

where k_i(l) is number of messages in the i-th QS at the l-th time interval,

¯®

­ d

! 0 , 0

, 0 , ) 1

( x

x x

u - Heavyside function.

Let us find expression for expected income of system S at the moment t. When_i realization of process k(t) is fixed, according to (13) and denotation (5), we can write:

^ `

««

¬

ª  

'

¦

ⁿ

j

j j

j ji ji i

i

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

d M

1 0

0 ( ( )) ( ( ))

) ( / )

( O P

) ( ))

( ( )) ( (

0

t o t c t p b l k u l

k _i

n

j ij ij i

i

i '  '  '

»»

¼

^P

¦

º ^{, i 1,n (14)}

Then taking into account m't t and (3) we obtain

^

^{V t k t}

` ¦

^M

^

^{d 't k t}

` 

^{a p c}



^t

M _{i i i}

m

l

il

i 0 0

1

) ( / ) ( )

( / )

( O

 '



¦

^{a p}

¦

^{m k l u k l t}

l

j j

j n

j

ji ji

1 1

)) ( ( )) ( P (

) ( )) ( ( )) ( (

1 0

t o t l k u l k p

b

m

l

i i i n

j ij

ij '  '



¦ ¦

^{P , i 1,n (15)}

When mof, 'to0

¦

^{m ' ' oto}

³

^{t j j j}

l

j j

j k l u k l t k s u k s ds

0 0 1

)) ( ( )) ( ( ))

( ( )) (

( P

P ^{, j 1,n(16)}

so

^

^{V t k t}

`

^{v }



^{c  a p}



^t

¦

^{a p}

³

^{k s u k s ds}

M

t

j j

j n

j

ji ji i

i i i i

1 0 0 0

0 ( ( )) ( ( ))

) ( / )

( O P

³ ¦

 ⁿ

j ij ij t

i i

i k s u k s ds b p

0 0

)) ( ( )) (

P ( ^{, i 1,n (17)}

Making average by k(t) and taking into account condition of normalization



^{( )}



¹

¦

k

k t k

P for expected income of system S we will have_i

^ `

^

¦   ^ `

k

i i

i

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

v ( ) ( ) ₀ ( ) ( )/ ( )

   

««

¬

ª 





 ^{c a p t}

¦

^{P k t k}

¦

^{a p}

³

^{k s u k s ds}

v

t

j j

j n

j

ji ji k

i i i i

1 0 0

0

0 O ( ) P ( ( )) ( ( ))

»»

¼



³

^{t i ki s u ki s ds}

¦

_jⁿ₀^bijpijº

0

)) ( ( )) (

P ( ^{, i 1,n (18)}

Let system S consists of _i m identical service lines, the service rate of messages _i in each line equals P_i, ^{i 1 n, .} In this case

), ), ( , min(

) ( ,

, ) ( ), )) (

( ( )) (

( _{i i i}

i i

i i

i i

i i i

i

i k s m

m s k m

m s k s s k

k u s

k P

P P P

¯®

­

!

d i 1,n (19)

Then from (18) it follows

^ `

^



^



^

¦  

^u

k i i i i i

i t M V t v c a p t Pk t k

v ( ) ( ) ₀ O _{0 0} ( )

»»

¼ º

««

¬

ª 

u

¦ ³ ³ ¦

ⁿ

j ij ij t

i i i

t

j j n

j

ji ji

ja p k s m ds k s m ds b p

0 0 1 0

) ), ( min(

) ), (

min( P

P ^{, i 1,n (20)}

It is natural to suppose that average of expression min(k_j(s),m_j) gives )

), (

min(N_j s m_j , i.e.

) ), ( min(

) ), (

min(k_j s m_j N_j s m_j

M , i 1,n (21)

where ^{Nj(s) M}

^ `

^kj(s) is average number of messages (waiting and servicing) in system S at time interval_i

>

t0^,t0s

@

^, i 1 n, . Therefore we receive from (20) the following relation



^



^{ }

 ^{c a p t}

¦

^{a p}

³

^{N s m ds}

v t v

t

j j n

j

ji ji j i

i i i i

1 0 0 0

0 min( ( ), )

)

( O P

³ ¦

 ⁿ

j ij ij t

i i

i N s m ds b p

0 0

) ), (

P min( ^{, i 1,n (22)}

As

³ ¦

¦

¦ ¦ ³

ⁿ

j ij ij t

i i n

i i n

i

t

j j n

j

ji ji

ja p N s m ds N s m ds b p

0 1 1

1 1 0

) ), ( min(

) ), (

min( P

P

so expected income of whole network equals

^ `

^{W t}

¦

_iⁿ _«^«_¬^{ªvi }



^{ci  a ip i}



^{t ibi pi}

³

^t



^{Ni s mi}



^ds_»^»_¼^º

M

1 0

0 0 0

0

0 min ( ),

)

( O P

For finding of N_i(t), i 1 n, , it can be applied recurrence by the time moments me- thod of analysis of average values for open QN [10].

4. Variations of incomes in Jackson HM-network

Taking into account (3), expression (22) can be rewritten as



^



^{ }

 ^{c a p t b p}

³

^{N s m ds}

v t v

t

i i i

i i i i i i i

0 0 0 0

0

0 min( ( ), )

)

( O P

»»

¼ º

««

¬

ª 



¦

^{n j ji ji}

³

^{t j j i ij ij}

³

^{t i i}

j

ds m s N p

a ds m s N p

a

0 1 0

) ), ( min(

) ), (

min( P

P ^{, i 1,n (23)}

From (13), (19), (9) it follows (similar as we found (11)):