batch arrival queue with n- policy and single vacation - CiteSeerX

BATCH ARRIV AL QUEUE WITH

N- POLICY AND SINGLE VACATION Soon Seok Lee ETRI Tae Jeon, KOREA 305-606 Ho Woo Lee, Seung Hyun Y oon Department of Industrial Engineering Sungkyunkwan University Su Won, KOREA 440-746 K. C. Chae of Management Science KAIST Tae Jeon, KOREA 305-701

Department

ABSTRACT We consider an as the system becomes When he returns value

MXI

ell

N-policy

he begins to serve the customers.

N

system size distribution

decomposes

into two random variables

ell

queue.

of random length

We derive the system size distribution

The interpretation

one of which is the system size of

of the other random

We also derive the queue waiting time distribution

and show that the

variab1e will also be provided.

of an arbitrary customer.

Finally we develop a

procedure to find the optimal stationary operating policy under a linear cost structure.

Key Words: MXI operating policy

ell

queue,

11:

If not, the server waits in the system until

or exceeds

MXI

As soon

if the system size is greater than or equal to a pretermined

the system size reaches

ordinary

and single vacation.

empty, the server leaves the system for a vacation

from the vacation,

N(threshold),

queueing system with

N-policy,

single

vacation,

system

Correspondence: Ho Woo Lee, Dept. of Industrial Engineering, Sung Kyun 440-746, (TEL) 82-331-290-5516, (FAX) 82-331-291-4502 (E-mail)[email protected]

size, waiting

Kwan

Univ.,

time, optimal

Su Won,

Korea

SCOPE AND PURPOSE

This paper concerns the modeling of a production

system in which the

production does not start until some specified number of raw materials, say are accumulated during an idle period.

N,

We assume that the machine undergoes

some extra operations (for example, machine repair, preventive maintenance, etc) when there are no raw materials to process. assume that the raw materials modeled by

To be more realistic, we also

arrive in batches.

This production

MX/ G/l/ N- policy queue with single vacation.

system is

We derive the

probability distributions of the number of materials and the waiting times of each material.

We

also suggest

a procedure

to obtain

the

optimal

minimizes the long-run average cost under a linear cost structure.

N* which

1.

INTRODUCTION Consider

a manufacturing

system

in which the production

does not start until some

specified number of raw materials, say N, are accumulated during an idle period.

We assume that

the operator of the machine performs some extra operations (for example, machine repair, preventive maintenance, etc) when there are no raw materials to process. assume that the raw materials arrive in batches.

To be more realistic, we also

This production system can be modeled by an

MX I GIl queue with N-policy and single vacation. We consider a queueing

system m which customers

X.

Poisson process with random arrival size leaves for a vacation of random length

arrive according to the compound

As soon as the system becomes empty the server

V(vacation period).

When he returns from the vacation and

the system size is greater than or equal to a predetermined value

N(threshold), the server begins to

serve the customers until there is no customers to serve (busy period).

If he finds fewer customers

N, he waits in the system until the system size reaches or exceeds

than

Thus, in our system,

N (dormant period).

a vacation period, a dormant period (the length of which is zero if the

returning server finds N or more customers) and a busy period constitute a cycle. this queueing system as MXI Gill N- policyl VAC(l) vacation'.

queue in which

We will denote

'VAC(1)'

implies 'single

The system is depicted in Figure 1. Vacation queues have attracted many attentions from numerous researchers since Levy and

Yechiali [11].

Fuhrmann

vacation queues.

and Cooper

[4] proved

the well-known

"decomposition

property" for

For application of vacation models to polling systems, see Takagi [13].

For

comprehensive survey on vacation queues, see Doshi [2] and Takagi [12]. Hofri [5] studied the two the

MI G/l queue with N-policy

problem.

N-policy

queues attended by a single server.

and vacations.

Kella [6] studied

He considered the optimal policy as a stopping

Their works both dealt with single-unit arrival systems. Batch arrival queues with threshold and with/without multiple vacations were first studied

by Lee and

Srinivasan[7].

They

derived

the mean

performance

measures

procedure to find the optimal operating policy under a linear cost structure. procedure to calculate the system size probabilities.

and deveoped

the

Lee [8] developed a

Lee et al. [10] studied the same queuemg

system and found that the system size decomposes into two random variables: one is the system size of the ordinary

MX I G/l

N-l

given by

period.

l~

queue and the probability generating function

(PGF) of the other is

. N-l 'J[

j Zl

I

l~

'J[

j

where

'J[

j

is the probability that the system state visits j during an idle

They also derived the condition under which the optimal threshold

N* is found under a

linear cost structure.

While Lee and Srinivasan [7] concentrated on the mean measures such as the

mean system size, and on the calculation of state probabilities with

N-policy, this paper concentrates on the development

[8] of the multiple vacation system

of the system size and waiting time

transform solutions. and its probabilistic interpretation of the single vacation system with threshold. In this study we derive the system size distribution and show that it decomposes into two random

variables

one

of which

is the system

size of the stationary

interpretation of the other random variable will also be provided. distribution of an arbitrary customer.

MXI G/l

queue.

We also derive the waiting time

We obtain other performance measures and the condition under

which the optimal operating policy is achieved under a linear cost structure. [Figure 1. 2. THE

here]

SIZE DISTRIBUTION

In this section we derive the system equations and the distribution.

N A

X gk ak

X(z) S V sex) vex) S' (e)

V'(e)

SoU) V°Ct) N(t)

yCt) RnCt)

PGF of the system size

Let us define following notations and probabilities:

threshold group arrival rate arrival size random variable Pr(X = k) probability that k customers arrive during a vacation the probability generating function( PGF) of X service time random variable vacation time random variable the probability density function of S the probability density function of V the Laplace - Stieltjes transform (LST) of S the LST of V remaining service time of the customer in service at time t remaining vacation time of the server in vacation at time t system size at time t ==

j

The

0 if server is on vacation, 1 if server is in dormancy, 2 if server is busy,

= Pr[ NCt) = n, yCt) = 1J.

n = 0, 1, ... , N - 1

P n(x, t) dt

=Pr[NCt)=n,

x~S°U)~x+dt,

YCt)=2],

n=1,2,

Qn(x, t) dt

=Pr[NCt)=n,

x~V°Ct)~x+dt,

YCt)=O],

n=O,1,2,

... ...

Then, we can easily set up the following steady-state system equations, (2-1)

0

- ARO + QO(O),

(2-2)

0

-ARn+

n

d

- -dx PI (x)

(2-3)

Qn(O) +A'f;lRn-k·gk'

= - API (x)

(n=1,2,

.. ,N-l),

+ Pz(O)s(x),

(2-4)

d --dxPm(X) (2-5)

m-l

= -APm(X) d

- -d-x P n(X)

+Pm+JO)S(x)

=

-'-APn(X)

+A'f;lPm-k(X)gk,

(m=2,3,"',N-l),

n-l

+ Pn+l(O)s(x)

+ A 'f;IPn-k(X)gk

N-I

+ AS(X) (2-6) (2-7)

- - d Q (x) dx 0 d

(2-8)

(2-9)

- AQO(X) -AQm(X)

- -dx Qm(x)

Taking the LSTs

'f;o Rk . g n- k + Qn(O) s(x),

(n~N)

+ PI (O)V(X), +

m

A'f;IQm-ix)gk

(m~l).

of both sides of the eqs. (2-3)-(2-7), we have

8~(8)

= AP~(8)

- PI (0)

- Pz(O)S* (8), m-l

1

= AP*",(8) - P m+ (O)S* (8) - A

8P*",(8) - Pm (0)

'f;1 P:,-k(8)gk>

(m= 2, 3, ''', N-l), n-l

(2-10)

8P:(8)

- Pn (0)

= AP:(8)

- Pn+ l(O)S* (8) - A 'f;l P:-A,(8)gk N-l

(2-11)

8Q~(8)

- Qo (0)

- AS*(8)

'f;o Rk'

gn-k-

= AQ~(8)

- Pl(O)

V' (8),

Qn(O)S* (8),

(m~l).

(2-12)

Let us define the following probability generating functions, 00

P*(z,8)

L p:(8)zn,

n=l

00

P(z,O)=

00

Q*(z,8)

L

n=O

L Pn(O)zn,

n=l 00

Q:(8)zn,

Q(z, 0)

=

L

n=O

Qn(O)zn.

(n~N),

After some manipulations with eqs. (2-11) and (2-12), it follows that

= Q(z,O) - PI(0)V'(9).

[9-J\+J\X(z)]Q*(z,9)

(2-13)

Letting 9 = J\ - J\X(z),

we have

Q(z, 0)

(2-14)

= PI(0) V'[J\ - J\X(z)].

Thus, eq. (2-13) becomes (2-15)

Q*(z,9)

PI(O){ V'[J\-J\X(z)] - V'(9)} 9 - J\ + J\X(z)

Similarly, from eqs. (2-8), (2-9), and (2-10), we have

(2-16)

[9-J\+J\X(z)]P*(z,9)

= P(z,0)-S*(9)[

P(~.O)

-PI(O)+Q(z,O)

From eqs. (2-1) and (2-2), we have

(2-17)

=-J\[X(z)-1]R(z) N-I

where R(z)

= ~

Rnzn.

n=O

Thus, eq. (2-16) becomes (2-18)

[9- J\+J\X(z)]P*(z,9) = P(z, 0) - S*(9) [ P(~.

0)

+ PI (O){ V'[J\

- J\X(z)] -l}

+ J\R(z)[X(z)

-1]].

e = 1\ -

Letting

1\X(z) in eq. (2-18),

(0) + 1\[X(z) -l]R(z)}

0) = z S*[1\ - 1\X(z)]{ [ V' [1\ - 1\X(z)] -l)PI

P(

(2-19)

we have

z,

z-

S*[1\-1\X(z)]

.

P*(z, e) becomes

Thus from eq. (2-18), (2

2

p*(z,e)=

z{S*[1\-1\X(z)]-s*(e)}·

0

{[ V'[1\-1\X(z)]-l)PI(O)+A[X(z)-l)R(z)}

{z-S*[1\-1\X(z)]}·

[e-1\+1\X(z)]

Finally the PGF of the system size distribution in steady-state becomes

= P*(z, 0) + Q* (z, 0) + R(z)

P(z)

(2-21)

_(_z_-~l~)S_*~r 1\_-~1\X_(~z~)1 _l_-_V'~r 1\_-_1\~X~(~z)~l . [. PI (0) z-s*[1\-1\X(z)] 1\-1\X(z)

+ R(z) ] .

n

gi

]I n = ~

• ]I n ~

]I n

i'

z= I

the

MX/

is the probability

that the system state visits

G/I! N- policy queue without vacations (Lee et al. [9]).

n during an idle period in

Then the system size given by

eq. (2-21) becomes

P(z)

(2-22)

where

p

= 1\E(X)E(S),

= (I-p)(z-I)s*r1\-1\x(z)l z-

\If

~z)

= P(z, ordinary MX/ G/l) . \IfN(Z)

,

and

n=O

=

nZ

n

+

N-I

+~

\If n

(2-14),

we

1\E( 11)

From

eq.

1- V'r1\-1\X(z)l

1\E(V) N-I

1\E( V)

n=O

PROOF.

(z) N

N-I "" \If

(2-23)

. \If

s*[1\-1\X(z)]

+~

\If n

E( V)[1\ - 1\X(z)]

.

n=O

have

Q n (0) = P I (0) a n'

First

we

show

that

n

1\R n = PI (0) ,~ eq. (2-2).

ai

. 1T

n_ ,= PI (0) II' n by mathematical

aH I

+

For

n = 1, it is obvious from

n = 2,3, ... ,k, let us see the case of n = k+ 1:

Assuming that it holds for

= PI (O){

induction.

k+Ik+I ~I

i

a j 1TH I -

J~

i- j

g ,}

k

=PI(O){aHI'

,~ai'

1TO+

1THI-,}

Multiplying both sides of the above equation by

1

R(z)

=

N-I

J: PI (0) n~o

II' nzn.

z n and summing over n from 0 to N- 1 yields

From eq. (2-21) and

PO)

=

1,

1\0- p) N-I

1\E( 11)

+ 2: n=O

II' n

Then the theorem follows. REMARK random

2.1.

From THEOREM

variables,

interpretation

of

THEOREM

2.2.

l1li

one

of which

II' j'

j

f 1,

Let

Conditioning

is the system

the other term (II' N(Z»

Ij =

size

of the

system

ordinary

will be given in THEOREM

= 0, 1, 2, ... , N - L is the probability

customers in the system) visits

PROOF.

2.1 we see that the stationary

size is the sum of two

MX / GIl

queue.

2.5 and REMARK

The

2.2.

that the system state (number of

j during a dormant period.

if state j is visited during a dormant period,

l0,

o/w.

on the arrival size during the vacation, we have j - I

Pr[ Ij where

Pr(Ij=l)=1Tj'

= 1] = a j

Letting

+ ~O

PrCIj=l)=lI'j

a k . Pr(Ij and

- k = 1) .

1I'0=ao.

we have

II'j=

toak'

1Tj-k"

Then

from

THEOREM

2.1,

the

result

follows.

N-l

THEOREM

2.3.

\IIj is the mean number of arrival groups during a dormant period.

J~

N-l

PROOF.

From THEOREM

2.2, we observe that

~ J=O

Ij

is the number

of states visited during a

dormant period which is is equal to the number of arriving groups during the same period. N-l

E(J~)=

N-l

N-l

N-l

J~E(Ij)=

J~Pr(Ij=l)=

J~

Then

\IIj•

II1II

THEOREM

2.4.

Let

dormant period).

PROOF.

2.3,

THEOREM

be the idle period

random variable

(idle period

vacation period

staying time in a state during an idle period is

expected

2.5.

length

of

a

dormant

E I be the event that the server is idle.

Let

either on vacation or in dormancy.

-}

period

and from THEOREM

1

becomes

I

J\E( V) +

Then,

= Pr(given

N-l

Ef, server zs zn dormancy with j customers),

\II j

J~

j=0,1,2, J\E(V)

= pr(given

N-l

J\E( V)

+

J~

\II j

J:

Note that the server is idle

\II.

and

+

Then

The expected

the

YON

... ,N-l

Ef, server zs idle due to vacation) .

N-l J~

\II j.

if

he is

From THEOREM

PROOF.

is the proportion

2.4,

of time that the system

state is j during a dom1ant period.

From the renewal reward theorem, the results follow.

REMARK

2.5, we can interprete

2.2.

From THEOREM

\If

Jlz)

given by eq. (2-23). A group \If

arriving during a dom1ant period finds

J customers with probability

J\E( V) PGF acconnts for the first tem1 of eq. (2-23). server

is

the

1- V'rJ\~J\X(z)l E( V)[1- J\X(z)]

,

define

vacation

number which

of occurs

customers with

l1li

1

+

Then its

N-l l~

\If j

The mcrease of the number of customers due to that

arrive

during

a

residual

J\E(V)

probability

J\E( V)

+

N-l l~

vacation

period,

Notationally,

let us

\If j

A j as the event that a group amvmg during a dormant period finds j customers and A v

as the event that a group aniving during an idle period finds that the server is idle due to vacation. The PGF of the system size distribution can then be rewritten as

.

P(z)

P(z,

This tells us that

X

ordznary M

\If

N(Z)

I G/l)·

IS the

N-l. { l~

Pr(A)

Zl

+

Pr(A

v)·

1-- V'TJ\-J\X(z)l

E( V)[J\ - J\X(z)]

} .

PGF of the conditional system SIze distribution during the server

idle period.

THEOREM

2.6.

Let

pt(z)

be the

PGF of the departure point system size distribution.

(1- p) (z- l)s*rJ\-J\X(z)l z- S*[J\-J\X(z)]

1- X(z) E(X)( 1- z)

(2-24)

pt(z)

PROOF.

We follow the arguement of Chaudhry and Templeton [3].

.

\If

Then,

N(Z)

A departing customer will see

j customers in the system just after a departure if and only if there were

j+ 1 customers in the

system just before the departure.

Thus we have

pt Let pt(z) be the

PGF of {Pt, j =

P-I-(z)=D· N

p(z.a)

=D'

we

a, 1, 2, ...

j'~

}.

a.

Then

il.O-p)s*ril.-il.X(z)l

z- S*[iI.-il.X(z)]

Z

ptO) = L

From

= D . P j -I- 1 (a) ,

1

have

D= il.E(X)

.

[X(z)--l]

and

the

. qiJz).

theorem

follows.

II1II

3. THE WAITING TIME In this section, we derive the

LST of the waiting time of an arbitrary customer.

Let us

define the following notations,

Tv,;(8):

the

LST of the queueing waiting time of an arbitrary customer,

WB( 8):

the

LST of the waiting time of an arbitrary customer who arrives when the server is

busy,

Wy(8):

the

LST of the waiting time of an arbitrary customer who a:rrives while the server is

on vacation, the

LST of the waiting time of an arbitrary customer who arnves when the server is

dormant,

WA(8)

:

the

LST of the sum of the serVIce times of those who precede the test customer m

the same group.

The waiting time of an arbitrary customer who arrives while the server is busy and sees customers in the system is composed of, 1) the residual service time of the customer in service, 2) the sum of the service times of those

n - 1 customers in the queue, and

3) the sum of the service times of those customers who precede him in the same group.

Thus after some laborious algebraic manipulation, we have

n ( :z 1)

f P~(8)

(3-1)

[5* (8)]

n~

WA(8)

1 •

n=l

F*rS*(8).81 5*(8)

l-X(S*(8)) E(X)[l5*(8)]

(1- p)r A ~ 8 - A

f

l

REMARK

~l

1- X( 5*(8)) E(X)[l5*(8)]

AX( 5*(8))1

+ AX(S*(8))

q1 j[

5* (8)]

AE( 11)+

~l

AE( V)

j

q1 j

+

AE( 11)+

~l

q1 j

•

11 ).

E( 11)[A - AX( 5* (8)]

1- X( 5* (8lL.

(1- p)rA-AX(S*(8))1 8 - A + AX( 5* ( 8))

3.1. In eq. (3-1),

of an arbitrary customer

11*r A - AX( 5* (8)

1-

E(X)[l-

5*(8)]

... 1S the wa1tmg time

who arrives while the server is busy in the ordinary

without N - policy and vacations.

MXI Gil

queue

But in our system, an arbitrary customer who arrives while the

server is busy would wait longer due to those customers who have arrived before our test customer but are in the waiting line because of server idleness. the vacation (this occurs with probability

Those customers, if they have arrived during

VZ _ 1

AE( AE( 11) +

J~

their countribution

),

to the additional

q1 j

waiting time of the test customer is the sum of the service times of those who have arrived during a residual vacation

time

(This property

was proved

AE(V)

accounts for the terms

AE( 11)

+

N-l J~

q1 j

in many

ordinary

1 - 11*r A - AX( 5* (8)) E( 1I)[A - AX( 5* (8))]

1

vacation

queues).

in the bracket.

This

It is easily

seen that the contribution of those customers who have arrived while the server is in dormancy is explained by the first term in the bracket. The waiting time of the customer who arrives while the server is dormant can be obtained from

the

waiting

time

of

the

customer

who

arrives

MXI Gill N- policy queue without vacations (Lee et al. [9]).

while

the

server

Thus we have,

is

idle

in

the

+ 'f r~,=]

:t [S'

e 8)] i -]

•

l . _~_g_r r

EeX)

}

N-]

(l-p)

~ n=O

Wn[S*(8)]n

,\Ee 11) + N-

n -]

{ r~]

where

r; (8)

is the

N-]

~

n=O

Wn

- e S* e 8» r] EeX)[ 1- S*(8)] g r[l

[TN-n-r(8)-1J+

1- XeS*(8» EeX)[l-S*(8)]

},

LST of the idle period of the MX I GIl queue with threshold j and without

vacations. The most complicated part of the waiting time comes from the case where the test customer arrives while the server is on vacation. server is on a vacation of length

Suppose that at the arrival epoch of the test customer, the

V = x and the residual vacation time is y (see Figure 2).

test customer belongs to a group of size

If the

r and sees k customers in the queue, his waiting time

consists of 1) the total service time of those

k customers in the queue,

2) the residual vacation time y, 3) the total service time of those who precede the test customer in the same group, and 4) the dormant period (the dormant period is not zero if vacation time for

m customers arrive during the residual

k+ r+ mI

Pr(Ax,)dydx= Then from eq. (3-5),

j

l

xvCx). E( 11)

•

dydx.

x

we have

Wv(9)

(3-6 a)

(1- p)"AE(X) "AE( 11) +

~coI Wve

=

9lx, y) Pr(Ax,y)

_0-p)"AEC1I) -

•.

+

"AE(1I)

+

lo le

+ :%1l[J

-eyN-l * D (9 k, ~

o - p) "AEC11)

{

----N--~l~·

y

x

x

1-XCS*(9))1 E(X)[l-

j

:~.:r r;, ~

xvCx).l dd E( 11)

Y)

=0

-{eyH.(x-y)[l-X{S'{e))]}.

co

+

l[J j

J~

II e

0-p)"AEC1I) "AE( 11)

"AE( 11)

N-l

dydx

5*(9)] ~dYdx E(1I)

(9) -1] . CP~-n(9)

}

l[J j

J~

+ 0- p)"AEC 11) N-l "AE( 11) + l[J j J~

V* r "A- "AXC5* C9)) 1- 11*C9)

1- XCS*(9))1 E(X)[l5*(9)]

E( 11)[9 -"A

+ "AX(S'

(9))]

where (3-6 b) cp* 9 - N-n N-n( )~l

We note that

cp~-n(9)

gJ1-(S*(9))r] E(X)[l-S*(9)]

is the

(CO

Jo

I

_eyN-n-r e

J~

. qj (x-y)[S

*

j

(9)] zN-n-r-/y)dy

LST of the time length that consisits of the residual vacation time,

the total service time of those customers in the queue and the total service time of those who precedes the test customer in the same group. Finally the

(3- 7 a)

LST of the waiting time is given by

dVCx) E(V)

0- p )8 8-A+AX(S*(8))

1- X( S*(8)) E(X)[ 1- S* (8)]

+

AEC V)

AE( V)

. 1-

+ N~l\If

V*(8)

8E( V)

n

)

n=O

where

From eqs. (3-7 a) and (3-7 b),

we have the mean waiting time (see Appendix), N-l

~ n=O

(3-8)

n\Ifn

+

N-l

AE( 11)

+~

\If

n=O

n

E( V2) 2E( V) .

AE(V) AE( 11)1 +

N-l

~

\If n

n=O

Combining eq. (3-8) with eqs. (2-22) and (2-23), we see that the celebrated

L q = AE( X) VVz

4.

Little's

formula

works.

OTHER PERFORMANCE

MEASURES

In this section we derive the distributions of the number of customers at a busy period initiation epoch (i.e., idle period termination point), the busy period distribution and the idle period distribution.

THEOREM

4.1.

PGF of Q~.

Let

Q~

be the queue size at a service

Then,

(4-1)

The first and second factorial

moments become

initiation

epoch.

Let

Q~(z)

be the

+

= E( X) [ "AE( V)

E( Q'Jy)

E(X)

j]

V2) + E(X2

="A 2 E2(X)E(

E( Q'Jy(Q'Jy-l»

~11JJ

-

Q'Jy, it is worthwhile to note that if

n(

j]

+ 2E(X)

~>IJJ

j


Conditioning on the vacation length and the number of customers that arrive during the

vacation, we have

if k = 0, 1, ... , N - 1, r;(8

I V=x, if k:? N.

Pr[N(x)

But

=

kl V=

k

x]

=

e- AX(J\X) igii) i!

= ,~

where

gii)

the

IS

i-fold convolution of

= 1. Just after the termination of the vacation, if k( < N) customers are

with itself and g~O)

{ g n}

k(X)

in the queue, the remaining idle period is stochastically identical to the idle period of

N - k.

queue with threshold

Finally d

5.

unconditioning f

f

on e

MX / GIl

Thus unconditioning on k, we have

x, we have the desired result. r

e

n

E( a

r

N)

IS

easily o

derived

by

n

OPTIMAL OPERATING POLICY

In this section, we find the optimal threshold

N* that minilnizes the average operating cost

under a linear cost structure. Let us consider following costs:

Cs

Turn - on cost per cycle.

Co

Operating cost per unit time.

Ch

Holding cost per unit time.

R

Reward due to vacation Per unit vacation. E( T c), is the sum of the expected idle period and the expected

Since the expected cycle length, busy period,

(5-1)

from COROLLARY 4.1 and THEOREM 4.2, we have

E( T c) = E(

r + E(B N)

N)

+ ~

E( V)

~lW

i+

EC~E~

S)

. [ l\E( V)

+

i]

~lW

N-l

l\E( V)

+

l~

Wi

1\(1- p) Then the total average cost with threshold

m becomes

l\ 2 E(X)E( V2)

2 . [l\E( V)

m-l

A

+ Ch l~

l\E( V)

+

i1¥ i m-l l~

1¥

i

+

7~11¥ i]

)

MX/ GIl queue.

size of the ordinary

THEOREM

5.1.

N* be the optimal threshold value that minimizes

Let

cost under the linear cost structure.

I

[AE( V) Note that

l~

V)

+ 1m]~E(

k such k[AE(V)+lk]-Mk>

first

i) \Iii > -C- ]. h

(m-

m.

\Ii i and M m=

+ 1m]{~E(

[AE( V)

A

m-I

mAE( V) + z~

m.

Let 1m=

PROOF.

operating

N* is given by:

Then

N* = min [ m?:.l

(5-2)

the stationary

+ 1m-I] V)

l~

i\Ii i.

Then

. {Ch[m(AE(

+ 1m-I] > 0 A

-C- . h

and

V)

+ 1m)-

m[AE( V)

Mm] - A}

+ 1m]-

Mm> O.

Let

m be the

Then m+ I

= (m+1)AE(V)+

(m+1)[AE(V)+lm+I]-Mm+I

1~(m+1-i)1IJi m

(m+ 1)AE( V)

+ l~(m-

nt·

i)w i+

m[ AE( V) + 1m]- M m+ AE( V) Thus

6.

TC( n) > TC(m)

NUMERICAL

for

n>m

and

the

z~

\Ii i

A + 1m> -C. h theorem

follows.

EXAMPLE

As an example,

suppose

N= 2, A = 1, gI = gz = 1/2, service time=O.1 (constant) and vacation

time

=

1.0 (constant).

E(X) = 1.5,

From a k

E(X2

= pr( k

-

Then

5*(8)

we have

X) = 1,

= e-O.1S,

V"(8)

E( V2) = 1,

p = 0.15,

E(52)=O.Ol,

-1 •

g2)+_e

-ii=

6

-1

. ,rl+_e -(2g1 2 From the definition of

TI: k

-1

•

g3)+_e6-(3~

in THEOREM 2.1,

3=gl'

we get

TI:o=3/4,

2+g2' TI:l=5/8,

TI:

TI:4= gl . TI:3+ g2 . TI:2= 11/16. Also from THEOREM 2.1, we obtain

+a1'1[O=e-1,

W1=aO

'TI:1

W2=aO

'1[2+al'1[I+a2

W3 = ao . 1[3+a1

W4=aO

6.1.

'1[4+al'

. 1[0=

13e-1/8,

. 1[2+ a2 . 1[1+ a3 . TI:o= TI:3+a2 '1[2+a3

Mean Waiting Time

'1[1+a4'

1ge-1/12 TI:o=

-1

. gZ)+_e

TI:1= gl . TI:o= 1/2,

1t

~(8)=

13e-1/48,

TI:o= 1,

TI:1+g2'

X(z) = z(1 +z)/2,

arrivals during a vacation), we have

. (2g1

TI:2=gl'

= e--s,

68ge-1/384.

-gi=73e-1/384.

24

1\

8+1\'

From eq. (3-2),

_ (1 - p)III0 - AE(1I)+IIJo+lIJ) (1- p)IIJ) S*(8) + AE(1I)+IIJo+lIJ)

(8) -1]

g) . [ ~

• {

+

E(X)

1 - X( S* (8» E(X)[1-S*(8)]

}

1- X*(S* (8» E(X)[1-S*(8)] (1-p). AE(1I)+IIJo+lIJ)

1-X*(S*(8» E(X)[1-S*(8)]

(1-p)IIJo g + AE( 11)+ IIJ0 + IIJ) . -E-(b-

[IIJ + IIJ)S*(GI)] o

(8) -1].

. [~

From eq. (3-6 a),

Wv(8)

_ (1- p)AE( 11) . [~(8)-1]· - AE( 11)+ IIJ0 + IIJ)

+

cD~(8)

1- X( S' (8»

(1- p)AE( V) AE( 11)+ IIJ0 + IIJ)

E(X)[1-

v*rA-AX(S* UJ»lE(1I)[8-A+AX(S*(8»]

S*(8)]

V*(8) .

From eq. (3-6 b),

""*(8) )

g)

=-E-(X-) ---g)-

-

I r= J Ie e r= -Ax.l l_

g_)_

E(X)

Jo e

gj

*(. ) 0 =

-8y

-

-qo ( x-y -A(X-Y)

o

= -E-(X-) . cD)

e

-8y

E(X)

__

Jr= o

8

V*(A)-

(

e

)

e

dV(x) . Zo ( Y )d Y E(V) dV(x) d Y E( 11)

-AY

-AX) dV(x) E( 11)

V*(8+A) 8E( 11)

Q)

AE(X)E(

11) .

Thus, we have (1- p)AE( 11) . cD~(8) AE( 11)+ IIJ0 + IIJ)

+

(1- p)AE( V) AE( 11)+ IIJ0 + IIJ)

Then from eq. (3-7 a),

. [~(8)-1]

1- X(S*(8» E(X)[1S*(8)]

r

V* A- AX( S* ( 8»1 - V* ( 8)

E( 11)[8 -A+AX(S*(8»]

(1-- p)8 1- X(5*(8)) 8 - A + AX(5*(8)) E(X)[1- 5*(8)] \If 0 +\If p$'* ( 8) . AE( 11) .•{ AE( 11) + \If0+ \If 1 + AE( 11) + \If0 + \If 1

1- V*(8)

8E(1I)

}

+ WN(8)

Now we use eqs. (3-7 a) and (3-7 b) (see also eqs. (A-I) and (A-2) of Appendix) for the mean waiting time rather than directly using eq. (3-8).

Using the values of aj,

6.2

1Ij,

and

\Ifj

we already obtained,

we easily see that

~=O.4774.

Mean System Size

From eq. (2-22) and (2-23),

Thus

we

see

that

0.8661=(1)(1.5)(0.4774+0.1)

which

confinns

the

Little's

formula

L= :\E(X) [ ~

6.3

+ E(5)].

Optimal operating policy

Suppose

we

have

costs,

C s = 10. 000.

C h = 1000. R = 500.

Then

we

get

A = 8825 and

A

-C- = 8.825. h

Using

l]J /s,

5.4372 for

we calculate

m

mJ\E( 11) +

m-l o~

(m - i)l]J

1 =

1.3679

for

m

==

L 3.1036 for

= 3. 8.3532 for m = 4 and 13.0635 for m = 5. Thus we get N* = 5.

m

= 2,

REFERENCES 1. Burke, P.l,

"Delays in Single-server Queues with Batch Input", Opns. Res., 23, 830-833, 1975

2.

Doshi, B.T., "Queueing Systems with Vacations: A Survey", Queueing Systems,

3.

Chaudhry, M.L. and Templeton, lG.C., A first Course in Bulk Queues, Wiley, 1983

4.

Fuhrmann, S.W. and Cooper, R.B., "Stochastic Decompositions Generalized Vacations", Opns. Res., 1117-1129, 1985

5.

Hofri, M., "Queueing Systems with a Procrastinating Server", Performance ACMSIGMETRICS 1980, Performance Evaluation Review, 14(1), 245-253, 1986

6.

Kella, 0., "The Threshold Policy in the MI G/l Logistics, 36, 111-123, 1989

7.

Lee, H.S. and Srinivasan, M.M., "Control Policies for the Sci., 35(6), 708-721, 1989

8.

Lee, H.S, "Steady state Probabilities for the Server Vacation Model with Group Arrivals and under Control-operating Policy (in Korean)", J. of the Korean ORjMS Soc., 16(2), 36-48, 1991

9.

Lee, H.W., Lee, S.S. and Chae, K.c., "Operating Characteristics of -Policy", To appear in Queueing Systems, 1994

MXI G/l

11. Levy, Y, and Yechiali, Y., "Utilization of Idle Time in an Sci., 22, 202-211, 1975

13. Takagi, H. Analysis

MI G/l

Queue with '86

and

Queue with Server Vacations", Naval Research

10. Lee, H.W., Lee, S.S., Park lO. and Chae, K.C., "Analysis of and Multiple Vacations", To appear in J. of Applied Prob., 1994

12. Takagi, H., Queueing Analysis: A Foundation Priority Systems, Part I, North-Holland, 1991

in the

1, 29-66, 1986

of Performance

of Polling Systems, the MIT Press, 1986

Queueing System", Mgmt.

MXI G/l

Queue with

N

MXI G/l Queue with N-policy

MIG/I

Queueing Systems", Mgmt.

Evaluation,

Vol I, Vacation and

Appendix (Derivation of mean waiting time

~

of eq. (3-8))

From eq. (3-7 a), (A-I)

~

~*(O)

=-

IGll!

~(MX

VAC(l»

N-l

p2;nWn

+

n=O

l\E( V) +

l\E(V)

+

N-l

2;

Wn n=O

N-l

l\E( V)

+ n=O 2; W n

From eg. (3-7 b),

=

W;;(O)

WN = -

(A-2)

(1 - p) N-I l\E( V)

N- l

+ 2;

~ W

N- n {

E(Tn)'

n n-l

[

~ r-l

rg r

•

- - -) WN-n-r+l\E(V)cDN-n(O) E (X

]}

n=O

By successive evaluation of

(A-3)

• _.k cDk(O)-~IE(X)

rgr

cD~(O)

fl oo

0

kak

= l\E(X)E(V)'

given by eg. (3-6 b), we have k-r l~Zi(X-Y)Zk-r-!(Y)dYE(V)

k=1,2,

dV(x)

....

Thus we have

By successive evaluation, it is easily seen that (A-5)

N-n [ ~_lrgrWN-n-r+ r

n-l (N-n)aN-n

] .

'5""'Jti=

l~

N-I

2;

n=O

nWn·

.

Thus we have N-l

(1- p) ~

n=O

(A-6)

nW

n

.

N-l

AE( V)

+ n=O ~ Wn

Finally the mean waiting time of an arbitrary customer becomes A

7

AE( V) AE( V)

+ ~lW n=O

E( n

V2)

2E( V)

III