BULK ARRIVAL QUEUEING MODEL WITH SETUP AND m OPTIONAL SERVICE UNDER BERNOULLI VACATION SCHEDULE AND SERVER FAILURE Текст научной статьи по специальности «Математика»

Binay Kumar RT&A, No 3 (74) BULK ARRIVAL QUEUEING MODEL WITH SETUP..._Volume 18, S^temfa 2023

BULK ARRIVAL QUEUEING MODEL WITH SETUP AND m OPTIONAL SERVICE UNDER BERNOULLI VACATION SCHEDULE AND SERVER FAILURE

Binay Kumar

Magadh Mahila College, Patna University, Patna bkmathasr@gmail. com

Abstract

In the present investigation, we consider a bulk queue model with the assumption that the server may stop working due to random failure during any stage of the service. As soon as the server fails, it is immediately sent for repair. The server offers all incoming units the first mandatory service and any one of the optional services as per the unit's requirements. For computation purposes, we assume that the server offers m+1 services, of which the first one is essential and the remaining are optional. The server may take a vacation in accordance with the Bernoulli vacation schedule with probability p as soon as both service phases of a unit are completed. As the system empties, the server idles and needs some time to set up before initiating the next service. In order to analyse the model and derive various steady-state queue length distributions, we incorporated the supplementary variables corresponding to service time, vacation time, and repair time and applied the probability generating function technique to determine the various system state distributions. Using these probability distributions, we derive the explicit form of various performance indices. To discuss the validity of the present model, we obtained some well-known results from the queueing literature as a special case of the present model by setting appropriate parameters. Finally, to analyse the sensitivity of several performance indices, a numerical demonstration is provided.

Keywords: queue, bulk, essential service, optional service, supplementary variable, queue length

I. Introduction

Most queueing literature makes the assumption that the server in the service station is always available and that the service station never fails. These presumptions, meanwhile, are notably irrational. In real-world systems, it frequently happens that service stations break down and need to be fixed. We frequently experience situations where the entire system pauses owing to a random failure of a unit in computer communication networks, flexible manufacturing systems, production systems, and other areas.

Due to the potential impact on system performance, these types of systems with a repairable service facility are highly worth investigating from both an operational and queueing theory perspective. For detailed related work on queueing models with unreliable servers, we may refer to the work done by Avi-Itzhak and Naor [2], Li et al. [13], Wang and Yang [22], etc. Chaudhury and Tadj [10] discussed the linear cost procedure to obtain the optimal stationary policy of an unreliable queueing model with a Bernaulli vacation schedule. Rajadurai et al. [17] investigated an unreliable queueing model with a modified vacation schedule and applied the supplementary variable technique to obtain the study state queue size distribution. Yang and Wu [23] discussed the M/M/1 queueing model with the assumption that there is a state-dependent breakdown rate under N policy. They assumed that as the system became empty, the server would take a working vacation. Further, Chakravarthy et al. [5] generalised the model of the working-repair-vacation queue by

assuming the concept of backup servers, which work at a relatively slow rate during the absence of the main server. Recently, Meena et al. [16] applied the supplementary variable technique to analyse the unreliable non-Markovian machine system, which comprised both operating and standby machines under N policy.

In some queueing situations, servers are unavailable for services for occasional intervals of time; such queueing models are termed vacation models. During vacation, the server may perform other types of service or may perform scheduled maintenance. Due to their variety of applications in computer systems, communication networks, and production and inventory systems, queuing systems with vacations have been extensively investigated. A comprehensive and detailed review of the vacation models can be found by Doshi [11], Choudhury [6] and Tian and Zhang [20] and Takagi [19]. Yang et al. [24] investigated a retrial queueing model with a constant retrial rate under the assumption that as orbit becomes empty, the server takes its first essential vacation. Further, the server may take additional option vacations after availing of the first essential vacation. Ayyapan and Karpagam [3] discussed an unreliable non Markovian queue model with a standby server under Bernoulli's schedule vacation policy. It is assumed that when the main server stops working due to random failure, a standby server starts serving the arriving unit. Ahuga et al. [1] applied the Runge-Kutta method to investigate a Markovian queueing system with multiple stages of service and vacation, where it is assumed that the server may breakdown during the busy period and vacation period. Recently, Rani et al. [18] applied recursive approach to find the steady-state queue size distribution of a finite population Markovian queueing model with vacation and discouragement factors. They apply the particle swarm optimisation technique to determine the optimal total cost.

It happens frequently in various queueing circumstances that when units use the first essential service, they subsequently need further services, or more than one service. For a better understanding, we will use the example of a car's service centre. Here, units arrive for routine maintenance, and if a serious problem is found with any element of the vehicle while it is being serviced, they go for repair or replacement of that component. For some comprehensive work in phase service, we may refer to Madan [14], Wang [21], Choudhury and Paul [8], Ke [12], etc. Choudhury and Deka [9] discussed a queueing model based on the assumption that units arrive one by one and the server is unreliable. But in real life situations where units arrive in groups of random size, units may demand more than one type of optional service apart from the essential one. Further, there may be a need for startup time to start the service again. Such situations motivated us to extend the model of Choudhury and Deka [9] by assuming that

• Units arrive in batches of random size.

• Second-phase services may choose among the available optional services.

• Server need start up time to start the service again.

• Server may go on vacation under Bernoulli's vacation schedule.

The remaining paper is organised as follows: In Section II, we describe the brief model description by making some basic assumptions. In Section III, the governing equations of the present model are described. In Section IV, we derive the steady-state queue size distribution function. In Section V, the performance measures of the present model are carried out. In Section VI, some well-known results are established as special cases of the present model. Finally, in Section VII, numerical illustration and sensitivity analysis of performance measures are done.

In the present model, we consider a non-Markovian queueing model with the assumption that units arrive in

batches of random size, according to poisson arrival fashion. There is a single server that provides the first

essential services as well as one of the optional services to each arriving unit. As soon as the system becomes

empty, the server gets turned off and needs startup time to start again when at least one or more units arrive.

The brief description of notations used for the present model is as follows:

2 : Batch arrival rate of the unit.

S(x) : Distribution function of set up time.

B0(x): Distribution function of essential service time.

Bj(x) : Distribution function of ith(i = l,2,...,m) optional service time.

V(x) : Distribution function of vacation time.

G0(x): Distribution function of repair when its fails during essential service of a unit . Gt(x): Distribution function of repair when its fails during ith(i = 1,2,...,m) optional service. g(k) :The kth moment of repair time when its fails during essential service of a unit

II. Medel description

Binay Kumar RT&A, No 3 (74) BULK ARRIVAL QUEUEING MODEL WITH SETUP..._Volume 18, S^temfa 2023

g(k' :The kth moment of repair time when its fails during ith (i = 1,2,...,m) optional service of a unit r : Probability to opt ith (i = 1,2,...,m) optional service after essential service. p : Probability to opt optional vacation after service completion of a unit. Nq (t): Denote the queue size in system at time t. S0(t) : Elapsed set up time at time t..

(t): Elapsed service time of essential service at time t. B0(t): Elapsed service time of ith(i = 1,2,...,m) optional service at time t. V0(t): Elapsed vacation time at time t..

g0 (t): Elapsed repair time at t time when its fails during essential service of a unit . G,°(t): Elapsed repair time at t time when its fails during ith(i = 1,2,...,m) optional service. Let y(t) denote the state of server at time t, where

0 if the serveris idle at timet,

1 if the serveris startup at timet,

2 if the serveris busy with essentialserviceat timet,

y(t) = <2+i if the serveris busy withith(i = 1,2,...m)optionalserviceat timet,

3 + m if the serveris on vacationat timet,

4 + m if the serveris under repair when it breakdown during essentialservice at timet, 4 + m + j if the serveris under repair when it breakdown jth(j = 1,2,...m)service.

The variables S0(t),B0(t),B,°(t) (i = 1,2,...,m), V0(t), G0(t) and Gf(t)(i = 1,2,...,m) are added as supplementary variable in order to obtain a bivariate markav process \Nq (t),X(t)\ where X(t) assumes values, 0, S 0(t), B00(t), B,0(t) if r(t) = 0, 1,2, 2+i (i = 1,2..m) respectively and values V0(t), G00(t), G,0(t) if y(t) = 3 + m, 4 + m, 4 + m + j (j = 1,2..m) respectively. To construct the model, we define the following probabilities

Ln(t) = Pr{Nq(t) = n,X(t) = 0}; n > 0, (2.1)

Sn (x,t) = Pr{Nq(t) = n, X(t) = S0(t); x < S0 (t) < x + dx}; x > 0, n > 1 (2.2)

pf) (x, t) = Pr{Nq (t) = n, X (t) = B0(t); x < B0 (t) < x + dx}; x > 0, n > 1, (2.3)

P(') (x,t) = Pr{Nq (t) = n, X(t) = B0(t); x < B°(t) < x + dx}; x > 0, n > 1,1 < i < m, (2.4)

Vn (y, t) = Pr {Nq (t) = n, X (t) = V 0(t); y < V 0(t) < y + dy}; y > 0, n > 1, (2.5)

R(n\ x, y, t) = Pr{Nq (t) = n, X (t) = R0 (t); y < R°(t) < y + dy/B°(t) = x};

x > 0, n > 1,

R()(x,y,t) = Pr{Nq(t) = n, X(t) = R0(t); y < R0(t) < y + dy/B0(t) = x}; x > 0,

n > 1,1 < i < m.

Further it is assume that

V(0) = 0, V(a>) = 1, S(0) = 0, S(<x>) = 1, Bi (0) = 0, Bi (») = 1, G{ (0) = 0, (») = 1.

Further it is assume that G (y) , V(y) functions are continuous at y = 0 ,while Bi(x) , S(x) are continuous at x = 0.

The hazard rate functions for present system is given by

. ,, dS( x) . . j dB, (x) _ dV(y) , dG (y)

v(x)dx = , m,(x)dx = , y(y)dy ^ , gi(y)dy = for 0 < i < m.

1 - S (x) ' 1 - Bi (x)' 1 - V (y)' 1 - Gi (y)

Further we define the following probability generating functions for i = 0,1,2,..., m as follows.

(2.6) (2.7)

R®(x,y,z) = £znR®(x,y) ; R(')(x,0,z) = ^znrn')(x,0) ; S(x,z) = £znSn(x)

n=1 n=1

n=1

S (0, z) = £ znSn (0)

n=l œ

V ( y, z) = £ znVn ( y)

n=l

P(i)( x, z) = ^ znP(i)( x)

n=l

œ

V (0, z) = £ znVn (0)

n=l

P (i)(0, z) = ^ znP(i)(0)

L(z) = ]T

n=l

znL„

n=0

III. Governing Equations

The governing equations of the system are ÂLq = q

ALi + q

TqX /U0( x) x)dx + I Ul( x)PP~\x)dx +....+ I um ( x)PSm\x)dx

J0 Jö Jö

/•œ

+ I v(y)Vi(y)dy, J0

<»œ ¡»œ ¡»œ

rA U0 ( x) P(x)dx + Ul (x)P,(l) (x)dx +....+ um (x)P,{m) (x)dx J0 J0 J0

= AclL0 ,

J»œ

v( y)Vi( y)dy = Ac-

0

n

ALn = ckLn-k , n > 2

k=1

dsn (X) + [A + n(x)]Sn (x) = A^CjSn-j (x); x > 0,n > l,

dx

d_

dx

d_

dy

d

j=l

P() (x) + [A + a, + u (x)W() (x) = A^ CjP(-j (x) + J g, (yW^ (x, y)dy

j=l 0

x > 0, y > 0, 0 < i < m,

V„ (y) + [A + v(y)]Vn (x) = A^cjVn-j (y); n > l,y > 0,

j=l

-^^(x,y) + [A + gi (y)]R(,)(x,y) = A^OjR^j (x,y);

n > 1, x > 0, y > 0, 0 < i < m, We will solve the equations (3.1)-(3.7) under the following boundary condition at x = 0 and y = 0 given by:

S{(0) = AL0, Sn(0) = 0; n > 2,

P™ (0) = q \r0 £ Mo (x)P£\ (x)dx + J" Mi ( x)PV (x)dx +... + J" Mm (x) (x)dx

i1" i*"

o x)Sn (x)dx + J v(y)Vn+l (y)dy; n > 1,

P(j)(0) = rt J"Mo(x)P(0)(x)dx; n > 1, 1 <i< m.

at y = 0 :

m

Vn (0) = p r0 Jo M0 (x)P(0) (x)dx + X J0 Mi (x)pn° (x)dx ; n > 1 _ i=i _ and at y = 0 for i = 0,1,2,..., m. and fixed value of x. R(x;0) = a.P^(x); n > 1, i = 0,1,2,...,m. The normalizing condition for present system is given by

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)

(3.9)

(3.10)

(3.11)

(3.12)

(3.13)

oo

yj

œ

n

CO

n

n

2Ln + 22 J0 pn)(x)dx + J J R«(x,y)dxdy + 2JS(x)dx + 2JVn(y)dy

= 1

(3.14)

IV. Mathematical Analysis

Apply summation formula after multiplying equation (3.2) and (3.3) by appropriate power of z , we get XL( z) + z

qj ro JJ" Mo (x)Pi(0) (x)dx + 2 JJ" M (x)Pi(i) (x)dx I + JJ" v(y)V (y)dy

(4.1)

qj ro JJ" Mo (x)Pi(0) (x)dx + 2 JJ" M, (x)Pi(i) (x)dx I + JJ" v(y)V (y)dy

Substitutes the value of (3.1) into equation (4.1) we get = LoO-f) ( ) 1 - X(z)

Solving equation (3.4), (3.6) and (3.7) in usual manner we get S (x, z) = S (o, z)[1 - S(x)] exp {-a (z)x}; x > o, V(y,z) = V(o,z)[1 - V(y)]exp{-az1(z)y}; y > o,

+ AX (z)L( z)

R(,) (x, y, z) = R(,) (x,o, z)[1 - G, (y)] exp{-fll (z) y};

, - »('■)

y > o, o < i <m.

(4.2)

(4.3)

(4.4)

(4.5)

On multiplying equations (3.8),(3.9) and (3.13) by appropriate power of z , then after little simplification, we get

R(i) (x,0,z) = atP(i) (x,z); i = 0,1,2,...,m,

S (0, z) = zAL0.

On simplifying equations (4.3) and (4.7) we have S (x, z) = zXL0 [1 — S (x)] exp { -a1 (z) x}; x > 0,

On simplifying equations (3.5) and (4.5), we have

d_

dx

P(,)(x, z) + (a (z) + at + M (x))P(,)(x, z) = R(,)(x,o, z)G, (a (z)); o < i <m

Solving equations (4.6) and (4.9), we get

P(i)(x, z) = P(i)(0, z)[1 - B' (x)]exp{-^ (z)x}; x > 0, 0 < i < m, where fa (z) = a1 (z) + at (1 - G, (a1 (z))) and a1 (z) = A(1 - X(z)). From equation (4.10), (4.6) and (4.5) we have

R(i) (x, y, z) = atP(i) (0, z)[1 - Bi (x)] exp{-fa (z)x}[1 - G, (y)] exp{-a1 (z)y};

(x,y) > 0, 0 < i <m.

Further, multiplying equations (3.11), (3.12) by suitable power of z and after simplification we have P(i) (0, z) = rP(0) (0, z)B0 (z)); 1 < i < m.

V (o, z) = pP(o) (o, z) B o (^ (z))

r + 2 r,

2r,Bi (<Pt (z))

Similarly, from equation (3.10) we have

m

ro Jo Mo(x)p(o)(x, z)dx + 2i0 M, (x)P(i)(x, z)dx

p(o) (o, z) = q

i=1

(4.6)

(4.7)

(4.8)

(4.9)

(4.10)

(4.11)

(4.12)

(4.13)

(4.14)

1 /»TO

0 x)S(x, z)dx + -J v(y)V(y, z)dy - ALt

Substituting the value of equation (4.4), (4.8), (4.10) in (4.14) and then using the value of equations (4.7), (4.12) - (4.13) we get

^0z[1 - zS(a1(z))] (4.15)

Bo(^o (z)){ro + 2 r,B, (<P, (z))}{q + pV(a (z))} - :

The limiting value of equation (4.15) when z —> 1, is given by

P(o) (o,1) =

AL0 [1 + AE( X )E (S)] (1 -P)

(4.16)

m

i=1

i=1

Binay Kumar RT&A, No 3 (74) BULK ARRIVAL QUEUEING MODEL WITH SETUP._Volume 18, S^temfa 2°23

1 m

where p = AE( X ){E(B0 )(1 + ^0 g ^ ) + X rEB )(1 + ag1 ) + pE (V )}.

i=1

Evaluating z —> 1 in equation (4.2),(4.4), (4.8)- (4.13) and using the equation (4.16) we have

L(1) = (4.17) E (X )

S(x,1) =AL0[1 - S(x)], (4.18)

P (0) ( x 1) = AL0[1 + AE(X )E (S XK1 - B0( x)] . x > 0 (4.19) ( , ) (1 -P) ' ,

pw = riAL0[1 + AE(X)E(S)][1 -Bi(x)] ; x > 0, 1 < i < m, (4.20)

(1 -P) ; ,

V( = PAL0[1 + AE(X)E(S)][1 - V(y)]. y > 0, (4.21)

(1 -P)

R(°)(x, y,V) = a°AL°^ + AE(X)E(S^ - B0(x)][1 - G0 (y)]. (x, y) > 0_ (4.22) , , (1 - P) , , ■

Ri\x,y, 1) = a'r'AL0[1 + AE(X)E(S)][1 - Bi (x)][1 - G, (y)]. ( ^ . Q (4.23)

, , (1-P) , , .

From equations (4.17)-(4.23) and normalizing condition (3.14), we have E ( X )(1 -p)

(4.24)

0 1 + AE (X )E (S )

Theorem 1: The joint probability distribution functions of system state and queue size, under stability condition, are given by

= zAE( X)(1 - p)[1 - S (x)] exp{-fli(z) x} (425) ( , 1+ AE(X)E(S) , '

P(0)(xz)= AzE(XX1 - p)[1 - zS(,a1 (z))][1 - B0 (x)]exp{-00 (z)x} (4.26)

^ 5 ' __m ____5

(1 + AE(X)E(S))[B0 (00 (z)){r0 + X r Bi (0, (z))}{q + pV(,a, (z))} - z]

P (i)(x, z) =-

rAzE (X)(l - p) B 0(00 ( z))[1 - zS g (z))][1 - B, (x)] exp{-0, (z) x}

(1 + AE(X)E(S))[B0(00 (z)){r0 + XrtB, (0, (z))}{q + pV(al (z))} - z]

i=1

1 < i < m,

____m _

pAzE (X )(1 - p)[1 - zS (a ( z))] B 0 (00 ( z)){r0 + X riBi 0 ( z))}[1 - V (y)] exp{-aj (z) y}

V(y, z) = - ^

(4.27)

(4.28)

__m ___

(1 + AE(X)E(S))[B0 (00 (z)){r0 + X rtBi (0, (z))}{q + pV(a1 (z))} - z]

R(0){xyz) = a0AzE (X )(1 -p)[1 - zS g ( z))][1 - B0 ( x)] exp{ -0O ( z) x}[1 - G0 ( y)] exp{-a1 ( z) y} (4.29)

__m ___

(1 + AE( X )E (S ))[ B 0 (00 ( z)){r0 + X r,Bi 0 ( z))}{q + pV a ( z))} - z]

(i) airiAzE (X X1 - p)[1 - zS (al(z))]B 0(^0 (z))[1 - Bi (x)]exp{-^i (z) x}[1 - Gt (y)]exp{-al(z) y}

R y)( x, y,z) =---m—1-1-- (4.30)

(1+ AE(X)E(S))[B0(Mz)){r +XriBi(6(z))}{q + pV(a^z))} - z]

i=1

1 < i < m,

Theorem 2: The marginal probability distribution function of system state queue size are given by S(z) = zAE(X)(1 -p)[1 - Sa(z))] (4.31)

( (1 + AE( X) E(S)) a1( z) '

P(0)(z) =_AzE(X)(1 - p)[1 - zS(at (z))][1 - B0 (0O (z))]__(4.32)

^ ' __m ____5

(1 + AE(X)E{S))[B0 (6 (z)){r0 + Xr.Bi (0, (z))}{q + pV(at (z))} - z]0, (z)

P(.i)(z)= 'AzE (X X1 - P) B 0 (00 (z))[1 - zS g ( z))][1 - Bi (0, (z))]_

^ ' __m ____:

(1 + AE( X ) E (S ))[ B 0 (00 ( z)){'0 + X r Bi 0 ( z))}{q + pV a ( z))} - z0 ( z)

i=1

1 < i < m,

(4.33)

i=1

i=1

i=1

Binay Kumar RT&A, No 3 (74) BULK ARRIVAL QUEUEING MODEL WITH SETUP._Volume 18, September 2023

pXzE(X)(1 - p)[1 - zS(a1 (z))]B0 (fa (z)){r, + £ rt Bi (fa (z))}[1 - V(a1 (z))]

V (z) = ■

0

i=1

(4.34)

__m ___

(1 + XE(X)E(S))[B0 (fa (z)){r + £r Bi (fa (z))}{q + pV(a (z))} - z]a1 (z)

^(0) (z)= ^0^zE(X)(1 - p)[1 - zS(a1 (z))][1 - B0 (fa (z))][1 - G0 (a1 (z))]__(4.35)

m 5

(1 + XE(X)E(S))[B0(fa(z)){r0 + £r.Bi (fa (z))}{q + pV(a1 (z))} - z]a1 (z)fa(z)

R (i)( z) =

O a^AzE (X )(1 - p)[1 - zS (V z))] B „(fa,( z))[1 - Bi (fa (z))][1 - Gi z))]

(1 + AE(X)E(S))[B0(fa(z)){r0 + £rtBi(fa(z))}{q + pVz))} - z]fa(z)a,(z)

i=1

1 < i < m,

Proof: See appendix A.

Theorem 3: The stationary queue size distribution at random epoch is given by

____m ___

1E(X )(1 - z)(1 - p){1 - zS (a (z))} [ B 0 (fa (z)){r + £ r.Bi (fa (z))}{q + pV (a1 (z))}]

(4.36)

P( z) =■

0

i=1

(4.37)

a1 (z)[1 + IE(X)E(S)][B0 (fa (z)){r + £ rtBi (fa (z))}{q + pV(a1 (z))} - z]

i=1

Proof: Adding the equations (4.31)-(4.36) we get required result. The equation (4.37) can be written as

P(z) = £(z) X coMXT! Vaca. (z) (4.38)

Where _ , AE(X){1 - zS(a1(z))}

4(z) =-

a1( z)[1 + AE( X )E(S )]

m

and (1 - z)(1 -P)[B 0(fa0(z)){r0 +£ riBi (fa (z))}{q + pV^ (z))}]

JoptwithVaca., \

__m ___

[B0(fa(z)){r0 +£riBi(fa(z))}{q + pV(a1(z))} -z]

(4.39)

Equation (4.38) shows that the queue size distribution divides into two independent random variables: the first (D'^'XctWfh Vaca. (z), the stationary queue size distribution of the unreliable bulk queue with optional

service including vacation and repair, and the second (z) is the number of arrivals during idle time including setup time.

Theorem 4: The stationary queue size distribution of system at departure epoch is given by

____m ___

(1 - p){1 - zS a (z))} [ B 0 (fa (z)){r0 + £ r,Bi (fa (z))}{q + pV a (z))}] tt( z) =-^-

v y __m ___

[1 + IE (X) E(S)][ B 0 (fa (z)){r0 + £ r Bi (fa (z))}{q + pV a (z))} - z]

i=1

Proof: See appendix B. Equation (4.39) can be written as

*( z) = 1 - X (z) X P(z) (4.40)

E(X )(1 - z)

*■( z) = £( z) X 1 - X (z) xrnJoptmthVaca(z) (4.41)

w bw E( X )(1 - z) mX / g/1 w v '

Thus, the queue size distribution at the departure epoch decomposes into three independent random

variables: (d'f^Q^'1™. (z) ,the stationary queue size distribution of the unreliable bulk queue with

optional service including vacation and repair; ^(z) the number of arrivals during idle time including setup

time; and the third independent random variable 1-X(z) , the number of customers placed before a

E( X )(1 - z)

tagged customer.

i=1

i=1

i=1

V. Performance measures

(a) System state probabilities

By considering limit z —> 1 in the marginal probability generating function of the server state queue distribution, it is possible to determine the system state probability of the server state.

• The probability that server is under startup is p _ AE(X)E{SX1 — p)

S {1 + AE(X)E(S)} '

• The probability that server is busy with essential service pB = AE(X)E(B0 ),

• The probability that server is busy in providing the ith (1 < i < m) optional service Pb = riAE(X)E(Bi),

• The probability that server is under optional vacation p = pAE(X)E(V),

• The probability that server is under repair when its fail during essential service PRt> = a0AE( X )E ( B0) g (0l)

• The probability that server is under repair when its fail during ith (1 < i < m) optional service Pr = r^AE (X) E (Bi) g?\

• Probability that server is idle is given by P =_(1 ~pP_

L 1+ AE (X )E (S)'

(b) Average queue length

(i) The mean system size (L ) at arbitrary epoch can be determined using

dP(z) ,

Lq = dz ^

2AE( X )E(S) + (AE( X ))2 E(S 2) + AE( X(2) )E(S)

Lq = P +

2(l + AE( X )E(S)) I m m

(AE(X))2|E(Bo2)(1 + a0g(1))2 + Jr,E(B2)(l + algjV) )2 +a0g(2)E(B0) + Jr^gi2E(Bt) + pE(V2)

i=l

+ 2E(Bo )1 + aog® rtE(Bt )(1 +at gf1) + 2pE(VrtE(Bt + a,g()) + 2pE(Bo )(1 + aog)E(V) [

1=1

2(1 -p) (5.1)

AE (X V )||e( Bo )(1 + ao g 01}) + JJ r,E(B% )l + a, gf) + pE(V) J

2(1 — p)

(ii) The mean system size (L ) at departure epoch can be determined using dn(z) |

ld = j |z=l

dz

l=l

l=l

Binay Kumar RT&A, No 3 (74) BULK ARRIVAL QUEUEING MODEL WITH SETUP._Volume 18, S^temfa 2°23

22E( X )E(S) + (2E( X ))2 E (S 2) + 2E( X(2) )E(S) ld = P +-

2(1 + 2E( X )E(S))

I m m

(2E( X ))2 J E( Bq2)(1 + «0 g 01) )2 + 2 riE(Bf )(1 + agf1 )2 + «q gQ2) E(B0) + 2 rI-aI-gI(2) E(B;) + pE(V 2)

i=1 i=1

-2E(Bq)(1 +«q g 01})2 riE(Bi )(1 + «ig®) + 2 pE(V )£ r,E(B% )(1 +«ig®) + 2 pE(Bq)(1 +«q g Q1))E(V)[

+

i=1 i=1

2(1 -p) (5.2)

2E( X(2) )| E(Bq )(1 + «o g 01}) + 2 r E(Bi )(1 + « g(1)) + pE(V) [

I i=1 | + E(X(2))

2(1 - p) 2E( X)

E( X(2))

From (5.1) and (5.2) we can easily observe that — = L +__(_-

D q 2E (X)

(c) Average waiting time

The average waiting time can be obtained as

E(Wq) = ——— (5.3)

q kE( X )

+

VI. Special cases

In this section, we evaluate some special case by setting appropriate parameter to validate our result with existing models.

Case (i): By setting P(S = 0) = 1, P(X = 1) = 1, r1 = 1, m = 1; equation (4.39) gives = (1 - p)(1 - z)[Bo (fa (z))B1 (fa (z)){q + pV(2(1 - z))}] Z [B o (fa (z)) B1 (fa (z)){q + pV (a (z))} - z]

where fa (z) = 2(1 - z) + at (1 - G, (2(1 - z))), i = 0,1. The present model reduces to the model studied by Chaudhury and Deka [9].

Case (ii): By settingP(S = 0) = 1, P(X = 1) = 1, r = 1,m = 1,^ = a2 = ... = am = 0; equation (4.39) gives

= (1 - pX1 - z)[ B 0 (2(1 -X ( z))) B1 (2(1 - X (z)—{q + pV (2(1 - X (z)))}] [B0 (2(1 - X(z)))B1 (2(1 - X(z))){q + pV(2(1 - X(z)))} - z] The present model reduces to the model studied by Chaudhury and madan [7]. Case (iii): By setting ax = a2 = ••• = CXm = 0, p = 0; equation (4.39) gives

(1 - p){1 - zS (a (z)) } [B o (2(1 - X (z))){ro + £ ri Bi (2(1 - X (z)))}]

n( z) =-i=1-

m

[1 + 2E( X )E(S )][B o (2(1 - X (z))){ro + 2 r,Bi (2(1 - X (z)))} - z]

The present model reduces to model investigated by Ke [12].

Case(iv):By setting ax = a2 = ... = &m = 0, p = 0, r = r2 = ... = = 0,; equation (4.39) gives

(1 - p){1 - zS(at(z))} [B0 (2(1 - X(z)))]

7T( z) = -=-

[1 + 2E(X)E(S)][B 0 (2(1 - X(z))) - z]

The present model reduces to the model studied by Choudhury [6]

Case(v):By setting a0 = ax = ... = am = 0, p = 0, r2 = ... = rm = 0, P(X = 1) = 1; equation (4.39) gives

n( z) =

(1 - p)(1 - z)Bo (A(1 - z)){r0 + Bi (A(1 - z))}

[B0(M1 - z)){r0 + Bi(A(1 - z))} - z] The present model reduces to the model studied by Medhi [15].

VII. Numerical illustration

In present section, we will provide the numerical illustration and sensitivity analysis of the various performance measures on different parameters of the model. For this, it assume that the first two moments of

T7iV\-b EY b(1 + b) U- 1

the batch size distribution are given by E(X ) = —, E(X ) =-;—• b = 1 - a- It is assumed that the server's

a a

start-up time will follow an deterministic distribution with first and second moments 1 ? 1

E(S) = —, E(S ) = —. The distribution of compulsory and elective service periods is assumed to be

s ' s2

exponential, and its first and second moments are therefore derived as 1 2

E(B. ) =_ E(B2) =_ • i = 012 where denote the service rate. Further, the distribution of vacation

' U ' i Mt2 '' ' time is assumed to be Erlangian-2 and has parameter y. (i = 1,2). The first and second moments of vacation

1 3

time distribution are E(V) = —, E(V2 ) =--The repair time distribution is further assumed to follow an

V 2v2

exponential distribution with a parameter g i and having the first two moments 1 2

g(1) =-, g =-; i = 0,1,2; Coding in MATLAB is used to create computer programmes. We now

' gi' ' gf present the numerical results in tables (1) -(5).

Table 1: E(X) = 2, Ju1 = M2 = 2u0,a0 = 0.01,a1 =a2 = 2a0, r0 = r1 = r1 = 1/3, v = 15,

s = 10, g 0 = 10, g1 = 15, g 2 = 15. Table2:E(X) = 2,uu = U = 2u ,a =a2 = 2a,r0 = r = h = 1/3,v = 15,

s = 10, g0 = 10, g1 = 15, g 2 = 15, A = 0.7, U0 = 2 Table 3: E(X) = 2, uu = M = 2M ,a = a = 2a, h = h = h = 1/3, v = 15, p = 0.5,

s = 10, g0 = 10, g1 = 15, g 2 = 15,a0 = 0.01. Table 4: E(X ) = 2, u1 = U2 = 2u0,a1 = a2 = 2a0, r0 = r1 = r1 = 1/3, v = 15, p = 0.5,

s = 10, g0 =10, g! = 15, g2 =15, A = 0.7, M0 = 2. Table 5: E(X) = 2,uu = U = 2u =a2 = 2a,ro = h = h = 1/3,v = 15,p = 0.5, s = 10, g0 = 10, g1 = 15, g2 = 15, A = 0.7,a0 = 0.01

Table 1: Effect of arrival rate and service rate on l (W ) for variation in p

u = 2 U = 2.1

P = 0.3 P = 0.7 P = 0.3 P = 0.7

A Lq W q Lq Wq Lq Wq Lq Wq

0.61 14.174 11.618 18.492 15.157 10.632 8.715 13.275 10.881

0.63 17.922 14.224 24.894 19.758 12.851 10.199 16.634 13.201

0.65 23.607 18.160 36.393 27.994 15.857 12.197 21.611 16.624

0.67 33.253 24.816 63.102 47.091 20.159 15.044 29.755 22.205

0.69 53.215 38.562 194.278 140.781 26.829 19.441 45.498 32.969

Binay Kumar RT&A, No 3 (74) BULK ARRIVAL QUEUEING MODEL WITH SETUP._Volume 18, S^temfa 2°23

Table 2: Effect of p on Lq ( Wq ) for variation in failure rate and m

m = - 2 m = 1 m = 0

P Lq W q Lq W q Lq W q

0.1 27.349 19.535 17.130 12.236 7.094 5.067

0.3 33.760 24.114 19.614 14.010 7.623 5.445

«0 = 0.01 0.5 43.930 31.379 22.869 16.335 8.218 5.870

0.7 62.558 44.684 27.328 19.520 8.894 6.353

0.9 107.754 76.967 33.814 24.153 9.668 6.906

0.1 28.516 20.368 17.578 12.556 7.172 5.123

0.3 35.519 25.371 20.188 14.420 7.710 5.507

«0 = 0.05 0.5 46.895 33.496 23.636 16.883 8.317 5.940

0.7 68.622 49.016 28.406 20.290 9.006 6.433

0.9 126.698 90.498 35.442 25.316 9.797 6.998

Table 3: Effect of arrival rate on system state probabilities

2 Pl Ps pb0 PB PB 1 B2 Pv Pr> PRi pr2

0.61 0.12934

0.63 0.10399

0.65 0.07882

0.67 0.05382

0.69 0.02900

0.01578 0.61000

0.01310 0.63000

0.01025 0.65000

0.00721 0.67000

0.00400 0.69000

0.10167 0.10167

0.10500 0.10500

0.10833 0.10833

0.11167 0.11167

0.11500 0.11500

0.04067 0.00061

0.04200 0.00063

0.04333 0.00065 0.04467 0.00067

0.04600 0.00069

0.00014 0.00014

0.00014 0.00014

0.00014 0.00014

0.00015 0.00015

0.00015 0.00015

Table 4: Effect of service rate on system state probabilities

M0 PL PS PB0 PB1 PB2 PV PR^ PR1 PR2

2 0.01666 0.00233 0.70000 0.11667 0.11667 0.04667 0.00070 0.00016 0.00016

2.1 0.05569 0.00780 0.66667 0.11111 0.11111 0.04667 0.00067 0.00015 0.00015

2.2 0.09117 0.01276 0.63636 0.10606 0.10606 0.04667 0.00064 0.00014 0.00014

2.3 0.12356 0.01730 0.60870 0.10145 0.10145 0.04667 0.00061 0.00014 0.00014

2.4 0.15326 0.02146 0.58333 0.09722 0.09722 0.04667 0.00058 0.00013 0.00013

Table 5: Effect of failure rate on system state probabilities

Pr

P.

Pn

Pr

Pn

Pr

P

R

P

P

2

2

0.01 0.01666

0.02 0.01577

0.03 0.01488

0.04 0.01400

0.05 0.01311

0.00233 0.70000

0.00221 0.70000

0.00208 0.70000

0.00196 0.70000 0.00184 0.70000

0.11667 0.11667

0.11667 0.11667

0.11667 0.11667

0.11667 0.11667

0.11667 0.11667

0.04667 0.00070

0.04667 0.00140

0.04667 0.00210

0.04667 0.00280

0.04667 0.00350

0.00016 0.00016

0.00031 0.00031

0.00047 0.00047

0.00062 0.00062

0.00078 0.00078

The impact of arrival rate and service rate on the average queue length (waiting time) Lq (Wq) is shown in

Table 1. The table clearly shows that the Lq (Wq) increases with rising arrivals, however, there is a

diminishing trend brought on by a rise in service rate. Additionally, there is an increasing tendency in Lq ( Wq

) with an increase in p for the fixed value of the arrival rate. Table 2 displays the impact of p on the average queue length (waiting time). The table clearly shows that there is an increasing tendency in Lq ( Wq ) as a

consequence of the growth in p. Additionally seen is a decline in Lq ( Wq ) as a result of a reduction in the

availability of optional services. The variation in system state probability caused by variations in arrival (service) rates is shown in Table 3(4). It is evident from the data that with an increase in arrival (service) rate PBo, P, PB2 and PV have growing (declining) trends, whereas PL and PS have decreasing (increasing)

Binay Kumar RT&A, No 3 (74)

BULK ARRIVAL QUEUEING MODEL WITH SETUP._Volume 18, S^temfa 2°23

trends. Table 5 demonstrates that as the failure rate rises, PL and PS tend to decline while PBo, PBi, PB2 and PV remain constants. Along with the rise in failure rates, increasing trends can be seen in Pr , Pr , and PR^ .

VIII. Conclusion

In the present article, we investigated a queueing model with an unreliable server under the provision of Bernoulli vacation, setup time, and two-phase service, where the first service is essential and the second is optional, and we had to choose among the available options. In the current study, we use the supplementary variable approach to build the model and assess several performance indices expressions. Our model may be useful in more flexible queueing circumstances that occur in many manufacturing and production systems, where some services may be optional based on the customer's desire and where the manufacture of the items must be done in phases, such as assembling, testing, packing, etc. The model studied can be further generalised by incorporating feedback services as well as some more features such as N-Policy, retrial, and extended vacation policies.

References

[1] Ahuja, A, Jain., A. and Jain, M. (2022). Transient analysis and ANFIS computing of unreliable single server queueing model with multiple stage service and functioning vacation. Mathematics and Computers Silulation, 192: 464-490.

[2] Avi-Itzhak, B. and Naor, P. (1963). Some queueing problems with the service station subject to breakdowns. Operations Research, 11(3): 303-320.

[3] Ayyapan, G. and Karpagam, S. (2019). Analysis of a bulk queue with unreliable server, immediate feedback, N-policy, Bernoulli schedule multiple vacation and stand-by server. Ain Shams Engineering Journal, 10(4): 873-880

[4] Banik, A.D.(2013). Stationary distributions and optimal control of queues with batch Markovian arrival process under multiple adaptive vacations. Computers & Industrial Engineering, 65(3): 455-465.

[5] Chakravarthy, S.R., Shruti and Kulshrestha, R.(2020). A queueing model with server breakdowns, repairs, vacations, and backup server. Operations Research Perspectives, 7: 100131

[6] Choudhury, G. (2000). An M[x]/G/1 queueing system with a setup period and a vacation period. Queueing systems, 36:23-38.

[7] Choudhury, G. and Madan, K.C. (2004). A two phases batch arrival queueing system with a vacation time under Bernoulli schedule. Applied Mathematics and Computation.149:337-349.

[8] Choudhury,G. and Paul, M.(2005). A two phase queueing system with Bernoulli feedback. International Journal of Information and Management Sciences. 16 (1): 35-52.

[9] Choudhury, G. and Deka, M. (2012). A single server queueing system with two phases of service subject to server breakdown and Bernoulli vacation. Applied Mathematical Modelling, 36(12): 6050-6060.

[10] Choudhury,G. and Tadj. L. (2011). The optimal control of an MAX/G/1 unreliable server queue with two phases of service and Bernoulli vacation schedule. Mathematical and Computer Modelling, 54(1-2): 673-688.

[11] Doshi, B.T. (1986). Queueing systems with vacations: a survey. Queueing Systems, 1:29-66.

[12] Ke, J.C.(2008). An MX/G/1 system with startup server and J additional options for service, Applied Mathematical Modelling, 32: 443-458.

[13] Li, W., Shi, D. and Chao, X. (1997). Reliability analysis of M/G/1 queueing system with server breakdowns and vacations. Journal of Applied Probability, 34(2):546-555.

[14] Madan ,K.C.(2000). An M/G/1 queue with second optional service. Queueing Systems, 34: 37-46.

Binay Kumar RT&A, No 3 (74) BULK ARRIVAL QUEUEING MODEL WITH SETUP._Volume 18, September 2023

[15] Medhi,J. (2002). A single server Poisson input queue with a second optional channel. Queueing Systems, 42(3): 239-242.

[16] Meena, K.,Jain, M., Assad, A., Sethi, R. and Garg, D. (2022). Performance and cost comparative analysis for M/G/1 repairable machining system with N-policy vacation. Mathematics and Computers in Simulation, 200: 315-328

[17] Rajadurai, P., Saravanarajan, M.C., and Chandrasekaran (2014). Analysis of an M[X]/(G1, G2)/1 retrial queueing system with balking, optional re-service under modified vacation policy and service interruption. Ain Shams Engineering Journal, 5(3): 935-950.

[18] Rani, S., Jain, M. and Meena, R.K.(2023). Queueing modeling and optimization of a fault-tolerant system with reboot, recovery, and vacationing server operating under admission control policy. Mathematics and Computers in Simulation, 209: 408-425.

[19] Takagi, H. Queueing Analysis' - A Foundation of Performance Evaluations, Amsterdam, 1991.

[20] Tian N, Zhang ZG. Vacation queueing models-theory and applications. SpringerVerlag, 2006.

[21] Wang, J. (2004). An M/G/1 queue with second optional service and server breakdowns. Computer & Mathematics with Application, 47: 1713-1723.

[22] Wang, K.H. and Yang, D.Y. (2009). Controlling arrivals for a queueing system with an unreliable server: Newton-Quasi method. Applied Mathematics and Computation, 213(1):92-101.

[23] Yang, D.Y. and Wu, C.H. (2015). Cost-minimization analysis of a working vacation queue with n-policy and server breakdowns. Computers & Industrial Engineering , 82:151-158.

[24] Yang, D.Y, Chang, F.M. and Ke, J.C.(2016). On an unreliable retrial queue with general repeated attempts and J optional vacations. Applied Mathematical Modelling, 40(4): 3275-3288.

Appendix A

Proof of theorem 1:

Integrating equations (4.25)-(4.27) with respect to X and ussing the result

n -

J e - (1 - M (x))dx = (A.1)

0

We get equations (4.31)-(4.33).

Similarly integrating equations (4.28) with respect to y and using (A.1) we get equation (4.34). On repeating the same process for equations (4.29) and (4.30) with variable X, y , and using equation (A.1), we get equations (4.35)-(4.36).

Appendix. B

Proof of theorem 2:

To obtain the queue size distribution at the departure epoch, on the line of Choudhury and Deka [9], we have

to toto

TTj = k 0 ^0 ¡Mo (x)P+ (x)dx + J n (x)j (x)dx +... + J Mm (xj (x)dx + J v( y)V]+l (y)dy \ (B^

I 0 0 0 0 J

where k0 is the normalizing constant and {n j ; j = 0,1,2,...} as the probability that there are j customers in the queue at a departure epoch.

Multiplying equation (B.1) by zj and using n(z) = Xn■ zj and after simplification,

j=0

We get

____m ___

k0L {1 - zS (ai (z))}[ B 0(00 ( z )){r0 + X rrB> (0, ( z ))}{q + pV a ( z))}] n( z) =-¿=--(B.2)

v y __m ___

[B0 (00 (z)){r0 + X r,B, (0, (z))}{q + pV(ax (z))} - z]

¿=1

Using the condition n(Y) = 1, we get

k0 =-^--(B.3)

0 XL0>{1 + XE (X )E (S)}

Using the value of equation (B.3) into (B.2), we get required result.

to