ANALYSIS OF MMAP/PH1,PH2/1 PREEMPTIVE PRIORITY INVENTORY RETRIAL QUEUEING SYSTEM WITH SINGLE VACATION, WORKING BREAKDOWN, REPAIR AND CLOSEDOWN Текст научной статьи по специальности «Компьютерные и информационные науки»

ANALYSIS OF MMAP/PH1,PH2/1 PREEMPTIVE PRIORITY INVENTORY RETRIAL QUEUEING SYSTEM WITH SINGLE VACATION, WORKING BREAKDOWN, REPAIR AND CLOSEDOWN

G. Ayyappan, S. M eena •

Department of Mathematics, Puducherr y Technological University, Puducherr y, India. ayy appan@ptuniv .edu.in, meenasundar2296@gmail.com

Abstract

This paper analyzes preemptive priority inventory retrial queueing system with a single vacation, working breakdown, repair, and closedown. We assume that an arrival follows the Marked Markovian arrival process and that the server will provide them with phase-type services. The (s, S) policy to replenish the items and the replenishing duration follow an exponential distribution. In this paper, we consider two types of customers: high-priority(HP) customers and low-priority(LP) customers. Arriving HP customers should get the service if the server is idle and has a positive inventory level; otherwise, they should wait in front of the service station. Arriving LP customers get service only if there is a positive inventory level and there are no high-priority customers in the system; otherwise, go for the finite capacity size of the orbit. After the completion of service, if no one is present in the high-priority queue and orbit, the server will close down the system and then go on a single vacation. The server is idle when the vacation period ends. When the server breaks down, it only serves the present customer and operates in slow mode while it is being repaired. The number of high-priority customers in the system, the number of low-priority customers in the orbit, the inventory level, and server status may all be determined in a steady state. Numerous key performance indicators are defined, and a cost analysis is obtained. To make our mathematical concept clearer, a few numerical examples are provided.

Keywords: Queueing-inv entory, (s, S) policy, Retrial, Preemptiv e Priority, Single Vacation, Working Breakdown, Repair, Closedo wn, Markovian Arrival Process, Phase-type distribution, Matrix Analytic Method.

AMS Subject Classification (2010): 60K30, 68M20, 90B05.

1. Introduction

The field of inventory retrial queueing systems has seen a rise in popularity in recent years due to developments in computer networking and communications technologies. In a queueing-inv entory model, each client receives a product from the inventory upon completion of the service. Neuts [19] presented the modified Markovian point process for the first time. A number of well-known techniques fall under the large category of point processes known as MAP, including PH-renewal, Markov-modulated Poissons, and Poisson. The Markovian arrival process with several correlated and non-correlated arrival types, as well as the phase-type distribution, were both extensively clarified by Chakra varthy [8]. Neuts [20] investigated the methods used in matrix-analytic queueing theor y.

Reordering products in a queueing order-demand inventory system is best done using the techniques described by Meliko v and Molchano v [16]. A study by Berman et al. [6] examined a

system for inventory control for a service center that uses one inventory item for each service render ed. Accor ding to their assumptions, ther e must alw ays be a shortage of items for the queue to form, and demand and service rates are predictable and steady. Berman and Kim [7] developed two types of queuing-inv entory models with service facilities. The first had an infinite one, while the other had a finite capacity for queuing. In their evaluation, Yadavalli et al. [25] made the assumption that reorders are readily available and that requests belong to a renewal process. The inventory system included a service station and an indefinite waiting area. Amirthakodi and Sivakumar [2] looked at retrial inventory queuing, in which there is a finite orbit size, a single server, and customer feedback. Sanjukta and Nabendu [21] looked into a carbon tax and an inventory queueing system with a partial replenishment strategy and a limited shelf life for perishable commodities.

Most of the time, it is believed that inventory and queueing models have not failed at service stations. In actuality , we regularly come across circumstances wher e service station malfunctions could occur. A server interruption inventory retry queueing system was covered by Krishnamoorthy et al. [11]. In their model, they took into consideration a (s, S) replenishment policy where the lead time and service time follow an exponential distribution, while the arrival follows a Poisson distribution. The retrial inventory queuing system with server failure was examined by Ushakumari [24]. When the server is processing requests or is idle, it could malfunction. If a ser ver failur e results in ser vice disruption, users are placed into an infinitely long orbit, obliged to retry service after an arbitrar y period, and so on unless the server is render ed inoperable.

The server could simply quit waiting for customers and remain unreachable in a variety of situations. It might also be completing other duties, such as maintenance or servicing more clients. Krishnamoorthy and Narayanan [12] consider ed a manufacturing inventory system including server vacations. They held that the manufacturing process adhered to the Markovian manufacturing method and that the service times for each customer were dispersed in a phasetype manner. The inventory queue for retrials with several vacations was analyzed by Sugany a and Sivakumar [23]. They took into account a pair of servers and a limited orbit size capacity in their model. The retrial queueing system incorporating a single server, Bernoulli feedback, and vacation has been examined by Ayyappan and Gowthami [4]. They took into account both the client's arrival based on MAP and the server's service delivery based on PH distribution. Melikov et al. [17] examined the retrial queueing system that incor porates Poisson arriv al, exponential service time, and delayed feedback. For their investigation, they employed both the (s,S) and (s, Q) replenishing policies.

An inventory queueing approach with MAP arrivals, PH offerings, and perishable goods was examined by Manuel et al. [15]. Additionally , they take into account their model, in which a positiv e customer adv ances one regular customer to the front of the line while a negativ e customer pushes one regular customer back. An inventor y retrial queueing system involving two commodities was presented by Anbazhagan and Jeganathan [3]. They think of their model as having a core item and a supplement item. Jeganathan and Selvakumar [9] examined a queueing system for inventory that employed a traditional retry rate. In their work, they present an optional oscillator y client arrival procedur e that is subject to Bernoulli testing and can pass through a waiting room or an infinite orbit. A two-component demand inventory retrial queueing system was examined by Abdul Reiyas and Jeganathan [1]. They took into account the (s, Q) replenishment policy while placing the order. The retrial inventory queueing model was examined by Jeganathan et al. [10] with two different kinds of clients. Mustapha and Majid [18] developed a two-phase production period production inventory model for non-immediately degrading products. For mixed demand with trade credit programs, Manisha et al. [14] have created the ideal replacement and conservation investment strategy. Ayyappan and Archana [5] discussed the non-preemptiv e priority queueing model with optional service and single vacation.

2. Description of the model

We take into consideration a single server with the preemptiv e priority inventory retrial queueing system, featuring inventory with maximum storage capacity of S units. Customers arrive via the Marked Markovian Arrival Process (MMAP) with depictions (Dq, D1, D2) with order size k, wher e the matrix D = D0 + D1 + D2. In the system, no arrival is governed by the squar e matrix Dq, the arrival of high priority customers is governed by the square matrix D1 and the arrival of low priority customers is governed by the square matrix D2. With n representing the probability vector of D, the mean arrival rate of HP customers is A1 = nD1 ek and the mean arrival rate of LP customers is A2 = nD2ek. The server provides high priority(HP) and low priority(LP) services that both follow a PH-distribution with depictions (7, P) and (v, M) of orders l1 and l2, respectiv ely. For HP and LP customers mean service rate is Z1 = [y(- P)-1 e\x]-1 and Z2 = [v(-M)-1 el2 ]-1.

Figure 1: A pictorial illustration of the model

If the server breaks down while serving HP or LP customers, it will first offer a slow service mode to the impacted customers before beginning the repair procedure. The PH-distribution is followed by the slower service for HP and LP customers, together with a representation of order l1 and l2, respectiv ely, represented by (j1,6 P) and (v1,6 M). The breakdown time has an exponential distribution with parameter a, and the repair process has a PH-distribution with a depiction (a, U) of order m2. When HP customers arrive, they only interrupt their regular service if LP customer services are still in progress, and the server serves HP clients. In the event that there are no pending requests in the HP queue, the server will serve LP customers. Arriving HP customers should get the service if the server is idle and has a positive inventory level; otherwise, they should wait in front of the service station. Arriving LP customers get service only if there is a positiv e inventory level and there are no high-priority customers in the system; otherwise, go for the finite capacity size of the orbit, say N. After the completion of service, if no one is present in the high-priority queue and orbit, the server will close down the system and then go on a single vacation. After the completion of the vacation period, the server is idle. The closedo wn times follo w an exponential distribution with parameter 5. The Vacation times follo w the PH-distribution with depiction (6, W) of order m1. The LP customers retrying for their service after the fixed times, the constant retrial rate follow an exponential distribution with parameter x

The average rate of repair and vacation is giv en by n and ty respectiv ely.

3. The Quasi Birth and Death Process for the Matrix Generations

We are going to discuss this part, which comprises the notation that forms the basis of the Quasi Birth and Death (QBD) process in our model.

• ® - Any two different order matrices can be multiplied to create a Kronecker product, and this can be founded on the research of Steeb and Hardy [22].

• ® - The Kronecker sum is the sum of any two of the different orders of matrices.

• 4 - The identity matrix has k dimensions.

• e - The column vector's appr opriate dimension for each of its elements is 1.

• ek - For every k elements in a column vector, the value is 1.

• ek(L) - The column vector with dimension L, where the kth element is 1 and remaining elements are 0.

• ek(L) - The transpose of ek(L).

• The arrival rate of HP and LP customers is represented by A, and described as A, = nD,ek, wher e i=1,2 respectiv ely.

• The service rate for HP customers is represented by Z1 and described as Z1 = [t(—P) —1 e;1 ]-1.

• The service rate for LP customers is represented by Z2 and described as Z2 = [v( —M)-1 e/2]-1.

• The vacation rate of the server is represented by ty and described as

ty =[fi(—W)—1 emi ]—1.

• The server's rate of repair is represented by n and described as n = [a( — U) 1 em2] 1.

• The number of HP customers in the system at time t can be represented by N1 (t).

• The number of LP customers in the orbit at time t can be represented by N2(t).

• Let V(t) be the state of the server at time t.

vacation state of the ser ver, idle state of the ser ver,

server is offering service for HP customers,

server is offering service for LP customers,

server is offering slow service for HP customers,

server is offering slow service for LP customers,

ser ver is under repair ,

server is under closedo wn process.

• Let I(t) be the level of inventory items at time t.

• J1 (t) denotes the phases of the vacation process.

• J2 (t) denotes the phases of the repair process.

• S(t) denotes the phases of the ser vice process.

• M(t) denotes the phases of the arrival process.

Let { N1 (t), N2(t), V(t), I(t), J1 (t), J2(t), S(t), M(t) : t > 0} indicate the Continuous Time Markov Chain (CTMC) with state-level independent QBD processes. The state space is as follows:

O = /(0) U~ =1 /(«),

wher e

V(t)

0, the

1, the

2, the

3, the

4, the

5, the

6, the

7, the

1(0) ={(0,u2,0, j,c) : 0 < m2 < N,0 < j < S, 1 < «1 < mi, 1 < c < k} U {(0, M2, 1, j, c) : 0 < M2 < N,0 < j < S, 1 < C < k}

U{(0, U2,3, j, d2, c) U{(0, U2,5, j, d2, c)

0 < M2 < N,1 < j < S,1 < d2 < 12,1 < c < k} 0 < m2 < N,1 < j < S,1 < d2 < 12,1 < c < k} U {(0,m2,6, j,«2,c) : 0 < m2 < N,0 < j < S, 1 < «2 < m2,1 < c < k} U {(0, m2, 7, j,c) : 0 < M2 < N,0 < j < S, 1 < c < k},

for M1 > 1,

1 (m1 ) ={(m1,m2,0, j,«1,c) : 0 < m2 < N,0 < j < S, 1 < «1 < m1,1 < c < k} U {(m1, m2, 1,0, c) : 0 < M2 < N,1 < c < k}

U {(m1,m2,2, j,d1,c): 0 < M2 < N,0 < j < S, 1 < d1 < 11,1 < c < k} U {(m1,m2,4, j,d1,c): 0 < M2 < N,0 < j < S, 1 < d1 < 11,1 < c < k} U {(m1, m2, 5, j, d2, c) : 0 < M2 < N,0 < j < S, 1 < d2 < 12,1 < c < k} U {(m1,m2,6, j,«2,c) : 0 < M2 < N,0 < j < S, 1 < «2 < m2,1 < c < k} U { (m1, m2, 7, j, c) : 0 < M2 < N,0 < j < S, 1 < c < k}.

The QBD procedur e generates an infinitesimal matrix, as provided by B00 B01 0 0 0 0 B10 A1 A0 0 0 0 0 A2 A1 A0 0 0 Q = 0 0 A2 A1 A0 0

The entries in Q's block matrices are specified as follows:

00

wher e

11 B00 R 12 B00 0 0 0 0

0 R 22 B00 R 23 B00 0 0 0

0 R 32 B00 33 B00 R 34 B00 0 B00

0 0 0 R 44 B00 45 B00 0

0 R 52 B00 0 0 55 B00 0

61 B00 0 0 0 0 B00

36

66

11

B00

C

001

C001 0

0 0

/1 0 0 /1

00 00

00

C002 C001

0

C002

0 0

00 00

/1 0 0 /2

00

C001 0

/31

/3

C002 C001 + C002.

/3

0

/2J

, C002 = Is+1 ® im1 ® D2,

Boo12 = In+i ® Is+i ® W0 ® Ik,

where Ji = W ® Do - t Im, k, J2 = W ® Do, J3 = t I,

mi k,

22

B00

C003 C004 0 0 C005 C004

C

003

C

005

C

006

00 00

J4 0 .

0 J4 .

0 0 ..

0 0 ..

.0 0 ...

J4 0 ...

0 J7 ...

0 0 ...

0 0 ...

0 0 ...

J9 0 ..

0 J10 ..

C

004

0 0 ... 0 0

e, (S + 1) ® D2" 0

0. 0.

00 00

J4 0 . .

0 J5 . .

0 0 ..

0 0 ...

0 0 ...

J7 0 . .

0 J8 ..

0 0 ..

00 00

C005 C004 0 C006 . J6 . J6

. J6 .0

J5

J6 J6

J6

0

J6 J6

J10 0 ... J6 0 J11 ... 0

J11.

wher e J4 = D0 - tIk, J5 = D0, J6 = tIk, J7 = D0 - (x + t)Ik,

D0 - XIk,

J9 = (D0 + D2) - tIk, J10 = (D0 + D2) - (x + t)Ik, J11 = (D0 + D2) - xIk,

23

B00

C

007

C007 0 C008 C007

00 00 0

Is ® v ® D2

C0

C

008 =

00 00

C007 0 C008 0 0

Is ® v ® X Im

R 33 B00

C009 0

C0010 C009

C

009

0 0

Jl2 0 0 Jl2

0

C0010

C009 C0010

0 C009 + C0010.

.. Jl4 ■

00

J12 0 0 J13

00

J14

J14 0

J13.

, C0010 = Is ® h2 ® D2

wher e J12 = M ® D0 - (a + t)Ihk, J13 = M ® D0 - aIhk, J14 = th

B0034 = in+i ® is ® e/2 ® viaik, B0032

00 0 In ® C0011

B0036 = [e1 (N + 1) ® C0011 0, C0011 = [Is ® M0 ® Ik 0] ,

44

B00

C

0012 C0010

C

0012 0010

C0

C

0012

0 0

J15 0 0 J15

0 . 0 .

00 00

C

0012

C

0010

0 C0012 + C0010. J14 ■

J14

J15 0 ... J14 0 J16 ... 0

0 0 ... 0 0 ... J16. wher e J15 = 0M ® D0 - tI/2k, J16 = 0M ® D0,

B

45

00

55

B00

IN+1 ® C0013, C0013 = [Is ® 0M0a ® Ik 0 ,

0013

C0014 C0015 0 0 C0014 C0015

C0014 0

C0015 C0014 + C0015.

B0052 = In+1 ® Is+1 ® U0 ® Ik,

k

0

C

0014

/l7 0 0 /l7

00

/l7 0 0 /l8

00

/19"

/l9

/19 0

/l8 —I

, C00l5 = Is + l ® Im2 ® °2>

wher e /l7 = U ® D0 - tim2k, /l8 = U ® D0, /l9 = tim2k,

00

66

C00l6 0

C00l7 C00l6

0

C00l7

B0061 = In+i ® Is+l ® P ® £Ik,

C

00l6 =

■/20 0 0 /20

/20 0 0 /21

C0016 0

/6 /6

/6

0

C0017 C0016 + C0017.

C0017 = is + 1 ® D2,

0 0 ... 0 0 ... /21

wher e /20 = d0 - (£ + t) ik, /21 = d0 - £ ik, B,

B,

01

wher e

11 01 0 0 0 0 0 0

0 22 B01 23 B01 0 0 0 0

0 0 33 B01 0 0 0 0

0 0 0 0 45 B01 0 0

0 0 0 0 0 56 B01 0

0 0 0 0 0 0 67 B01

B0111 = 1n+i ® Is+l ® V ® D1, B0122 = In+i ® el (S + l) ® D1,

R 23 B0

01

JN+1

® C

011

C

011

0

is ® 7 ® D1

01

33

1n+i ® Is ® e/2 ® 7 ® D1, B0145 = IN+I ® Is ® i/2 ® D1,

In+i ® is+l ® D1,

56 B01 = In+i ® is+l ® im2 ® Dl, 67 B01 =

00 0 0 0 0

00 0 0 0 0

0 B1032 0 0 0 36 B10

B10 = 00 0 0 45 B10 0 ,

00 0 0 0 0

00 0 0 0 0

00 0 0 0 0

wher i

R 32 B10

0 0 0 IN ® C101

, B

36

10

C101 0 00

, c101 = [is ® p0 ® ik 0] ,

B10 45 = In+1 ® C102, C102 = [Is ® 0P0a ® Ik 0] ,

Ai

"Ai11 0 0 0 0 0

Ai71

wher e

A112 A122 0 0 0

A162 0

A113 A123 A133 0 0

A163 0

0 0

A134 A144 0 0 0

0 0 0 0

A155 0 0

0 0 0 0

A156 A166 0

0 0 0 0 0 0

A177

A1

A1

11

= B00

A112 = In+1 ® e1 (S + 1) ® W0 ® Ik,

13

IN+1 ® C111, C111

D0 - TIk

A1

22

0

D2

D0 - TIk

0

Is ® W0 7 ® 0

D2 .

A123 = In+1 ® eS(S) ® 7 ® tIk, A1

A1

33

C112 0

C113 C112

C

112

0 0

/22 0

0

C113

34

0 0

D0 - TIk D2

0 (D0 + D2) - tIk.

In+1 ® Is ® e^ ® 71 ® oIk,

0 0

C113 = Is ® I/1 ® D2,

0

J22

0 0

C112 C113

0 C112 + C113 J

... /24

... /24

J22 0 ... J24

0 /23 ... 0

00

00

/23-

wher e /22 = P © D0 - (o + t)I/1 k, /23 = P ® D0 - o^ k, /24 = TIhk

A1

44

C114 C113 0 0 C114 C113

C114 C113

0 C114 + C113

11

C

114

725 0 0 725

724'

724

725 0 ... J24

o 726 ... 0

o 0 ... 0 0 ... 726J

wher e 725 = 0P ® D0 - TIhk, 726 = 0P ® D0,

55 44 56 45 66 55

Ai = B00 , Ai = B00 , Ai = B00 ,

A162 = In+1 ® ei (S + 1) ® U0 ® Ik, A163

C = i 0 C115 Is ® U0 7 ® Ik

I

N+1

® C

115

A177 = 66 R00 , A171 61 = R00 ,

rA 11 A0 0 0 0 0 0 0

0 A 22 A0 0 0 0 0 0

0 0 A 33 A0 0 0 0 0

A0 = 0 0 0 A 44 A0 0 0 0 ,

0 0 0 0 A 55 A0 0 0

0 0 0 0 0 A 66 A0 0

0 0 0 0 0 0 A 77 A0

wher e A 11 = A0 = R 11 R01 A 22 A0 = In+1 ® D1, A 33 A0 = In+1 ® Is ® I/1

A 44 = A 0 = A 33 A 0 , A 55 A0 45 = R01 A 66 A0 = R01 56, A , A 0 77 67 = R01 ,

A

A233 0 0 0

0 0 A246

0 0 0 0 0

0 0 0 0 0

0 A232 2 = 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

where A232 = In+1 ® e1 (S) ® P0 ® Ik, A233 = In+1 ® C211,

C

211

00 Is-1 ® P0Y ® Ik 0

A246

45 R10 .

4. Stationary Analysis

We analyze our model in a few consistent system configurations.

4.1. Criteria for stability

Let us define the matrix A as follows: A = A0 + A1 + A2, signifying that it is an irreducible infinitesimal generator matrix with dimensions of ((N + 1)(S + 1)m1 k + (N + 1)k + 2(N + 1 )Sl1 k + (N + 1)Sl2 k + (N + 1)(S + 1)m2 k + (N + 1)(S + 1)k).

The vector x represents the stationar y probability vector of A that achieving the criteria xA = 0 and xe = 1. The vector x is divided by x = (x0, k1, x2, x3, x4, x5, x) = (x000, x001,

..., K00S, к010, к011, ..., K01S, ..., K0N0, x0N1, ..., x0NS, к100, ^^ ..., K0N0, x201, ..., x20S, x211, . . . , x21S, . . . , x2N1, . . . , K2NS, к301, . . . , x30S, x311, . . . , x31S, . . . , K3N1, . . . , K3NS, к401, . . . , K40S, x411, ..., x41S, ..., x4N1, ..., x4NS, к500, к501, ..., x50S, к510, x511, ..., x51S, ..., K5N0, x5N1, . . . , K5NS, к600, к601, . . . , x60S, к610, к611, . . . , K61S, . . . , K6N0, x6N^ . . ., x6NS ), wher e x0 has a dimension of (N + 1)(S + 1)m1 k, x1 has a dimension of (N + 1)k, x2 has a dimension of (N + 1 )S/1 k, x3 has a dimension of (N + 1 )S/1 k, x4 has a dimension of (N + 1)S12k, x5 has a dimension of (N + 1)(S + 1)m2k and K6 has a dimension of (N + 1)(S + 1)k. When examining the Markov process within the framework of QBD, our model's stability should satisfy the essential and sufficient requirements of xA0 e < xA2 e. Upon performing certain algebraic reductions, the stability condition xA0e < xA2e is determined to be

N S N N S

E E K0u2j(e«1 ® D1 ek)+ E K1„20(D1 ek)+ E E (e\l ® D1 ek)

«2 =0 j = 0 U2 =0 U2 =0 j=1

N S N S N S

+ E E X3«2j (e/1 ® D1 ek)+ E E K4u2j (e/2 ® D1 ek)+ E E x5«2j (em2 ® D1 ek)

«2 =0 j=1 «2 =0 j=1 «2 =0 j = 0

N S N N N N

+ E E X6«2j(D1 ek) < E E X2«2j(P0 ® ek)+ E E X3«2j(0P0 ® ek).

«2 =0 j = 0 «2 =0 j = 1 «2 =0 j=1

4.2. Analysis of Stationar y Probability Vector

Let < represent the stationar y probability vector for Q, and this is divided as < = (<^0,<^2,...). Mention that <0 has a dimension of (N + 1)(S + 1)m1 k + 2(N + 1)(S + 1)k + 2(N + 1)S/2k + (N + 1)(S + 1)m2 k and <1, <2,... have a dimension of (N + 1)(S + 1)m1 k + (N + 1)k + 2( N + 1)S11 k + (N + 1 )S12k + (N + 1)(S + 1)m2k + (N + 1)(S + 1)k and the vector < satisfies <Q = 0 and <e = 1.

Additionally , after the stability requir ement of the model is met, the stationar y probability vector < can be obtained by applying the follo wing equation:

<«1 = <1R«1 -1, «1 > 1.

The matrix quadratic equation R2 A2 + RA1 + A0 = 0 is satisfied by the minimal non-negativ e solution R based on Neuts [20]. The matrix quadratic equation yields the rate matrix. The order of the rate matrix R is given by ((N + 1)(S + 1)m1 k + (N + 1)k + 2(N + 1)S11 k + (N + 1)S12k + (N + 1)(S + 1)m2k + (N + 1)(S + 1)k) and it fulfills the condition RA2e = A0e.

By solving the follo wing equations, the sub vectors <0 and <1 can be deter mined.

<0 B00 + <1B10 = 0,

<0 B01 + <1 (A1 + RA2) = 0, Subject to the normalizing condition

<0e0 + <1 (I - R)-1 e1 = 1,

wher e e0 = e(N+1)(S+1)m1 k+2(N+1)(S+1)k+2(N+1)S/2k+(N+1)(S+1)m2k and

e1 = e(N+1)(S+1)m1 k+(N+1)k+2(N+1)S/1 k+(N+1)SZ2 k+(N+1)(S+1)m2k+(N+1)(S+1)k.

According to Latouche and Ramasw ami [13], by utilizing important stages in the logarithmic reduction process, the R matrix can be produced analytically .

5. Measures of System Performance

• Average size of the HP customers in the system

TO

Esys = E «10«1 e. «1=1

• Average size of the LP customers in the orbit

TO N S m k NSk

Eorb = E E E E E «20«!«20j«1c + E EE «2^0«21jc

«1 =0 «2 = 1 j = 0 «1 =1 C=1 «2 = 1 j=0 c=1

TO Nk TO N S /1 k

+ E E E «20«1 «210c + E E E E E «20«1 «22jd1 c

«1 =1 «2 = 1 C=1 «1=1 «2 = 1 j=1 d =1 C=1

N S /2 k TO N S /1 k

+ E E E E «200«23jd2c + E E E E E «20«1 «24jd1 c

«2 = 1 j=1 ^2 = 1 c=1 «1 =1 «2 = 1 j=1 d =1 c=1

TO N S /2 k TO N S m2 k

+ E E E E E «20«1 «25jd2c + E E E E E «2 0«1 «26j«2c

«1 =0 «2 = 1 j=1 d2 = 1 c=1 «1 =0 «2 = 1 j=0 a2 = 1 c=1

to NSk

+ EE EE «2 0«1 «27jc.

«1 =0 «2 = 1 j=0 c=1

• Expected size of the inventory items

TO N S m k NSk

= E E E E Ej0«1 «20j«1 c + E EEM^^c

«1 =0 «2=0 j=1 «1 = 1 c=1 «2=0 j=1 c=1

to N S /1 k N S /2 k

+ E E E E E j0«1 «22jd1 c + E E E E c

«1 =1 «2 =0 j = 1 ¿1 =1 c=1 «2 = 0 j=1 d2 = 1 c=1

to N S /1 k to NS/2 k

+ E E E E E j0«1 «24jd1 c + E E E E E j0«1 «25^2c

«1 = 1 «2=0 j=1 d1 = 1 c=1 «1 =0 «2=0 j=1 d2 = 1 c=1

to N S m k to NSk

+ E E E E Ej0«1 «26j«2c + E E EEj0«1«27;c.

«1 =0 «2=0 j=1 «2 = 1 c=1 «1 =0 «2=0 j=1 c=1

• Expected reorder rate

N /2 k TO N /2

ER = E EE 00«23(s+1)d2 c (M0 ® Ik )e + E E E «25(s+1)d2 c (0 M0 a ® Ik )e

«2 =0 ¿2 =1 c=1 «1 =0 «2 =0 ¿2 = 1

N /1 k TO N /1 k

+ E E E 01«22(s+1)d1 c(P0 0 Ik)e + E E E E <?«i «22(s+1)dic(P07 0 Ik)e

«2=0 d1 =1 c=1 «1 =2 «2 =0 d1 =1 c=1

TO N /1 k

+ E E E E «22(5+1)^1 c(op0«0 Ik)e.

«1 = 1 «2 =0 d1 = 1 c=1

• Probability for the vacation state of the server

TO N S mi k

Pvac = E E E E E «2 0;«ic.

«i =0 «2=0 ;=0 ai = i c=i

Probability for the idle state of the server

N S k TO N k

E EE ^0«2i;c + E E E ^«i«2i0c.

«2=0 ;=0 c=i «i = i «2=0 c=i

• The probability that HP customers receive normal mode service from the server

m N S ¡1 k

PHNB = E E E E E <«1 «22jd1 c.

«1 = 1 «2=0 j=1 d1 = 1 c=1

• The probability that LP customers receive normal mode service from the server

N S ¡2 k

PLNB = E E E E <0«23jd2c.

«2=0 j=1 d2 = 1 c=1

• The probability that HP customers receive slow mode service from the server

m N S ¡1 k

PHSB = E E E E E <«1 «24jd1 c.

«1 = 1 «2=0 j=1 d1 = 1 c=1

• The probability that LP customers receive slow mode service from the server

m N S ¡2 k

Plsb = E E E E E <«1 «25jd2c.

«1 =0 «2 =0 j=1 d2 = 1 c=1

• Probability of the server is in repair process

m N S m2 k

Prep = E E E E E <«1 «26j«2c.

«1 =0 «2 =0 j=0 «2 = 1 c=1

• Probability of the server is in closedo wn process

m N S k

Pcd E E E E <«1 «27jc.

«1 =0 «2=0 j=0 c=1

• The rate of effectiv e retrials

N S k

R = x E EE<0«21 jc

«2 = 1 j=1 c=1

6. Analysis of Cost function

We have assumed that every cost factor (per unit of time) correlates to a distinct system measur e while de veloping the expense function for our model.

• Ei - The cost of inventory for retaining each unit of goods.

• Eh1 - Keeping a HP customer 's cost in the system for each unit of time.

• Eh2 - Keeping a LP customer 's cost in the system for each unit of time.

• Es - Initial costs for each order.

TC(S, S) = ElEinv + EH1 Esys + EH2 Eorb + ESER

7. N umerical Results

Using both numerical and graphical illustrations, we will be studying the behavior of the models in the section that follows. The next three are various MAP representations with the same mean value of 1 across all arrival processes. Chakravarthy [8] used these three arrival value sets as input data in their literatur e.

• A-ER(Arrival in Erlang):

Do =

A-EX(Arrival in Exponential):

Do = [-1],

A-HE(Arrival in Hyper-exponential):

-1.90 0 0 -0.19

-2 2 D1 — 0 0 , D, — 0 0

0 -2 1.2 0 0.8 0

Di = [0.6] , D2 = [0.4] .

D

0 —

Di

D,

0.684 0.0684

0.076 0.0076

1.026 0.114 0.1026 0.0114

The service, vacation, and repair processes each have three distinct phase-type distributions that we should take into consideration. We will use the notations X-ER, X-EX and X-HE respectiv ely for Erlang, exponential and hyper -exponential cases dealing with X-type distribution wher e X — S, V, R depending on whether the services, vacations or repairs are under consideration. • X-ER(Erlang):

(1,0),

P — M — W — U

Y — v — $ — a X-EX(Exponential):

Y — v — $ — a X-HE(Hyper-exponential):

Y — v — $ — a — (0.8,0.2 ), P — M — W — U

-2 2 02

P — M — W — U — [1].

-2.8 0 0 0.28

7.1. Illustrativ e Example 1

We explored the effects of repair rate (n) versus the average size of HP customers in the system( Esys). In order to attain system stability, we fix A = 1, Z1 = 10, Z2 = 8, ty = 8, a = 1, t = 5, X = 4, S = 4, Q = 0.6, s = 3, S = 6, N = 5.

• We combine the arrival and service time categories in Tables 1 through 3 to investigate the repair rate versus the average size of HP customers in the system.

• When the repair rate (n) rises, the corresponding the average size of HP customers in the system( Esys) reduces.

• When comparing arrival times to all other arrivals, the Esys drops quickly for hyper-exponential arrivals, and slowly for Erlang arrivals. Similarly , for service durations, the Esys decreases more slowly in Erlang services than it does with hyper-exponential services.

7.2. Illustrativ e Example 2

We explored the effects of HP service rate (Z1) versus the Total Cost(TC) of the system. In order to attain system stability, we fix A = 1, Z2 = 8, ty = 8, n = 6, a = 1, t = 5, X = 4, S = 4, Q = 0.6, s = 3, S = 6, N = 5, Ei = 50, EH1 = 200, EH2 = 180, ER = 220.

• We combine the arrival and service time categories in Tables 4 through 6 to investigate the HP ser vice rate versus the total cost of the system

• When the HP service rate (Z1) rises, the corresponding the total cost of the system( TC) reduces.

• When comparing arriv al times to all other arriv als, the TC drops quickly for hyper -exponential arrivals, and slowly for Erlang arrivals. Similarly, for service durations, the TC decreases more slowly in Erlang services than it does with hyper-exponential services.

7.3. Illustrativ e Example 3

We explored the effects of retrial rate (x) versus the average size of LP customers in the orbit( Eorb). In order to attain system stability, we fix A = 1, Z1 = 10, Z2 = 8, ty = 8, a = 1, t = 5, n = 6, S = 4,

0 = 0.6, s = 3, S = 6, N = 5.

• We combine the arrival and service time categories to investigate the rate of retrial versus the average size of LP customers in the orbit, using Figures 2 through 4.

• When the retrial rate (x) rises, the corresponding average size of LP customers in the orbit( Eorb) reduces.

• When comparing arrival times to all other arrivals, the Eorb,f drops quickly for hyper -exponential arriv als, and slo wly for Erlang arriv als.

Table 1: Repair rate(n) vs Esys - X-ER

X-ER

n A-ER A-EX A-HE

6.0 0.10813513 0.12190924 0.14286803

6.5 0.10786026 0.12152199 0.14226519

7.0 0.10763579 0.12120252 0.14176864

7.5 0.10744949 0.12093492 0.14135332

8.0 0.10729269 0.12070781 0.14100134

8.5 0.10715912 0.12051288 0.14069960

9.0 0.10704411 0.12034388 0.14043834

9.5 0.10694416 0.12019608 0.14021010

10.0 0.10685657 0.12006582 0.14000914

10.5 0.10677924 0.11995019 0.13983096

Table 2: Repair rate(n) vs Esys - X-EX

X-EX

n A-ER A-EX A-HE

6.0 0.10901178 0.12413720 0.14698476

6.5 0.10867336 0.12369032 0.14629182

7.0 0.10839815 0.12332339 0.14572339

7.5 0.10817062 0.12301736 0.14524978

8.0 0.10797985 0.12275870 0.14484984

8.5 0.10781790 0.12253750 0.14450816

9.0 0.10767894 0.12234641 0.14421327

9.5 0.10755857 0.12217984 0.14395643

10.0 0.10745340 0.12203347 0.14373094

10.5 0.10736082 0.12190393 0.14353155

•10-

A-ER

A-EX

6

6 7

x

0.12

w

8 • 10

45678 x

Figure 2: Rein'«/ rafe(x) vs Eorb - A-ER

Figure 3: Retr'a/ rate(%) vs Eorb - A-EX

2

8

4

5

8

4

Table 3: Repair rate(n) vs Esys - X-HE

X-HE

n A-ER A-EX A-HE

6.0 0.11918175 0.13742625 0.16861900

6.5 0.11850292 0.13665603 0.16748394

7.0 0.11795708 0.13603251 0.16656203

7.5 0.11751074 0.13551942 0.16580136

8.0 0.11714044 0.13509118 0.16516511

8.5 0.11682931 0.13472934 0.16462655

9.0 0.11656496 0.13442027 0.16416588

9.5 0.11633814 0.13415374 0.16376814

10.0 0.11614181 0.13392192 0.16342189

10.5 0.11597051 0.13371874 0.16311819

Table 4: HP service rate(Z1) vs TC - X-ER

X-ER

Zi A-ER A-EX A-HE

10 287.97326332 298.59933362 316.43100759

11 287.74939522 298.08848910 315.15693318

12 287.58577777 297.70317492 314.18547489

13 287.46248681 297.40469160 313.42429235

14 287.36716153 297.16827941 312.81429994

15 287.29182156 296.97748060 312.31615614

16 287.23113722 296.82099764 311.90277542

17 287.18144377 296.69085871 311.55497162

18 287.14015651 296.58130309 311.25881669

19 287.10541130 296.48808035 311.00398146

Table 5: HP service rate(Z1) vs TC - X-EX

X-EX

Z1 A-ER A-EX A-HE

10 288.98571417 299.98654358 318.45145147

11 288.70159130 299.40909492 317.07970441

12 288.49311376 298.97229601 316.03161949

13 288.33556222 298.63313169 315.20927312

14 288.21349742 298.36397016 314.54968265

15 288.11689338 298.14638167 314.01074509

16 288.03902055 297.96767769 313.56338397

17 287.97523158 297.81888191 313.18695269

18 287.92223650 297.69349308 312.86643407

19 287.87765496 297.58670481 312.59067408

Table 6: HP service ra£e(Zi) vs TC - X-HE

X-HE

Zi A-ER A-EX A-HE

i0 295.87738733 307.40357235 328.4i423547

ii 295.3i869946 306.5886i243 326.77655347

i2 294.89650447 305.96029062 325.5003i090

i3 294.569i7993 305.464i0267 324.4826246i

i4 294.30990735 305.0643550i 323.65537379

i5 294.i007594i 304.7368i683 322.97i8766i

i6 293.92937439 304.464532i8 322.399i9670

i7 293.78699896 304.23532420 32i.9i350697

i8 293.66729077 304.04024987 32i.497i9i85

i9 293.56556ii8 303.8726ii47 32i.i3697796

A-HE

Figure 4: Reír/«/ raíe(x)

vs Eorb - A-HE

8. Conclusion

The present study investigated a retrial inventory queuing system that incorporates MMAP arrivals for HP and LP customers, services, vacations, and repairs, all of which follow phase-type distribution, (s,S) replenishment inventory policy, working breakdown, and closedown. We examined the system's stability criteria as well as the invariant probability vector. We analyzed the active period and also offered cost evaluations and system performance measur es. Employing numeric values of arrivals and services in this model, we computed the average size of HP customers in the system for different values of repair rate and the total cost of the system for different values of service rate. The two-dimensional plots show the average size of LP customers in the orbit for different values of retrial rate. The average size of HP customers in the system for various values of vacation and service rates is depicted in the three-dimensional graphs. Every table and graph shows the stability of the system. We also expand our resear ch to include multi-ser vers with two commodity inventory queueing systems.

R eferences

[1] Abdul Reiyas, M. and Jeganathan, K. (2023). A classical retrial queueing inventory system with two component demand rate, Internationa/ 7o«rna/ of Operation«/ Research, 47(4):508-533.

[2] Amirthakodi, M. and Sivakumar, B. (2015). An inventory system with service facility and finite orbit size for feedback customers, OPSEARCH, 52(2):225-255.

[3] Anbazhagan, N. and Jeganathan, K. (2013). Two-commodity Markovian inventory system with compliment and retrial demand, British 7onrna/ of Mathematics and Computer Science, 3(2):115-134.

[4] Ayyappan, G. and Gowthami, R. (2019). Analysis of MAP/PH/1 retrial queue with constant retrial rate, Bernoulli schedule vacation, Bernoulli feedback, breakdown and repair, Re/iabi/ity: Theory and App/ications, 14(2):86-103.

[5] Ayyappan, G. and Archana Gurulakshmi, G. (2023). Analysis of MMAP/PH/1 Classical Retrial Queue with Non-pr eemptiv e priority, Second optional service, Differentiate breakdowns, Phase type repair, Single vacation, Emergency vacation, Closedown, Setup and Discouragement, Re/iabi/ity: Theory and App/ications, 18(3):528:551.

[6] Berman, O., Kaplan, E. H. and Shimshak, D. G. (1993). Deterministic approximations for inventory management at service facilities, IIE Transactions, 25(5):98-104.

[7] Berman,O. and Kim, E. (1999). Stochastic models for inventory management at service facility, Comm«nications in Statistics. Stochastic Mode/s, 15(4):695-718.

[8] Chakravarthy, S. R. (2010). Markovian arrival process, Wiley Encyclopaedia of Operations Resear ch and Management, John Wiley and Sons, Inc., USA.

[9] Jeganathan, K. and Selvakumar, S. (2022). An optional arrival process of the oscillatory demands in the inventory queueing system with queue dependent service rate, Internationa/ 7o«rna/ of Mathematics in Operationa/ Research, 22(2):162-194.

[10] Jeganathan, K., Vidhya, S., Hemavathy, R., Anbazhagan, N., Joshi, G. P., Kang, C. and Seo, C. (2022). Analysis of M/M/1/N Stochastic Queueing- Inventory System with Discretionary Priority Service and Retrial Facility, S«stainabi/ity, 14(10), 6370.

[11] Krishnamoorthy , A., Nair, S. S. and Narayanan, V. C. (2011). An inventory model with server interruptions and retrials, Operationa/ Research, 12(22):151-171.

[12] Krishnamoorthy , A. and Narayanan, V. C. (2011). Production inventory with service time and vacation to the server, IMA 7o«rna/ of Management Mathematics, 22(1):33-45.

[13] Latouche, G. and Ramasw ami, V. (1999). Introduction of Matrix Analytic Methods in Stochastic Modeling, Society for Industrial and Applied Mathematics, Philadelphia.

[14] Manisha Pant, Seema Sharma and Anand Chauhan (2022). Optimal replenishment and preser vation investment policy for hy brid demand with trade credit schemes, Internationa/ 7o«rna/ of Mathematics in Operationa/ Research, 23(2):232-258.

[15] Manuel, P., Sivakumar, B. and Arivarignan, G. (2007). A perishable inventory system with service facilities, MAP arrivals and PH-Service times, Journal of Systems Science and Systems Engineering, 16(1):62-73.

[16] Meliko v, A. Z. and Molchano v, A. A. (1992). Stock optimization in transportation/storage systems, Cybernetics and System Analysis, 28(1):484-487.

[17] Meliko v, A., Aliyeva, S., Nair, S. S. and Kumar, B. K. (2022). Retrial Queuing-Inv entory Systems with Delayed Feedback and Instantaneous Damaging of Items, Axioms, 11(5), 241.

[18] Mustapha Lawal Malumfashi and Majid Khan Majahar Ali (2023). A production inventory model for non-instantaneous deteriorating items with two-phase production period, stock-dependent demanand and shortages, International Journal of Mathematics in Operational Research, 24(2):173-193.

[19] Neuts, M. F. (1979). A versatile Markovian point process, Journal of Applied Probability, 16(4):764-779.

[20] Neuts, M. F. (1984). Matrix-analytic methods in queuing theory, European Journal of Operational Research, 15(1):2-12.

[21] Sanjukta Malakar and Nabendu Sen (2023). An inventory model to study partial replacement policy and finite shelf life for deteriorating items with carbon tax, International Journal of Mathematics in Operational Research, 24(2):286-299.

[22] Steeb, W. H. and Hardy, Y. (2011). Matrix Calculus and Kronecker Product: A Practical Appr oach to Linear and Multilinear Algebra, World Scientific Publishing, Singapor e.

[23] Sugany a, C. and Sivakumar , B. (2019). MAP/P H(1), P H(2)/2 finite retrial inventory system with service facility, multiple vacations for servers, International Journal of Mathematics in Operational Research, 15(3):265-295.

[24] Ushakumari, P. V. (2017). A retrial inventory system with an unreliable server, International Journal of Mathematics in Operational Research, 10(2):190-210.

[25] Yadavalli, V. S. S., Sivakumar, B. and Arivarignan, G. (2008). Inventory system with renewal demands at service facilities, International Journal of Production Economics, 114(1):252-264.