Steady State Analysis of A Non-Markovian Bulk Queueing Model with Multiple Vacation , Accessible Batches, Setup Times with N-Policy and Closedown Times

Steady State Analysis of A Non-Markovian Bulk Queueing Model with Multiple Vacation , Accessible Batches, Setup Times with N-Policy and Closedown Times

J. Ganeshsivakumar 1, M. Revathy 2

1( Department of Mathematics, CMS college of Science and Commerce, Coimbatore)

2( Department of Maths(CA), Dr.N.G.P. Arts and Science College,Coimbatore)

Abstract

In this paper, a generalized non-Markovian bulk arrival service queueing system is considered with multiple vacations, setup times and closedown time. The service starts only if minimum of a customers are available in the queue. At the service completion epoch,

if the number of customers is ,where a d-1 (db)

then the server takes the entire queue for batch service and admits the subsequent arrivals for service till the service of the current batch is over, or the accessible limit d is reached, whichever occurs first. At the service initiation epoch, if the number of customers waiting in

the queue is atleast d (adb), then the server

takes min( ,b) customers for service and does not allow further arrival into the batch. On completion of a service, if the queue length is less than a, then the server performs a closedown work such as, shutting down the machine, removing the tools etc. Following closedown work, the server leaves for a vacation of random length irrespective of queue length. When the server returns for a vacation and if the queue length is still less than a, he leaves for another vacation and so

on until he finds atleast a customers waiting for service in the queue. That is, if the server finds atleast N customers waiting for service, then he requires a setup time R to start the service. After the setup he serves a batch of b customers, where ba. Various characteristics of queueing system and a cost model are presented.

Keywords: Markovian Bulk, Multiple Vacation, Setup time, Accessible Batches and Closedown times.

1. Introduction

   Many researchers have concentrated on bulk service queueing models, in which once the service is started arriving customers, can not enter the service station though enough space is available to accommodate them. It can be observed in many practical situations that arriving customers will be considered for service with current batch in service with some restriction. The concept of non accessibility while receiving service, has been studied by Weiss[6], Sivasamy[7] analyzed a Markovian single arrival bulk service queue with accessible and non accessible batches. R. Arumuganathan and S. Jeyakumar[1] had given results for setup times with N-Policy. Sharma and Jain[8] obtained results for average queue length and waiting time distribution for state dependent Markovian single arrivl bulk service queueing system with accessible and non accessible batches. Sharma et al.[9] established the expression for average queue length for state dependent Mx/M(a,d,b)/1 queue with accessible and non accessible batches without vacations. In the literature, only less attention is given for general because of the complexity in getting a closed form solution.

   In this paper, a generalized non-Markovian bulk arrival service queueing system is considered with multiple vacations, setup times and closedown time. The service starts only if minimum of a customers are available in the queue. At the service completion

   epoch, if the number of customers is ,where a d-

   1 (db) then the server takes the entire queue for batch

   service and admits the subsequent arrivals for service till the service of the current batch is over, or the accessible limit d is reached, whichever occurs first. At the service initiation epoch, if the number of customers

   waiting in the queue is atleast d (adb), then the

   server takes min( ,b) customers for service and does not allow further arrival into the batch. On completion of a service, if the queue length is less than a, then the server performs a closedown work such as, shutting down the machine, removing the tools etc. Following closedown work, the server leaves for a vacation of random length irrespective of queue length. When the server returns for a vacation and if the queue length is still less than a, he leaves for another vacation and so on until he finds atleast a customers waiting for service in the queue. That is, if the server finds atleast

   N customers waiting for service, then he requires a setup time R to start the service. After the setup he serves a batch of b customers, where ba.

   Q a

   Service Accessibility a n d-1

   Non-Accessibility d n b

   Q < a

   Closedown work

   Closedown work

2. Mathematical model

   Let X be the group size random variable, be the Poisson arrival rate, gk be the probability that k customers arrive in a batch and X(z) be its probability generating function. Let S(.),V(.), R(.) and C(.) be the cumulative distributions of the service time, vacation time, setup time and closedown time, respectively. Let s(x), v(x), r(x) and c(x) be the probability density functions of service time, vacation time, setup time and closedown time respectively. S0(t), V0(t), R0(t) and C0(t) denote the remaining service tine of a batch, the remaining vacation time, setup time and closedown time of a server at an arbitrary time t, respectively. Let us denote the Laplace transform (LT)of s(x), v(x), r(x)

   ~ ~ ~ ~

   and c(x) as S , V , R and C respectively.

   The number of customers in the queue and the number of customers in service are denoted by Ns(t), Nq(t), respectively. The different states of the server at time t are defined as follows:

   Y(t) = 0, if the server is busy with bulk service

   = 1, if the server is doing closedown work

   = 2, if the server is on vacation

   and define Z(t) = j, if the server is on jth vacation starting from the idle period.

   To obtain system equations, the following probabilities are defined. Let

   Pij(x,t)dt = P{Ns(t)=i, Nq(t)=j, xS0(t)x+dt, Y(t)=0},

   aib, j0,

   which means that there are i customers under service, j

   Multiple Vacations Q a

   Setup job

   Q < a

   customers in the queue, the server is busy with remaining service time of x.

   In a similar manner, it is defined,

   Cn(x,t)dt = P{Nq(t)=n, xC0(t)x+dt, Y(t)=1},

   n0,

   Qjn(x,t)dt = P{Nq(t)=n, xV0(t)x+dt, Y(t)=2, Z(t)=j},

   Fig 1: Schematic Representation of Queueing Model

   For the proposed model, the probability generating function of the number of customers in the queue at an arbitrary time epoch is obtained using supplementary variable technique. The complexity of general service accessible batch queueing system involving LST of unknown probability functions is overcome by providing a recursive epoch. Expression

   for expected queue length, expected length of idle

   n0, j1, Rn(x,t)dt = P{Nq(t)=n, xR0(t)x+dt, Y(t)=1},

   nN

3. Analysis

   b

   b

   The steady state queue size equations are obtained as

   P' (x) P (x) P (0)s(x) Q (0)s(x)

   period, expected length of busy period and expected waiting time are derived. A cost model of the queueing system is discussed.

   i0 i0

   ia

   • Pik ,0

     k 1

     mi

     md

     (x)gk ,

     li

     l 1

     a i d

     (1)

     P' (x) P

     (x) P

     (0)s(x)

     ~ ( ) P (0) ~ ( ) P

     (0) ~( ) Q (0) ~( )

     b

     b

     i 0 i 0

     p>

     mi

     md

     Pi0

     b

     b

     io

     ia

     Pi 0

     mi S

     md

     li S

     l 1

     Qli (0)s(x) ,

     l 1

     d 1 i b

     (2)

     Pik ,0 ( )gk ,

     k 1

     a i d

     (15)

     P' (x) P

     1. P

        (x)g

        ~ ( ) P

        (0) ~ ( ) P

        (0) ~( )

        n

        n

        dn dn

        k 1

        d ,nk k

        Pi0 i 0

        Pi0

        mi S

        b

        b

        md

        d a

        • P

     (x)g

     , n 1

     (3)

     Q (0) ~( ) ,

     d 1 i b

     (16)

     d k ,0

     k 1

     k n

     li S

     l 1

     n

     n

     P' (x) P (x) P

     (x)g ,

     ~

     ~

     ~

     in in

     k 1

     i,nk k

     Pdn ( )

     Pdn (0)

     Pdn ( )

     n

     n

     k 1

     Pd ,nk (

     ) gk

     d i b 1, n 1

     b

     (4)

     d a

     ~

     ~

     Pd k ,0

     ( )g

     k n ,

     n 1

     (17)

     P' (x) P

     (x) P (0)s(x) Q

     (0)s(x)

     k 1

     bn bn

     m,b n

     l ,b n ~

     ~ n ~

     md

     n

     n

     l 1

     P ( ) P (0) P ( ) P

     ( )g ,

     P (x)g ,

     n 1

     (5)

     in in

     in i,nk k k 1

     b,nk k

     k 1

     d i b 1, n 1

     (18)

     b

     b

     C ' (x) C (x) P

     (0)C(x)

     (6)

     ~

     ~

     ~ ~

     0 0 m0

     ma

     Pbn ( )

     Pbn

     Pbn ( )

     b

     b

     md

     Pm,bn (0)S ( ) Ql ,bn (0)S ( )

     l 1

     b

     b

     C ' (x) C (x) P

     (0)C(x)

     ~

     n n

     n

     n

     • C

     mn

     md

     (x)g ,

     1 n a 1

     (7)

     ~

     Pb,nk (

     n

     n

     k 1

     ~

     ) gk , n 1

     b

     b

     ~

     (19)

     k 1

     nk k

     C0 ( )

     C0 (0)

     C0 ( ) Pm0 (0)C( )

     ma

     (20)

     n

     n

     C ' (x) C (x) C

     (x)g ,

     n a

     (8)

     ~

     ~ ~

     n n

     k 1

     nk k

     Cn ( )

     Cn (0)

     Cn ( )

     b

     b

     md

     Pmn (0)C( )

     Q' (x) Q (x) C (0)v(x)

     (9) n ~

     10 10 0

     C ( )g ,

     1 n a 1

     (21)

     Q' (x) Q (x) C (0)v(x)

     nk k

     1n 1n n

     n

     n

     k 1

     n

     n

     ~ ( ) C (0) ~ ( ) ~

     ( )g ,

     k 1

     Q1,nk (x)gk ,

     n 1

     (10)

     Cn n

     Cn

     k 1

     Cnk k

     • Q

     • Q

     j 0

     j 0

     j 0

     j 0

     ' (x) Q

     1. Q

        j1,0

        (0)v(x),

        j 2

        (11)

        ~

        n a

        ~ ~

        (22)

        Q' (x) Q

        1. Q

           (0)v(x)

           Q10 ( ) Q10 (0) Q10 ( ) C0 (0)V ( ),

           (23)

           jn jn

           n

           n

           j 1,n

           ~ ( ) Q (0) ~ ( ) C (0) ~( )

           • Qj ,nk (x)gk ,

     1 n a 1, j 2

     (12)

     Q1n

     1n Q1n n V

     n ~

     k 1

     Q1,nk ( )gk ,

     n 1

     (24)

     Q' (x) Q

     (x) Q

     (x)g ,

     k 1

     ~ ~ ~

     n

     n

     jn jn

     k 1

     j,nk k

     Qj 0 ( ) Qj 0 (0) Qj 0 ( ) Qj 1,0 (0)V ( ),

     n a, j 2

     n n N

     (13)

     ~ ( ) Q

     (0) ~

     ( ) Q

     j 2

     (0) ~( )

     (25)

     R' (x) R (x) Q (x)r(x) R

     (x)g ,

     Qjn

     jn Qjn

     j 1,n V

     n n

     k 1

     j ,n

     n N

     k 1

     nk

     k

     (14)

     n

     n

     k 1

     ~

     Qj ,nk

     ( )gk ,

     1 n a 1, j 2 (26)

     Qjn

     Qjn

     Taking LT on both sides of the equations (1)- (14), we get

     ~

     Qjn

     ( ) Qjn

     (0) ~

     n

     n

     ( )

     k 1

     ~

     Qj ,nk

     ( )gk ,

     n a, j 2 (27)

     ~

     ~

     n N

     ~

     ~

     ( X (z)) ~(z, ) C(z,0)

     Rn ( )

     Rn (0)

     Rn ( )

     n

     n

     l 1

     Ql ,n

     (0)R( )

     k 1

     Rnk (

     ) gk ,

     C

     ~ b

     a1 b

     n n

4. Queue Size Distribution

n N

(28)

C( ) Pm0 (0)z

ma

Pmn (0)z

n1 md

(32)

To obtain the probability generating function of the queue size at an arbitrary time epoch, the following probability generating functions are defined.

Multiplying the equation (15) by z0 with i=d, (17) by zn (n 1) and taking summation from n=0 to and using (29), we get

~ ~ b

~ ij

( X (z))Pd (z, ) Pd (z,0) S ( ) Pmd (0) Qld (0)

Pi (z, ) Pij ( )z

j0

d 1 ~

md

i

l 1

ij

ij

Pi (z,0)

j0

P (0)z j ,

d i b

z d

d 1i

ia

Pi0 ( )z

* X (z) Gi (z)

(33)

~ ~ n

where Gi(z)= g z k

k

k

Qj (z, ) Qjn ( )z k

n0

Qj (z,0) Qjn

n0

(0)zn ,

j 1

(29)

Multiplying the equation (16) by z0, (18) by zn (n 1) and taking summation from n=0 to and using (29), we get

~ ~ n

C (z, ) Cn ( )z

~ ~ b

n0

( X (z))Pi (z, ) Pi (z,0) S ( ) Pmi (0) Qli (0)

C(z,0) C (0)z

C(z,0) C (0)z

n

n

n0

md l 1

d 1 i b 1 (34)

~ ~ n 0 n

R(z, ) Rn ( )z

Multiplying the equation (16) by z with i=b, (19) by z

n N

(n 1) and taking summation from n=0 to and using

R(z,0)

Rn

(0)zn

(29), we get

~

n N

( X (z)) ~ (z, ) P (z,0) S ( )

Multiplying the equation (23) by z0 and (24) by zn

Pb b

b

zb

b1

(n 1) taking summation from n=0 to and using

P (z,0) P

m

m

(0)zn

(29), we get

md

mn

n0

(35)

b1

~ ~ Q (z,0) Q

(0)zn

( X (z))Q (z, ) Q (z,0) V ( )C(z,0)

(30)

l

ln

1 1

l 1

n0

Multiplying the equation (25) by z0 , (26) by zn (1 n a-1) and (27) by zn (n a) taking summation

Multiplying the equation (28) by z0 (n 1) and taking summation from n=0 to and using (29), we get

from n=0 to and using (29), we get ~ ~

( X (z)) ~ (z, ) Q (z,0)

( X (z))R(z, ) R(z,0) R( )

Qj j

~ a n

b Q (z,0) N Q (0)zn

(36)

V ( )Qj 1,n (0)z

n0

(31)

l

l 1

ln

n0

Multiplying the equation (20) by z0 , (21) by zn (1 n a-1) and (22) by zn (n a) taking summation from n=0 to and using (29), we get

Substituting X (z) in (30) through (36), we get

Q (z,0) ~( X (z))C(z,0)

(37)

1 V

~ ~ a1 n

( ~( X (z)) ~( ))

Q (z,0) V ( X (z))Q (0)z

(38) V V

j

n0

j 1,n

a1

Q

~

j 1,n