M|G|∞ queue system transient behaviour and busy period

M

_{½G½¥ QUEUE SYSTEM TRANSIENT BEHAVIOUR}

AND BUSY PERIOD

FERREIRA Manuel Alberto M., (Portugal)

Abstract. This paper main subject is the M_{½G½¥ queue system transient probabilities} study as time functions. We achieve it completely when the time origin is an unoccupied system instant. But we do not get such a goal when the time origin is a busy period beginning instant. We shall see that, in this last situation, the service time length distribution hazard rate function plays a very important role. And so the results got may be useful in the survival analysis field. As the M_{½G½¥ queue system can be applied in} the modelation of many social problems: sickness, unemployment, emigration, ...(see, for instance, Ferreira (1995 and 1996)), in these situations it is very important to study the busy period length distribution of that system. We show, in this work, that the solution of the problem may be in the resolution of a Ricatti equation generalizing the work of Ferreira (1998) as a consequence of the transient behaviour study.

Key words: M_{½G½¥, transient behaviour, hazard rate function, busy period, Ricatti} equation

1 Introduction

We call M½G½¥ a queue system at which the customers arrive according to a Poisson process at rate l, receive a service whose time length is a positive random variable with distribution function G and mean a . So a . Upon its arrival each customer finds immediately a server available. Each customer service is independent from the other customers services and from the arrival process. The traffic intensity is

()

. =

ò

¥

[

-G

( )

v dv

0 1

]

la r = .

Being the occupied servers number, that is the same that the being served customers number, at time t in a M

( )

½G½¥ system, according to Carrillo (1991) we have, putting

t N

( )

[

( )

n

( )

0 0

]

n=0,1,2,...,

0 t t = N =

( )

(

[

( )

]

)

[

( )

]

, 0,1,2,... ! 1 _{0 1} 0 0 = - ò = ò - -n e n dv v G t p t Gv dv n t n l l (1.1).

So, the transient distribution, being the time origin an empty system instant, is Poisson with mean lò_{0 1}t

[

-G

( )

v

]

dv.

The stationary distribution is the limit one:

( )

, 0,1,2,... ! 0 lim = - = ¥ ® n e n n t n p t r r

being Poisson with mean r.

Be p₁¢n

( )

t =P

[

N

( )

t =nN

( )

0 =1¢

]

,n=0,1,2,..., meaning that the time origin is the one of a customer arrival at the system, making the number of served customers jump from 0to1. That is: a busy period begins.

( )

0 =1¢

N

At t³0, possibly:

The customer that arrived at the time origin abandoned the system, with probability , or goes on being served, with probability 1 ;

( )

t

G -G

( )

t

The other servers, that were unoccupied at the time origin, are still unoccupied or occupied with 1,2,... customers, with probabilities given by p_0n

( )

t ,n=1,2,...

Both subsystems, the one of the initial customer and the other of the initially unoccupied servers, are independent and so we have

( )

t p

( ) ( )

t Gt

p_1¢0 = ₀₀ (1.2)

( )

₀

( ) ( )

_{0 1}

( )

(

1

( )

)

, 1,2,... 1¢n t = p n t G t + p n- t -G t n=

p

For the M½M½¥ system (exponential service times) (1.2) is valid even if , that is: since the time origin in an instant at which there is one only customer in the system, simply, owing to the exponential distribution lack of memory. So

( )

0 =1 N

( )

÷ ÷ ø ö ç ç è æ -÷ ÷ ÷ ø ö ç ç ç è æ -= -a r a t e e t e t M p 1 1 10

( ) ( )

, 1,2,... 2 1 1 1 1 1 ! 1 1 1 = ÷÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ø ö çç ç ç ç ç ç ç ç ç è æ -+ ÷ ÷ ÷ ø ö ç ç ç è æ -÷ ø ö ç è æ -÷ ÷ ÷ ø ö ç ç ç è æ -= -n t e n t e e e n t e n n t M n p t a r a r a r a

( )

, 0,1,2,... ! lim _1¢ = - = ¥ ® p t n e n n n t r r

Calling m

(

and the mean values associated to the distributions given by (1.2) and (1.1), respectively, we have that

)

)

]

)

t , 1¢ m

(

0,t

( )

( )

( ) ( )

¥

( )

(

-

( )

)

= = -+ ¥ = = ¢ ¥ = = ¢ å å å t G t n np n t p n nG t t n n np t 1 1 0 1 00 1 1 1 , 1 m

( ) ( )

(

( )

) (

¥

) (

= = + -+ å p _j t j j t G t t G ₀ 0 1 1 , 0 m

)

( )

0,t + 1

(

-G

( )

t

)

. So

( )

1¢,t =1-G

( )

t +l_{ò0 1}t

[

-G

( )

v dv m (1.3)

We intend to present some results about and behaviours as

time functions. We will show too that the study induces a Ricatti equation important to the determination of a M ½G½¥ systems collection with practically exponential busy period.

( )

, 0,1,2,..., 0 t n= p _n

( )

t p_1¢0

( )

t p_1¢0 m

(

1¢,t

2 _p0n

( )

t , n=0,1,2,... Behaviour as Time Function

This section main result is:

Proposition 2.1

If G

( )

t <1, t>0, continuous and differentiable a) p₀₀

( )

t , t>0 is a decreasing function,

b) p_0n

( )

t , n³r, t >0 is an increasing function, c) p_0n

( )

t , 0< n<r, r _>1

i) increases in

]

0,t_n

[

being given by t_n

( )

[

]

dv _ln t_{n -G v =}

ò_{0 1} (2.1),

ii) decreases in

]

t_n,_¥

[

and iii) the p_0n

( )

t maximum is

( )

_e n n n n n t n p = -! 0 (2.2)

Dem: a) is evident since

( )

_{t =e}- òt

[

-G

( )

v

]

dv p 0 1 00 l For n³ 1

( )

( )

(

( )

)

( )

[

]

1 , 0 0 1 1 0 0 > ÷÷ ÷ ø ö çç ç è æ -= ò t t _{G v dv} n t G t n p t n p dt d l l . As lò_{0 1}t

[

-G

( )

v

]

dv<r,if n³r, d p0n

( )

t >0,t>0and we conclude b). If

< , p_0n

( )

t =0 dt d n r Û

[

( )

]

l n dv t _{-G v =} ò_{0 1} and we have c). Notes:

- Although t , given by (2.1), depends on the arrival rate and on the service time length distribution, that does not happen with given by (2.2),

n

(

tn n

p0

)

- For a certain arrival rate and service time length distribution we have, evidently,

n t n t ₊₁³ and, as ( )

( )

(

(

)

)

(

)

1 1 1 ! ! 1 1 1 ₁ 1 0 1 1 0 + ÷ ø ö ç è æ + + = + + = + - - -+ + n e n n n e n n e n n t p p n n n n n n n n t n = , 1 1 1 ÷ 1£ 1= ø ö ç è æ + = e- ee -n p0n+1

(

tn+1

)

£ pon

(

tn

)

Under Proposition (2.1) conditions, but with 1-G

( )

t =0,t³t_l, (1.1) becomes

( )

(

[

( )

]

)

[

( )

]

l t _{Gv dv} n t n _n e t t dv v G t p = ò - - ò - , £ ! 1 _{0 1} 0 0 l l and 0

( )

_! , l, n n t _n e t t p = r - r > 2 , 1 , 0 ,n= ,...

and, so, the Proposition (2.1) conclusions are still valid, but the values , 0,1,2,...

! = - _n e n n r r

occur after t =t_l. Evidently, t_n <t_l,0<n<r,r >1.

3 p_1¢0

( )

t Behaviour as Time Function

For the it is not possible to perform such a complete study as for the . But the results for are very interesting as we will see. Now the important result is

( )

, 0,1,2,... 1¢_n t n= p ,... 2 , 1 , 0 =

( )

, 0 t n p _n p_1¢0

( )

t Preposition 3.1

If G

( )

t < t1, >0,continuous, differentiable and

( )

t ³ G

( )

t ,t >0

h l (3.1), being h

( )

t the hazard rate function associated to G

()

. , _{p 01¢}

( )

t is non-decreasing.

( )

( )

(

( )

)

_ççèæ

( )

_{( )}

-

( )

_÷÷øö -= ¢ Gt t G t g t G t p t p dt d l 1 1 00 0 1 where g

( )

t = _dtd G

( )

t and h

( )

t =_1-g_G

( )

t

_{( )}

_t . Notes: Note that

( )

t ³l h (3.2) is a sufficient condition for (3.1) and so if the rate at which the services end is greater or equal than the customers arrival rate _{p 01¢}

( )

t is non-decreasing,

- For the M½M½¥ system (3.2) is equivalent to

1 £ r

- Evidently these results may be useful in the survival analysis fields. Putting

( )

t G

( )

t IR h -l =b,bÎ

we have a Ricatti equation whose solution is (note that G

( )

t =1,t³0 is a solution)

( )

₍

(

₎

)

1 , 0 , 1 1 1 -£ £ -³ + ÷ø ö çè æ + -+ ÷ø ö çè æ _- -= r l b l l b l r l b l r e t t e e e t G (3.3),

see Ferreira (1998). For a M½G½¥ system with this service time length distribution

( )

(

)

1 , 0 , 1 0 1 -£ £ -³ + -÷ø ö çè æ _- -= ¢ _l l b l b _rl b r r e t t e e e t p Concretely -b -= l we get

( )

1, 0 0 1¢ t = t³ p .

In fact, in this situation, . So is degenerated at the origin. That is: every customer has null service time length. So the system is never occupied,

( )

0 , 1 ³ = t t G G

()

. -b =0

( )

, 0 0 1¢ t =e- t³ p r and so p_1¢0

( )

t ,t³0is constant, -1 -= _rl b e

( )

1

(

)

, 0 0 1¢ t =e- çèæ -e- + tö÷ø t³ p r l b

( )

(

)

(

)

_, 1 ! 1 log 0 çèæ - +çèæ - - ÷øö - + ÷øö ÷ ø ö ç è æ ÷ø ö çè æ - + ÷ø ö çè æ _- -+ -= e e e t n n t e e e t n p r r l b b l r r 1 , 0 , -£ £ -³ r l b l e t .

Calling T the random variable associated to given by (3.3) we have, see Ferreira (1998

a)

()

. G

(

)

(

)

[ ]

(

)

, ! 1 ! 1 1 1 -+ -£ £ + -n n n n e T E n e e b l l b l l r r r , 1,2,. 1 , = -£ < - n er l b l .. (3.4) Notes:

- The expression (3.4) giving bounds for E_êëéTn_úûù,n=1,2,... proves its existence,

- For n (3.4) is unuseful because we know that . Curiously, the upper bound is 1 = E

[ ]

T =a l r -e 1

, the M½G½¥ system busy period mean value, - For b =-l,E_êëéTnù_úû=0,n=1,2,... evidently.

See however that (3.3) may take the form

( )

₍

(

₎

)

1 , 0 , 1 1 1 1 -£ £ -³ + -÷ø ö çè æ -+ -÷ø ö çè æ -+ = l b _rl b l r b l r l b e t t e e t e e t G

and, since r <log2,

( )

1

( )

1 ( )

( )

1 ( ) , 0 t k k k t _{e e} e e t G r l b r l b l b ¥ - + = + - _{÷ å} -ø ö ç è æ ₊ -= 1 , 0 , -£ £ -³ l b _rl e t (3.5)

After (3.5) we can easily compute the T Laplace transform for r<log2. And then get

(

)

(

)

, 1, log2, 1,2,... 1 1 ! 1 < = -£ < -¥ = + ÷ø ö çè æ -÷ ø ö ç è æ + -= úû ù êë é _{å n} e k k n k e n n T E l b _rl r b l r l b

Notes: -

[ ]

( )

(

)

(

+

)

( )

- = + -= + -÷ ø ö ç è æ + -= å ¥å = ¥ =1 1 1 . 1 1 k k k k e k e T E r r b l l b l b l l b a l r r l r l ÷ø = = = ö çè æ -¥ = + -å e k k e k k 1 1_log 1 1 ) 1 ( 1 =

- For n³2 we must truncate the infinite sum. Taking only M terms, to get an error lesser or equal than we must have simultaneously e

. 1 -1 + > b l M .

(

)

1 ! log 1 ₊ -> ÷ø ö çè æ _{- l} l_b r r _n e M e

4 A M½G½¥ Systems Collection with Exponential Busy Period Putting now h

( )

t -lG

( ) ( )

l =b t (b

()

. is any time function) we get

( )

_G

_{( )}

_t

₍

_{( )}

_t

_{) ( ) ( )}

_G

_t

_t

dt

t

dG

b

l

b

l

-

-

+

-=

2

that is a Ricatti equation about G .

()

.

Solving it, after noting that G

( )

t = t1, ³0 is a solution again, we get

( )

(

)

( )

( )

(

₁

)

( )

,

1

1

1

0 0 0 0 0

ò

¥ - -ò -

ò

- -ò ò

-=

dw

e

e

dw

e

e

e

t

G

t w u du du u w du u t w w t b l r b l b l r

l

( )

1

,

0

,

0

-£

£

-³

l

ò

b

_r

l

e

t

du

u

t

t (4.1). Putting (4.1) in

( )

( )

[

]

_÷

÷

÷

ø

ö

ç

ç

ç

è

æ

-+

=

ò

¥ - - ò

-dt

e

s

s

B

_t dv v G st 0 1 1 0

1

1

l

l

( )

(

)

(

( )

)

( )

( )

(

)

( )

( )

1

,

0

1

1

0

1

1

0 0 0

-£

£

-ú

û

ù

ê

ë

é

-ú

û

ù

ê

ë

é

-+

-=

ò

ò -ò -r b l b l

l

b

l

l

l

e

t

du

u

e

L

G

e

L

G

s

s

B

t du u t du u t t t (4.2)

where means Laplace transform and

L

( )

( )

( )

_dw

e

e

dw

e

G

_w w du u w du u w

ò

ò

¥ - -_ò -¥ - -_ò

₊

-=

0 0 0 0

₁

0

b l r b l

l

l

.

After (4.2) we can compute

s

1

B

( )

s whose inversion gives

( )

(

( )

)(

( )

( )

)

* ÷÷ø ö ççè æ _- -+ -= G e t ò u du òte w ò u dudw t B 1 1 0 l 0tb _{l 0} l 0wb

(

( )

)

( )

( )

1 0 , 0 1 0 0 -£ £ -* ÷÷ø ö ççè æ - -¥ = * å ò ò r l b l b l l e t t _{u du} n du u t e n G n n t _(4.3)

for the M½G½¥ busy period d.f., where

*

is the convolution operator. If b =

( )

t b (constant), we get (3.3) and

( )

(

)

1 , 0 , 1 1 -£ £ -³ + -÷ø ö çè æ _- -+ -= r - l b l b _rl l b l b r e t t e e e t B

So, if the service time d.f. is given by (3.3) the M½G½¥ busy period d.f. is the a mixture of a degenerate distribution at the origin and an exponential distribution.

Finally note that, for

1

-=

_r

l

b

e

,

( )

t e

e

t

B

₌

1

_-

- r -1 l

b _,

_t

_{(purely exponential). And}

, given by (4.3) satisfies

0

³

( )

t

B

( )

_t

_e

e t

B

_³

1

_-

- r-1 l ,

t

³

0

,

( )

1

0

-£

£

ò

b

_r

l

l

e

t

du

u

t

-5 m

(

1¢t

)

Behaviour as Time Function

Proposition 5.1

If G

( )

t < t1, >0, continuous, differentiable and

( )

t £l

h (5.1) m

(

1¢,t

)

is non-decreasing.

Dem: After (1.3) we have

( )

t

(

G

( )

t

)(

h t

)

dt d -= ¢ l m 1, 1

( )

)

. Notes:

- If the rate at which the services end is lesser or equal than the customers arrival rate is non-decreasing,

(

1¢,t

m

- For the M½M½¥ (5.1) is equivalent to

1 ³ r

- These results may be useful in the survival analysis field. Putting

( )

t =l h obviously we get

( )

t =1-e- t,t³0 G l

and, for this M½M½¥ system,

( )

1,t =1

m

6 Concluding Remarks

In queues practical applications often it is used the populational process stationary distribution. This happens generally because the transient distribution is very complex and unuseful. And so the stationary distribution is used as a good transient one approximation. But in various situations this is not true. So it is necessary to know as well as possible the transient behaviour.

The M½G½¥ systems transient behaviour, with an unoccupied system instant time origin, is very well known and not too complex. We deduced the time origin at the beginning of a busy period transient distribution.

We presented here a transient behaviour study, with some interesting results, for the M½G½¥ systems with a lot of possible applications, namely in survival analysis.

It was done more exhaustively for the than for the , but in

the former situation everything is easier than in the other. But the study leads to very interesting results even that they are looked only from the mathematical point of view. And, no less important, it allows through the resolution of a Ricatti equation the determination of a M½G½¥ infinite systems collection with a very simple busy period distribution: a mixture of a degenerate distribution at the origin and an exponential distribution.

( )

t n

p0 p_1¢n

( )

t ,n=0,1,2,...

( )

t p 01¢

References

1. Carrilho, M. J. (1991): “Extensions of Palm’s Theorem: A review”. Management Science. Vol. 37. N.º 6. 739-744.

2. Ferreira, M. A. M. (1995), “Aplicação do Sistema M½G½¥ ao Estudo do Desemprego numa certa actividade”, Revista Portuguesa de Gestão, ISCTE, IV/95

3. Ferreira, M. A. M. (1996), “Comportamento Transeunte do Sistema M½G½¥ - Aplicação em Problemas de Doença e de Desemprego”, Revista de Estatística, INE, Vol. 3, 3.º Quadrimestre.

4. Ferreira, M. A. M. (1998): “Aplicação da Equação de Ricatti ao Estudo do Período de Ocupação de Sistemas M½G½¥”. Revista de Estatística. Vol. I. INE. 23-28.

5. Ferreira, M. A. M. (1998 a): “Momentos de Variáveis Aleatórias com Funções de

Distribuição dadas pela Colecção

( )

₍

(

₎

)

1 , 0 , 1 1 1 -£ £ -³ + ÷ø ö çè æ + -+ ÷ø ö çè æ _- -= r l b l l b l r l b l r e t t e e e t G ”. Revista Portuguesa de

Gestão. II/98. INDEG/ISCTE. 67-69.

6. Stadje, W. (1985), “The Busy Period of the Queueing System M½G½¥”. J. A. P. 22.

Current Address

Manuel Alberto Martins Ferreira, Associate Professor, Iscte, Departamento de Métodos Quantitativos, Av. Das Forças Armadas, 1649-026 Lisboa, Telefone: + 351 21 790 30 46, Fax: + 351 21 790 39 64, e-mail: vice.presidente@iscte.pt