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Analysis of MAP/PH1, PH2, PH3/1 Queueing-Inventory System with Two Commodities
S. Meena, N. Arulmozhi*, G. Ayyappan, K. Jeganathan
•
Department of Mathematics, Puducherry Technological University, Puducherry, India. Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai 600005, India. meenasundar2296@gmail.com, arulmozhisathya@gmail.com, ayyappan@ptuniv.edu.in, kjeganathan@unom.ac.in
Abstract
In this work, a single server implements a two-commodity inventory queueing system. We assume that both commodities have a finite capacity. Customers arrive by a Markovian Arrival Process, there is a need for a single item, and either or both types of commodities are required, and this requirement is modeled using certain probabilities. The lead times are exponentially distributed, and the service times have a PH distribution. We use matrix analytical techniques to investigate the queueing inventory system and adopt an (s, S)-type replenishment policy that is dependent on the type of commodity. In the steady state, the joint and individual probability distribution of the Esystem, inventory level, and server status is obtained. A few significant performance measures are attained. Our mathematical concept is then illustrated with a few numerical examples.
Keywords: Queueing-inventory; Markovian Arrival Process; Phase-type distribution; (s, S)-type policy; Two-Commodity.
AMS Subject Classification (2010): 60K25, 68M30, 90B22.
1. Introduction
Many researchers have been interested in the study of queueing inventory systems, and proposals involving two commodities have been made. Sigman and Levi [18] presented the M/G/1 queueing-inventory model with exponentially distributed lead time under light traffic in 1992. Several models with various ordering criteria have been developed to operate such systems. Balintfy [5] and Silver, E.A., [19] both contributed to the development of the joint ordering policy. A two-commodity inventory system with zero lead time and an equal demand process was examined, according to Krishnamoorthy et al. [12] and Anbazhagan and Arivarignan [2].
Neuts [15] developed, studied, and instructed MAP in 1984. Chakravarthy [8] derived the Markovian arrival process by depicting matrix (D0, D1) as the guideline for the MAP at the dimension m, where D0 governs for no arrival, where D1 governs for arrival. The generator of the matrix Q defined by D = D0 + D1 is an irreducible stochastic matrix. A single-server inventory system using Markovian Arrival Process (MAP)-based arrivals were studied by Paul Manuel et al. [16].
Yadavalli et al. [22] considered a two-commodity stochastic inventory system with joint and individual ordering policies, Poisson arrivals and lost sales. Anbazhagan et al. [3] for their
consideration of a two-commodity continuous review inventory system with substitutable items and Markovian demands. When the sum of the two commodities' on-hand inventory levels reaches a certain level s, reordering for supply is initiated. A two-commodity inventory problem was studied by Krishnamoorthy and Varghese [13] with no lead time and Markovian shifts in demand for the first, second, and both commodities.
Binitha Benny et al. [6] considered a total cost inventory system with a single server and the buffer capacity will be limited. Customers arrive through a Poisson process, and the probabilities used to determine the demand for each commodity or both commodities depend on which commodity is being purchased. Sivakumar et al. [20] investigated a total cost continuous review inventory system with a demand renewal and ordering policy, a policy combination known as ordering individual commodities and ordering both commodities jointly.
A two-commodity model with a compliment and regular working vacations is examined by Lakshmanan et al. in their study [14]. Each customer orders service at a convenient moment, and both commodities are independent of their ordering procedures. Each customer is given a finite retry orbit when the requested item is out of stock or the server is overloaded. Schwarz et al. [17] looked into a brand-new type of stochastic network that shows a product from steady-state distribution. There, integrated models for networks of service stations and inventories were constructed using stochastic networks. They assume that even though a server with associated inventory stops accepting new customers when the stock is out, lost sales are still recorded in the system.
According to Yadavalli et al. [23], the three types of demand for the two goods are comparable. They looked at a system with a phase-type distributed lead time and perishable items. A Markovian arrival process governs the occurrence of all three different kinds of demands. Each commodity's lifetime has an exponential distribution with unique properties. A continuous-time Markov chain that identified the system was used to give a stability analysis and identify individual ordering strategies. Amirthakodi [1] thought of an inventory system with one server service facility and a limited number of trial feedback customers. An inventory system with a single server, two commodities, queue-dependent services for a finite queue, and an optional retrial facility was examined by Jeganathan et al. [10].
Federgruen et al. [9] investigated a continuous review multi-item inventory system with demands generated by independent compound Poisson processes using the (S, c, s) ordering strategy. One consequence of implementing this approach is the requirement to find three optimal variables for each item. Kalpakam and Arivarignan [11] proposed a policy with fewer variables for making decisions and for an (s, S) policy generated by [11] that is appropriate for related but non-substitutable items, a single reorder level s is determined. The total cost is determined by the average inventory, a customer in queue, and reorder rates, according to Berman [7], who provided a deterministic approximation for their inventory system with a service facility.
The demand for each commodity occurs in independent Poisson processes with a variety of parameters in two-commodity retrial inventory systems with varied ordering strategies has been studied by Sivakumar [21] and Jeganathan and Anbazhagan [4]. The constant retrial policy was taken into consideration in both experiments. In other words, a signal is sent out when there are i demands in the orbit according to an exponential distribution that is independent of the orbit's number.
1.1. Motivation for the proposed model
The main motivating factor for our model is the Textile scenario. Buyers usually go to a Textile shop to purchase one or more (like churidar, sarees, shirts, kurtas, and so on) items or goods. Let's say there are n various items and people are shopping for the product i with probability, Pi, 1 < i < n. Customers shop for objects i1,..., ik, for 2 < k < n with probability pi1,...,ik, where i'l,..., ik is an element of the set of integers 1,2,..., n. Customers will be served only those products
that are in stock of the ones requested if all of the requested different goods are not in stock. If a Buyer is unable to obtain any product, they will be disappointed. A customer has a 2n — 1 different possibility to shop for the products and we will concentrate on the case where n = 2.
1.2. Research Gap
Benny et al. [6] worked with two-commodity in the single server queueing inventory system and arrival follows the Poisson process and service follows an exponential distribution. This article examines two-commodity in the inventory with arrival following MAP and service times following Phase-type distributions. The authors handle (s, S) policy, and both individual and joint orders are obtained. In this article, we develop (s,S) policy, both individual and joint orders, and numerical implementation of 2D using Matlab software.
1.3. Viewpoint for This Work
The manuscript for this work is synchronized as follows: A brief explanation of our model is provided in Section 2. Our model's notations and matrix generation are described in Section 3. Section 4 contains our model's steady-state probability. Section 5 provides performance measures. Numerical illustrated in Section 6. The conclusion is given in Section 7.
2. Model Description
Consider a single server queueing model subject to a two-commodity. Customers arrive according to a MAP and each commodity has a single item demand. The MAP is specified by two m x m matrices (Do, Di), D = Do + Di, which is an irreducible infinitesimal generator. The matrix Do means no arrival similarly, the matrix Di means arrival.
There is a need for a single unit, and either or both types of commodities are required, and this requirement is modeled using certain probabilities. The lead times are exponentially distributed, and the service times have a PH distribution. Customers may want both commodities or only one, depending on some predetermined probability. Only when services are being offered are the customers' needs disclosed. If the requested item is not available, the customer permanently exits the system. When only one of the requested items is available and both are demanded, the customer is given the one that is in stock. In the case where both commodity inventory levels are 0, customers are not allowed to join the system. However, customers join the system even when the server is operating and no more inventory is available. For the customer's needed item to be provided at the time the item is taken for service, it is planned that the items will be replenished during the current service. When a customer cannot get the commodity they need at the time of service, the customer is also lost.
When taken for service, the customer requests item Ii with probability ci, for i = 1,2 or both I1 and I2 with probability c3 such that c1 + c2 + c3 = 1. After a random period of service, the requested item is delivered to the customer. The service times for processing orders for I1,I2 or both I1 andI2 are PH- distribution with represented by (av, Tv), 1 < v < 3. Whose matrix is order nv with Tv0 + Tve = 0 implies that T° = — Tve . Here A is the arrival rate, which is signified as A = n1 D1e, where n1 is the steady-state probability vector. The mean service rate is denoted by Uv = K (—Tv)—1 env ]—1.
For both commodities, the system has a maximum capacity of Si items. We utilize a (si, Si) replenishment strategy for the commodity Ii, where i = 1,2. That is, an order is placed for just that item to raise the inventory level of commodity Ii back to Si, i = 1, 2 at the time of replenishment, anytime it drops to si. For parameters, fti, for i = 1,2, the lead time has an
exponential distribution.
3. The QBD process's infinitesimal generation matrix
The following notations and assumptions are used to explain our model of producing QBD processes in this section.
Notations
We will define the following notations:
• ® -Kronecker product of two matrices of various dimensions resulting in a block matrix.
• ® - Kronecker sum of two matrices of various dimensions resulting in a block matrix.
• Im stand for identity matrix of m rows and m columns.
• e - A column vector of the suitable order. Each of its entries is one.
• N(t) represents the total number of customers in the queue.
• V(t) represents the server's status at epoch t.
V (t)
0, if server is idle
1, if server is busy with 4
2, if server is busy with I2
3, if server is busy with and 12
• Lj (t) stands for the excess inventory level of commodity ij, i = 1,2.
• S(t) stands for phases of the service.
• M(t)- The Markovian arrival process is considered in phases.
• Let Y={Y(t) : t > 0}, where Y(t) = {N(t), V(t),I1 (t),I1 (t),S(t),M(t)} is a CTMC with state space
0(0) u <Ki)-i=1
(1)
where
0(0) = {(0,0,a1,a2,k) : 0 < a1 < S1, 0 < a2 < S2, 1 < k < m}
u{(0,v,ai,a2,jv,k) : 1 < v < 3, 0 < ai < Si, 0 < a2 < S2, 1 < j < n0, 1 < k < m} and for p > 1,
p) = {(p, v,a1,a2,jv,k) : 1 < v < 3, 0 < a1 < S1, 0 < a2 < S2, 1 < jv < nv, 1 < k < m}.
3.1. The Infinitesimal Generator Matrix
The infinitesimal generator matrix of the Markov chain is given by:
Q
B00 B01 0 0 0 0 0
B10 Ai Ao 0 0 0 0
B20 A2 A1 Ao 0 0 0
B30 A3 A2 A1 Ao 0 0
B40 A4 A3 A2 A1 A0 0
(2)
The following describes Markov chain transitions and the corresponding rates: The matrix Boo governs,
• (0, v, a1, a2, jv, k) ^ (0,0, a1, a2, k) with rate Tv0 ® Im for 1 < v < 3, 0 < a1 < S1, 0 < a2 < S2, 1 < jv < nv, 1 < k < m,
• (0,0, a1, a2, k) ^ (0,0, S1, a2, k) with rate ^ Im for 0 < v < 3, 0 < a1 < s1, 0 < a2 < S2, 1 < k < m,
• (0, v, a1, a2, jv, k) ^ (0, v, S1, a2, jv, k) with rate Invm for 1 < v < 3, 0 < a1 < s1, 0 < a2 < S2,1 < jv < nv, 1 < k < m,
• (0,0, a1, a2, k) ^ (0,0, a1, S2, k) with rate Im for 0 < v < 3, 0 < a1 < S1, 0 < a2 < s2, 1 < k < m,
• (0, v, a1, a2, jv, k) ^ (0, v, a1, S2, jv, k) with rate j82 Invm for 1 < v < 3, 0 < a1 < S1, 0 < a2 < s2,1 < jv < nv, 1 < k < m,
• (0,0,0,a2,k) ^ (0,2,0,a2 — 1,j2,k) with rate a2 ® (c2 + c3)D1 for 1 < a2 < S2,1 < j2 < n2, 1 < k < m,
• (0,0, a1,0, k) ^ (0,1, a1 — 1,0, j1, k) with rate a1 ® (c1 + c3)D1 for 1 < a1 < S1, 1 < j < n1, 1 < k < m,
• (0,0, a1, a2, k) ^ (0,1, a1 — 1, a2, ji, k) with rate a1 ® c1 D1 for 1 < a1 < S1, 1 < a2 < S2, 1 < j1 < n1,1 < k < m,
• (0,0,a1,a2,k) ^ (0,2,a1,a2 — 1,j2,k) with rate a2 ® c2D1 for 1 < a1 < S1, 1 < a2 < S2, 1 < j2 < n2,1 < k < m,
• (0,0, a1, a2, k) ^ (0,3, a1 — 1, a2 — 1, j3, k) with rate a3 ® c3D1 for 1 < a1 < S1, 1 < a2 < S2, 1 < j3 < n3,1 < k < m.
The matrix B(p+1)0, p > 1, governs
• (p, v, 0,0, jv, k) ^ (0,0,0,0, k) with rate Tv0 ® Im for 1 < v < 3, 1 < j < nv, 1 < k < m,
• (p, v, 0, a2, jv, k) ^ (0,0,0, a2, k) with rate Tv0c1 p ® Im for 1 < v < 3, 1 < a2 < S2, 1 < jv < nv, 1 < k < m,
• (p, v, 0, a2, jv, k) ^ (0,2,0, a2 — 1, j2, k) with rate Tv %p—1(c2 + c3 )a2 ® Im for 1 < v < 3, 1 < a2 < S2,1 < jv < nv, 1 < k < m,
• (p, v, a1,0, jv, k) ^ (0,0, a1,0, k) with rate Tv0c2p ® Im for 1 < v < 3, 1 < a1 < S1, 1 < jv < nv, 1 < k < m,
• (p,v,a1,0,jv,k) ^ (0,2,a1 — 1,0,j1,k) with rate Tv0c2p—1(c1 + c3)a1 ® Im for 1 < v < 3, 1 < a1 < S1,1 < jv < nv, 1 < k < m,
• (1, v, a1, a2, jv, k) ^ (0,1, a1 — 1, a2, ji k) with rate Tv0c1a1 ® Im for 1 < v < 3, 1 < a1 < S1, 1 < a2 < S2,1 < jv < nv, 1 < k < m,
• (1, v,a1,a2, jv,k) ^ (0,2,a1,a2 — 1, j2,k) with rate Tv0c2a2 ® Im for 1 < v < 3, 1 < a1 < S1, 1 < a2 < S2,1 < jv < nv, 1 < k < m,
• (1, v,a1,a2, jv,k) ^ (0,3,a1 — 1,a2 — 1,j3,k) with rate Tv0c3a3 ® Im for 1 < v < 3,1 < a1 <
51, 1 < a2 < S2,1 < jv < nv, 1 < k < m.
The matrix Ai, p > 1, governs
• (p, v, a1, a2, jv, k) ^ (p, v, S1, a2, jv, k) with rate Invm for 1 < v < 3, 0 < a1 < s1, 0 < a2 <
52, 1 < jv < nv, 1 < k < m,
• (p, v, ai, a2, jv, k) ^ (p, v, ai, S2, jv, k) with rate ^2Invm for 1 < v < 3, 0 < ai < Si, 0 < a2 < s2, 1 < jv < nv, 1 < k < m.
The matrix A/+i, 1 < l < p — 1, p > 3, governs
• (p, v,0,a2, jv,k) ^ (p - 1,2,0,a2 - 1,j2,k) with rate Tv0c11 1(c2 + c3)a2 ® Im for 1 < v < 3, 1 < a2 < S2,1 < jv < nv, 1 < k < m,
• (p, v,a1,0, jv,k) ^ (p - 1,1,a1 - 1,0,j1,k) with rate Tv0c21-1(c1 + c3)a1 ® Im for 1 < v < 3, 1 < a2 < S2,1 < jv < nv, 1 < k < m,
• (p, v, a1, a2, jv, k) ^ (p - 1,1, a1 - 1, a2, j1, k) with rate Tv0c1 a1 ® Im for 1 < v < 3,1 < a1 < S1,1 < a2 < S2,1 < jv < nv, 1 < k < m,
• (p, v, a1, a2, jv, k) ^ (p - 1,2, a1, a2 - 1, j2, k) with rate Tv0c2a2 ® Im for 1 < v < 3,1 < a1 < S1,1 < a2 < S2,1 < jv < nv, 1 < k < m,
• (p, v, a1, a2, jv, k) ^ (p - 1,3, a1 - 1, a2 - 1, j3, k) with rate Tv0c3a3 ® Im for 1 < v < 3, 1 < a1 < S1,1 < a2 < S2,1 < jv < nv, 1 < k < m.
The matrix B01, governs
• (0, v, a1, a2, jv, k) ^ (1, v, a1, a2, jv, k) with rate Inv ® D1 for 1 < v < 3, 0 < a1 < S1, 0 < a2 < S2, 1 < jv < nv, 1 < k < m.
The matrix A0, p > 1, governs
• (p, v, a1, a2, jv, k) ^ (p + 1, v, a1, a2, jv, k) with rate Inv ® D1 for 1 < v < 3, 0 < a1 < S1, 0 < a2 < S2, 1 < jv < nv, 1 < k < m.
4. Analysis of Steady-State
The nonsingularity of B00 and A1 is need for Q to be irreducible. Consider the matrix A = £/= A/• Let the unique stationary distribution of A be 0. Under the condition (Neuts [15]),
00
0A0e < £(/ - 1)0A/e, /=2
an irreducible Markov chain with generator Q possesses a unique stationary solution vector Y = ^y^...) satisfying
YQ = 0, Ye = 1. Partitioning Y as Y = (y0, y1, y2,...) where
y0 = (y0(v, a1, a2, jv, k) : 0 < v < 3, 0 < a1 < S1, 0 < a2 < S2, 1 < < nv, 1 < k < m),
yp = (yp (v, a1, a2, jv, k) : 1 < v < 3, 0 < a1 < S1, 0 < a2 < S2, 1 < < nv, 1 < k < m), for p > 1,
where y0 is of dimension 1 x (S1 + 1)(S2 + 1)m + (S1 + 1)(S2 + 1)n1 m + (S1 + 1)(S2 + 1)n2m + (S1 + 1)(S2 + 1)n3m and yp for p > 1, is of dimension 1 x (S1 + 1)(S2 + 1)n1m + (S1 + 1)(S2 + 1)n2m + (S1 + 1)(S2 + 1)n3m. Then Y is obtained from
yp = y1 Rp-1, p > 2
where R is the minimal nonnegative solution of the matrix equation £j=0 YjAj = 0. The boundary equations are given by
00
£ ypBp0 = 0
p=0
00
y0B00 + £ ypAp = 0
p=1
The normalizing condition Ye = 1 gives
y0e + y1 [I - R]-1e = 1 R matrix is obtained using the algorithm:
R(0) = 0
R(p + 1) = -A0A1-1 - R2(p)A2A1-1 - R3(p)A3A1-1 - ..., p > 0
5. Performance Measure
• Expected number of customers in the system, EN = E~=1 Wp
• Expected number of customers demanding Ii alone, Ej1 = ciEn
• Expected number of customers demanding I2 alone, Ej2 = c2EN
• Expected number of customers demanding both I1 and I2, EIi2 = c3EN
• Expected rate of replenishment for item Ii,
si S2 m tc 3 si S2 nv m
eriI = M E E E yo(0,^a2,k) + E E E E E E yp(val,a2,^k)}
ai=0 a2=0 k=i p=0 v=i ai=0 a2=0 jv=i k=i
• Expected rate of replenishment for item I2,
Si s2 m TC 3 Si s2 nv m
eri2 = M E E E y0(0,al,a2,k) + E E E E E E yp(val,a2,nv,k)}
ai=0 a2=0 k=i p=0 v=i ai=0 a2=0 jv=i k=i
• Expected reorder rate of commodity Ii,
TC S2 ni m
ERi = m E E E E yp(i si +1, a2, nl, k)
p=0 a2=0 ji =i k=i
• Expected reorder rate of commodity I2,
TC Si n2 m
ER2 = ^2 E E E E yp (2, ^ s2 + 1, n2, k)
p=0 ai=0 j2=i k=i
• Expected reorder rate of commodity Ii and I2,
TC n3 m
Er12 = F3 E E E yp(3, si + i, s2 + i, n3, k)
p=0 j3=i k=i
6. Numerical Implementation
In this section, we examine the outcome of our system using numerical and graphical representations. The three different MAP representations are distinct with the following variance and correlation structures and their mean values are i.
Arrival in Erlang of order 2(ERL-A):
D0
—2 2 Di = 0 0
0 —2 2 0
Arrival in Exponential(EXP-A):
D0 = [—i] Di = [i] Arrival in Hyper exponential(HYP-EXP-A):
D0
—1.90 0 0 —0.i9
Di
i.7i0 0.i90 0.i7i 0.0i9
Let us consider PH-distributions for the service process as follows: ERL-S (Service in Erlang of order 2):
ai = x2 = 1x3 = [1,0] Ti = T2 = T3
-2 2 02
EXP-S(Service in Exponential):
Xi = X2 = X3 = [1] Ti = T2 = T3 = [-1] HYP-EXP-S(Service in Hyper exponential):
x1 = x2 = x3 = [0.8,0.2] Ti = T2 = T3
-2.8 0 0 -0.28
6.1. Illustration
In the following tables 1, 2 and 3, we have examined the impact of the arrival rate A on the expected system size. Fix S1 = 8, S2 = 10, s1 = 2, s2 = 3, = 2, ^2 = 3, = 4, = 2, j82 = 3, c1 = 0.1, c2 = 0.1, c3 = 0.8.
Table 1: Arrival r«te(A) vs En
ERL-S
A ERL-A EXP-A HYP-EXP-A
i 0.038353086 0.090270325 0.i9755986i
i.i 0.050688987 0.ii378i945 0.256893077
i.2 0.065624277 0.i4i308387 0.329498505
i.3 0.083549669 0.i734008i7 0.4i7940725
i.4 0.i04926605 0.2i07i5786 0.525249i33
i.5 0.i30305758 0.25404i436 0.654975645
i.6 0.i6035ii57 0.30433i9i3 0.8ii26i47
i.7 0.i95872i59 0.362753i8 0.998927098
i.8 0.23786654 0.430744844 i.223609249
i.9 0.287579538 0.5i0i0493 i.49i978i6i
2.0 0.346586293 0.603i08i57 i.8i20776ii
Table 2: Arrival raie(A) vs En
EXP-S
A ERL-A EXP-A HYP-EXP-A
i 0.062592538 0.i20376635 0.2544i6346
i.i 0.0820737i 0.i5i739687 0.328650275
i.2 0.i055002i7 0.i884649i9 0.4i8427098
i.3 0.i33449i38 0.23i29i358 0.526393629
i.4 0.i66604203 0.28i099839 0.6556i6534
i.5 0.205783572 0.338948i74 0.80964334i
i.6 0.25i976048 0.406ii725i 0.992584752
i.7 0.306389033 0.484i72226 i.20923i96
i.8 0.3705i3054 0.575044848 i.465227i37
i.9 0.4462i00i6 0.68ii45853 i.767309959
2 0.535836027 0.80552094 2.i23668949
Table 3: Arrival rate(A) vs En
HYP-EXP-S
A ERL-A EXP-A HYP-EXP-A
1 0.249031469 0.32018599 0.571239637
1.1 0.319730773 0.403892787 0.721444007
1.2 0.403462978 0.502016937 0.897044873
1.3 0.502065778 0.616543171 1.10107116
1.4 0.617713186 0.749810152 1.336958686
1.5 0.752987938 0.904583472 1.608618699
1.6 0.910971089 1.084145482 1.920517083
1.7 1.095353193 1.292405822 2.27776399
1.8 1.310572349 1.534037132 2.686212657
1.9 1.561985236 1.814640848 3.152564574
2 1.856077719 2.140947812 3.684475578
We observe that from the above tables 1, 2 and 3:
• As arrival rate (A) increases, the variety of arrangements of arrival and service times then the corresponding En also increases.
• Observe the arrival times, En rises more quickly in the case of HYP — EXP — A and more slowly in the case of ERL — A. Similarly, it rises gradually in the case of ERL — S and rapidly in the case of HYP — EXP — A.
6.2. Illustration
We have investigated the consequence of the arrival rate A against the Expected to reorder rate of commodity I1 (ERl)in the obeying table 4, 5 and 6. Fix S1 = 8, S2 = 10, s1 = 2, s2 = 3, ^ = 2, p2 = 3, p3 = 4, fa = 2, = 3, c1 = 0.1, c2 = 0.1, c3 = 0.8.
Table 4: Arrival rate(A) vs Er1
ERL-S
A ERL-A EXP-A HYP-EXP-A
1.0 0.000025 0.003443 0.000174
1.1 0.000036 0.003781 0.000228
1.2 0.000050 0.004121 0.000290
1.3 0.000069 0.004461 0.000360
1.4 0.000092 0.004804 0.000437
1.5 0.000121 0.005148 0.000522
1.6 0.000155 0.005495 0.000612
1.7 0.000196 0.005846 0.000707
1.8 0.000245 0.006199 0.000806
1.9 0.000302 0.006557 0.000908
2.0 0.000367 0.006918 0.001013
Table 5: Arrival rate(A) vs Er1
EXP-S
A ERL-A EXP-A HYP-EXP-A
1.0 0.003780 0.015642 0.004010
1.1 0.004109 0.017110 0.004381
1.2 0.004435 0.018566 0.004750
1.3 0.004758 0.020010 0.005117
1.4 0.005080 0.021444 0.005481
1.5 0.005402 0.022870 0.005844
1.6 0.005723 0.024289 0.006203
1.7 0.006047 0.025702 0.006560
1.8 0.006372 0.027112 0.006914
1.9 0.006701 0.028519 0.007264
2.0 0.007034 0.029924 0.007610
Table 6: Arrival rate(A) vs Er1
HYP-EXP-S
A ERL-A EXP-A HYP-EXP-A
1 0.000059 0.002861 0.000272
1.1 0.000087 0.003140 0.000356
1.2 0.000124 0.003421 0.000453
1.3 0.000170 0.003705 0.000561
1.4 0.000228 0.003993 0.000679
1.5 0.000297 0.004286 0.000808
1.6 0.000378 0.004585 0.000947
1.7 0.000473 0.004888 0.001093
1.8 0.000583 0.005197 0.001247
1.9 0.000707 0.005512 0.001408
2 0.000845 0.005832 0.001573
We observe that from the above table 4, 5 and 6:
• As arrival rate (A) increases, the variety of arrangements of arrival and service times then
the corresponding ER1 also increases.
• Observe the arrival times, Er1 rises faster in the case of EXP - A and more gradually in the case of HYP - EXP - A. Comparably, it rises gradually in the case of HYP - EXP - S and significantly in the case of EXP-S.
6.3. Illustration
In the 2D image, the influence of arrival rate(A) on the expected number of customers demanding both i1 and 112 has been examined. Fix S1 = 8, S2 = 10, s1 = 2, s2 = 3, ^ = 2, = 3, = 4, = 2, j82 = 3, c1 = 0.1, c2 = 0.1, c3 = 0.8 so that the stability condition is satisfied.
From Figures 1 to 9,
• we can visualize that as the arrival rate (A) maximizes, both the value of Ei1 and Ei12 maximizes.
• Furthermore, the rate of an increase of Ei1 and Ei12 for HYP - EXP - A is rapid and slow for ERL — A. It is also faster for HYP — EXP — S and shorter for ERL — S.
1.4 1.6 A
Figure 1: Arrival rate(A) vs both Ej1 and Ej12 - Ek /Ek /1
1.4 1.6 A
Figure 2: Arrival rate(A) vs both Ej1 and Ej12 - M/Ek/1
Figure 3: Arrival rate(A) vs both Ej1 and Ej12 - H/Ek/1
Figure 4: Arrival rate(X) vs both Ej1 and Ej12 - Ek/M/1
Figure 5: Arrival rate(X) vs both Ej1 and Ej12 - M/M/1
Figure 6: Arrival rate(X) vs both Ej1 and Ej12 - H/ M/1
1.4 1.6 A
Figure 7: Arrival rate(A) vs both Ej1 and Ej12 - Ek / H /1
1.4 1.6 A
Figure 8: Arrival rate(A) vs both Ei1 and Ei12 - M/H /1
tu
N
Sg S= T3
c
tc
(U T3
T3
(U
(J
(U
sp
X
w
1.4 1.6 A
Figure 9: Arrival rate(A) vs both Ej1 and Ej12 - H /H /1
3
2
1
0
7. Conclusion
We looked at an inventory problem with two commodities and MAP demand arrival. When being taken for service, customers express their needs. If the requested item is unavailable, the customer is permanently removed from the system. When taken for service, if both goods are demanded, and when there is only one thing left, it is served to the customer. Depending on the type of demand, service times are distributed using a phase-type parameter. With parameter ^ for Ii, i = 1,2, the lead times for each commodity are exponentially distributed. It is determined that the continuous-time Markov chain is of type GI/M/1. The stability of the system is demonstrated. Many system performance indices are developed, along with numerical examples and numerical studies.
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