N-Policy for State-Dependent Batch Arrival Queueing System with l ...

Quality Technology & Quantitative Management

Vol. 7, No. 3, pp. 215-230, 2010

QTQM

© ICAQM 2010

N-Policy for State-Dependent Batch Arrival Queueing System with l-Stage Service and Modified Bernoulli Schedule Vacation Madhu Jain1 and Praveen Kumar Agrawal2 1

2

Department of Mathematics, IIT- Roorkee, Roorkee, India Department of Mathematics, Institute of Basic Science, Dr. B.R. Ambedkar University, Agra, India (Received November 2007, accepted May 2009)

______________________________________________________________________ Abstract: This paper deals with a batch arrival queueing system with modified Bernoulli vacation under N-policy, where the server starts the service of the customers only when the queue size becomes at least N ( t 1 ) otherwise remains idle. The customers arrive in batches to the system in Poisson fashion but may also balk in case of long queue, when server is in working state. The customer needs l-stage of service in succession i.e. the first stage service (FSS) is followed by the second stage service (SSS), the second stage service followed by third stage service (TSS) and so on up to l-stages of service. It is assumed that after completion of lth stage service either the server goes on vacation with probability p or decides to stay in the system with probability (1  p ) to give the service to the next customer. We determine the queue size distribution and other performance indices by using the generating function method to solve the governing equations constructed after introducing the supplementary variables. We propose a method to find the optimal value of the threshold parameter to minimize the total expected cost.

Keywords: Bernoulli schedule, generating function, MX/G/1 queue, N-policy, queue size, state dependent rate, supplementary variable technique, vacation time.

______________________________________________________________________ 1. Introduction

I

n many real world queueing systems, the server may become unavailable for a random period of time when there is no customer in the waiting line at a service completion instant. This random period in which the server is absent, is often called a server vacation. The classical vacation scheme with Bernoulli schedule discipline was introduced and studied by Keilson and Servi [20]. There is extensive literature on M / G /1 queue with Bernoulli schedule vacation which has been contributed in various forms by several authors including Harris and Marchal [16], Ramaswami and Servi [28], Choi and Park [8], Selvam and Sivasankaran [30] and Feng et al. [15]. Atencia et al. [5] considered an M / G /1 queue with blocked customers who either with probability q join the infinite waiting room or with complementary probability p leaves the service area. Excellent contributions on vacation models have been made by Anabosi and Madan [1], Artalejo and Choudhury [2], Choudhury and Madan [11], Madan and Abu Al-Rub [26] and many others. Atencia and Moreno [4] obtained the steady state solution of a single server queue with Bernoulli schedule. Choudhury [10] discussed a two-phase batch arrival retrial queueing system with Bernoulli vacation schedule. Ke [19] considered a batch arrival queue under vacation policies with server breakdown and startup/closedown times. Choudhury et al. [14] gave steady state analysis of an M X / G /1 queue with two-phase service and Bernoulli vacation schedule under multiple vacation policy.

216

Jain and Agrawal

For some controllable queueing systems with general vacations, it is usually assumed that the server is available, or unavailable completely depending upon the number of customers present in the system. Every time when the system is empty, the server goes on vacation. The instance at which the server returns back from a vacation and finds at least N customers in the system, it begins serving immediately and exhaustively. This type of control policy is also called N policy for the queueing systems with vacations. Kella [21] and Lee and Srinivasan [22] first provided detailed discussions concerning N-policy for M / G /1 and M X / G /1 queueing systems with vacations, respectively. Single server queueing model with N -policy and vacation have many applications including flexible manufacturing systems, service systems, telecommunication systems, transportation systems, etc. In this direction, the notable contributions have been made by several researchers including Medhi and Templeton [28], Lee et al. [23], Chae and Lee [7], Choudhury [9], Bacot and Dshalalow [6] and Ke [17]. Choudhury and Madhuchanda [13] analyzed a batch arrival queueing system with an additional service channel under N-policy. Choudhury and Madan [12] considered a batch arrival queueing system, where the server provides two stages of heterogeneous service with a modified Bernoulli schedule under N-policy. Tadj and Ke [32] discussed a control policy of a hysteretic bulk queueing system. Arumuganathan and Jeyakumar [3] considered a queueing system with multiple vacations, setup time, N-policy and closedown times. Following closedown period, the server leaves for vacation of random length irrespective of queue length. Ke [18] studied the vacation policies of an M/G/1 queueing system with server breakdowns, startup and closedown times under NT-policy in which the length of the vacation period is controlled either by the number of arrivals during the vacation period or by a timer. Recetly, Wang et al. [33] dealt with an N policy M/G/1 queueing system with a single removable and unreliable server where arrivals form a Poisson process.

In this paper an M X / (G1 , G2 , G3 ,...,Gl ) /VS /1( BS ) / N state dependent batch arrival queueing system with l-stages of service and Bernoulli vacation schedule under N -policy is analyzed, where Gi (i 1, 2, 3,..., l ) represents the service of a customer done in i th -stages and is general distributed, VS represents single vacation time, BS represents Bernoulli schedule. An attempt has been made to generalize a batch arrival two stage service system with a modified Bernoulli schedule vacation under N-policy, studied by Choudhury and Madan [12], by including state dependent rates and l-stages of service. The rest of the paper is structured as follows. In order to develop the mathematical model, some assumptions and notations are provided in section 2. Using supplementary variables, the steady state equations are constructed in section 3. Furthermore, the probability generating functions of the queue length distributions at arbitrary epoch as well as at departure epoch are established. In section 4, some performance characteristics are obtained. Some earlier works are deduced as special cases of our general model in section 5. A cost function to find the optimal threshold value has been developed in section 6. Final section 7 includes the conclusions of the work.

2. Model Description An optimal batch arrival queueing system with l-stage service under modified Bernoulli schedule vacation is considered. To formulate the mathematical model, the following assumptions are made. z

The service discipline is FCFS (First Come First Serve).

z

The customers arrive in batches of size ‘ X ’ according to Poisson process with state dependent rates given as:

N-Policy for State-Dependent Batch Arrival Queueing System

j

217

­O ; server is in idle state ° th ®Oi ; server is providing i phase service (i 1, 2,...., l ). °O ; server is on vacation ¯ v

k ) ak , k 1, 2, 3,... with ¦ fk 1 ak

Also P ( X

1.

z

When the queue size becomes N ( N t 1), the server starts service of the customers otherwise remains idle.

z

It is assumed that the service time of the server of each stage is general distributed.

z

After becoming the queue size N ( N t 1), the server instantly starts working and provides l-stages of service in succession to each customer i.e. the first stage service is followed by the second stage service, the second stage service by the third stage service and so on up to l-stages of service.

z

As soon as all the l-stage services of the customer are completed, the server with probability (1  p ) may opt for the next customer to serve or else with probability p (0 d p d 1), he may leave the system and go on vacation.

z

The server provides the service to the customers according to FCFS discipline.

Table 1 describes the notations used for some r.v.’s along with their probability distribution functions, Laplace-Stieljes transform (LST) and k th moment ( k t 1). Table 1. Notations. r.v.

pdf

LST

k th moments

i th phase service time

Bi (i 1, 2, 3,..., l )

Bi (t )

Bi * ( s )

E ( Bik )

Vacation time

V

V (t )

V * (s)

E (V k )

Required service time

G

G (t )

G * (s)

E (G k )

Elapsed service time of i th phase

Bi0 (i 1, 2, 3,..., l )

Bi0 (t )

-

-

-

-

Time

Elapsed vacation time

V0

0

V (t )

3. Queue Size Distribution Let N (t ) denotes the number of customers in the queue at time t . Define a random variable ] (t ) denoting the server’s state as

] (t )

­0, ° ®i , °l  1, ¯

server is idle server is busy with i th stage service, 1 d i d l . server is on vacation

The time taken by a customer to complete service cycle, called a modified service time is given

218

Jain and Agrawal

G

­ B1  B2  B3  ...........Bl  V , ® ¯ B1  B2  B3  ...........Bl ,

with probability p, with probability 1  p.

Thus,

G * ( s ) (1  p ) B1 * ( s ) B2 * ( s ).......Bl * ( s )  pB1 * ( s ) B2 * ( s )..........Bl * ( s )V * ( s ). We now define the joint distributions of the server state and queue size in steady state as follows Qn Pi , n ( x )dx R ( x )dx

lim Prob[N (t ) n, ] (t ) 0], n

0, 1, 2,......, (N ),

t of

n, ] (t ) i , x  Bi0 (t ) d x  dx ], x ! 0, n t 1, 1 d i d l ,

lim Prob[N (t ) t of

lim Prob[N (t ) n, ] (t ) l  1, x  V 0 (t ) d x  dx ], x ! 0, n t 0. t of

Let Pi ( x ) and X ( x ) be the conditional completion rates (at time x ) for the service and vacation times respectively, with Bi (0) 0, Bi (f) 1, (i 1,2,3,......, l ), V (0) 0, V (f) 1. Then

dBi ( x ) , 1 d i d l, 1  Bi ( x )

Pi ( x )dx

dV ( x ) . 1 V ( x )

X ( x )dx

(1)

(2)

Let us define [ n (n 1,2,3,......,( N  1)) which is the probability that a batch of customers finds at least n jobs in the system during the idle period. Thus, n

] 0 1, [n

¦ ak [ n 1 , 0 d n d N  1.

(3)

k 1

Using appropriate transition rates we can easily construct the following steady state equations governing the model

O Q0

f

f

0

0

³ X ( x ) R0 ( x )dx  (1  p ) ³ Pt ( x ) Pl , 1 ( x )dx ,

O Qn

(4)

n

O ¦ a j Qn  j , n 1, 2, 3,..., ( N  1), j 1

d Pi , n ( x )  [Oi  Pi ( x )]Pi , n ( x ) dx

(5)

n

Oi ¦ a j Pi , n  j ( x ), x ! 0, n t 1, 1 d i d l ,

(6)

j 1

d R 0 ( x )  [Ov  X ( x )]R 0 ( x ) dx

0, x ! 0,

n d R n ( x )  [Ov  X ( x )]R n ( x ) O ¦ a j R n  j ( x ), x ! 0, n t 1. dx j 1

(7)

(8)

N-Policy for State-Dependent Batch Arrival Queueing System

219

The above steady state equations (4)-(8) are to be solved under the following boundary conditions at x 0 given by f

p ³ Pl ( x )Pl , n 1 ( x )dx , n t 0,

Rn (0)

(9)

0

f

f

0

0

P1, n (0) (1  p ) ³ Pl ( x ) Pl , n 1 ( x )dx  ³ X ( x ) Rn ( x )dx , n 1, 2, 3,......, ( N  1), f

f

n

0

0

j 1

P1, n (0) (1  p ) ³ Pl ( x ) Pl , n 1 ( x )dx  ³ X ( x ) Rn ( x )dx  O ¦ a j Qn  j , n t N , f

³ Pl 1 ( x ) Pl 1, n ( x )dx , n t 1, 2 d i d l .

Pi , n (0)

(10) (11)

(12)

0

The normalizing condition is given as N 1

f

l

f f

¦ Qn  ¦ ¦ Pi , n ( x )dx  ¦ ³ Rn ( x )dx 1.

n 0

i 1n 1

(13)

n 10

In order to solve the system of equations (4)-(8) with boundary conditions (9)-(12), we define the following generating functions Q(z )

N 1

n ¦ Qn z ,| z | 1,

(14)

n 0

Pi ( x , z ) Pi (0, z ) R( x, z )

f

n ¦ Pi , n ( x ) z ,| z | 1, x ! 0, 1 d i d l ,

(15)

n 1 f

n ¦ Pi , n (0) z , | z | 1, 1 d i d l ,

(16)

n 1 f

n ¦ Rn ( x ) z , | z | 1, x ! 0,

(17)

n 1

R (0, z )

f

n ¦ Rn (0) z ,| z | 1,

(18)

n 1

X (z )

f

k ¦ ak z , | z | 1 .

(19)

k 1

The following theorem provides the solution of the system of equations (4)-(8) with boundary conditions (9)-(12), in terms of the generating functions. Theorem 1: The marginal probability generating functions for the idle, busy with i-stage service and vacation states, respectively of the server are given as Q ( z ) (1  U )S ( z ),

(20)

220

Jain and Agrawal i

O z (1  U )S ( z )[1  Bi (Oi  Oi X ( z ))]– B j (O j  O j X ( z )) j 1 j zi

f

³ Pi ( x , z )dx

Pi ( z )

l

OEi [((1  p )  pV (Ov  Ov X ( z ))– Bi (Oi  Oi X ( z ))  z ]

0

, 1di d l,

(21)

i 1

l

R( z )

O p (1  U )S ( z )[1  V (Ov  Ov X ( z ))]– Bi (Oi  Oi X ( z ))

f

³ R ( x , z )dx

i 1

l

v X ( z )) – Bi ( i i 1

Ov [((1  p )  pV (Ov  O

0

O  Oi X ( z ))  z ]

,

(22)

N 1

n ¦ [n z

where S ( z )

n 0 N 1

¦ [n

.

n 0

Proof: See Appendix A. Theorem 2: The marginal probability generating function P ( z ) of queue size distribution at an arbitrary epoch is given by

^

l

l

S ( z )[(1  U )[[((1  p )  pV (Ov  Ov X ( z ))– Bi (Oi  Oi X ( z ))  z ]  O z ¦ [1  Bi (Oi i 1

i 1

i

 Oi X ( z ))] – B j (Oi  O j X ( z ))  j 1 j zi

P (z )

½

O ° pz[1  (V (Ov  Ov X ( z ))]– Bi (Oi  Oi X ( z ))]]¾ Ov i 1 ° l

l

¿ . (23)

v X ( z )) – Bi ( i i 1

[((1  p )  pV (Ov  O

O  Oi X ( z ))  z ]

l

Proof: Using P ( z ) Q ( z )  ¦ Pi ( z )  zR ( z ), we obtain P ( z ). i 1

Theorem 3: The marginal probability generating function P  ( z ) of queue size distribution at a departure epoch is

^

l

l

(1  U )S ( z )[[ E [ X ]{Ov pE [V ]  ¦ Oi E [ Bi ]}  1]  O[ E [ X ]{ pE [V ]  ¦ E [ Bi ]}]]



P (z )

`

i 1

l

u[1  [ E [ X ][((1  p )  pV ( v  v G ( z )) – Bi ( i  i X ( z ))] i 1 l [ E [ X ][((1  p )  pV ( v  v X ( z )) – Bi ( i  i X ( z )) i 1

O

O

O

O

O

O

O

O

i 1

.

(24)

 z]

Proof: See Appendix B.

4. Performance Characteristics In this section, we derive some performance indices using probability generating functions as follows: 4.1. Long Run Probabilities of the Server States

(i)

Long run fraction time when server is in idle state is PI

lim Q ( z ) (1  U ). z o1

(25)

N-Policy for State-Dependent Batch Arrival Queueing System

221 th

(ii) Long run fraction time when server is busy in i (i 1, 2, 3,..., l ) stage of service, is

lim Pi ( z )

PBi

z o1

O E [ Bi ]E [ X ] l

l

i 1

i 1

[ E [ X ]{Ov pE [V ]  ¦ Oi E [ Bi ]}  1]  O[ E [ X ]{ pE [V ]  ¦ E [ Bi ]}]

, 1 d i d l. (26)

(iii) Long run fraction time when server is on vacation, is given by PV

O E [V ]E [ X ]

lim R ( z )

, l [ E [ X ]{Ov pE [V ]  ¦ Oi E [ Bi ]}  1]  O[ E [ X ]{ pE [V ]  ¦ E [ Bi ]}] l

z o1

i 1

(27)

i 1

l

where U

O[ E [ X ]{ pE [V ]  ¦ E [ Bi ]}] i 1

l

l

i 1

i 1

[ E [ X ]{Ov pE [V ]  ¦ Oi E [ Bi ]}  1]  O[ E [ X ]{ pE [V ]  ¦ E [ Bi ]}]

.

4.2. Average Queue Length

(i)

Average queue length at departure epoch N 1

n[ Acc(1) E [ X ]  Ac(1) E [ X ( X  1)] n¦0 n  N 1  ( Ac(1)  1). 2 Ac(1) E [ X ] [ ¦ n

L

(28)

n 0

(ii) Average queue length at arbitrary epoch N 1

¦ n[ n

n 0 N 1

L

¦ [n

U

O[ Acc(1) B c(1)  Ac(1) B cc(1)] , 2[ Ac(1)  O B c(1)]2 (1  U )

(29)

n 0

where Ac(1)

l

pOv E [V ]¦ [ Oi E [ Bi ]E [ X ], i 1

Acc(1)

l

¦[ O i 1

2 2 i E [ Bi ] 

l

pOv2 E [V 2 ]]E [ X ]2  ¦ Oi [ E [ Bi ]  pOv E [V ]]E [ X ( X  1)]

l

i 1

 pOv E [V ]¦ [ Oi E [ Bi ]E [ X ] , 2

i 1

B c(1) B cc(1)

l

 ¦ [ E [ Bi ]  pE [V ]] E [ X ] , i 1 l

l

 ¦ [ E [ Bi ]  pE [V ]] E [ X ( X  1)]  ¦ Oi [ E [ Bi2 ]  pOv E [V 2 ]]E [ X ( X  1)] i 1

i 1

l

l

i 2

i 1 j zi

l

 2 ¦ [ E [ Bi ]¦ Oi [ E [ Bi ]E [ X ]  2 pE [V ]¦ [ Oi E [ Bi ]E [ X ]2 . 2

i 1

222

Jain and Agrawal

4.3. Mean Busy Period

The length of time for which service station is turned off or removed from the system is known as the idle period and the expected idle period is denoted by E [ LI ], the busy period (B) is the length of time for which service station is rendering service and its expectation is denoted by E [ LB ], the expected vacation period is denoted by E [ LV ] and the mean length of cyclic period is denoted by E [T ]. According to the memoryless property of the Poisson process, the length of the idle period is the sum of N exponential random variables each having mean 1/ O and ¦ nN 01[ n (cf. Appendix A) gives the mean number of batches arriving during an idle period with arrival rate O. Thus N 1

¦ [n

E [ LI ]

n 0

O

.

(30)

Since E [T ] E [ LI ]  E [ LB ]  E [ LV ],

(31)

we obtain PI

E [ LI ] , PB E [T ]

E [ LV ] . E [T ]

E [ LB ] , PV E [T ]

(32)

Expected length of the cyclic period is now obtained as N 1

E [T ]

E [ LI ] PI

¦ [n

n 0

O (1  U )

.

(33)

The mean busy period is equal to l

E [ LB ]

U 1 U

E [ LI ] u

¦ E [ Bi ]

t 1

l

.

(34)

.

(35)

{ pE [V ]  ¦ E [ Bi ]} i 1

Now, expected length of vacation period E [ LV ]

U 1 U

E [ LI ] u

E [V ] l

{ pE [V ]  ¦ E [ Bi ]} i 1

5. Some Special Cases In the following section, we present some existing results appeared in the literature, which are special cases of our model:

N-Policy for State-Dependent Batch Arrival Queueing System

223

5.1. Two Stage Service and Batch Arrival Queueing System with Vacation under N-policy

Putting OV O, E [ Bi ] 0, 3 d i d l in equation (23), our results agree with the results of M X / (G1 , G2 ) /VS /1( BS ) / N -policy given by Choudhury and Madan [12]. In this case equation (23) converts to 2

(1  z )Q ( z ) [((1  p )  pV (O  O X ( z )) – Bi (O  O X ( z ))] i 1

P (z )

[((1  p )  pV (O  O

2

X ( z )) – Bi ( i 1

O  O X ( z ))  z ]

.

(36)

5.2. MX /G/1 Queue with N-policy

If we put p 0 (i.e., there is no server on vacation), OV O , E1 E 2 1, E [ B1 ] E [ B ], E [ Bi ] 0, 2 d i d l , in equation (23), our results coincide with the results of M X / G /1 queue with N-policy obtained by Lee et al. [23]. 5.3. MX /G/1 Queue with Two Stage Service and Vacation

Taking p 0, OV O, E1 E 2 1, E [ B1 ] E [ B ], E [ Bi ] 0, 2 d i d l , N 1 in equation (23), the results coincide with the results of Madan [24] for a single server queue with two stage general heterogeneous service and vacation. 5.4. M/G/1 Queue with Two Stage Service and Vacation

Putting p 0, OV O , E1 E 2 1, E [ B1 ] E [ B ], E [ Bi ] 0, 2 d i d l , N 1 and [ X 1] 1 (i.e. single arrival) in equation (22), our results agree with the results given by Madan [25].

6. Cost Function In this section, we construct a cost function for expected total cost per unit time in which N is a decision variable. Our objective is to determine the optimal policy to minimize the cost function, while maintaining the minimal service quality to the customers. Let C h , C S , C I and C bi , respectively be the holding cost per unit time, startup cost per unit time, cost per unit time of server being idle and cost per unit time of server being busy in i th (i 1, 2, 3,..., l ) stage of service, for each customer present in the system. Based on the definitions of each cost element, and its corresponding system characteristics, the total expected cost per unit time is given by E{TC [ N ]} C h L  C s

l 1  C I PI  ¦ C bi PBi E [T ] i 1

N 1

¦ n[ n t O[ Acc(1) B c(1)  Ac(1) B cc(1)] O (1  U ) ]  C s N 1 C h [ nN 01 U  C I PI  ¦ C bi PBi 2 i 1 2[ Ac(1)  O B c(2)] (1  U ) ¦ [n ¦ [n n 0

n 0

N 1

OC s (1  U )  C h ¦ n[n n 0

N 1

¦ [n



t OC h [ Acc(1) B c(1)  Ac(1) B cc(1)]  C h U  C I PI  ¦ C bi PBi , 2 i 1 2[ Ac(1)  O B c(2)] (1  U )

n 0

(37)

224

Jain and Agrawal

where PI and PBi are given in equations (25) and (26), respectively. We can determine the optimal value of N (i.e. N ) by considering the following differences ' E{TC [ k ]} { E{TC [ k  1]}  E{TC [ k ]}

[k E (k ) H ( k  1) H ( k )

,

where ' is the difference operator. k

E ( k ) C h [ kH ( k )  M ( k )]  OC s [1  E ( X )( ¦ Oi E ( Bi )  pOv E (V ))] i 1

k 1

k

j 0

i 1

(38)

C h [ ¦ ( k  j )[ j ]  OC s [1  E ( X )( ¦ Oi E ( Bi )  pOv E (V ))], where H ( k )

¦ kj 0 [ j and M ( k )

¦ kj

0

j[ j .

We note that k 1

[k

j 0

H ( k  1) H ( k )

C h [ ¦ ( k  j )[ j ] ! 0 and

! 0.

(39)

By using equation (38), we have 'E{TC [ k  1]} ! 0. Now setting k m, we see that E (m  1) C h [(m  1) H (m  1)  M (m  1)]  OC s [1  E ( X ) ( ¦ ik 1 Oi E ( Bi )  p Ov E (V ))] E (m ) C h H (m  1), such that E (m  1) ! E (m ), hence for some n ! m, we have E{TC [n ]} ! E{TC [m ]}. In order to obtain the optimal value of threshold parameter N * we minimize the objective function (38). Thus

N*

k ­ ½ OC s [1  E ( X )( ¦ Oi E ( Bi )  pOv E (V ))] ° k 1 °° ° i 1 min ®k t 1 ¦ ( k  j )[ j ! ¾. Ch j 0 ° ° °¯ °¿

(40)

Denoting J ( k ) ¦ ik 01 ( k  i )[i and A OC s [1  E ( X )( ¦ ik 1 Oi E ( Bi )  pOv E (V ))], equation (39) can be written as N*

­ A½ min1d k d N ®k : J ( k ) t ¾. Ch ¿ ¯

(41)

By setting the best positive value of k, the optimal value of threshold parameter can be determined. We also observed that if k

Ch ! Cs

O[1  E ( X )( ¦ Oi E ( Bi )  pOv E (V ))] i 1

then the optimal threshold value should be 1.

J (k )

,

N-Policy for State-Dependent Batch Arrival Queueing System

225

7. Conclusion In this paper an optimal policy for l-stage service queue with batch arrival and state dependent rates is presented. Because of wide applications, especially in computer and communication networks, manufacturing process and transportation, Bernoulli schedule vacation incorporated in the model makes our study more realistic and more versatile in comparison to other existing models. Various operating characteristics have also been derived in explicit form by using probability-generating function approach which is computationally tractable as shown by taking numerical illustration. The optimal value of the threshold parameter N , that minimizes the expected total cost per unit time is determined which may give an insight to improve the grade of service by selection of appropriate system parameters.

References 1.

Anabosi, R. F. and Madan, K. C. (2003). A single server queue with two types of service, Bernoulli schedule server vacations and a single vacation policy. Pakistan Journal of Statistics, 19(3), 331-342.

2.

Artalejo, J. R. and Choudhury, G. (2004): Steady state analysis of an M/G/1 queue with repeated attempts and two-phase service. Quality Technology and Quantitative Management, 1(2), 189-199.

3.

Arumuganathan, R. and Jeyakumar, S. (2005). Steady state analysis of a bulk queue with multiple vacations, setup times with N-policy and closedown times. Applied Mathematical Modelling, 29, 972-986.

4.

Atencia, I. and Moreno, P. (2005). A single-server retrial queue with general retrial times and Bernoulli schedule. Applied Mathematics and Computation, 162, 855-880.

5.

Atencia, I., Bouza, G. and Rico, R. (2002). A queueing system with constant repeated attempts and Bernoulli schedule. Proceeding on Fifth International Conference Operations Research, La Habana.

6.

Bacot, J. B. and Dshalalow, J. H. (2001). A bulk input queueing system with batch-gated service and multiple vacation policy. Mathematical and Computer Modelling, 34(78), 873-886.

7.

Chae, K. E. and Lee, H. W. (1995). Mx/G/1 vacation models with N-policy heuristic interpretation of the mean working time. Journal of Operations Research Society, 46(2), 258-264.

8.

Choi, B. D. and Park, K. K. (1990). The M/G/1 retrial queue with Bernoulli schedule. Queueing Systems, 7, 219-228.

9.

Choudhury, G. (1997). A Poisson queue under N-policy with a general setup time. Indian Journal of Pure and Applied Mathematics, 28, 1595-1608.

10. Choudhury, G. (2007). A two phase batch arrival retrial queueing system with Bernoulli vacation schedule. Applied Mathematics and Computation, 188(2), 1455-1466. 11. Choudhury, G. and Madan, K. C. (2004). A two phase batch arrival queueing system with a vacation time under Bernoulli schedule. Applied Mathematics and Computation, 149, 337-349. 12. Choudhury, G. and Madan, K. C. (2005). A two-stage batch arrival queueing system with a modified Bernoulli schedule vacation under N-policy. Mathematical and Computer Modelling, 42, 71-85. 13. Choudhury, G. and Madhuchanda, P. (2004). A batch arrival queue with an

226

Jain and Agrawal

additional service channel under N-policy. Applied Mathematics and Computation, 156, 115-130. 14. Choudhury, G., Tadj, L. and Paul, M. (2007). Steady state analysis of an Mx/G/1 queue with two-phase service and Bernoulli vacation schedule under multiple vacation policy. Applied Mathematical Modelling, 31(6), 1079-1091. 15. Feng, W., Kowada, M. and Adachi, K. (1998). A two-queue model with Bernoulli service schedule and switching times. Queueing Systems, 30, 405-434. 16. Harris, C. M. and Marchal, W. G. (1988). State dependence in M/G/1 server vacation models. Operations Research, 36, 560-565. 17. Ke, J. C. (2003). The optimal control of an M/G/1 queueing system with server vacations, startup and breakdowns. Computer and Industrial Engineering, 44(4), 567-579. 18. Ke, J. C. (2006). On M/G/1 system under NT policies with breakdowns, startup and closedown. Applied Mathematical Modelling, 30, 49-66. 19. Ke, J. C. (2007). Batch arrival queues under vacation policies with server breakdowns and startup/closedown times. Applied Mathematical Modelling, 31(7), 1282-1292. 20. Keilson, J. and Servi, L. D. (1986). Oscillating random walk models for GI/G/1 vacation systems with Bernoulli schedule. Journal of Applied Probability, 23, 790- 802. 21. Kella, O. (1989). The threshold policy in the M/G/1 queue with server vacations. Naval Research Logistics Quarterly, 36, 111-123. 22. Lee, H. S. and Srinivasan, M. M. (1989). Control policies for the MX/G/1 queueing system. Management Sciences, 35, 708-721. 23. Lee, H. W., Lee, S. S. and Chae, K. C. (1994). Operating characteristics of MX/G/1 queue with N-policy. Queueing Systems, 15, 205-219. 24. Madan, K. C. (2000). On a single server queue with two stage general heterogeneous service and binomial schedule server vacations. Egypt Statistical Journal, 44, 39-55. 25. Madan, K. C. (2001). On a single server queue with two stage general heterogeneous service and deterministic server vacations. International Journal of System Sciences, 32, 837-844. 26. Madan, K. C. and Abu Al-Rub, A. (2004). Transient and steady state solution of a single server queue with modified Bernoulli schedule vacations based on exhaustive service and a single vacation policy. Revista Investigación Operacional, 25(2), 158-165. 27. Medhi, J. and Templeton, J. G. C. (1992). A Poisson input queue under N-policy and with a general start-up time. Computers and Operations Research, 19, 35-41. 28. Ramaswami, R. and Servi, L. D. (1988): The busy period of the M/G/1 vacation model with a Bernoulli schedule. Stochastic Modelling, 4(3), 507-521. 29. Selvam, D. D. and Sivasankaran, V. A. (1994). A two-phase queueing system with server vacations. Operations Research Letters, 15, 163-168. 30. Tadj, L. and Ke, J. C. (2005). Control policy of a hysteretic bulk queueing system. Mathematical and Computer Modelling, 41, 571-579. 31. Wang, K. H., Wang, T. Y. and Pearn, W. L. (2007). Optimal control of the N-policy M/G/1 queueing system with server breakdowns and general startup times. Applied Mathematical Modelling, 31(10), 2199-2212.

N-Policy for State-Dependent Batch Arrival Queueing System

227

Appendix A Proof of Theorem 1:

Multiplying equations (6)-(8) by appropriate powers of z and taking summation over all values of n and using equations (15) and (17) respectively, we get Pi ( x , z )

Pi (0, z )[1  Bi ( x )]e  ( Oi Oi X ( z )) x , x ! 0, 1 d i d l ,

(A1)

R (0, z )[1  V ( x )]e  ( Oi Oi X ( z ) x ) , x ! 0,

(A2)

R( x, z )

Again multiplying equations (10)-(11) by appropriate powers of z and taking summation over all values of n and using equation (16), we get l

l

P1 (0, z )[(1  p ) z 1 – Bi (Oi  Oi X ( z ))  pz 1 – Bi ( Oi  Oi X ( z ))

P1 (0, z )

i 1

i 1

f

N 1

n N

k 0

(A3)

uV (Ov  Ov X ( z ))]  O ¦ z n ¦ an  k Qk  OQ0 , where Bi (Oi  Oi X ( z )) ³0f e  ( Oi Oi X ( z )) x dBi ( x ), 1 d i d l is the z-transform of V (OV  OV X ( z )) ³0f e  ( OV OV X ( z )) x dV ( x ) is the z-transform of V .

Bi

and

Now using equations (14) and (A3), we get P1 (0, z ) (1  p ) z 1 P1 (0, z ) B1 (Oi  O1 X ( z ))  R (0, z )V ( OV  OV X ( z ))  OQ ( z )[ X ( z )  1], (A4) where O ¦ fn

N

z n ¦ uN 01 an u Qu OQ ( z )[ X ( z )  1]  OQ0 .

In the similar manner using equations (9), (12), (16), (18), we get Pi (0, z )

Pi 1 (0, z ) Bi 1 (Oi 1  Oi 1 X ( z )), 2 d i d l , R (0, z )

(A5)

z 1 pP1 (0, z ) B1 (Oi  O X ( z )),

(A6)

Now using equations (A5), (A6) in (A4) and after simplification, we obtain

P1 (0, z )

O zQ ( z )[1  X ( z )]

l

[((1  p )  pV (OV  O

O  Oi X ( z ))  z ]

.

(A7)

* V G ( z )) – Bi ( i i 1

Hence Pi ( z )

f

³ Pi ( x , z )dx

0

i

O zQ ( z [1  Bi (Oi  Oi X ( z ))] – B j (O j  O j X ( z ))

j 1 j z1 l

V X ( z )) – Bi ( i i 1

Oi [((1  P )  pV (OV  O

O  Oi X ( z ))  z ]

(A8) , 1 d i d l,

228

Jain and Agrawal

R( z )

f

³ R ( x , z )dx

0

l

O pQ ( z )[1  V (Ov  Ov X ( z ))]– Bi (Oi  Oi X ( z )) i 1 l

Ov [((1  p )  pV (Ov  Ov X ( z ))– Bi (Oi  Oi X ( z ))  z ]

(A9) .

i 1

In order to obtain the value of Q(z), equations (3) and (14) yield Qn \ 0[ n , n 1, 2, 3,..., ( N  1),

(A10)

where \ 0 is the normalizing constant. Now N 1

Q ( z ) \ 0 ¦ [n z n .

(A11)

n 0

Using normalizing condition (13), the value of normalizing constant \ 0 is obtained as 1 (1  U ) u N 1 , ¦ [n

\0

(A12)

n 0

where

O ª« E [ X ]{ pE [V ]  ¦ E [ Bi ]}º» l

U

i 1 ¬ ¼ . ª E [ X ]{O pE [V ]  l O E [ B ]}  1º  O ª E [ X ]{ pE [V ]  l E [ B ]}º ¦ i ¦ i v i «¬ »¼ «¬ »¼ i 1 i 1

ʳ

(A13)

From equations (A11) and (A12), we get Q ( z ) as N 1

Q ( z ) (1  U )

n ¦ [n z

n 0 N 1

¦ [n

.

(A14)

n 0

Here U is the utilization factor and ¦ nN 01 [ n is the mean number of batches arriving during an idle period. Let sn be the probability of idle period when there are n customers in the system, then by the conditioning on the number of arrivals in equation (A10), we obtain sn

Qn

[

N 1

n N 1

n 0

n 0

¦ Qn

¦ [n

,n

0, 1, 2, 3,......, ( N  1).

(A15)

The PGF of the number of units present in the system during an idle period {sn ; n 1, 2,......, N  1} is given by

0,

N 1

S (z )

N 1

¦ sn z

n 0

n

n ¦ [n z

n 0 N 1

¦ [n

n 0

.

(A16)

N-Policy for State-Dependent Batch Arrival Queueing System

229

Using equations (A14) and (A16), we get Q ( z ) (1  U )S ( z ),

(A17)

where S ( z ) be the additional queue size distribution due to N-policy.

Appendix B Proof of Theorem 3:

We employ embedded Markov chain technique to analyze the queue size distribution at departure epoch. Since there are j customers in the queue at departure point, if ( j  1) customers are present in the queue just before the departure, then f

f

0

0

{Pj ; j t 0} :0 (1  p ) ³ P1 ( x ) P1, j 1 ( x )dx :0 ³ X ( x ) R j ( x )dx ,

(B1)

where :0 is the normalizing constant. Define the probability generating function P  (z )

f

 j ¦ Pj z .

(B2)

j 0

Now multiplying (B1) by appropriate powers of z and summing over all the values of n and using (B2), we get f

f

0

0

P  ( z ) :0 (1  p ) z 1 ³ P1 ( x ) P1 ( x , z )dx :0 ³ X ( x ) R ( x , z )dx ,

(B3)

Using equations (A1) and (A2) in (B3), we get after simplification l

:0 O [1  [ E [ X ][(1  p )  pV (OV  OV X ( z )) – Bi (Oi  Oi X ( z ))]

P  (z )

l

N 1

i 1

n 0

i 1

u[ E [ X ]{OV pE [V ]  ¦ Oi E [ Bi ]}  1] u ¦ [ n z n l

l

[[ E [ X ]{OV pE [V ]  ¦ Oi E [ Bi ]}  1]  O[ E [ X ]{ pE [V ]  ¦ E [ Bi ]}]] i 1

l

i 1

.

(B4)

N 1

u [((1  p )  pV (OV  OV X ( z ) – Bi (Oi  Oi X ( z ))  z ] u ¦ [ n n 0

i 1

Now using the normalizing condition P  (1) 1 we get l

:0

l

[[ E [ X ]{OV pE [V ]  ¦ Oi E [ Bi ]}  1]  O[ E [ X ]{ pE [V ]  ¦ E [ Bi ]}]] i 1

i 1

O E[ X ]

Hence, we have P  ( z ) as given in equation (24).

.

(B5)

230

Jain and Agrawal

Authors’ Biographies: Dr. Madhu Jain is a Professor of Mathematics, Dr. B. R. Ambedkar University, Agra and presently faculty in the Department of Mathematics, IIT Roorkee, Roorkee, India. She received her M.Sc., M.Phil., Ph.D. and D.Sc. degrees in Mathematics from University of Agra. She has been a Gold Medalist of Agra University at M.Phil. level. There are more than 180 research publications in refereed International/National Journals and more than 15 books to her credit. She was recipient of the young scientific award, SERC visiting follow ship of Department of Science and Technology (India) and Career award of University Grants Commission (India). She has successfully completed six sponsored major research projects of Department of Science and Technology (India), University Grants Commission (India) and Council of Scientific and Industrial Research (India). Her current research interest includes Performance Modelling, Soft Computing, Bio-informatics, Reliability, Engineering and Queueing Theory. Twenty five candidates have received their Ph.D. degrees under her supervision. She has visited more than 25 reputed Universities/Institutes in USA, Canada, UK, Germany, France, Holland, and Belgium. She has participated and presented her research works in more than 30 Internationals and 75 Nationals Conferences/Seminars. Praveen Kumar Agrawal is a research scholar in the Department of Mathematics at Institute of Basic Science, Khandari, Dr. B. R. Ambedkar University, Agra, India. After completing his Master’s degree in first division, he received his M. Phil. degree in Mathematics from Dr. B. R. Ambedkar University, Agra, India. His research interest includes Queueing Theory and Reliability Analysis. He has participated and presented his research work in 11 International/National Conferences/Seminars.