CC BY
39
NUMERICAL INVESTIGATION OF RETRIAL QUEUEING INVENTORY SYSTEM WITH A CONSTANT RETRIAL RATE, WORKING VACATION, FLUSH OUT, COLLISION AND IMPATIENT CUSTOMERS
G. Ayyappan, N. Arulmozhi •
Department of Mathematics, Puducherr y Technological University, Puducherr y, India. ayy appan@ptuniv .edu.in, arulmozhisathy [email protected],
Abstract
The retrial queueing inventory system with working vacation, flush out, balking, breakdown, and repair, as well as a constant retrial rate and orbital client collision are all examined in this study. We made the assumption that customers arrive through a Markovian arrival process and that they would get phase-type services from the server. The inventory is replenished using a (s, S) and (s, Q) strategy, and it is expected that the replenishment time will follow an exponential distribution. If there are zero inventory items, no customers in the orbit, or both, the server will go into working vacation mode. When a customer retries an orbit while the server is serving arriving customers, the orbital customer may collide with the arriving customer during that retry, in which case both of them will be shifted back into orbit; otherwise, the orbital customer may avoid colliding with the arriving customer and may rejoin the orbit for another retry. The number of customers in the orbit and the inventory level may be found in the steady state. A cost analysis is produced along with the establishment of various important performance measures. Moreover, some numerical examples are provided to clarify our mathematical notion.
Keywords: Markovian arrival process, PH-distribution, working vacation, collision of orbital customers, flus out.
AMS Subject Classification (2010): 60K25, 68M30, 90B22.
1. Introduction
Retrial queues occur when initial consumers identify all servers and/or waiting space full. They may choose to try again after a random length of time or abandon the system per manently. RQ models have been thoroughly researched in a significan number of papers. Artalejo et al. [3] introduced the concept of retrial requests for inventory. They assumed that demand points are Poisson processes, wher eas lead and retrial time points are exponential. They thought that the orbit's size is limitless. Manuel et al.[8] proposed a retrial inventory system that includes a service facility. They assumed clients come according to a Markovian arrival process (MAP), that service time for each client follows a phase-type distribution (PH), that lead time, lifetime of each item, and retrial times follow an exponential distribution.
Customers arrive at the single server retrial queueing-inv entory system under consideration in this study using a Markovian Arrival Process, also known as the flexibl point process. The MAP tries to accomplish significan generalisation of the Poisson process while keeping it tractable. Many real-world applications do not require a renewal procedur e before arriving. As a result, the most useful tool for simulating renewal and non-renewal appearance situations is
the MAP. We can have realistic arriv al patter ns in this model because of the MAP, which also accounts for correlations and dependencies between arrivals. Further more, the continuous-time case is necessar y, even though the MAP is define for both discr ete and continuous periods. See Chakravarthy [5] and Neuts [10] for further details on the MAP and its properties.
The notion of server vacation was firs presented in the retry inventory system by Sivakumar[ 17]. For lead, inter-trial, inter-demand, and server vacation durati ons, he made the assumption that the distributions would be exponential. He also believed that these incidents are unrelated to one another. He instituted a programme of repeated vacations. A two-commodity substitutable retrial inventory system with a shar ed ordering strategy was examined by Sivakumar [15]. Sivakumar [16] examined a system of perishable inventory that had requests for retrials. The exponentially distributed lead periods for orders, the finit sour ce of requests, the exponentially distributed life durations for stored objects, and the exponentially distributed inter-retrial intervals have all been assumed by the author. A two-commodity stochastic inventory technique with a complement item was proposed by Jeganathan et al. [11] in the context of a traditional retrial facility. When the primar y item is out of supply , each new client will immediately enter an orbit of infinit capacity.
A M/ M/ 1 retrial queue under (s, S) policy with a storage system was examined by Shajin and Krishnamoorthy [14]. The authors use the assumption that when the server is inactive, the exter nal arriv als immediately enter an orbit and that the time betw een two successiv e retrials has an exponential distribution. Only the client at the head of the orbit is allowed to reach the server. In contrast to the traditional method of employing just one vendor, Chakravarthy and Hayat [6] established the idea of multiple vendors responsible for replacing inventories. This way, replenishment happens via two vendors. The authors used the MAM to analyse the model in steady-state under the assumptions of a two-vendor system, wher e the lead times are exponentially distributed with a parameter that depends on the vendor, the demands occur according to a MAP, and the service times are PH. There are also interesting numerical examples given, such as a comparison of the systems with one and two vendors.
A queueing inventory model in which a new customer comes and waits for service when the server is unavailable due to vacation was examined by Y Zhang et al. [19]. The model included the server's multiple vacations and dissatisfie clients. They were able to extract some significan perfor mance metrics and fin the matrix geometric solution of the steady-state probability by using the truncated approximation approach. Using numerical analysis, the impact of the probability and impatience rate on a few performance metrics was examined. Using the genetic algorithm, the authors calculated the best possible policy and cost and arrived at the ideal service rate. Ayyappan et al. [4] studied the notions of working breakdown, collision, vacation, and reneging in a non-pr eemptiv e priority retrial queueing system with immediate feedback. They applied the supplementar y variable technique to their model and also provided particular cases.
Service interruptions were originally implemented in an inventory model by Krishnamoorthy et al. [7]. They also believed that orders are processed instantly and that there is no limit to the amount of disruptions that can happen during a single service. Ushakumari [18] examined a (s, S) inventory system with recurrent demands for unfulfille requests from the orbit and a random lead time. In their paper [1], Amirthakodi and Sivakumar spoke about retrial inventory queueing with a single server and customer feedback, wher e the orbit size is finite.Th retrial queueing model with exponential service time, Poisson arrival, and delayed feedback was examined by Meliko v et al. [9]. They used both (s, S) and (s, Q) replenishment policies for their study. In their analysis of an M/ M/ 1/ N queuing system with reverse balking, Kumar et al. [13] incorporate the idea of reverse reneging. Customers' input is used by Kumar and Som [?] in an M/ M/ 1/ N queuing system with reverse balking, reverse reneging, and retention of reneged customers. They calculate the system size stationar y probability .
2. Model Description
• We examine a single-ser ver retrial queueing inventory model in which customers arrive at the system as represented by MAP, with D0 and D1 matrices as its dimension m. The service times, denoted as (j, U) of order n, are assumed to follow the PH-distribution with U0 + Ue = 0.
• If the server is available, he serves the customer right away upon their arrival. If not, the customer must enter the orbit of infinit . Every customer retries from the orbit at a constant rate, despite the size of the orbit. The inter-retrial times follow an exponential distribution with parameter 5.
• If the orbit is empty, the inventory is zero, or both, then the server goes on vacation after serving the customer. Additionally , the vacation periods are expected to follow a ^-parameter exponential distribution. In the event that a customer arrives during vacation time, the server will start charging the customer less for services than usual. Additionally , it is expected that the service times throughout the vacation period follow the PH distribution, denoted as (j, 9U), with 0 < 0 < 1. If the server examines the customer who is waiting in the system after completing this vacation, he will begin a normal busy period. Otherwise, he is dor mant.
• The incoming customer may enter the orbit for a retry with probability q1 or balk the system with probability p1 during the service delivery, repair, and no inventory items, ensuring that p1 + q1 = 1.
• When a customer retries an orbit while the server is servicing incoming customers, there is a chance that the orbital customer and the incoming customer will collide and be shifted to the orbit with a probability of q2; if not, the orbital customer may not collide and will rejoin the orbit for a subsequent retry with a probability of p2, such that p2 + q2 = 1.
• During regular busy periods, the server may get breakdo wn. As a result, the customer getting service at the moment must enter the orbit of limitless capacity. The server goes into idle mode when the repair operation is complete d. The breakdo wn times are exponentially distributed with parameter wher eas the repair times are PH-distributed with rate (a, T).
• All the customers in the orbit are flushe out periodicall y and the flus out times follow exponential distribution with parameter a. The schematic picture of this model is provided in Figur e 1.
• ® - 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. 0 - It denotes zero matrices in the suitable order.
3. A nalysis
In the following section, we establish the queueing-inv entory system's transition rate matrix. Assume that N(t), J(t), I(t), R(t), S(t), A(t) describe the total customers in the orbit, status of server, stock level, repair phases, service phases, arrival phases, respectively.
Consider X(t)
0, server is idle in normal service mode,
1, server is busy in normal service mode, J(t) = < 2, server is idle in WV mode,
3, server is busy in WV mode,
4, server is repair mode.
{N(t), J(t), I(t), R(t), S(t), A(t)} is a CTMC with state space
*=<ko) u $(t).
i=1
(1)
wher i
Figure 1: Schematic representation
and for i > 1,
0(0) ={(0,0, Mi, M4) : 0 < Mi < S, 1 < M4 < m}
U {(0, 1, M1, M3, M4) : 1 < M1 < S, 1 < M3 < n, 1 < M4 < m} U {(0,2,M1,m4) : 0 < m1 < S, 1 < m4 < m} U {(0,3, M1, M3, M4) : 1 < M1 < S, 1 < M3 < n, 1 < M4 < m} U {(0,4, M1, M2, M4) : 1 < M1 < S, 1 < M2 < l, 1 < M4 < m}
0(i) ={(i, 0, M1, M4) : 0 < M1 < S, 1 < M4 < m}
U {(i, 1, M1,M3,M4) : 1 < M1 < S, 1 < M3 < n, 1 < M4 < m} U {(i, 2, M1, m4) : 0 < m1 < S, 1 < m4 < m} U {(i,3, M1,M3,M4) : 1 < M1 < S, 1 < M3 < n, 1 < M4 < m} U {(i,4, M1, M2, M4) : 1 < M1 < S, 1 < M2 < l, 1 < M4 < m}
3.1. Construction of the QBD process for Model 1
The generator matrix of the Markov chain under (s, S) policy is given by:
Q
A00 A01 0 0 0 0
A10 F1 F0 0 0 0
A F2 F1 F0 0 0
A 0 F2 F1 F0 0
A 0 0 F2 F1 F0
The entries in the block matrices of Q are define as follows,
wher e
A
00
An
A00
A22 A00
All A12 00 0
A 31
A00
A 51
A00
00
A22 A23 0
A00 A00 0
(1 4 33 A34
0 A00 A00
A42 a43 a 44
■firiri Siriri ^im
00
0
00
0
00
0
0 0 0
0
A55 A00
C1 0 0 . . 0 0 . . 0 C2
0 C3 0 . . 0 0 . . 0 C2
0 0 C3 . . 0 0 . . 0 C2
0 0 0 . . C3 0 . . 0 C2
0 0 0 . . 0 C4 . . 0 0
0 0 0 . . 0 0 . . C4 0
0 0 0 . . 0 0 . . 0 C4
C5 0 0 . . 0 0 . . 0 C6
0 C5 0 . . 0 0 . . 0 C6
0 0 C5 . . 0 0 . . 0 C6
0 0 0 . . C5 0 . . 0 C6
0 0 0 . . 0 C7 . . 0 0
0 0 0 . . 0 0 . . C7 0
0 0 0 . . 0 0 . . 0 C7
wher e C1 = (Do + pD1) - ftIm, C2 = ftIm, C3 = Do - ftIm, C4 = Do,
C5 = U ® (Do + P1D1) - ($ + ft) Inm, C6 = ft Inm, C7 = U ® (Do + D1) - $ I,
A20 = Is ® Uo ® Im, Aoj = Is+1 ® qIm,
A
A
Cs 0 0
0 0
0 0
0
C1o 0
0 0
C1o
0 0
0 0
0
Is ® Y ® D1
0 0 0
C1o 0
0 0
0 0 0
0
C11
0 0
C9 C9 C9
C9 0
C11 0 0 C11
wher e Cs = (Do + p D1) - (q + ft) Im, C9 = ft Im, Qo = Do - (q + ft) Im, Cn = Do - q I, At22 = Is+1 ® qInm, A43 = Is ® QU° ® Im,
A 34
Aoo
0
is 0 j 0 Di
A 44 , Aoo
C 2 0 0 . . 0 0 .. . 0 C13
0 C 2 0 . . 0 0 .. . 0 C13
0 0 Ci2 . . 0 0 .. . 0 C13
0 0 0 . . Ci2 0 .. . 0 C13
0 0 0 . . 0 Ci4 . . . 0 0
Ci4 0
0 Ci4
where C12 = 9U ® (Do + pi Di) - (n + /) inm, CB = p inm, CH = 6U ® (Do + pi Di) - n in
= [0 Is 0 T0 0 im], A55 =
Ci5 0 0
0 0
0 0
00
Ci5 0 0 Ci5
Ci5 0 0 Ci7
where Ci5 = T ® (Do + pi Di) - p i/m> Ci6 = / i/„, Ci7 = T ® (Do + pi Di).
0 0 0
0 0
Ci7 0
Ci6 Ci6 Ci6
Ci6 0
0
Ci7
Aii Aoi
qi Di 0
A
0i
A0i 0 0 0 0
A 22 Aoi
0 0 0
0 0
A 33 Aoi 0 0
0 0
0
A 44 Aoi
A 25 Aoi 0
0
A 55
A 22 , Aoi
Is 0 In 0 qiDi, A
Is 0 en a 0 $im, A33 =
A0i = Is 0 In 0 qi Di, A5i = is 0 i/ 0 qi Di,
qi Di
0
A110 A12 Aio 0 0 0
A21 Aio 0 0 0 0
Aio = 0 0 A33 Aio A34 A1o 0
0 0 A43 Aio 0 0
0 0 0 0 A55 A1o
wher e
A i 0 = is+i 0 vim, A i 0 =
, A20 = [0 is 0 en 0 vim], A33 = is+i 0 vi,
A34 = A 0=
0
Is 0 Sj 0 im
is 0 Sj 0 Im_ , A43 = [0 is 0 en 0 vim], A50 = is 0 vl/m.
Fi
F i
F2 i f3 i 0
lF^1
Fii2 F22
0
F42 F
0
F 23
F33 F43
00
0
0
F34
44
F
0
0 0 0
0
F55 rl ■
0
wher i
F11
C18 0 0 . . 0 0 . . 0 C19
0 C2o 0 . . 0 0 . . 0 C19
0 0 C2o . . 0 0 . . 0 C19
0 0 0 . . C2o 0 . . 0 C19
0 0 0 . . 0 C21 . . 0 0
0 0 0 . . 0 0 . . C21 0
0 0 0 . . 0 0 . . 0 C21
wher e Ci8 = (Do + pi Di) - (a + ft)Im, C
19
C21 = Do - (S + a)Im. F/
12
0
Is ® y ® D1
= ft im> C 20
F21 To 1
Do - (S + a + ft)I„
0
0 Is-1 ® U0 ® I„
F23
U0 ® Im 0 00
, F2
22
C22 0 0 . . 0 0 . . 0 C23
0 C22 0 . . 0 0 . . 0 C23
0 0 C22 . . 0 0 . . 0 C23
0 0 0 . . C22 0 . . 0 C23
0 0 0 . . 0 C24 . . 0 0
0 0 0 . . 0 0 . . C24 0
0 0 0 . . 0 0 . . 0 C24
wher e C22 = U ® (Do + P1D1) + [(92S - S) - (f + a + ft)]I„„, C23 = ftI
C
24
f34 1
0
Is ® Y ® D1
, F3
33
(f + a)]Inm, F31 = Is+1 ® q Im,
C25 0 0 .. 0 0 .. . 0 C26
0 C27 0 .. 0 0 .. . 0 C26
0 0 C27 .. 0 0 .. . 0 C26
0 0 0 . . C27 0 .. . 0 C26
0 0 0 .. 0 C28 . . 0 0
0 0 0 .. 0 0 .. . C28 0
0 0 0 .. 0 0 .. . 0 C28
wher e C25 = (Do + P1D1) - (a + q + ft)Im, C26 = ftIm, C27 = Do - (a + S + q + ft)Im
C28 = Do - (a + S + q)Im, Ff = Is+1 ® q Inm, F43 1 = [Is ® 9Uo ® im 0,
C29 0 0 ... 0 0 ... 0 C3o
0 C29 0 ... 0 0 ... 0 C3o
0 0 C29 . . . 0 0 ... 0 C3o
F44 = 1= 0 0 0 ... C29 0 ... 0 C3o
0 0 0 ... 0 C31 . . . 0 0
0 0 0 ... 0 0 ... C31 0
0 0 0 ... 0 0 ... 0 C31
where C29 = 0U ® (Do + P1D1) + [(92 S - S) - (a + q + ft)] I„„, C3o = ft I, C31 = 9U ® (Do + p1 D1) + [(92S - S) - (a + q)]I„„.
o
F51 = [0 is © T0 © im], Ff
55 _
C32 0 0. .. 0 0 . .0 C33
0 C32 0. .. 0 0 . .0 C33
0 0 C32 . .. 0 0 . .0 C33
0 0 0. . . C32 0 . .0 C33
0 0 0. .. 0 C34 . . .0 0
0 0 0. .. 0 0 . . C34 0
0 0 0. .. 0 0 . .0 C34
where C32 = T © (Do + pi Di ) - (a + §) ijm, C33 = jSijm, C34 = T © (Do + pi Di ) - aijB
F11 F0
F33 0
wher e F212
rF11 0 0 0 0 0
F21 F0 F22 F0 0 0 F25 F0
0 0 F33 0 0 0
0 0 F43 0 F44 F0 0
0 0 0 0 F55 0
qi Di 0 00 qi Di 0 00
, F02i = [0 is © e„ © P2Sim], F022 = is © In © qi Di, F025 = is © e„a © $i„ [0 is © e„ © P2 S im ], F044 = is © i„ © qi Di, F055 = is © ii © qi Di.
, F
43
0 Fi2 F2 0 0 0
0 0 0 0 0
F2 = 0 0 0 F34 2 0
0 0 0 0 0
0 0 0 0 0
0
is © Sj © iff
, F2
34
0
is © Sj © i„
Aii 0 0 0 0
A21 0 0 0 0
0 0 A33 0 0
0 0 A43 0 0
0 0 0 0 A55
A
wher e A11 = is+i © aim, A2i = [0 is © e„ © aim], A33 = is+i © vim, A43 = [0 is © e„ © aim], A53 = [0 is © ej © aim]. A55 = is © aijm,
F
0
Stability condition for Model I
To discuss the stability condition, we firs consider the generator matrix F = F0 + F1 + F2. If X = Xl, X2, Xз, X4 ) = (X00, Xo^ . . . , X0s, X0s+^ . . . , XoS, Xi^ Xi2^ . . , Xls, Xis+i^ . . , XlS, X20, X21, . . . , X2s,X2s+1, . . . , X2S,X31,X32,. . . , X3s, X3s+1,. . . , X3S, X41, X42, . . . ,X4s, X4s+1, . ...... , X4S ).
The vector x represents the invariant vector of matrix F. Consequently , we have the relations XF = 0 and xe = 1. For the Markov process with a QBD structur e to exhibit stability, our model must satisfy the condition xF0e < XF2 e. This condition is both necessar y and sufficien for the stability of the queueing model under study and reduces to the inequality A < p.
3.2. QBD process for Model II
In accordance with the assumptions outlined in the "Model Description" section, we will now examine Model II, while solely modifying the ordering policy from (s, S) to (s, Q). The generator matrix of the process for the (s, Q) policy takes on the following form:
Q
Aoo A 01 0 0 0 0
A 10 F1 Fo 0 0 0
A F2 F1 Fo 0 0
A 0 F2 i?1 Fo 0
A 0 0 F2 i?1 Fo
The entries in the block matrices of Q are define as follows,
A
00
'A11
A00
A31
A00
451 LA00
A12
A00
0
0
¿22 123 n
A 00 A 00 0
(1 433 4 34
0 A 00 A 00
A42 A43 A44
00 00 00
0 0 0 0
A 55 A 00
A
00
"C1 0 0 . . 0 0 . . C2 0 .. . 0 0
0 C3 0 . . 0 0 . . 0 C2 .. . 0 0
0 0 C3 . . 0 0 . . 0 0 .. . 0 0
0 0 0 . . C3 0 . . 0 0 .. . 0 C2
0 0 0 . . 0 C4 . . 0 0 .. . 0 0
0 0 0 . . 0 0 . . C4 0 .. . 0 0
0 0 0 . . 0 0 . . 0 C4 .. . 0 0
0 0 0 . . 0 0 . . 0 0 .. . C4 0
0 0 0 . . 0 0 . . 0 0 .. . 0 C4
/412 —
A00 —
0
is ® Y ® D
a~22 A 00
/423 — Is ® U0 ® Im, A31 -
00
00
is+1 ® nim,
C5 0 0 . .0 0 . . C6 0 .. . 0 0
0 C5 0.. . 0 0 . . 0 C6 .. . 0 0
0 0 C5 . . 0 0 . . 0 0 .. . 0 0
0 0 0.. . C5 0 . . 0 0 .. . 0 C6
0 0 0 . . 0 C7 . . 0 0 .. . 0 0
0 0 0.. . 0 0 . . C7 0 .. . 0 0
0 0 0 . . 0 0 . . 0 C7 .. . 0 0
0 0 0.. . 0 0 . . 0 0 .. . C7 0
0 0 0 . . 0 0 . . 0 0 .. . 0 C7
A33 A00
"Cs 0 0 . . 0 0 . . C9 0 . . 0 0
0 C10 0 . . 0 0 . . 0 C9 . . 0 0
0 0 C10 . . 0 0 . . 0 0 . . 0 0
0 0 0 . . C10 0 . . 0 0 . . 0 C9
0 0 0 . . 0 C11 . . 0 0 . . 0 0
0 0 0 . . 0 0 . . C11 0 . . 0 0
0 0 0 . . 0 0 . . 0 C11 . . 0 0
0 0 0 . . 0 0 . . 0 0 . . C11 0
0 0 0 . . 0 0 . . 0 0 . . 0 C11
A34 _ A00 _
0
is 0 y 0 Di
, A02 _ Is+1 0 nInm, A43 _ Is 0 9U0 0 Im, _ [0 Is 0 T0 0 im],
A
C12 0 0 . . 0 0 . . C13 0 . . 0 0
0 C12 0 . . 0 0 . . 0 C13 . . 0 0
0 0 C12 . . 0 0 . . 0 0 . . 0 0
0 0 0 . . C12 0 . . 0 0 . . 0 C13
0 0 0 . . 0 C14 . . 0 0 . . 0 0
0 0 0 . . 0 0 . . C14 0 . . 0 0
0 0 0 . . 0 0 . . 0 C14 . . 0 0
0 0 0 . . 0 0 . . 0 0 . . C14 0
0 0 0 . . 0 0 . . 0 0 . . 0 C14
C15 0 0 . . 0 0 . . 0 C16
0 C15 0 . . 0 0 . . 0 C16
0 0 C15 . . 0 0 . . 0 C16
0 0 0 . . C15 0 . . 0 C16
0 0 0 . . 0 C17 . . 0 0
0 0 0 . . 0 0 . . C17 0
0 0 0 . . 0 0 . . 0 C17
¡1
rjPi1 ¡21 FF31 0
JT
£22 ¡1 0
£42 1
0
F£123
£33 1
£43 1
00
0
0
¡£34 1
¡£44 1
0
0 0 0
0
¡£55 1
wher i
F12
Fi11
CO u 0 0 . . 0 0 . . C19 0 . . 0 0
0 C20 0 . . 0 0 . . 0 C19 . . 0 0
0 0 C20 . . 0 0 . . 0 0 . . 0 0
0 0 0 . . C20 0 . . 0 0 . . 0 C19
0 0 0 . . 0 C21 . . 0 0 . . 0 0
0 0 0 . . 0 0 . . C21 0 . . 0 0
0 0 0 . . 0 0 . . 0 C21 . . 0 0
0 0 0 . . 0 0 . . 0 0 . . C21 0
0 0 0 . . 0 0 . . 0 0 . . 0 C21
0
Is ® y ® D1
, -2
21
F31 = Is+1 ® n Im, -Ff4 =
00
0 Is-1 ® U0 ® Im 0
is ® Y ® D
, -2
23
U0 ® Im 0' 00
F2
22
C22 0 0 . . 0 0 . . C23 0 .. . 0 0
0 C22 0 . . 0 0 . . 0 C23 . . 0 0
0 0 C22 . . 0 0 . . 0 0 .. . 0 0
0 0 0 . . C22 0 . . 0 0 .. . 0 C23
0 0 0 . . 0 C24 . . 0 0 .. . 0 0
0 0 0 . . 0 0 . . C24 0 .. . 0 0
0 0 0 . . 0 0 . . 0 C24 . . 0 0
0 0 0 . . 0 0 . . 0 0 .. . C24 0
0 0 0 . . 0 0 . . 0 0 .. . 0 C24
r33 F1
C25 0 0 . .0 0 . . C26 0 .. . 0 0
0 C27 0.. . 0 0 . . 0 C26 . . 0 0
0 0 C27 . . 0 0 . . 0 0 .. . 0 0
0 0 0.. . C27 0 . . 0 0 .. . 0 C26
0 0 0 . . 0 C28 . . 0 0 .. . 0 0
0 0 0.. . 0 0 . . C28 0 .. . 0 0
0 0 0 . . 0 0 . . 0 C28 . . 0 0
0 0 0.. . 0 0 . . 0 0 .. . C28 0
0 0 0 . . 0 0 . . 0 0 .. . 0 C28
F?l42 = is+1 ® ninm, i?43 = [is ® OU0 ® im 0 ,
£44 F1
F51
£55 F1
C29 0 0 . .0 0 . . C30 0 .. .0 0
0 C29 0 . .0 0 . .0 C30 . . .0 0
0 0 C29 . .0 0 . .0 0 .. .0 0
0 0 0 . . C29 0 . .0 0 .. .0 C30
0 0 0 . .0 C31 . .0 0 .. .0 0
0 0 0 . .0 0 . . C31 0 .. .0 0
0 0 0 . .0 0 . .0 C31 . . .0 0
0 0 0 . .0 0 . .0 0 .. . C31 0
0 0 0 . .0 0 . .0 0 .. .0 C31
® Im ,
C32 0 0 . .0 0 . . C33 0 .. .0 0 "
0 C32 0 . .0 0 . .0 C33 . . .0 0
0 0 C32 . .0 0 . .0 0 .. .0 0
0 0 0 . . C32 0 . .0 0 .. .0 C33
0 0 0 . .0 C34 . .0 0 .. .0 0
0 0 0 . .0 0 . . C34 0 .. .0 0
0 0 0 . .0 0 . .0 C34 . . .0 0
0 0 0 . .0 0 . .0 0 .. . C34 0
0 0 0 . .0 0 . .0 0 .. .0 C34
Stability condition for Model II
To discuss the stability condition, we firs consider the generator matrix F = F0 + F\ + F2. If X = (Xo, Xi, X2, X3, X4 ) = (Xoo , X01,. . . , X0s, X0s+1,. . . , X0Q,. . . , X0S, X11, X12,. . ., X1s, X1s+1,. . . , X1Q . . . ,
X1S, X20, X21, . . . , X2s, X2s + 1, . . . , X2Q, . . . , X2S, X31, X32, . . . , X3s, X3s+1, . . . , X3Q, . . . , X3S, X41, X42, . . . ,
X4s, X4s+i,..., X40,..., X4S ). Considering the QBD structure of the Markov process, stability exists in our model if it satisfie the condition xF0e < XF2 e. This condition is both necessar y and sufficien for the stability of this queueing model under study, and it reduces to A < p.
3.3. The stationar y probability vector
Let X be the stationar y probability vector of the infintesima generator Q of the process {X(t): t > 0}. The subdivision of X = (x0,x1,x2,...), wher e x0 is of dimension 2(S + l)m + 2Snm and x1,x2,... are of dimension 2(S + l)m + 2Snm + Slm. As X is a vector satisfie the relation XQ = 0 and Xe = 1. The probability vector X follows a matrix geometric structur e under the steady state is
Xj = x1 Rj-1, j > 2 (2)
wher e R is the quadratic equation's lowest non-negativ e solution R2F2 + RF1 + F0 = 0 and the vector x0, x1 are obtained with the help of succeeding equations:
co
x0A00 + x1 A10 + E xiA = 0 (3)
i=2
x0 A01 + xi [Fi + RF2 ]= 0, (4)
subject to a condition normalization
x0 e2(S+1)m+2Snm + x1[ ^ — R] e2(S+1)m+2Snm+SZm = 1. (5)
The rate matrix R can be computed with the help of the following iteration formula which has been suggested by Neuts [10] R(n + l) = —F0F—1 — R2(n)F2F—1 for n > 0 wher e R(0) = 0. Since F—1 and (F0 + R2F2) are positiv e, the rate matrix R will converge and so the entries of R will increase monotonically in the successiv e iterations. Iteration may be terminated when the condition maxij[Rij(n + 1) — Rij(n)] < e is attained. Here, e denotes the degr ee of accuracy and R(n) indicates the value of the rate matrix at the n-th iteration.
4. System characteristics
Probability that the server is idle in regular process PINM = Ei=0 =0 E«4 = 1 xi0u1 u4.
Probability that the server is idle in working vacation process
PIWV = Ei=0 E«1 =0 E«4 = 1 xi2u1 u4.
Probability that the ser ver is busy in regular process
PBNM = Ei=0 Eu1 = 1 Eu3 = 1 E«4 = 1 xi1u1 u3 u4. Probability that the ser ver is busy in working vacation
P _ v i v S pn r^m x
P BWV = Ei=0 Eu1 = 1 Eu3 = 1 Eu4 = 1 xi3u1 u3u4.
Probability that the server is breakdown
PBD = Ei=1 Es1=1 Eu2=1 E«4=1 xi4u1 u2 uj.
Expected number of customers in the orbit
Eorbit Ei=1 ixie.
Probability that the ser ver is busy PBusy = PBNM + PBWV.
Expected number of customers in the system
Esystem = Eorbit + PBusy.
Expected number of items in the inventory level
EiL = Ei=0 Es1 = 1 E«4 = 1 u1 xi0u1 u4 + Ei=0 Eu1 = 1 En3 = 1 E«4 = 1 u1 xi1u1 u3 u4 + Ei=0 Es1 = 1 E«4 = 1 u1 xi2u1 u4 + Ei=0 Eu1 = 1 Eu3 = 1 E«4 = 1 u1 xi31u1 u2u3u4 + Ei =1 Eu1 =1 Eu2 = 1 E«4 = 1 u1 xi4u1 u2 u4 .
Expected reorder rate
ER = Ei=0 En3 = 1 E^ = 1 xi1(s+1)u3u4 (U0 ® )e + Ei=0 Eu3 = 1 E^=1 xi3(s+1)u3u4 ® ^m)e.
The effective retrial rate
A = $ Ei=1 Eu1 = 1 E«4 = 1 xi0u1 u4 + $ Ei=1 Eu1 = 1 E«4 = 1 xi2u1 uj.
5. Cost Analysis
The total cost for our model is given below, with the cost elements (per unit time) related to various system measur es.
TC = cwEsystem + chEIL + csER
wher e
• TC: Total cost (per unit time)
Ch: The inventory holding cost (per unit time)
Cw: Waiting cost of a customer in the system (per unit time)
Cs: Setup cost (per order)
6. N umerical Implementation
To compute numerical outcomes, we have employed diverse MAP demonstrations for the incoming arrival in a manner that ensur es their mean values are 1, as recommended by [5]. • Erlang arrival (ERA):
Dn
Exponential arrival (EXA): Hyper exponential arrival (HEXA):
Do =
-2 2 " Di = 0 0"
0 -2 2 0
Do = [-1]Di = [1]
-1.90 0 0 -0.19
Di
1.710 0.190 0.171 0.019
Consider the following PH-distributions for the service and repair progression: • Erlang service (ERS):
— 2 2
Y =[1,0] U = 2 2
Erlang repair (ERR):
a =[1,0] T
0 -2
-2 2 " 0 -2
Exponential service (EXS): Exponential repair (EXR): Hyper exponential service (HEXS):
Y = [0.8, 0.2] U Hyper exponential repair (HEXR):
a = [0.8, 0.2] T
7 =[1] U =[-l] a =[l] T =[-l]
-2.8 0 0 -0.28
-2.8 0 0 -0.28
Illustration 1
For this both policies, it was assumed that values of all parameters of the QIS were fixe except the service rate p: A = 1, q = 3, d = 0.6, t = 2, ft = 2, ty = 1, 5 = 3, a = 0.5, p = p2 = 0.6, q1 = q2 = 0.4, s = 5, S = 15.
Here, we compar e and analyse the two policy (s,S) and (s, Q) as follows in tables 1-6:
• First, we observe that both Esystem and Eorbit in Table 1-6 under varying service rate p , it is gradually decreases as p increase for both (s,S) and (s, Q) but the notable is (s,S) policy give the minimum for both Esystem and Eorb,f.
• Obser ve the service times, Esystem and E0rb/f are decreases highly in HEXS and slowly decrease in ERS than all other service times. Likewise, from the view point of arrival times, Esysfem and Eorb/t are decreases highly for HEX A compar ed to other arrival times.
Table 1: Service rate (y) vs Esystem and Eorbn - ERA
ERS EXS HEXS
y Esystem Eorbit Esystem Eorbit Esystem Eorbit
15 0.081396697 0.047355675 0.116583261 0.046932851 0.060209747 0.031898478
16 0.075864324 0.043882684 0.109224350 0.043648505 0.057304727 0.030565025
17 0.071064407 0.040901610 0.102739862 0.04079181 0.054674483 0.029325579
18 0.066854402 0.038311241 0.096982359 0.038284564 0.052279029 0.028171817
19 0.063128015 0.036037241 0.091835886 0.036066509 0.050086560 0.027096253
20 0.059803954 0.034023538 0.087207964 0.034090466 0.048071247 0.02609211
21 0.056818744 0.032226894 0.083023918 0.032318974 0.046211764 0.025153247
22 0.054121959 0.030613354 0.079222779 0.030721913 0.044490267 0.024274106
23 0.051672939 0.029155827 0.075754267 0.029274789 0.042891663 0.023449656
24 0.049438484 0.027832404 0.072576547 0.027957481 0.041403065 0.022675351
Table 2: Service rate (y) vs Esystem and Eorbn - EXA
ERS EXS HEXS
y Esystem Eorbit Esystem Eorbit Esystem Eorbit
15 0.093658859 0.057831180 0.125620027 0.057370051 0.077226462 0.047628434
16 0.087616380 0.053656465 0.117884279 0.053393412 0.073004243 0.044783511
17 0.082319352 0.050040468 0.111041640 0.049917853 0.069231262 0.042257589
18 0.077636049 0.046878488 0.104946594 0.046856002 0.065837924 0.039999559
19 0.073464370 0.044090388 0.099483485 0.044139417 0.062768568 0.037968788
20 0.069723889 0.041613800 0.094559255 0.041713775 0.059978093 0.036132537
21 0.066350351 0.039399467 0.090098203 0.039535447 0.057429527 0.034464111
22 0.063291771 0.037407970 0.086038129 0.037569008 0.055092251 0.032941515
23 0.060505619 0.035607397 0.082327459 0.035785419 0.052940673 0.031546455
24 0.057956750 0.033971637 0.078923079 0.034160655 0.050953225 0.030263586
Table 3: Service rate (y) vs Esystem and Eorbn - HEXA
ERS EXS HEXS
y Esystem Eorbit Esystem Eorbit Esystem Eorbit
15 0.130072755 0.085272901 0.140741030 0.072324015 0.085907558 0.047067013
16 0.118644770 0.076620377 0.131673874 0.066854199 0.080218556 0.043713552
17 0.109278270 0.069644502 0.123722961 0.062135811 0.075394726 0.040903528
18 0.101432497 0.063889862 0.116692238 0.058026255 0.071233209 0.03850289
19 0.094745436 0.059054418 0.110429349 0.054416549 0.06759258 0.036419856
20 0.088964620 0.054929433 0.104814044 0.051222021 0.064370838 0.034589272
21 0.083908013 0.051365737 0.099750091 0.048375934 0.061492291 0.032963458
22 0.079440679 0.048253720 0.095159543 0.045824997 0.058899399 0.031506611
23 0.075460231 0.045510930 0.090978563 0.043526164 0.056547500 0.030191241
24 0.071887408 0.043074081 0.087154360 0.041444300 0.054401305 0.02899583
Table 4: Service rate (y) vs Esystem and Eorbu - ERA ERS EXS HEXS
y Esystem Eorbit Esystem Eorbit Esystem Eorbit
15 0.082004602 0.047355519 0.116584109 0.046933442 0.060824563 0.031913956
16 0.076429926 0.043882824 0.109225231 0.043649085 0.057874696 0.030578216
17 0.071593705 0.040901916 0.102740765 0.040792375 0.055206267 0.029336964
18 0.067352119 0.038311641 0.096983274 0.038285111 0.052777843 0.028181749
19 0.063597952 0.036037694 0.091836807 0.036067038 0.050556568 0.027104999
20 0.060249224 0.034024019 0.087208889 0.034090977 0.048515832 0.026099873
21 0.057241938 0.032227388 0.083024844 0.032319470 0.046633719 0.025160188
22 0.054525258 0.030613851 0.079223704 0.030722393 0.044891928 0.024280350
23 0.052058203 0.029156322 0.075755191 0.029275254 0.043275004 0.023455306
24 0.049807313 0.027832894 0.072577470 0.027957933 0.041769774 0.022680491
Table 5: Service rate (y) vs Esystem and Eorfrit - EXA
ERS EXS HEXS
y Esystem Eorbit Esystem Eorbit Esystem Eorbit
15 0.094342828 0.057831783 0.125622511 0.057371230 0.077912574 0.047638686
16 0.088262387 0.053657353 0.117886912 0.053394652 0.073653585 0.044793161
17 0.082931496 0.050041572 0.111044412 0.049919149 0.069847634 0.042266700
18 0.078217783 0.046879757 0.104949496 0.046857350 0.066424553 0.040008185
19 0.074018636 0.044091787 0.099486509 0.044140813 0.063328223 0.037976979
20 0.070253213 0.041615303 0.094562394 0.041715216 0.060513165 0.036140334
21 0.066856921 0.039401053 0.090101450 0.039536929 0.057942099 0.034471552
22 0.063777495 0.037409625 0.086041476 0.037570530 0.055584149 0.032948633
23 0.060972173 0.035609109 0.082330902 0.035786976 0.053413507 0.031553280
24 0.058405612 0.033973398 0.078926612 0.034162246 0.051408424 0.030270142
Table 6: Service rate (y) vs Esystem and Eorbit - HEXA ERS EXS HEXS
y Esystem Eorbit Esystem Eorbit Esystem Eorbit
15 0.131245994 0.085194321 0.14075261 0.072332391 0.087390312 0.047182656
16 0.119735103 0.076569695 0.131688802 0.066864727 0.081577421 0.043823642
17 0.110296386 0.069612267 0.123740628 0.062148038 0.076649406 0.041007718
18 0.102387591 0.063870343 0.116712188 0.058039850 0.072399139 0.03860134
19 0.09564524 0.059043966 0.110451236 0.054431263 0.068682030 0.036512935
20 0.089815602 0.054925634 0.104837597 0.051237669 0.065393700 0.034677418
21 0.084715599 0.051366937 0.099775098 0.048392369 0.062456649 0.033047113
22 0.080209444 0.048258756 0.095185831 0.045842105 0.059811936 0.031586194
23 0.07619406 0.045518963 0.091005993 0.043543856 0.057413806 0.030267136
24 0.072589624 0.043084494 0.087182817 0.041462502 0.055226098 0.029068385
Illustration 2
We picturise the consequences of the breakdo wn rate ty against the Pbusy. Fix A = 1, y = 15,
d = 0.6, n = 3, t = 5, ft = 2, S = 3, a = 0.5, p1 = p2 = 0.6, q1 = q2 = 0.4, s = 5, S = 15,
these values satisfy the condition for stability. From the figu es 2 - 4: we can explore that while increasing the server's breakdown rate (ty), Pbusy decreases for all feasible provisions of incoming arrival and service patterns. As increase in breakdo wn rate indicates that customers will frequently be unable to access the server, which is decreases of Pbusy is higher for HEXA and lower for ERA. Likewise, it is higher for ERS and lower for HEXS.
Illustration 3
To investigate the impact of the TC on both the service (y) and repair (t) rates in the Figures 5-13. Fix A = 1, a = 0.2, d = 0.6, ft = 3, 5 = 3, p1 = p2 = 0.6, q1 = q2 = 0.4, s = 5, S = 15, Ch = 70, Cj = 110, Cr = 120, such that the system leftovers stable.
From the viewpoint of Figures 5-13, we maximize both the service and repair rates for all possible groups of arrival and service times, we notice that the TC decreases. Consider the service times, TC decreases exceedingly for ERS and decreases moderately for EXS. Therefore, TC decreases slowly for ERA and rapidly for HEX A.
•10
-2
(a) ERA
Ek / Ek/ 1 Ek/ M/ 1 Ek/ Hk/ 1
—*—»—*-
1.5 2 2.5 3 IP
10
-2
(a) EXA
M/ Ek/ 1 M/ M/ 1 M/ Hk/ 1
1.5 2 2.5 3
P
Figure 2: Breakdown rate vs. Pbusy
Figure 3: Breakdown rate vs. Pbusy
10-
(a) HEXA
RH 5
Hk / Ek/ 1 Hk/ M/ 1 Hk/ Hk/ 1
1.5 2 2.5 3
P
Figure 4: Breakdown rate vs. Pbusy
Figure 5: Service and repair rates vs. TC
Figure 6: Service and repair rates vs. TC
Figure 7: Service and repair rates vs. TC
6
6
4
4
2
7
6
4
Figure 8: Service and repair rates vs. TC
Figure 9: Service and repair rates vs. TC
Figure 10: Service and repair rates vs. TC
Figure 11: Service and repair rates vs. TC
Figure 12: Service and repair rates vs. TC
Figure 13: Service and repair rates vs. TC
7. Conclusion
A retrial inventory model with MAP arrivals, PH-distributed service, working vacations, collision of orbital customers, flus out, balking, breakdo wn and repair has been investigated. The peculiarity of this model is that the server can offer service even in the vacation period and the system is alw ays stable because of the flus out of the system. We have consider ed MAP for arrivals and would like to extend our models by considering BMAP for arrivals which is best suited for modelling arriv als which come in batches.
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