Analysis of a bulk queue with unreliable server, immediate feedback, N-policy, Bernoulli schedule multiple vacation and stand-by server

Ain Shams Engineering Journal xxx (xxxx) xxx

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Engineering Physics and Mathematics

Analysis of a bulk queue with unreliable server, immediate feedback, N-policy, Bernoulli schedule multiple vacation and stand-by server G. Ayyappan, S. Karpagam ⇑ Department of Mathematics, Pondicherry Engineering College, Puducherry, India

a r t i c l e

i n f o

Article history: Received 2 November 2018 Revised 18 January 2019 Accepted 5 March 2019 Available online xxxx Keywords: Unreliable server Stand-by server Loss and feedback N-policy Bernoulli schedule multiple vacation policy

a b s t r a c t We have studied the behaviour of a non-Markovian bulk service queueing model with unreliable server, stand-by server, loss and feedback, N-policy and Bernoulli schedule multiple vacation. The stand-by server is employed only during the main server’s repair period. At each service completion the main server may have an option to go for a short vacation. After each short vacation completion or service completion, if the system size is less than ‘a’, the server takes a sequence of vacation until the queue size reaches at least ‘N’. The concept of loss and feedback is also incorporated in this model. The PGF of queue size and some important performance measures are derived. An extensive numerical result is illustrated. Ó 2019 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Ain Shams University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/ by-nc-nd/4.0/).

1. Introduction Many authors have contributed to the theory of queues with batch arrival and bulk service. Neuts [1] first introduced the GBSR, which states that ‘‘the server will start the service only when minimum ‘a’ units are present in the queue, and maximum capacity of service is ‘b’ ”. The book by Chaudhry and Templeton [2] explains the bulk service queueing systems in-depth. The real time system’s efficiency and availability are balanced by the queueing models with stand-by’s support. The queueing system maintain the smooth functioning of the system that are supported with the provision of stand-by. The main applications of queueing models with stand-by are noticed in the field of distribution and service systems, computer and communication systems, manufacturing systems, etc. Kolledath et al. [3] made a excellent survey on queueing systems with stand-by support. Khalaf [4] studied the queuing system together with a stand-by server that works when main server is under repair or on vacation. Khalaf et al. [5] studied the stand-by

⇑ Corresponding author. E-mail address: [email protected] (S. Karpagam). Peer review under responsibility of Ain Shams University.

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server queuing system with random breakdown, vacation, and delay time. Madan [6] reviewed a stand-by server with three fluctuating modes, balking and random breakdown. Murugeswari and Maragatha Sundari [7] discussed a queuing model with standby server, bulk arrival and compulsory server vacation. The concepts of loss of a queueing model was discussed by many authors. The concept of feedback was introduced by Takacs [8]. ‘‘If the server is busy at the time of the arrival of customer, then due to impatient behaviour of the customer, customer may or may not join the queue. This is called loss in queueing theory”. ‘‘At the completion of service, the customers may join the queue with probability ‘q’, if they are dissatisfied with the service or with probability ‘1
q’ they may leave the system. This is called feedback in queueing theory”. Ayyappan et al. [9] discussed loss and feedback queue with batch service and multiple vacation, Ayyappan et.al [10] studied loss and feedback of Markovian Retrial queueing system with non-pre-emptive priority service. Vignesh et al. [11] analysed a non-Markovian batch arrival queueing model with multi stages of service of restricted admissibility, feedback service and three optional vacations in production and manufacturing. At each completion of service the server may have a option to go for a vacation or start new service. This type of model is called Bernoulli vacation model which was introduced by Kelsion et al. [12]. Recently, Ayyappan et al. [13,14] investigated single server non-Markovian queueing system with Bernoulli schedule vacation and Goswami [15] analysis Bernoulli schedule working vacations and vacation interruption. A batch arrival unreliable server delaying repair queue with two phases of service and Bernoulli vacation

https://doi.org/10.1016/j.asej.2019.03.008 2090-4479/Ó 2019 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Ain Shams University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article as: G. Ayyappan and S. Karpagam, Analysis of a bulk queue with unreliable server, immediate feedback, N-policy, Bernoulli schedule multiple vacation and stand-by server, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.03.008

2

G. Ayyappan, S. Karpagam / Ain Shams Engineering Journal xxx (xxxx) xxx

under multiple vacation policy was analysed by Choudhury and Deka [16]. Bouchentouf and Yahiaoui [17] investigated a feedback queueing system with reneging and retention of reneged customers, multiple working vacations and Bernoulli schedule vacation interruption. Strategic behaviour and optimal strategies in an M/G/1 queue with Bernoulli vacations was examined by Zhu and Wang [18]. We consider a soft flow dyeing machine with 1080 kg capacity in dying industries. This machine is used for dyeing the cloth by using chemicals. The dying machine is operated to the maximum capacity of 1080 kg called its upper limit (b) and a minimum capacity of 800 kg called its lower limit (a). If the supply of cloth is less than 800 kg, dying process is not possible. During this process, sometimes the machine may failure then the dying process is stopped and the machine sent for repair. After repair completion the dying process continued. If the quantity of cloth for dying is less than the required batch quantity, the machine wait until it finds enough quantity for processing. We consider a system which has, apart from the regular channel, the provision of a stand-by which is employed only during the repair times of the regular service channel. The stand-by may not be as efficient as the main service channel, but still may contribute a lot to avoid the queue becoming out of bounds during the failure times of the main service channel. Such situations are not uncommon. The stand-by support is essential in order to achieve high reliability performance and availability of every queueing system that are operating in machine environment.

ð0Þ

ð0Þ

S0v ðtÞ; S0u ðt Þ; V ð1Þ ðtÞ and V ð2Þ ðtÞ represent the stand-by server’s remaining service time, main server’s remaining service time, remaining short vacation and multiple vacation time at ‘t’ respectively. e e ð1Þ ð/Þ and V e ð2Þ ð/Þ represent the Laplace Stieltjes S v ð/Þ; e S u ð/Þ; V

transform (LST) of Sv ; Su ; V ð1Þ and V ð2Þ respectively. For the further development, we define the following. rðtÞ ¼ 1; 2; 3; 4 and 5 denote main server busy, on short vacation, on multiple vacation, stand-by server is idle and busy respectively. N 1 ðt Þ and N 2 ðtÞ = number of customers in service station and queue at time ‘‘t” respectively. we define the probabilities as follows: Se ðt ÞDt ¼ Pr fN 2 ðtÞ ¼ e; rðt Þ ¼ 4g;0 6 e 6 a
1; n o Mr;e ðw; tÞDt ¼ Pr N1 ðt Þ ¼ r; N 2 ðtÞ ¼ e; w 6 S0u ðt Þ 6 w þ Dt; rðt Þ ¼ 1 ; a 6 r 6 b; e P 0;

n o ð0 Þ V ðe1Þ ðw; t ÞDt ¼ Pr N 2 ðt Þ ¼ e; w 6 V ð1Þ ðt Þ 6 w þ Dt; rðt Þ ¼ 2 ; e P 0; n o ð0 Þ V ðe2Þ ðw; t ÞDt ¼ Pr N 2 ðt Þ ¼ e; w 6 V ð2Þ ðt Þ 6 w þ Dt; rðt Þ ¼ 3 ; e P 0; n o Lr;e ðw; t ÞDt ¼ Pr N 1 ðt Þ ¼ r;N 2 ðt Þ ¼ e;w 6 S0v ðt Þ 6 w þ Dt; rðt Þ ¼ 5 ;a 6 r 6 b;e P 0:

3. Queue size distribution The steady state equations are: The main server is in the busy state Md;0 ðw
Dt;t þ DtÞ ¼ ð1
kDtÞð1
aDtÞMd;0 ðw; tÞ

b X þ ð1

ÞkMd;0 ðw; tÞDt þ ð1
pÞq Mr;d ð0; tÞsu ðwÞDt r¼a Z 1 Ld;0 ðy; tÞdysu ðwÞDt þ pMd;0 ð0; tÞsu ðwÞDt þ g

2. Model description In this paper we consider the arrival follows a compound Poisson process with arrival rate ‘k’. Both server’s service time, short vacation and multiple vacation time follows general distributions. Breakdown and repair times of main server follow exponential distributions with rate ‘a’ and ‘g’ respectively. The probability of an arriving customer enter into the system with probability
and balk the system with probability ð1

Þ. At each service completion if the batch of customers is dissatisfied with his service and get new service again immediately with probability p otherwise the satisfied customer leave the system with probability ð1
pÞ. The main server is brokendown at any instant during the service. In such case, he immediately sent for repair and current batch service is exchanged to stand-by server and he starts service to that batch afresh. The stand-by server is stay in the system until the main server’s repair completion. At the instant of main server’s repair completion, if the stand-by is busy then the current service is interrupted and the batch of customers is exchanged to the main server who starts service to that batch afresh. At each service completion (served by the main server) the server may have an option to go for a short vacation with probability ‘p’. otherwise he may retain the system with probability ‘qð¼ 1
pÞ’, with the same probability he goes for a multiple vacation if the queue length is less than ‘a’. This multiple vacation process was repeated until he finds minimum ‘N’ in the queue. After this vacation time he starts the service to a batch of ‘b’ customers.

0

ð1Þ

þ V d ð0;t Þsu ðwÞDt; a 6 d 6 b;

Dividing both sides by Dt , and letting the limit Dt ! 0 , the steady state equation is obtained as


M 0d;0 ðwÞ ¼
ðk þ aÞMd;0 ðwÞ þ ð1

ÞkM d;0 ðwÞ þ ð1
pÞq þg

b X Mr;d ð0Þsu ðwÞ þ pMd;0 ð0Þsu ðwÞ r¼a

Z

1 0

ð1Þ

Ld;0 ð yÞdysu ðwÞ þ V d ð0Þsu ðwÞ; a 6 d 6 b;

We use the following notations: Let the batch size random variable be X and the probability of ‘k’ customers arrive in a batch be g k , whose PGF is X ðzÞ. Sv ð:Þ; Su ð:Þ; V ð1Þ ð:Þ and V ð2Þ ð:Þ represent the (CDF) of stand-by server’s service time, main server’s service time, short vacation time and multiple vacation time with corresponding pdf are sv ðxÞ; su ðxÞ; v 1 ðxÞ and v 2 ðxÞ respectively.

ð2Þ

Similarly, the remaining steady state equations are obtained as


M 0d;e ðwÞ ¼
ðk þ aÞM d;e ðwÞ þ ð1

ÞkM d;e ðwÞ e X þ pM d;e ð0Þsu ðwÞ þ
M d;e
k ðwÞkg k k¼1 Z 1 þg Ld;e ð yÞdysu ðwÞ; e P 1; a 6 d 6 b
1;

ð3Þ

0


M 0b;e ðwÞ ¼
ðk þ aÞM b;e ðwÞ þ ð1

ÞkM b;e ðwÞ þ pMb;e ð0Þsu ðwÞ e b X X þ
M b;e
k ðwÞkg k þ ð1
pÞq Mr;bþe ð0Þsu ðwÞ r¼a k¼1 Z 1 ð1Þ þg Lb;e ð yÞdysu ðwÞ þ V bþe ð0Þsu ðwÞ; 0

2.1. Notations

ð1Þ

1 6 e 6 N
b
1;

ð4Þ


M 0b;e ðwÞ ¼
ðk þ aÞM b;e ðwÞ þ ð1

ÞkM b;e ðwÞ þ pMb;e ð0Þsu ðwÞ e b X X þ
M b;e
k ðwÞkg k þ ð1
pÞq Mr;bþe ð0Þsu ðwÞ þg þ

k¼1 Z 1

Lb;e ð yÞdysu ðwÞ 0 ð2 Þ V bþe ð0Þsu ðwÞ; e P N

r¼a

þ

ð1Þ V bþe ð0Þsu ðwÞ;


b;

ð5Þ

Please cite this article as: G. Ayyappan and S. Karpagam, Analysis of a bulk queue with unreliable server, immediate feedback, N-policy, Bernoulli schedule multiple vacation and stand-by server, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.03.008

3

G. Ayyappan, S. Karpagam / Ain Shams Engineering Journal xxx (xxxx) xxx ð2Þ

The Laplace-Stieltjes Transform of Mr;e ; Lr;e ; V ðe1Þ and V l;e are defined

The stand-by server is in busy state


L0d;0 ðwÞ ¼
ðk þ gÞLd;0 ðwÞ þ ð1

ÞkLd;0 ðwÞ

as,

b X þ ð1
pÞ Lr;d ð0Þsv ðwÞ þ pLd;0 ð0Þsv ðwÞ

þa

r¼a

Z

1

0

þ

M d;0 ð yÞdysv ðwÞ

Z 1 Z 1   M r;e ð/Þ ¼ e
/w Mr;e ðwÞdw; Lr;e ð/Þ ¼ e
/w Lr;e ðwÞdw; 0 0 Z Z 1 1   ð2Þ ð1 Þ ð2 Þ V e ð/Þ ¼ e
/w V ðe1Þ ðwÞdw; V l;e ð/Þ ¼ e
/w V l;e ðwÞdw: 0

a
1 X

Sk kg d
k sv ðwÞ; a 6 d 6 b;

ð6Þ

k¼0

0

While appling the LST to the Eqs. (2)–(14), we get,

e d;0 ð/Þ
M d;0 ð0Þ ¼ ðk þ aÞ M e d;0 ð/Þ
ð1

Þk M e d;0 ð/Þ /M b X
ð1
pÞq Mr;d ð0Þe S u ð/Þ


L0d;e ðwÞ ¼
ðk þ gÞLd;e ðwÞ þ ð1

ÞkLd;e ðwÞ e X þ
Ld;e
k ðwÞkg k þ pLd;e ð0Þsv ðwÞ þa

k¼1 Z 1 0

r¼a

S u ð/Þ
g
pM d;0 ð0Þe

M d;e ð yÞdysv ðwÞ; e P 1; a 6 d 6 b
1;

0

e X





0

M b;e ð yÞdysv ðwÞ;

ð18Þ

e P 1;

ð8Þ e b;e ð/Þ
ð1

Þk M e b;e ð/Þ e b;e ð/Þ
M b;e ð0Þ ¼ ðk þ aÞ M /M e X e b;e
k ð/Þkg S u ð/Þ

M
pM b;e ð0Þe k

The main server is in the short vacation state ð1Þ 0


V 0 ðwÞ ¼
kV 0 ðwÞ þ ð1
pÞp ð1 Þ

b X Mr;0 ð0Þv 1 ðwÞ;

ð9Þ
g

r¼a

0


V ðe1Þ ðwÞ ¼
kV ðe1Þ ðwÞ þ ð1
pÞp

b X

e X

r¼a

k¼1

Mr;e ð0Þv 1 ðwÞ þ

ð1Þ

V e
k ðwÞkg k ; e P 1;

The main server is in the multiple vacation state ð2Þ 0

ð2 Þ

b X M r;0 ð0Þv 2 ðwÞ r¼a

þ gS0 v 2 ðwÞ þ

v

ð1 Þ V 0 ð0Þ 2 ðwÞ

0


V ðe2Þ ðwÞ ¼
kV ðe2Þ ðwÞ þ ð1
pÞq

þ V 0 ð0Þv 2 ðwÞ ð2 Þ

ð11Þ

þ

v

ð19Þ

e b;e ð/Þ
M b;e ð0Þ ¼ ðk þ aÞ M e b;e ð/Þ
ð1

Þk M e b;e ð/Þ /M e X e b;e
k ð/Þkg S u ð/Þ

M
pM b;e ð0Þe k Z

k¼1 1

Lb;e ð yÞdye S u ð/Þ

r¼a

ð12Þ

ð2 Þ S u ð/Þ; e P N
b;
V bþe ð0Þe

ð20Þ

/e L d;0 ð/Þ
Ld;0 ð0Þ ¼ ðk þ gÞe L d;0 ð/Þ
ð1

Þke L d;0 ð/Þ ð13Þ




a
1 X Sk kg d
k e S v ð/Þ S v ð/Þ
pLd;0 ð0Þe k¼0

b X S v ð/Þ
ð1
pÞ Lr;d ð0Þe

k¼1 e X ð2Þ V e
k ðwÞkg k ; e P N:

r¼a ð1Þ S u ð/Þ; 1 6 e 6 N
b
1;
V bþe ð0Þe

b X ð1 Þ
ð1
pÞq Mr;bþe ð0Þe S u ð/Þ
V bþe ð0Þe S u ð/Þ

0


V ðe2Þ ðwÞ ¼
kV ðe2Þ ðwÞ þ V ðe2Þ ð0Þv 2 ðwÞ e X ð2 Þ V e
k ðwÞkg k ; a 6 e 6 N
1 þ 0

Lb;e ð yÞdye S u ð/Þ

0

v

V ðe1Þ ð0Þ 2 ðwÞ þ V ðe2Þ ð0Þ 2 ðwÞ e X ð2 Þ V e
k ðwÞkg k ; 1 6 e 6 a
1 k¼1


V ðe2Þ ðwÞ ¼
kV ðe2Þ ðwÞ þ

k¼1 1

b X
ð1
pÞq Mr;bþe ð0Þe S u ð/Þ


g

b X M r;e ð0Þv 2 ðwÞ þ gSe v 2 ðwÞ r¼a

þ

Z 0

ð10Þ


V 0 ðwÞ ¼
kV 0 ðwÞ þ ð1
pÞq

e d;e
k ð/Þkg ; e P 1; a 6 d 6 b
1; M k

k¼1

r¼a

k¼0

ð17Þ

e d;e ð/Þ
ð1

Þk M e d;e ð/Þ e d;e ð/Þ
M d;e ð0Þ ¼ ðk þ aÞ M /M Z 1
g Ld;e ð yÞdye S u ð/Þ
pMd;e ð0Þe S u ð/Þ

k¼1

þa

Ld;0 ð yÞdye S u ð/Þ

ð1Þ
V d ð0Þe S u ð/Þ; a 6 d 6 b;

ð7Þ

a
1 b X X Sk kg bþe
k sv ðwÞ þ ð1
pÞ Lr;bþe ð0Þsv ðwÞ þ 1

1

0


L0b;e ðwÞ ¼
ðk þ gÞLb;e ðwÞ þ ð1

ÞkLb;e ðwÞ e X þ
Lb;e
k ðwÞkg k þ pLb;e ð0Þsv ðwÞ

Z

Z

ð14Þ

k¼1


a

Z

r¼a 1

0

M d;0 ð yÞdye S v ð/Þ; a 6 d 6 b;

ð21Þ

Stand-by server is in idle state

ðk þ gÞS0 ¼ ð1
pÞ

b X Lr;0 ð0Þ;

ð15Þ

r¼a

L d;e ð/Þ
ð1

Þke L d;e ð/Þ /e L d;e ð/Þ
Ld;e ð0Þ ¼ ðk þ gÞe Z 1 e
a M d;e ð yÞdy S v ð/Þ
pLd;e ð0Þe S v ð/Þ 0

b e X X ðk þ gÞSe ¼ ð1
pÞ Lr;e ð0Þ þ Se
k kg k ; 1 6 e 6 a
1; r¼a

k¼1

ð16Þ





e X

e L d;e
k ð/Þkg k ; e P 1; a 6 d 6 b
1;

ð22Þ

k¼1

Please cite this article as: G. Ayyappan and S. Karpagam, Analysis of a bulk queue with unreliable server, immediate feedback, N-policy, Bernoulli schedule multiple vacation and stand-by server, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.03.008

4

G. Ayyappan, S. Karpagam / Ain Shams Engineering Journal xxx (xxxx) xxx

  S v ð g ð z ÞÞ e ð/
g ðzÞÞ 1
pe L d ðz; /Þ " b a
1   X X e d ðz; 0Þ þ ¼ e S v ð g ðzÞÞ
e S v ð/Þ ð1
pÞ Lr;d ð0Þ þ a M Sk kg

/e L b;e ð/Þ
Lb;e ð0Þ ¼ ðk þ gÞe L b;e ð/Þ
ð1

Þke L b;e ð/Þ
ð 1
pÞ

b X Lr;bþe ð0Þe S v ð/Þ
pLb;e ð0Þe S v ð/Þ r¼a





e X

e L b;e
k ð/Þkg k
a

Z

k¼1

r¼a

1

0

ð23Þ

k¼0 b X e ð1Þ ð/Þ
ð1
pÞp M r;0 ð0Þ V e ð1Þ ð/Þ; e ð1Þ ð/Þ
V ð1Þ ð0Þ ¼ k V /V 0 0 0

k¼0

e ð1Þ ð/Þkg ; e P 1; V k e
k

e¼0 r¼a

ð34Þ

e¼b

b   X e ð1Þ ðz; /Þ ¼ V e ð1 Þ ð h ð zÞ Þ
V e ð1Þ ð/Þ ð1
pÞp Mr ðz; 0Þ; ð/
hðzÞÞ V

r¼a e X

r¼a

# a
1 1 X X þ k Sk zk g e
k ze
k ;

r¼a




ð33Þ

    S v ð g ðzÞÞ
ð1
pÞe ð/
g ðzÞÞ zb 1
pe S v ð g ð zÞÞ e L b ðz; /Þ " " # b
1 b
1 X b   X X e b ðz; 0Þ þ ð1
pÞ S v ð /Þ z b a M Lr ðz; 0Þ
Lr;e ð0Þze ¼ e S v ð g ðzÞÞ
e

ð24Þ

b X e ð1Þ ð/Þ
ð1
pÞp M r;e ð0Þ V e ð1Þ ð/Þ e ð1Þ ð/Þ
V ð1Þ ð0Þ ¼ k V /V e e e

r¼a

ð25Þ

ð35Þ

k¼1

e ð2Þ ðz; /Þ ð/
hðzÞÞ V

b X e ð2Þ ð/Þ
ð1
pÞq Mr;0 ð0Þ V e ð2Þ ð/Þ e ð2Þ ð/Þ
V ð2Þ ð0Þ ¼ k V /V 0 0 0



r¼a

e ð2Þ ð/Þ e ð2 Þ ð h ð zÞ Þ
V ¼ V

e ð2Þ ð/Þ
V ð1Þ ð0Þ V e ð2Þ ð/Þ
gS0 V 0 ð2 Þ e ð2Þ ð/Þ
V 0 ð0Þ V

;

k¼0

a 6 d 6 b
1;

Mb;e ð yÞdye S v ð/Þ

a
1 X S v ð/Þ; e P 1; Sk kg bþe
k e


# d
k

" N
1  X

a
1 X

e¼0

e¼0

v ðe2Þ ð0Þze þ

!#

ð26Þ þ gSe ze þ v ðe1Þ ð0Þze

b X e ð2Þ ð/Þ
ð1
pÞq Mr;e ð0Þ V e ð2Þ ð/Þ e ð2Þ ð/Þ
V ð2Þ ð0Þ ¼ k V /V e e e r¼a

e ð2Þ ð/Þ
V ð1Þ ð0Þ V e ð2Þ ð/Þ
V ð2Þ ð0Þ V e ð2Þ ð/Þ
g Se V e e e X ð2Þ e ð/Þkg ; 1 6 e 6 a
1 V ð27Þ
k e
k

ð1
pÞq

b X M r;e ð0Þze r¼a

:

ð36Þ

f ðzÞ ¼
k þ a

kX ðzÞ; g ðzÞ ¼
k þ g

kX ðzÞ; hðzÞ ¼ k
kX ðzÞ: 4. Probability generating function of queue size

k¼1

e ð2Þ ð/Þ
V ð2Þ ð0Þ ¼ k V e ð2Þ ð/Þ
V ð2Þ ð0Þ V e ð2Þ ð/Þ /V e e e e e X ð2Þ e ð/Þkg ; a 6 e 6 N
1 V
k e
k

4.1. The PGF of the queue size at an arbitrary time epoch

ð28Þ

Let PðzÞ be the PGF of the queue size at an arbitrary time epoch. Then,

ð29Þ

P ð zÞ ¼

k¼1

e ð2Þ ð/Þ
V ð2Þ ð0Þ ¼ k V e ð2Þ ð/Þ
/V e e e

e X

e ð2Þ ð/Þkg ; e P N V k e
k

b X

e d ðz; 0Þ þ M

d¼a

k¼1

b X

e e ð1Þ ðz; 0Þ þ V e ð2Þ ðz; 0Þ þ SðzÞ: L d ðz; 0Þ þ V

d¼a

ð37Þ

Let us define following Probability Generating Function (PGF):

e d ðz; /Þ ¼ M

1 X

e d;e ð/Þze ; Md ðz; 0Þ ¼ M

e¼0

e L d ðz;/Þ ¼

1 X

(

e¼0

1 X e Ld;e ð0Þze ; a 6 d 6 b; L d;e ð/Þze ; Ld ðz;0Þ ¼

e¼0 1 X

e¼0 1 X

e¼0

e¼0

ðiÞ V~ e ð/Þze V ðiÞ ðz; 0Þ ¼

e ðiÞ ðz;/Þ ¼ V

By substituting / ¼ 0 in Eqs. (31)–(36) then Eq. (37) becomes

1 X M d;e ð0Þze ; a 6 d 6 b;

A1 ð z Þ

ð30Þ

b
1 X 

"

d¼a

  e d ðz; /Þ S u ð f ðzÞÞ M ð/
f ðzÞÞ 1
pe " # b   X ð1Þ ¼ e S u ð f ð zÞ Þ
e L d ðz; 0Þ ; S u ð/Þ ð1
pÞq M r;d ð0Þ þ V ð0Þ þ ge

ð31Þ

e¼0

r¼a

d



qd þ #

d¼a

a
1 X

! Sk kg d
k

k¼0

n¼0

k¼0

; ð38Þ

where

cd ¼ md þ v d ; ð1 Þ

md ¼

b X M r;d ð0Þ;

v ðd1Þ ¼ V ðd1Þ ð0Þ;

r¼a

v ðd2Þ ¼ V ðd2Þ ð0Þ;

qd ¼

b X Lr;d ð0Þ r¼a

and the expressions for A1 ðzÞ; A2 ðzÞ; A3 ðzÞ; A4 ðzÞ and Y 1 ðzÞ are defined in Appendix A.

b
1 N
1 b
1 b
1 X b X X X X V ðe1Þ ð0Þze
V ðe2Þ ð0Þze þ ð1
pÞq M r ðz; 0Þ
M r;e ð0Þze e¼0

z
z b

hðzÞY 1 ðzÞ

d

r¼a




b
1 X 

n¼0

P ð zÞ ¼

By multiplying Eqs. (17)–(29) by appropriate powers ‘z’ and then taking summation over ‘e’ ðe ¼ 0 to 1Þ, using (30) and after some mathematical manipulations, we get,

    e b ðz; /Þ S u ð f ðzÞÞ
ð1
pÞqe ð/
f ðzÞÞ zb 1
pe S u ð f ðzÞÞ M  h ð1 Þ b e e e ¼ S u ð f ðzÞÞ
S u ð/Þ z g L b ðz; 0Þ þ V ðz; 0Þ þ V ð2Þ ðz; 0Þ

d

a
1 N
1 a
1   X X X e ð2Þ ðhðzÞÞ A3 ðzÞ c n zn þ v ðn2Þ zn þ A4 ðzÞ Sk zk þ 1
V

V ðeiÞ ð0Þze ;i ¼ 1;2:

a 6 d 6 b
1;



z
z c d þ A2 ð z Þ b

!# ;

e¼0 r¼a

ð32Þ

4.2. Steady state condition The PGF of the queue length has to satisfy P ð1Þ ¼ 1, for that we apply L’ Hopital’s rule and evaluating limz!1 P ðzÞ, then equating to

Please cite this article as: G. Ayyappan and S. Karpagam, Analysis of a bulk queue with unreliable server, immediate feedback, N-policy, Bernoulli schedule multiple vacation and stand-by server, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.03.008

5

G. Ayyappan, S. Karpagam / Ain Shams Engineering Journal xxx (xxxx) xxx b
1  X  b  hðzÞK 1 ðzÞ 1
e S u ð f ð z ÞÞ z
zd md

1, we have, H ¼ H13 . where cd ; qd ; v d and Sd represents the ‘d’ number of customers in the queue, it follows that H > 0. The condition Pð1Þ ¼ 1 is satisfied iff H13 > 0. If ð2Þ

q¼

bag

h

d¼a b
1 a
1   X X  b  þhðzÞ 1
e S v ð g ð z ÞÞ K 2 ð z Þ z
zd qd þ Sk kg d
k

h i   
kX 1 ða þ gÞ eS u ðaÞ
1
pð1
pÞeS u ðaÞV 11 eS v ðgÞ
1         i e S u ðaÞ
1 peS v ðgÞ
1 þ eS v ðgÞ
1 þ eS v ðgÞ
1 peS u ðaÞ
1 þ eS u ðaÞ
1

then q < 1 is the condition stability condition for our model under consideration.

d¼a

!

k¼0

a
1  h  i X e ðhðzÞÞ K 3 ðzÞ þ hðzÞK 1 ðzÞ e þ 1
V S u ð f ð z ÞÞ
1 ðm n z n þ v n z n Þ n¼0

P ð zÞ ¼

h h  i   e ðhðzÞÞ
g ðzÞhðzÞ 1
e S v ð g ð z ÞÞ K 2 ð z Þ þSðzÞ JðzÞ hðzÞ þ g 1
V  i e ðhðzÞÞK 3 ðzÞ 1
e S u ð f ð z ÞÞ þghðzÞ V

ð40Þ

Eq. (38) has ‘2b þ N’ unknowns c0 ; c1 ; . . . ; cb
1 ; qa ; . . . ; qb
1 ; S0 ;

S1 ; . . . ; Sa
1 and v 0 ; v 1 ; . . . ; v N
1 . We can express v d in terms of cd and Sd such that numerator has 2b constants only. Now Eq. (38) involves only ‘2b’ unknowns. By Rouche’s theorem, it can be proved that Y 1 ðzÞ has 2b
1 zeros inside and one on the unit circle jzj ¼ 1. Since PðzÞ is analytic within and on the unit circle, the numerator must vanish at these points, which gives 2b equations in 2b unknowns. We solve these equations by appropriate numerical methods. ð2Þ

ð2Þ

ð2Þ

which coincides with Ayyappan et al. [20] and the expressions K 1 ðzÞ; K 2 ðzÞ, and K 3 ðzÞ are defined in Appendix A. 5. Some performance measures 5.1. Expected queue length Mean number of customers in the queue is obtained by differentiating PðzÞ at z ¼ 1 and is given by



f 1 X;Su ;Sv ;V

4.4. Result-1

v is the probability that at the main server’s, multiple vacation completion epoch, there are ‘d’ ð0 6 d 6 N
1Þ customers in the queue, it can be expressed as sum of the probabilities of ‘d’ customers in the queue during stand-by server’s idle and main server’s busy period as, n X

bn þ

Kn ¼

þf 2

" # b
1  X ½bðb
1Þ
dðd
1Þcd

þf 3

b
1 a
1 X X X;Su ;Sv ; V ð1Þ ðbðb
1Þ
dðd
1ÞÞ qd þ Sk kg d
k



þf 4



b
1 a
1 X X X;Su ;Sv ; V ð1Þ ðb
dÞ qd þ Sk kg d
k d¼a

k¼0

; n ¼ 1; 2; . . . ; a
1;

þf 6

a
1 N
1  X X X;Su ;Sv ; V ð1Þ ; V ð2Þ cn þ v ðn2Þ



a
1 N
1  X X ð2Þ X;Su ;Sv ; V ð1Þ ; V ð2Þ ncn þ nv n

n¼0

n¼0

and b0d s represents the probabilities of ‘d’ customers arrive during multiple vacation.



þf 7 X;Su ;Sv ; V

ð1 Þ

;V

Let wn ¼

d¼0

bn
d þ

a
1
d P

"

n
a X

wn þ

v

ð 2Þ n

¼

d¼1 1
b0

the expressions for

# ; n ¼ a þ 1;. . . ; N
1; where v

ð2Þ a

wa ¼ 1
b 0

P ð zÞ ¼

b
1 X  d¼a

s are defined in Appendix B.

By Little’s formula,

EðW Þ ¼

Case 1: When there is no breakdown, loss and feedback, short vacation (p = 0) and N-policy then Eq. (38) becomes e S u ðhðzÞÞ
1

0 fi

5.2. Expected waiting time

ð2Þ

4.6. Particular case



n¼0

ð41Þ

K j bn
j
d ðcd þ gSd Þ; n ¼ a; a þ 1; . . . ;N
1; then

bd v n
d

n¼0

a
1 a
1 X  X Sn þ f 8 X;Su ;Sv ; V ð1Þ ; V ð2Þ nSn

3ðH13 Þ2

!

j¼0

!

n¼0

EðQ Þ ¼

4.5. Result-2 aP
1

ð2 Þ

!

!



n¼0

bd K n
d

!

k¼0

" # b
1  X ð1 Þ X;Su ;Sv ; V ðb
dÞcd d¼a

þf 5

d¼1 1
b0



d¼a

b0 K d ðgSn
d þ cn
d Þ; n ¼ 0; 1; 2; . . . ; a
1; where K 0 ¼ 1
b ; 0

d¼0 n X

ð1 Þ

d¼a

ð2Þ d

v ðn2Þ ¼

;

hðzÞK 3 ðzÞ

4.3. Computational aspects

ð2Þ

!

! a
1  X    e ðhðzÞÞ
1 zb
zd c d þ zb
1 V ðmn zn þ v n zn Þ   ð
hðzÞÞ zb
e S u ðhðzÞÞ

n¼0

EðQ Þ ; kEð X Þ

ð42Þ

6. Numerical example A numerical example is examined with the following assumptions:

; ð39Þ

which coincides with Senthilnathan et al. [19]. Case 2: When there are no loss and feedback, short vacation and N-policy then Eq. (38) becomes

1. Batch arrival follows Geometric distribution with mean 2. 2. Service time follows Erlang-2 distribution (for both servers). 3. Main server’s both vacations are exponential with parameters c ¼ 2 and f ¼ 5. 4. Let main servers service rate be l1 and l2 be stand-by server’s service rate respectively.

Please cite this article as: G. Ayyappan and S. Karpagam, Analysis of a bulk queue with unreliable server, immediate feedback, N-policy, Bernoulli schedule multiple vacation and stand-by server, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.03.008

6

G. Ayyappan, S. Karpagam / Ain Shams Engineering Journal xxx (xxxx) xxx

The numerical technique was used to compute the unknown probabilities of the queue size distribution. MATLAB software was used for finding the zeros of the function Y 1 ðzÞ and solving the simultaneous equations. The mean queue length and waiting time are calculated for both server’s service rate, various arrival rate and main server’s repair rate and the results are tabulated (see Figs. 1–4).

Fig. 4. Main server’s repair rate (vs) Performance measures.

 From Table 1, it is investigated that (q, E(Q) and E(W)) increases as k increases.  From Table 2–4 it is found that (q, E(Q) and E(W)) decreases as l1 l2 and g respectively. Fig. 1. Arrival rate (vs) Performance measures.

Table 1 Arrival rate vs performance measures. Let a ¼ 5; b ¼ 8; N ¼ 10; a ¼ 1; g ¼ 2; c ¼ 2 and f ¼ 5.

l1 ¼ 10; l2 ¼ 7;

k

q

EðQ Þ

EðW Þ

3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 7.00 7.50 8.00 8.50

0.0923 0.1077 0.1231 0.1385 0.1538 0.1692 0.1846 0.2000 0.2154 0.2308 0.2462 0.2615

11.0276 17.0081 24.4203 33.3803 44.0056 56.4542 70.9384 87.7208 107.125 129.502 155.417 185.412

1.8379 2.4297 3.0525 3.7089 4.4006 5.1322 5.9115 6.7478 7.6518 8.6335 9.7136 10.9066

Fig. 2. Main server’s service rate (vs) Performance measures. Table 2 Main server’s service rate vs performance measures. Let a ¼ 5; b ¼ 8; N ¼ 10; k ¼ 5; l2 ¼ 5; a ¼ 1; g ¼ 2; c ¼ 2 and f ¼ 5.

Fig. 3. Stand-by server’s service rate (vs) Performance measures.

l1

q

EðQ Þ

EðW Þ

6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75 10.00

0.1786 0.1782 0.1774 0.1764 0.1752 0.1739 0.1725 0.1710 0.1696 0.1681 0.1666 0.1651 0.1636 0.1622 0.1607 0.1593 0.1580

86.4362 84.6400 83.0128 81.5030 80.1043 78.8102 77.6417 76.6126 75.7487 75.0324 74.4318 73.5884 71.5265 68.0697 64.9664 62.9700 61.6029

8.64362 8.46400 8.30128 8.15030 8.01043 7.88102 7.76417 7.66126 7.57487 7.50324 7.44318 7.35884 7.15265 6.80697 6.49664 6.29700 6.16029

Please cite this article as: G. Ayyappan and S. Karpagam, Analysis of a bulk queue with unreliable server, immediate feedback, N-policy, Bernoulli schedule multiple vacation and stand-by server, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.03.008

G. Ayyappan, S. Karpagam / Ain Shams Engineering Journal xxx (xxxx) xxx Table 3 Stand-by server’s service rate vs performance measures. Let a ¼ 5; b ¼ 8; N ¼ 10; k ¼ 4; l1 ¼ 7; a ¼ 1; g ¼ 2; c ¼ 2 and f ¼ 5.

l2

q

EðQ Þ

EðW Þ

4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00 5.10 5.20 5.30 5.40 5.50

0.1435 0.1433 0.1430 0.1427 0.1424 0.1421 0.1417 0.1414 0.1410 0.1406 0.1402 0.1397 0.1392 0.1387 0.1381 0.1375

57.8989 57.6179 57.3359 57.0904 56.8469 56.6287 56.4168 56.2428 56.0670 55.9161 55.7807 55.6775 55.5882 55.5073 55.4514 55.4049

7.23736 7.20224 7.16699 7.13629 7.10586 7.07859 7.05210 7.03035 7.00837 6.98951 6.97259 6.95968 6.94853 6.93842 6.93143 6.92561

Table 4 Main server’s repair rate vs performance measures. Let a ¼ 5; b ¼ 8; N ¼ 10; k ¼ 5; l1 ¼ 7; l2 ¼ 6; a ¼ 1; g ¼ 2; c ¼ 2 and f ¼ 5.

g

q

EðQ Þ

EðW Þ

3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50

0.1447 0.1433 0.1419 0.1407 0.1395 0.1384 0.1373 0.1363 0.1354 0.1345 0.1336 0.1328 0.1321 0.1313 0.1306 0.1300

78.2610 77.0475 76.2562 75.7096 75.2628 75.0274 74.7240 74.5843 74.4710 74.3297 74.2514 74.1808 74.1532 74.1453 74.1479 74.1279

7.8261 7.7047 7.6256 7.5709 7.5263 7.5027 7.4724 7.4584 7.4471 7.4330 7.4251 7.4181 7.4153 7.4145 7.4148 7.4128

7. Conclusion In this investigation, we have studied a single server bulk queueing system with unreliable server, immediate feedback, Npolicy, Bernoulli schedule multiple vacation and stand-by server. The condition for the system to be stable is obtained. The analytical results that are validated with the help of numerical illustrations may be useful in various real-life situations. The probability generating functions for the numbers of customers in the queue at an arbitrary epoch is found using the supplementary variable technique. Some important system performance measures are also obtained. The motivation for this model comes from wide range applications in many real-time systems, for example in production and manufacturing system. This work can be further extended by incorporating the concepts of working breakdowns and multi stage service. From our numerical study, we found that while increasing the stand-by server’s service rate the performance measures are decreasing. Therefore the provision of stand-by during machine breakdown in a manufacturing system is very much useful to reduce the heavy loss. Appendix A The expressions which are used Eqs. (38) and (40) are given below:

7

A1 ðzÞ ¼ g ðzÞA5 ðzÞA7 ðzÞ þ A8 ðzÞ; A3 ðzÞ ¼ Y 1 ðzÞ
A1 ðzÞ    S v ð g ðzÞÞ
1 f ðzÞhðzÞA6 ðzÞ þ zb gA5 ðzÞ ; A2 ð z Þ ¼ e h  i e ð2Þ ðhðzÞÞ
1 þ g V e ð2Þ ðhðzÞÞA1 ðzÞ A4 ð z Þ ¼ Y 1 ð z Þ h ð z Þ
g V þA2 ðzÞð
k
g þ kX ðzÞÞ;   S u ð f ð zÞ Þ
1 where A5 ðzÞ ¼ hðzÞ e   e ð1Þ ðhðzÞÞ
1 ; S u ð f ðzÞÞ V þð1
pÞpf ðzÞe     e ð1Þ ðhðzÞÞ ; A6 ðzÞ ¼ zb pe S u ð f ð zÞ Þ q þ p V S u ð f ðzÞÞ
1 þ ð1
pÞe   A7 ðzÞ ¼ zb pe S v ð g ðzÞÞ; S v ð g ðzÞÞ
1 þ ð1
pÞe    S v ð g ð zÞ Þ
1 ; S u ð f ðzÞÞ
1 e A8 ðzÞ ¼ zb ahðzÞ e    A9 ðzÞ ¼ z2b ag 1
e S u ð f ð zÞ Þ 1
e S v ð g ðzÞÞ ; Y 1 ðzÞ ¼ f ðzÞg ðzÞA6 ðzÞA7 ðzÞ
A9 ðzÞ;     S v ð g ð zÞ Þ þ g ð zÞ zb
e S v ð g ð zÞ Þ ; K 1 ð zÞ ¼ zb a 1
e     K 2 ð zÞ ¼ zb g 1
e S u ð f ðzÞÞ þ f ðzÞ zb
e S u ð f ðzÞÞ ;    S u ð f ð zÞ Þ zb
e S v ð g ð zÞ Þ K 3 ðzÞ ¼ f ðzÞg ðzÞ zb
e   
z2b ag 1
e S u ð f ð zÞ Þ 1
e S v ð g ðzÞÞ :

Appendix B The expressions which are used Eq. (41) are given below:

  f 1 X; Su ; Sv ; V ð1Þ ¼ 3H1 H13 ;   f 2 X; Su ; Sv ; V ð1Þ ¼ 3H4 H13 ;   f 3 X; Su ; Sv ; V ð1Þ ¼ 3H2 H13
2H1 H14 ;   f 4 X; Su ; Sv ; V ð1Þ ¼ 3H5 H13
2H4 H14 ;   f 5 X; Su ; Sv ; V ð1Þ ; V ð2Þ ¼
3H13 ðV 22 H7 þ V 21 H8 Þ þ 2V 21 H7 H14 ;   f 6 X; Su ; Sv ; V ð1Þ ; V ð2Þ ¼
6V 21 H7 H13 ;   f 7 X; Su ; Sv ; V ð1Þ ; V ð2Þ ¼ H12 H13
H11 H14 ;   f 8 X; Su ; Sv ; V ð1Þ ; V ð2Þ ¼ 3H11 H13 ;    where E1 ¼ e S u ðaÞ
1 e S v ðgÞ
1 ;     E2 ¼ Su1 e S v ðgÞ
1 þ Sv 1 e S u ðaÞ
1 ;     S v ðgÞ
1 þ 2Su1 Sv 1 þ Sv 2 e S u ðaÞ
1 ; E3 ¼ Su2 e     E5 ¼ F 1 e S v ðgÞ
1 þ F 4 e S u ðaÞ
1 ;     E4 ¼ Su3 e S v ðgÞ
1 þ 3Su2 Sv 1 þ 3Su1 Sv 2 þ Sv 3 e S u ðaÞ
1 ;   S v ðgÞ
1 ; E8 ¼ F 7 e     E6 ¼ F 2 e S v ðgÞ
1 þ 2F 1 F 4 þ F 5 e S u ðaÞ
1 ;   S v ðgÞ
1 ; E9 ¼ 2F 4 F 7 þ F 8 e     E7 ¼ F 3 e S v ðgÞ
1 þ 3F 2 F 4 þ 3F 1 F 5 þ F 6 e S u ðaÞ
1 ;   E10 ¼ 3F 5 F 7 þ 3F 4 F 8 þ F 9 e S v ðgÞ
1 ;   S u ðaÞ ; E11 ¼ gF 7 þ akX 1 1
e

Please cite this article as: G. Ayyappan and S. Karpagam, Analysis of a bulk queue with unreliable server, immediate feedback, N-policy, Bernoulli schedule multiple vacation and stand-by server, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.03.008

8

G. Ayyappan, S. Karpagam / Ain Shams Engineering Journal xxx (xxxx) xxx





E12 ¼ akX 2 1
e S u ðaÞ þ gðF 8 þ 2bF 7 Þ h  i S u ðaÞ ;
2kX 1 aF 1 þ
kX 1 1
e   h  i S u ðaÞ
1 þ 3kX 1 aF 2
2
kX 1 F 1 þ
kX 2 1
e S u ðaÞ E13 ¼ akX 3 e h  i S u ðaÞ
g½bðb
1ÞF 7 þ 3bF 8 þ F 9 ; þ 3kX 2 aF 1 þ
kX 1 1
e   S u ðaÞ
1 þ Su1 þ ð1
pÞpe S u ðaÞV 11 ; F 1 ¼ b pe   S v ðgÞ
1 þ Sv 1 ; F 4 ¼ b pe   S u ðaÞ
1 þ 2bpSu1 F 2 ¼ bðb
1Þ pe h i þ Su2 þ ð1
pÞp e S u ðaÞV 12 þ 2Su1 V 11   S u ðaÞ
1 þ 3bðb
1ÞpSu1 þ 3bpSu2 þ Su3 F 3 ¼ bðb
1Þðb
2Þ pe h i þ ð1
pÞp e S u ðaÞV 13 þ 3Su1 V 12 þ 3Su2 V 11 ;   S v ðgÞ
1Þ þ 2bpSv 1 þ Sv 2 ; F 5 ¼ bðb
1Þ pe   S v ðgÞ
1 ðb
1Þðb
2Þ þ 3ðb
1ÞbpSv 1 þ 3bpSv 2 þ Sv 3 ; F 6 ¼ b pe   S u ðaÞ þ ð1
pÞape S u ðaÞV 11 ; F 7 ¼ kX 1 1
e   S u ðaÞ
2kX 1 Su1 F 8 ¼ kX 2 1
e h  i S u ðaÞV 12 þ 2V 11 aSu1

kX 1 e S u ðaÞ ; þ ð1
pÞp ae h   S u ðaÞV 13 S u ðaÞ
3ðkX 2 Su1 þ kX 1 Su2 Þ þ ð1
pÞp ae F 9 ¼ kX 3 1
e  i   S u ðaÞ þ 3V 12 aSu1

kX 1 e S u ðaÞ ; þ 3V 11 aSu2
2
kX 1 Su1

kX 2 e F 10 ¼
akX 1 E1 ; F 11 ¼
a½ðkX 2 þ 2bkX 1 ÞE1 þ 2kX 1 E2 ; F 12 ¼
a½ðkX 3 þ 3bkX 2 þ 3bðb
1ÞkX 1 ÞE1 þ 3ðkX 2 þ 2bkX 1 ÞE2 þ 3kX 1 E3 ; F 13 ¼ ag½2bE1 þ E2 ; F 14 ¼ ag½2bð2b
1ÞE1 þ 4bE2 þ E3 ; F 15 ¼ ag½2bð2b
1Þð2b
2ÞE1 þ 2bð2b
1ÞE2 þ 6bE3 þ E4 ; F 16 ¼

kX 1 ða þ gÞE1 þ agE5
F 13 ;   F 17 ¼ 2ð
kX 1 Þ2

kX 2 ða þ gÞ E1
2
kX 1 ða þ gÞE5 þ agE6
F 14 ; F 18 ¼ ½

kX 3 ða þ gÞ þ 6
kX 1
kX 2 E1
3
kX 1 ða þ gÞE6 h i þ 3 2ð
kX 1 Þ2

kX 2 ða þ gÞ E5 þ agE7
F 15 ;

H ¼ 2H1

b
1 b
1 a
1 X X X ðb
dÞcd þ 2H4 ðb
dÞ qd þ Sk kg d
k d¼a


2V 21 H7

a
1 N
1 X X cn þ v ðn2Þ n¼0

H1 ¼ F 10 þ gE8 ;

!

d¼a

þ H11

!

k¼0 a
1 X

n¼0

Sk ;

k¼0

H2 ¼ F 11 þ gE9
2
kX 1 E8 ;

H3 ¼ F 12 þ gE10
3
kX 1 E9
3
kX 2 E8 ;     H4 ¼ e S v ðgÞ
1 E11 ; H5 ¼ e S v ðgÞ
1 E12 þ 2Sv 1 E11 ;   H6 ¼ 1
e S v ðgÞ E13 þ 3Sv 1 E12 þ 3Sv 2 E11 ;

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H7 ¼ F 16
H1 ; H8 ¼ F 17
H2 ; H9 ¼ F 18
H3 ;     H10 ¼ g V 21 e S u ðaÞ þ H1 þ kX 1 e S v ðgÞ
1
gH4 ; H11 ¼
2ðkX 1 þ gV 21 ÞF 16 þ gð2V 21 H1 þ H2 Þ þ kX 1 H4
gH5 ; H12 ¼
3ðkX 2 F 16 þ kX 1 F 17 Þ þ gð3V 22 H1 þ 3V 21 H2 þ H3 Þ þ3kX 2 H4 þ 3kX 1 H5
gH6 ; H13 ¼
2kX 1 F 16 ; H14 ¼
3½kX 2 F 16 þ kX 1 F 17 ; Su1 ¼
kX 1 e S 0u ðaÞ; Sv 1 ¼
kX 1 e S 0v ðgÞ; Su2 ¼ e S 0 ðaÞ; S 00 ðaÞð
kX 1 Þ2
kX 2 e u

u

S 0v ðgÞ; Sv 2 ¼ e S 00v ðgÞð
kX 1 Þ2
kX 2 e      2 V ði1Þ ¼ kX 1 E V ðiÞ ; V ði2Þ ¼ kX 2 E V ðiÞ þ k2 X 21 E V ðiÞ ; i ¼ 1; 2:

Please cite this article as: G. Ayyappan and S. Karpagam, Analysis of a bulk queue with unreliable server, immediate feedback, N-policy, Bernoulli schedule multiple vacation and stand-by server, Ain Shams Engineering Journal, https://doi.org/10.1016/j.asej.2019.03.008