CC BY
9
A MAP/PHi, PH2/2 INVENTORY QUEUEING SYSTEM WITH TWO COMMODITY, MULTIPLE VACATION, SERVER FEEDBACK, WORKING BREAKDOWN, REPAIR AND EMERGENCY REPLENISHMENT
G. Ayyappan, N. Arulmozhi •
Department of Mathematics, Puducherry Technological University, Puducherry, India. ayyappan@ptuniv.edu.in, arulmozhisathya@gmail.com,
Abstract
We investigate a continuous review inventory queuing system in the present study that has two heterogeneous servers: Server-2, which is reliable, and Server-1, which is unreliable. An exponentially distributed random time is used to describe the repair process when server-1 has an interruption. On the other hand, server-2 is completely dependable, but it goes on vacation when the system is empty. These two goods can be reordered under ordering regulations. To ensure customer satisfaction, an emergency replenishment of one item with no lead time occurs when the on-hand inventory level falls to zero. We use the matrix analytic approach for the QBD process under a steady-state probability vector. We also take into account the overall cost and the busy time. Furthermore, numerical data shows the benefits of the suggested approach in a range of random circumstances.
Keywords: Markovian arrival process, PH-distribution, multiple vacation, two commodity, working breakdown, emergency replenishment.
AMS Subject Classification (2010): 60K25, 68M30, 90B22.
1. Introduction
Substitution methods are essential for reducing client losses in an inventory-based organization. While the demand-driven item's stock-out time, emergency replenishment may be utilized. Stockouts may be managed and frequently avoided by employing excellent inventory management methods such as accurate demand forecasts, establishing reorder points, utilizing buffers and safety stock, and discovering stockout trends and emergency orders using inventory management software. In circumstances where stockouts are beyond a retailer's control, it's necessary to take efforts to save expenses and prevent dissatisfied consumers, such as proposing or inventing a product substitute or finding an alternate supplier.
The study [8] examined a (0, S) in a service facility with multiple server vacations and impatient clients.The combined probability distribution of the inventory level and the number of customers in the waiting area is calculated under steady-state scenarios. A few system performance metrics are obtained, and the total estimated cost rate is computed. The two-commodity (TC) inventory system, in which commodities are split into primary and supplemental products, was examined by Jeganathan et al. [13]. They examined an individual reordering strategy for a single commodity inventory with no replenishment period and another commodity utilised for the (s,Q) policy. An inventory queuing system with Poisson arrival, randomly
distributed service times, zero lead times, requiring one item per client, and limited waiting room capacity was studied by Berman and Sapna [3].
Kalyanaraman and Senthilkumar [7] studied two heterogeneous server Markovian queues, with the first server's service mode altering at a threshold. If both servers are idle, the consumer will be served by the faster server. Using MAP arrival and group services, Chakravarthy et al. [4] investigated the multi-server finite capacity model. They proved that the invariant distribution of the sojourn time is phase-type if the inter-arrival durations are as well. They also developed mathematical approaches for determining the invariant density of waiting queue size. Yang et al. [15] examined a Markovian queue with two servers, malfunctions, and leaves of absence. They used the usual particle swarm optimization technique for numerical analysis, developed a heuristic cost model, and investigated their system using the matrix approach.
In a recent article, Laxmi and Soujanya [11] discussed QIS with multiple server working vacations. They consider both working vacation and multiple vacation, and numerically analyze the system to determine the best strategy. An (s, Q) replenishment technique was used by Manikandan and Nair [12] to assess a single server QIS while taking breaks and interruptions into consideration. Performance data are examined, the steady-state probability vector is calculated, and the stability condition of the system is established. A busy period analysis and a derived stationary waiting time distribution for the queue are included in the study.
Yadavalli et al. [16] used a finite source QIS with interruption to conduct research on two heterogeneity servers. Jeganathan et al. [6] studied a Markovian inventory system with two queues and customers jockeying. Yadavalli and Jeganathan [17] investigate the impact of two servers and partial vacations on inventory systems. Suganya et al. [14] proposed an inventory system with numerous server vacations, as opposed to partial ones.They did, however, take into account the Markovian Arrival Process (MAP) of clients. A queueing model subject to a single server that can offer two types of heterogeneous services, closedown, vacation, setup, immediate feedback, breakdown, and repair was determined by Ayyappan and Gowthami [2]. Ayyappan and Karpagam [1] have discussed a classical model with unreliable server, vacation and immediate feedback. They have assumed that the unsatisfied customers will be given re-service immediately without any delay. They have found the PGF of the size of the waiting line and some interesting performance measures.
2. Model Description
We investigate the two server inventory queueing model, where customers arrive at the system based on MAP. The parameter matrices Do and D1 indicate that no customers have arrived at the system and that customers have arrived, respectively. Dimension m is used by the matrices Do and D1. We examine two different kinds of heterogeneous servers here: One among them does not go on vacation and the other one goes for a vacation after service completion. That is, the server-1 will always be there in the system but the server-2 will move forward to vacation after service completion to the customers. The service rendering by the server-1 during normal mode with representations (71, U1) of dimension n1 such that U0 + U1 e = 0. Likewise, The service rendering by the server-2 during normal mode is follows the PH-type with representation (?2, U2) and of dimension n2 such that U0 + U2e = 0. During the server-1 providing service to the customers who may struck with breakdown and the system has an option: continue delivering slow service to current customer. During the working breakdown period, the server provide slow service to current customer and service times are phase type distributed with notation (71,d1U1); 0 < 01 < 1 of order n1 and rate is ^d = [l1(S1U1 )-1 e]-1.
After service completion during this working breakdown period, the system automatically enters a repair phase. The server will begin a rejuvenating service for the customer, after the repair process. After service completion, no customer in the system server-2 go for multiple vacation. At the end of a vacation, when the customers are staying in the system then the server-2 is provided normal service. Otherwise, server-2 takes another vacation immediately. After received service
from the service station, the satisfied customer would left out of the system with probability pi, for i = 1,2 and if the customer is not satisfied with probability qi, for i=l,2 then they will get feedback immediately that is, the instantaneous feedback will offer by the same server. The breakdown and repair times of server-i follows an exponentially distributed with parameter ty and t respectively. We analyse a stochastic inventory system with two distinct goods in stock: server-1 (I-commodity) and server-2 (Il-commodity). The maximal storage capacity for the commodity ith is Si (i = 1,2). The initial commodity with zero lead time follows a (0, Si) ordering strategy. When the on-hand inventory level of the second commodity falls below a predetermined level, s2, an order for S2 units is made. The lead time for this order is exponentially distributed with parameter ft. Furthermore, suppose an emergency replenishment of one item with zero lead time occurs when the on-hand inventory level falls to zero. Emergency replenishment is included into the system to ensure client happiness. The server-2 does not offer feedback service during emergency replenishment. The schematic picture of this model is provided in Figure 1.
Figure 1: Schematic representation
3. Analysis
In the following section, we establish the queueing-inventory system's transition rate matrix. Assume that N(t), J1(t), J2(t), I1 (t), I2(t), S1 (t), S2(t), M(t) described total customers in the system, status of server-1, status of server-2, stock level for commodity I, stock level for commodity II, service phase for server-1, service phase for server-2, arrival phases, respectively.
J1(t)
0, server-
1, server-
2, server-
3, server-
J2 (t)
Consider X(t)
0, server-2 is vacation,
1, server-2 is busy in normal mode,
{N(t), J1 (t), J2(t), I1(t), I2(t), S1 (t), S2(t), M(t)} is a CTMC with state space
O = 0(0) u 0(1) U 0(i).
i=1
(1)
where
0(0) ={(0,0,0,u1,u2,u5) : 1 < u1 < S1, 1 < u2 < S2, 1 < u5 < m}
U {(0,3,0,u1,u2,u5) : 1 < u1 < S1, 1 < u2 < S2, 1 < u5 < m}
0(1) ={(1,0,1,u1,u2,u4,u5) : 1 < u1 < S1, 1 < u2 < S2, 1 < u4 < n2, 1 < u5 < m}
U {(1,1,0, u1, u2, u3, u5) : 1 < u1 < S1, 1 < u2 < S2, 1 < u3 < n1, 1 < u5 < m} U {(1,2,0, u1, u2, u3, u5) : 1 < u1 < S1, 1 < u2 < S2, 1 < u3 < n1, 1 < u5 < m} U {(1,3,0,u1,u2,u5) : 1 < u1 < S1, 1 < u2 < S2, 1 < u5 < m} U {(1,3,1, u1, u2, u4, u5) : 1 < u1 < S1, 1 < u2 < S2, 1 < u4 < n2, 1 < u5 < m} and for i > 2,
0(i) = {(i, 1,0, u1, u2, u3, u5) : 1 < u1 < S1, 1 < u2 < S2, 1 < u3 < n1, 1 < u5 < m}
U {(i, 1,1,u1, u2,u3,u4,u5) : 1 < u1 < S1, 1 < u2 < S2, 1 < u3 < n1, 1 < u4 < n2, 1 < u5 < m} U {(i,2,0, u1, u2, u3, u5) : 1 < u1 < S1, 1 < u2 < S2, 1 < u3 < n1, 1 < u5 < m} U {(i,2,1,u1,u2,u3,u4,u5) : 1 < u1 < S1, 1 < u2 < S2, 1 < u3 < n1, 1 < u4 < n2, 1 < u5 < m} U {(i, 3,0, u1, u2, u5) : 1 < u1 < S1, 1 < u2 < S2, 1 < u5 < m} U {(i,3,1,u1,u2,u4,u5) : 1 < u1 < S1, 1 < u2 < S2, 1 < u4 < n2, 1 < u5 < m}
Notations:
• ® - Kronecker product of two matrices of different dimensions.
• ® - Kronecker sum of two matrices of different dimensions.
• e - Column vector has an suitable size with each of its entries as 1.
• e0 - e2S1 S2m.
• e1 - e2S1 S2n1m+2S1S2n2m+S1 S2m.
• e2 - e2S1 S2n1m+2S1S2n1n2m+S1S2n2m+S1S2m.
• Ij - Square matrix with jxj size with diagonal entries as 1.
• 0 - It denotes zero matrices in the suitable order.
3.1. Construction of the QBD process for our Model
The generator matrix of the Markov chain under (s, S) policy is given by:
Q
A00 A01 0 0 0 0
A10 A11 A12 0 0 0
0 A21 F1 F0 0 0
0 0 F2 F1 F0 0
0 0 0 F2 F1 F0
The entries in the block matrices of Q are defined as follows:
0 , A** = Isi 0 pv a02 = ISi 0 P2,
Aoo
Aoo
A 21 A 22 A oo A oo
rc? 0 0 . .0 0. .0 C2] "C5 0 0 . .0 0 . .0 Cf|
0 C? 0 . .0 0. .0 C2 0 C5 0 . .0 0.. .0 C6
0 0 C? . .0 0. .0 C2 0 0 C5 . . .0 0 . .0 C6
Pi = 0 0 0 . . C? 0. .0 C2 , P2 = 0 0 0 . . C5 0.. .0 C6
0 0 0 . .0 C3 . .0 0 0 0 0 . .0 C7 . .0 0
0 0 0 . .0 0. . C3 0 0 0 0 . .0 0.. . C7 0
0 0 0 . .0 0. .0 C3. 0 0 0 . .0 0 . .0 C7.
Ci = Do - ftIm, C2 = ftIm, C3 = Do, C4 = Is2 0 TIm, C5 = Do - (t + ft)Im, C6 = ftIm C7 = Do - TIm,
A21 Aoo
A1o21
Cs = Is2 0 71 0 Di, C9 = Is2 0 Di,
C4 0 0 . .0 0
0 C4 0.. .0 0
0 0 C4 . . .0 0 , Aoi = 0 0 Aio2i 0 00 0 A2o4i 0" 0 ,
0 0 0 . . C4 0
0 0 0.. .0 C4.
Cs 0 0 . .0 0" C9 0 0 ... 0 0
0 Cs 0.. .0 0 0 C9 0 ... 0 0
0 0 Cs . . .0 0 , A24 = , Aoi = 0 0 C9 . . . 0 0
0 0 0 . . Cs 0 0 0 0 ... C9 0
0 0 0.. .0 Cs 0 0 0 ... 0 C9
A
1o
A11 A1o
Aio 0 0 0
0 0
A32
Aio
0
A52 Aio.
Aio = Isi 0 Cio, Aio
io
Aio = Isi 0 Ci3, Cio = Ci2 = Is2 0 6iUo 0 Im, Ci3 =
lA-i
0 Cii
Cii
0
0 Im Is2-i 0 q2U° 0 Im 0 Im Is2-i 0 q2U° 0 Im
A32 Aio
Cii = Is2 0 qiUo 0 Im,
0 Ci2 _Is*-i 0 Ci2 0
A
ii
Aii 0 0 0
Aii
0
0
A22 A23 Aii Aii A??
0
a??
A44 A45 Aii Aii
0 a??.
, Aii = Is? 0 P3, A?? =
P4 0 0 . .0 P5
P5 P4 0 . .0 0
0 P5 P4 . . .0 0
0 0 0 . . P4 0
0 0 0 . . P5 P4
0
P3
C14 О О C16 C14 О О C16 C14
C14
О
C16 C17
О C15 О C15 О C15
О C15
ОО
C17 О C16 C17
, Cl4 = U2 e Do — ßIn2m, C15 = ßIn2m,
Cl6 = p2U°72 ® Im, C17 = U2 e Do, P4
[Cls О О . .О О . .О C19
О Cls О.. .О О . .О C19
О О Cls . .О О . .О Cl9
О О О.. . Cls О . .О C19
О О О . .О C2o . . .О О
О О О.. .О О . . C20 О
О О О . .О О . .О C20
Cls = Ul e Do — (ß + 0)In1 m, C19 = ßIn1 m, C20 = Ul e Do, P5 = Is2 ® plU0Yl ® Im
A2131
C21 О О
О О
О
C21 О
О О
О О
C21
О О
C21 О О C21.
, C21 = Is2 ® en1 ® Y1 ® 0Im,
All = Isi ® P6, A42
C25 О О О C25 О О О C25
C25 О О C25
, All = Isi ® P7
C22 О О ... О О .. C23 C26 О О . .О О. . C27
О C22 О ... О О .. C23 О C26 О . .О О. . C27
О О C22 . . . О О .. C23 ОО C26 . . .О О. . C27
P6 = О О О ... C22 О .. C23 , P7 = ОО О . . C26 О. . C27
О О О ... О C24 . . . О ОО О . .О C2s . .О
О О О ... О О .. О ОО О . .О О. .О
О О О ... О О .. C24 ОО О . .О О. . C2s
C22 = 0Ul e Do ß Inlm, C23 = ß Inlm, C24 = 0Ul e Do, C25 = Is2 ® Yl ® тIm,
C26 = Do — (т + П + ß)Im, C27 = ß Im , C2s = Do — (т + П)Im,
All
C29 о о о C29 о о о C29
C29 о о C29.
, All
C30 о о о C30 о о о C30
C29 = Is2 ® 72 ® П Im, C30 = Is2 ® In2 ® ТIm, A55 = Isl ® P7,
C30 о о C30
P7
C31 о о
C33 C3l о о C33 C3l
о C32 о C32 о C32
C31 о
C33 C34
C32
о
C34 о C33 C34
where
C31 = U2 © Do - (т + ß)In2m, C32 = ßIn2m, C33 = p2U°72 ® Im, C34 = U2 © Do - ТIn2m,
Al2
Al2
A33 = Al2 =
о Al2 о о о о
Ai2 о о о о о
о о A33 a12 о о о
о о о о A45 a12 о
о о о о о A 55 Al2-I
C36 о о о о "
о C36 о о о
о о C36 о о
о о о C36 о
о о о о C36
C37 о о о о "
о C37 о о о
о о C37 о о
о о о C37 о
о о о о C37
Al2
■C35 о о
о о
о
C35 о
о о
о о
C35
о о
C35
о
о о о
о
C35J
, C35 = Is2 ® Yi ® In2 ® Di, C36 = Is2 ® Inl ® Di,
A45 , Al2
C3s о о о C3s о о о C3s
C3s
о
о о о
о
C3s
C37 = Is2 ® Ini ® Di, C3S = Is2 ® Di, A45 =
C39 о о о C39 о о о C39
о о о
C39
о
о о о
о
C39
0 A21 0 0 0
A21 A21 0 0 0
Is2 © In2 © Dl, A21 = 0 0 0 0 0 A43 A21 A34 A21 0 0 A45 A21 , A21 = 0 C40" Is1 -1 © C40 0_
0 0 0 0 0
0 0 0 0 A65 A2lJ
A21 = Is! © C41, A21
em © U20Yl © Im
0 C42
Js1-1 © C42 0
C41
0
Is2-1 © ещ © q2U°Yl © Im 0_ A21 = Isj © C44, Aff
In! © U0 © Im
, C40 = Is2 © ql U°Yl © Im, , C42 = Is2 © en2 © ql U°Y2 © Im, A34
0 C45' Is!-1 © C45 0
C44
0
Is2-1 © Inl © q2U0 © Im 0_
Is!-1 © C43
, A21 = Is! © C46, C43 = Is2 © 0U0 © Im,
U Y2 © Im
, C45 = Is2 © 0U° © In2m, C46 =
Is2-l © q2U0Y2 © Im
Fl
Fi1 F112 F113 0 F22 F23 0 0" 000 00 0 F162 0
33 Fl
p51 F1
0
0
F34
Fl44 0 0
P8
C47 0 0 0 C47 0 0 0 C47
0 0 0
0
F55
0 " P8 0
0 P9 P8
0 0 , Fl11 = 0 P9
f56 F1 F66 00 00
0 ... 0 C48
0 ... 0 C48
0 ... 0 C48
0 P9 00 00
P8 0 P9 P8
C47 0 0 C49
0 C48 00
C49 0 0 C49
, P9 = Is2 © PlUf Yl © Im,
C43
0
C47 = Ul © D0 - (n + ß + 0)Inlm, C48 = ßInlm, C49 = Ul © D0 - (n + 0)I
nl m/
0
F112
Fl22
C50 0 0 ... 0 0 C51 0 0. .. 0 0
0 C50 0 ... 0 0 0 C51 0. .. 0 0
00 C50 . . . 0 0 , Fl13 = 0 0 C51 . .. 0 0
0 0 0 ... C50 0 0 0 0. . C51 0
00 0 ... 0 C50 0 0 0. .. 0 C51-
Is2 © In1 Y2 © n Im, C51 = Is2 © en1 © Yl © 0Im,
"P10 0 0 ... 0 P11 " C57 0 0 . .0 0"
Pll P10 0 ... 0 0 0 C57 0.. .0 0
0 Pll P10 ... 0 0 , F24 = 0 0 C57 . . .0 0
0 0 0 ... P10 0 0 0 0 . . C57 0
00 0 ... Pll P10. 0 0 0.. .0 C57
Pic
where
C52 0
C54 C52
C54 C5:
0 0 . .. 0 C53 C56 0 0 . .0 0 "
0 0 . .. 0 C53 0 C56 0.. .0 0
0 0 C56 . . .0 0
C52 0. .. 0 C52 , Pii =
C54 C55 . .. 0 0
0 0 0 . . C56 0
0 0. . . C55 0 0 0 0.. .0 C56
0 0. . . C54 C55
C52 = Ui ® U2 ® Dc - + ft)i„i„2m, C53 = ftini„2m, C54 = ini ® P2U°72 ®
C55 = Ui ® U2 ® Dc - $inin2m, C56 = Is2 ® PiUCYi ® i„2m, C57 = Is2 ® eni ® Yi ® $i„2m,
Fi33 = iSi ® Pi2,
Pi2
C58 0 0 . .0 0
0 C58 0.. .0 0
0 0 C58 . . .0 0
0 0 0 . . C58 0
0 0 0.. .0 C6
0 0 0 . .0 0
0 0 0.. .0 0
C58 = OUi ® Dc - (7 + ft) inim, C59 = ft Ini
C6i 0 0 ... 0 0 "
0 C6i 0 ... 0 0
F34 = Fi = 0 0 C6i ... 0 0
0 0 0 ... C6i 0
0 0 0 ... 0 C6i.
C59 C59 C59
C59 0
0
C6c
C62 0
C64 C62
Fi44
0 0
C64 C62
C62 0
C64 C65
where
C6i = Is2 ® ini ® Y2 ® 7Im, C62 = OUi ® U2 ® Dc - ftini„2m, C63 = ftinin2m, C64 = ini ® P2UCY2 ® Im, C65 = OUi ® U2 ® Dc, "C66 0 0 ... 0 0
5i Fi
0
C66
0 0
56 Fi
where
C66 0 0 C66J
, Fi55 = ISi ® Pi3,
C7c 0 0 . .0 0
0 C7c 0.. .0 0
0 0 C7c . . .0 0
0 0 0 . . C7c 0
0 0 0.. .0 C7c
, Pi3
C67 0 0 0 C67 0 0 0 C67
C67 0
0
C69
C65
C63"
C63
C63
C63 0
0
C65
C69 0
C68 C68
C68 0
0
C69_
C66 = Is2 ® Yi ® TIm, C67 = Dc - (t + 7 + ft)Im, C68 = ftIm, C69 = Dc - (7 + t)
C
0
0
0
53
0
0
0
F6
C71 О О О О
О C71 О О О
О О C71 О О
Is2 ® Y2 ® n Imz Fi2 =
О О О C71 О
О О О О C71
C72 О О ... О О О C73
C74 C72 О ... О О О C73
О C74 C72 . . . О О О C73
О О О ... C72 О О C73
О О О ... C74 C75 О О
О О О ... О О C75 О
О О О ... О О C74 C75
z C71 = Is2 ® Yl ® Т J^mz
where
C72 = U2 © Do - (т + ß)In2mz C73 = ßIn2mz C74 = P2U°Y2 ® Im C75 = U2 © Do - Тi^mz
rpll p0 О О О О О О " C76 О О . .О О
p22 p0 О О О О О О О C76 О.. .О О
Fo = О О p33 p0 О О po44 О О О О z p11 = z po = О О C76 . . .О О
О О О О p55 po О О О О . . C76 О
О О О О О 66 po О О О.. .О C76
p33 p0
55 p0
C8o
C77 О О. .. О О
О C77 О. .. О О
О О C77 . .. О О z C76 = Js2 ® D1, C77 = ÍS2 ® ^«2 ®
О О О. . . C77 О
О О О. .. О C77
C78 О О. .. О О" C79 О О .. О О"
О C78 О. .. О О О C79 О .. О О
О О C78 . .. О О z Po44 = О О C79 . . . О О
О О О. . . C78 О О О О .. C79 О
О О О. .. О C78 О О О... О C79
s2 ® I И1 ® Dlz C79 = Js2 ® ínln2 ® Dl
C8o О О. .. О О" C81 О О .. О О"
О C8o О. .. О О О C81 О... О О
О О C8o . .. О О P66 = , po = О О C81 . . . О О
О О О. . . C8o О О О О .. C81 О
О О О. .. О C8o О О О... О C81
rs2 ® Dlz C8l = Is2 ® In2 ® D 1z
F2
F22
where
F^ 0 0 0 0 0
0 F222 0 0 0 0
0 0 0 0 F35 F2 0
0 0 0 F244 0 0 ,
0 0 0 0 0 0
0 0 0 0 0 F266
"Pi4 0 0 .0 Pi5 "
Pi5 Pi4 0 .0 0
0 Pi5 Pi4 .0 0
0 0 0 . Pi4 0
0 0 0 . Pi5 Pi4_
Fii
F2
_iSl _i ® Cs2
C82 0
Cs2 = Is2 ® qiU07i ® Im, Pi4 =
ini ® U0 72 ® Im _is2_i ® ini ® q2U072 ® Im
, Pi5 = Is2 ® qiU07i ® i„2m,
0 Cs3
_iSi_i ® Cs3 0
, F44
0 Cs4"
_iSi_i ® Cs4 0
r35 _ F2 =
where
Cs3 = Is2 ® 0U0 ® Im, Cs4 = Is2 ® 0U0 ® i„2m, Cs5 =
p66 , F2
= Is2 ® Cs5,
U072 ®Im
Is2_i ® q2U072 ® Im 0_
3.2. Stability condition
To discuss the stability condition, we first consider the generator matrix F = F0 + Fi + F2. The vector x is the invariant vector of the matrix F. Then, relations xF = 0 and xe = i and The LIQBD fashion with infinitesimal generator Q is stable if and only if
xFoe < xF2e.
The stability obtained after some mathematical rearranging is shown below:
x0[esis2ni ® Diem] + xi[esis2nin2 ® Diem] + x2[esis2n ® Diem] + x3[esis2nin2 ® Diem] + x4[esis2 ® Diem] + xs[esis2n2 ® Diem] < x0[es2 ® qiU0 ® em] + xi(esi ® [(e«i ® U0 ® em + qiU° ® e„2m) + es2_i ® (eniq2U0 ® em + qiU° ® e„2m)]
+ x2[esis2 ® OU° ® enim] + x3[esis2 ® OU° ® e„2m] + xs (esi ® [U0 ® em + es2_i ® q2U0 ® em]).
3.3. The steady state probability vector
Let X be the steady state probability vector of the infintesimal generator Q of the process {X(t): t > 0}. The subdivision of X = (x0,xi,x2,...), where x0 is of dimension 2(SiS2m), xi is of dimension 2(SiS2n2m) + 2(SiS2nim) + SiS2m and x2,x3,... are of dimension 2(SiS2nim) + 2(SiS2wiw2m) + SiS2m + SiS2n2m. As X is a vector satisfies the relation
XQ = 0 and Xe = i.
The probability vector X follows a matrix geometric structure under the steady state is
x = x2Rj_i, j > 3 (2)
0
0
where R is the quadratic equation's lowest non-negative solution
R2 F2 + RFi + Fo = 0
and the vector xo, x1 andx2 are obtained with the help of succeeding equations:
xo A oo + xi Aio = 0, (3)
xo Aoi + xi Aii + X2 A21 = o, (4)
xi Ai2 + x2[Fi + RF2 ] = o, (5)
subject to a condition normalization
xoeo + xiei + x2[I - R]-ie2 = i. (6)
Computing the rate matrix R is necessary before attempting to solve the set of equations mentioned above. However, [9] used Logarithmic reduction approach, an algorithm that makes it simple to produce R.
4. System characteristics
• Probability of the system is empty:
Pempfy = xo eo.
• The probability of the server-i is idle:
Si S2 m Si S2 «2 m
e e e xooo«i «2«5 + e e e e
xioi«i«2«4«5.
«i=i «2=i «5=i «i=i «2=i «4=i «5=i
• The probability of the server-2 is on vacation:
Si S2 m Si S2 m
Pv«c = e e e xooo«i«2«5 + e e e xo3o«i«2«5
«i =i «2=i «5 =i «i =i «2=i «5=i
Si S2 ni m Si S2 ni m
+ xiio«i«2«3«5 + ^e ^e ^e ^e xi2o«i«2«3«5
«i =i «2=i «3=i «5=i «i=i «2=i «3=i «5=i
Si S2 m œ Si S2 «i m
+ ^e ^e ^e xi3o«i«2«5 + ^e ^e ^e ^e ^e xiio«i«2«3«5 «i =i «2=i «5=i i=2 «i=i «2=i «3 =i «5 =i
œ Si S2 «i m œ Si S2 m
+ e e e e e xi2o«i«2«3«5 + e e e e xi3o«i«2«5.
i=2 «i=i «2=i «3=i «5=i i=2 «i=i «2=i «5=i
• The probability of the server-i is offering service in normal mode:
œ Si S2 «i m œ Si S2 «i «2 m
PSi B = ee e e e xiio«i «2«3«5 + e e e e e e xiii«i «2«3«4«5.
i=i «i=i «2=i «3=i «5=i i=2 «i=i «2=i «3=i «4=i «5=i
• The probability of the server-i is offering service in working breakdown:
œ Si S2 «i m œ Si S2 «i m
PSiWBD = e e e e e xi2o«i
«2«3«5 + ee e e e xi2i«i«2«3«4«5.
i=i «i=i «2=i «3 =i «5 =i i=2 «i=i «2=i «3=i «5 =i
The probability of the server-2 is offering service in normal mode:
Si S2 «2 m to Si S2 «2 m
PS2 B = e e e e xioiui «2«4«5 + e e e e e xi3i«i «2«4«5 «i =i «2 = i «4 = i «5 =i i = i «i =i W2 = i «4 = i «5 =i
to Si S2 «i «2 m
+ ee e e e e xi2i«i«2«3«4«5.
i=2 «i=i «2=i «3=i «4=i «5=i
The probability of the server-i is on Repair:
Si S2 m to Si S2 m
PSi R = e e e x030«i «2«5 + e e e e xi30«i«2«5
«i=i «2=i «5=i i=i «i =i «2=i «5 =i
to Si S2 «2 m
+ ee e e e xi3i«i«2«4«5.
i=i «i =i «2=i «4=i «5=i
Expected number of customers in the system:
to
Esystem = e iXie = xiei + X2[2(I _ R)_i + R(I _ R)_2]e2. i=i
Expected first inventory level:
to Si S2
EiLi = e e e «ixi(«i, «2)
i=i «i=i «2=i
Expected first inventory level:
to Si S2
eil2 = e e e «2xi(«i, «2)
i=i «i=i «2=i
Expected reorder rate with first commodity
to S2 «i m
ERRi = e e e e [Xii0i«2«3«5 (U07i ® Im)e + Xi20i«2«3«5 (OU0 ® Im)e] i=i «2=i «3=i «5=i
to S2 «i «2 m
+ e e e e e xiiii«2«3«4«5 (U17i ® i«2m)e i=2 «2=i «3=i «4=i «5=i
to S2 «i «2 m
+ e e e e e Xi2ii«2«3«4«5 (U1 7i ® i«2m)e. i=2 «2=i «3=i «4=i «5=i
Expected reorder rate with second commodity
Si «2 m
ERR2 = e e e xi0i«i(s2+i)«5 (U2 ® Jm)e «i =i «4=i «5=i
to S2 «i «2 m
+ ee e e e [xiii«i (s2+i)«3«4«5 + xi2i«i (s2+i)«3 «4 «5 ]( ^i ® U072 ® im )e i=2 «2=i «3=i «4=i «5=i
to S2 «2 m
+ e e e e xi3i«i(S2+i)«4«5 (U072 ® i«2m)e. i=i «2=i «4=i «5=i
5. Cost Analysis
The cost function for our model was created with the premise that each cost element (per unit of time) correlates to a distinct system measure.
• Cjj - the first item in inventory with a cost per unit
• Cl2 - the second item in the inventory with a cost per unit
• Ch - storing a customer's cost in the system for each unit of time.
• CRl -setup costs for each order of the primary item
• CR2 - Setup costs for each order of complimentary items
TC = Ci1 EiL1 + Ci2 EiL2 + CHEsystem + CR1ERR1 + CR2 ERR2
6. Numerical Implementation
To compute numerical outcomes, we have employed diverse MAP demonstrations for the incoming arrival in a manner that ensures their mean values are 1, as recommended by [5].
Erlang arrival (ERA):
Do
-2 2 D1 = 0 0"
0 -2 2 0
Exponential arrival (EXA):
Do = [-1]Di = [1] Hyper exponential arrival (HEXA):
Do
-1.90 0 0 -0.19
Di
1.710 0.190 0.171 0.019
MAP-Negative Correlation arrival (MNCA):
D0
-1.00243 1.00243 0 0 -1.00243 0 0 0 - 225.797
D1
MAP-Positive Correlation arrival (MPCA): D0 =
-1.00243 0 0
1.00243 0 -1.00243 0 0 225.797
D1
0 0 0' 0.01002 0 0.99241 223.539 0 2.258
0 0 0 0.99241 0 0.01002 2.258 0 223.539
Consider the following PH-distributions for the service and repair progression: • Erlang service (ERS):
Y =[1,0] U =
-2 2 0 -2
Exponential service (EXS):
Y =[1] U =[-1]
Hyper exponential service ( HEXS):
Y =[0.8,0.2] U
-2.8 0 0 -0.28
6.i. Illustrative i
We examine the consequence of the service rate of server-2 (^2) versus the expected system size (Esystem) in the Table i-3. We fix A = i, = i2, n = i, ft = i, $ = i, t = 2, Si = 4, S2 = 6, s2 = 3, pi = 0.5, qi = i _ pi, p2 = 0.5, q2 = i _ p2, O = 0.6, such that the system remains stable. The observation from Table i-3 as follows:
• While we maximize the service rate of the server-2 then the corresponding Esystem decreases with the combination of arrival and service times.
• From the point of view of arrival times, Esystem decreases highly for HYPA and decreases slowly for ERLA while an increase the server-2's service rate. However, consider the service times, Esystem decreases highly in ERLS and decreases slowly in HYPS with the combination of EXPA, MNCA and MPCA, but in the case of ERLA and HYPA, Esystem decreases slowly in EXPS and decreases fastly in HYPS.
With the support of the Tables 4-6, we visualize the influence of the service rate (^2) of the server-2 upon the probability of server-2 is undergoing vacation (Pvac). Fix A = i, = i2, q = i, ft = i, $ = i, t = 2, Si = 4, S2 = 6, s2 = 3, pi = o.5, qi = i - pi, p2 = o.5, q2 = i - p2, 9 = o.6, so that the stability condition is satisfied.
• Observation of Tables 4-6 discloses the fact that Pvac maximizes while maximizing the service rate of the server for the distinct feasible ordering of service and arrival times.
• This is because an increase in service rate leads to a decrease in the duration of service time respectively. As a result, the server will be getting more chances to go on vacation. Besides, the increment rate is higher in the case of HYPA and lower in the case of ERLA. In the same way, from the service time point of view, the increment rate is rapid for HYPS and gradual for ERLS.
With the support of Tables 7-9, we visualize the impact of the repair rate of the server-i(T) on the Probability of server-i being busy(PSi B). Fix A = i, ^ = i2, ^2 = i0, n = i, ft = i, $ = i, Si = 4, S2 = 6, s2 = 3, pi = 0.5, qi = i _ pi, p2 = 0.5, q2 = i _ p2, O = 0.6, so that the stability condition is satisfied.
• From Table 7-9, we may view that as the repair rate of server-i(t) increases, (PSiB) increases for all possible groupings of service and arrival times.
• An increase in repair rate implies that the server takes minimum time to complete the repair process.
• As a result, the server will get more time to serve the customer and so (PSi B) increases. Hence, the system can focus on decreasing the time taken to repair the failed server for its optimal utilization. Moreover, the speed of increment of (Psi b ) is high for ERLA and low for HYPA. In the same way, it is high for ERLS and low for HYPS.
6.2. Illustration 2
6.3. Illustrative 3
Table 1: Server-2 service rate (^2) vs Esystem - ERLS
^2 ERLA EXPA HYPA MNCA MPCA
8.0 0.197384249 0.224815927 0.2713795337 0.290089474 4.308746051
8.5 0.196918323 0.224009689 0.269865612 0.288694105 4.167282560
9.0 0.196498144 0.223275819 0.268486466 0.287407096 4.034807543
9.5 0.196117354 0.222604963 0.267225347 0.286215133 3.910580674
10.0 0.195770689 0.221989289 0.266068055 0.285107103 3.793926042
10.5 0.195453765 0.221422194 0.265002498 0.284073649 3.684228530
11.0 0.195162900 0.220898080 0.264018333 0.283106828 3.580929431
11.5 0.194894989 0.220412169 0.263106672 0.282199845 3.483521781
12.0 0.194647394 0.219960366 0.262259844 0.281346850 3.391545721
12.5 0.194417864 0.219539139 0.261471205 0.280542773 3.304584052
Table 2: Server-2 serv/ce rate 2) vs E system - EXPS
H2 ERLA EXPA HYPA MNCA MPCA
8.0 0.202436650 0.229325675 0.276190307 0.288189790 4.284049580
8.5 0.201923153 0.228463897 0.274610523 0.286720757 4.143969323
9.0 0.201458764 0.227678126 0.273168421 0.285367819 4.012769799
9.5 0.201036863 0.226958737 0.27184722 0.284116698 3.889718360
10.0 0.200651935 0.226297636 0.270632665 0.282955451 3.774147068
10.5 0.200299357 0.225687981 0.269512585 0.281874005 3.665448646
11.0 0.199975227 0.225123948 0.268476541 0.280863797 3.563071842
11.5 0.199676236 0.224600554 0.267515540 0.279917499 3.466516647
12.0 0.199399561 0.224113511 0.266621796 0.279028795 3.375329599
12.5 0.199142780 0.223659113 0.265788546 0.278192214 3.289099329
Table 3: Server-2 serv/ce rate (^2) vs E system - HYPS
H2 ERLA EXPA HYPA MNCA MPCA
8.0 0.227728762 0.247910753 0.291707349 0.281256741 4.144213513
8.5 0.226937445 0.246779024 0.289865528 0.279466679 4.010955831
9.0 0.226211855 0.245738665 0.288170867 0.277819490 3.886146349
9.5 0.225544345 0.244779141 0.286606470 0.276298631 3.769067923
10.0 0.224928388 0.243891462 0.285157942 0.274889987 3.659073727
10.5 0.224358381 0.243067914 0.283812936 0.273581450 3.555580430
11.0 0.223829490 0.242301839 0.282560799 0.272362579 3.458061772
11.5 0.223337523 0.241587463 0.281392290 0.271224328 3.366042632
12.0 0.222878828 0.240919754 0.280299352 0.270158822 3.279093637
12.5 0.222450210 0.240294308 0.279274929 0.269159176 3.196826301
Table 4: Server-2 serv/ce rate vs PvAc - ERLS
Ц2 ERLA EXPA HYPA MNCA MPCA
8.О О.992696356 О.988326139 О.981О73741 О.978176264 О.96О252482
8.5 О.993О59887 О.988853514 О.981898129 О.979О3277О О.961666866
9.О О.993389158 О.989334618 О.982654586 О.979817153 О.962989215
9.5 О.993688677 О.989775249 О.98335О966 О.98О538758 О.964227717
1О.О О.993962221 О.99О18О294 О.983993994 О.9812О5329 О.965389685
1О.5 О.994212971 О.99О553889 О.984589456 О.981823341 О.966481656
11.О О.994443622 О.99О899561 О.985142349 О.982398253 О.9675О9488
11.5 О.994656468 О.99122О336 О.985657О14 О.9829347О8 О.968478435
12.О О.994853476 О.991518816 О.986137231 О.98343668О О.969393223
12.5 О.995О36336 О.991797261 О.9865863О9 О.9839О7592 О.97О2581О4
Table 5: Server-2 service rate (ц2) vs Pvac - EXPS
Ц2 ERLA EXPA HYPA MNCA MPCA
8.О О.991933ОО9 О.987434189 О.98ОО744О5 О.976954919 О.96О264988
8.5 О.9923248О8 О.987994543 О.98О928958 О.977881661 О.961679429
9.О О.99268О715 О.9885О6494 О.981713965 О.978729839 О.963ОО1556
9.5 О.993ОО5319 О.988976О19 О.98243743О О.9795О9541 О.964239645
1О.О О.9933О2482 О.9894О8162 О.9831О6196 О.98О229156 О.9654О1О72
1О.5 О.993575474 О.9898О72О6 О.983726147 О.98О89572О О.96649242О
11.О О.993827О78 О.99О176811 О.9843О2365 О.981515186 О.967519581
11.5 О.994О59676 О.99О52О121 О.984839265 О.982О92622 О.968487832
12.О О.994275317 О.99О839849 О.98534О695 О.982632376 О.9694О1917
12.5 О.994475769 О.991138351 О.98581ОО23 О.9831382ОО О.97О266О99
Table б: Server-2 serv/ce rate ) vs Pvac - HYPS
Ц2 ERLA EXPA HYPA MNCA MPCA
8.О О.9873О1352 О.982924О32 О.975917875 О.9715ОО4О8 О.96ООО7628
8.5 О.987838969 О.983624О68 О.97687О15О О.972668334 О.961435754
9.О О.988334О16 О.984269283 О.977749342 О.973743753 О.962769344
9.5 О.988791273 О.984865824 О.978563672 О.974737О72 О.964О17128
1О.О О.989214835 О.985418958 О.97932О15О О.97565723О О.965186832
1О.5 О.9896О8227 О.98593322О О.98ОО2479О О.976511937 О.966285315
11.О О.9899745О6 О.986412541 О.98О682791 О.9773О7876 О.967318682
11.5 О.99О316333 О.98686О343 О.981298668 О.978О5О865 О.96829238О
12.О О.99О636О36 О.987279615 О.981876366 О.97874599 О.969211284
12.5 О.99О935659 О.987672986 О.982419347 О.979397714 О.97ОО79769
Table 7: Server-l repair raie (t) vs Ps1 b - ERLS
t ERLA EXPA HYPA MNCA MPCA
2.1 0.141008388 0.13790536 0.13246213 0.130940643 0.075845503
2.2 0.141253797 0.138188589 0.132887094 0.131195807 0.075919075
2.3 0.141473036 0.138443996 0.133273312 0.131426853 0.07598541
2.4 0.141669716 0.138675233 0.133625511 0.131636941 0.076045499
2.5 0.141846844 0.138885360 0.133947705 0.131828717 0.076100165
2.6 0.142006944 0.139076961 0.134243323 0.132004409 0.076150095
2.7 0.142152144 0.139252229 0.134515310 0.132165904 0.076195868
2.8 0.142284249 0.139413038 0.134766207 0.132314815 0.076237974
2.9 0.142404799 0.139560993 0.134998218 0.132452523 0.076276831
3.0 0.142515115 0.139697482 0.135213260 0.132580217 0.076312796
Table 8: Server-l repair rate (t) vs Ps1 b - EXPS
t ERLA EXPA HYPA MNCA MPCA
2.1 0.138021661 0.134843332 0.129303375 0.127808856 0.074550912
2.2 0.138263883 0.135118502 0.129710507 0.128054649 0.074624807
2.3 0.138480662 0.135366864 0.130080690 0.128277310 0.074691540
2.4 0.138675482 0.135591923 0.130418429 0.128479865 0.074752084
2.5 0.138851247 0.135796620 0.130727549 0.128664843 0.074807246
2.6 0.139010396 0.135983435 0.131011317 0.128834378 0.074857702
2.7 0.139154988 0.136154477 0.131272540 0.128990278 0.074904020
2.8 0.139286770 0.136311545 0.131513638 0.129134086 0.074946683
2.9 0.139407235 0.136456186 0.131736709 0.129267126 0.074986104
3.0 0.139517664 0.136589733 0.131943580 0.129390538 0.075022635
Table 9: Server-l repair rate (t) vs Ps1 b - HYPS
T ERLA EXPA HYPA MNCA MPCA
2.1 0.121762987 0.118860529 0.113611432 0.112325780 0.067357513
2.2 0.121967364 0.119083143 0.113924156 0.112523046 0.067428853
2.3 0.122151232 0.119284568 0.114208672 0.112702075 0.067493742
2.4 0.122317344 0.119467554 0.114468443 0.112865232 0.067553018
2.5 0.122468001 0.119634405 0.114706401 0.113014495 0.067607384
2.6 0.122605135 0.119787070 0.114925047 0.113151535 0.067657429
2.7 0.122730382 0.119927203 0.115126525 0.113277768 0.067703651
2.8 0.122845133 0.120056218 0.115312681 0.113394403 0.067746476
2.9 0.122950579 0.120175328 0.115485115 0.113502478 0.067786270
3.0 0.123047742 0.120285582 0.115645216 0.113602890 0.067823346
7. Conclusion
In this study, we explain inventory management at service facilities using two types of servers: reliable servers and unreliable servers. Under steady-state conditions, matrix analytic techniques are used to determine the number of customers in the system, the server status, and the inventory level. Measures of important system features are derived in the steady state. We determined the optimality of this model by numerical analysis. As a result, this approach is appropriate for situations involving working vacation allocation when the server is reliable and for service disruption, or emergency vacation where the other server is unreliable.
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