Next Article in Journal
Dynamic Model for Biomass and Proteins Production by Three Bacillus Thuringiensis ssp Kurstaki Strains
Next Article in Special Issue
A Novel Prediction Process of the Remaining Useful Life of Electric Vehicle Battery Using Real-World Data
Previous Article in Journal
Recycling of Waste Oils: Technology and Application
Previous Article in Special Issue
Approaching Sustainability Transition in Supply Chains as a Wicked Problem: Systematic Literature Review in Light of the Evolved Double Diamond Design Process Model

Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

# Steady State Analysis of Impulse Customers and Cancellation Policy in Queueing-Inventory System

by
V. Vinitha
1,
N. Anbazhagan
1,
S. Amutha
2,
K. Jeganathan
3,
4,
Woong Cho
5,* and
Suseok Seo
6,*
1
Department of Mathematics, Alagappa University, Karaikudi 630003, India
2
Ramanujan Centre for Higher Mathematics, Alagappa University, Karaikudi 630003, India
3
Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai 600005, India
4
Department of Computer Science and Engineering, Sejong University, Seoul 05006, Korea
5
Department of Software Convergence, Daegu Catholic University, Gyeongsan 38430, Korea
6
Department of e-Business, Yuhan University, Bucheon-si 14780, Korea
*
Authors to whom correspondence should be addressed.
Processes 2021, 9(12), 2146; https://doi.org/10.3390/pr9122146
Submission received: 30 October 2021 / Revised: 23 November 2021 / Accepted: 24 November 2021 / Published: 28 November 2021

## Abstract

:
This article discusses the queueing-inventory model with a cancellation policy and two classes of customers. The two classes of customers are named ordinary and impulse customers. A customer who does not plan to buy the product when entering the system is called an impulse customer. Suppose the customer enters into the system to buy the product with a plan is called ordinary customer. The system consists of a pool of finite waiting areas of size N and maximum S items in the inventory. The ordinary customer can move to the pooled place if they find that the inventory is empty under the Bernoulli schedule. In such a situation, impulse customers are not allowed to enter into the pooled place. Additionally, the pooled customers buy the product whenever they find positive inventory. If the inventory level falls to s, the replenishment of Q items is to be replaced immediately under the (s, Q) ordering principle. Both arrival streams occur according to the independent Markovian arrival process (MAP), and lead time follows an exponential distribution. In addition, the system allows the cancellation of the purchased item only when there exist fewer than S items in the inventory. Here, the time between two successive cancellations of the purchased item is assumed to be exponentially distributed. The Gaver algorithm is used to obtain the stationary probability vector of the system in the steady-state. Further, the necessary numerical interpretations are investigated to enhance the proposed model.

## 1. Introduction

In a queueing-inventory system, customers arrive at the service system on an individual basis if the item is needed. Inventory must be present if customers are to pick up the service. Once the service is completed, the items are removed from inventory for those who have earned the services. This system has emerged as an immense aspect in the mathematical modelling of predicament commotions issuing in a tremendous application, transportation system, grocery store, computer network system, and many other systems. Many researchers have worked with queueing-inventory systems in the last few decades. When ordered products are plentiful, backlogged demands are fulfilled quickly; but, when the item is replenished, backlogged demands may have to wait for fulfillment. This type of backlogged demand is called postponed demand [1].
Customers that buy on the spur of the moment are referred to as impulse customers. Customers who buy on the spur of the moment have little regard for planning, budgeting, or the necessity for a certain item. It is, in reality, a sudden event in which our emotions take over our brain and drive us to go ahead and purchase that object, even though we may not need it right now or even immediately. For instance, perfume and body spray, their attractive smell often encourages customers to buy them even if they do not have any need for it, dress, shoes, and cosmetics for other examples.
Cancellation is a very frequent occurrence that happens in life. For instance, after purchasing any garment or dress, later, if it is found to be unfit or not liked, we go for cancellation. Similarly, we cancel an advance travel reservation, including Bus, Train, and Flight, if we happen to come across some unexpected and sudden circumstances that don’t allow us to make our journey as planned.
Aspects of this work are as follows: The model representation of the queueing-inventory with the cancellation policy and impulse customers governed by the Markovian arrival process portrayed in Section 2. An investigation has been made in Section 3 to show the joint probability distribution of the number of pool customers and stock level in a steady-state case. Section 4 is a delineated discussion of a few numerical instances.

#### Literature Review

Sigman and Levi [2] and Melikov and Molchanov [3] commenced to discourse on the queueing-inventory problem. The MAP is one of the emerging trends in the queueing-inventory system, and Neuts [4] has put forward the Markovian arrival process. Berman et al. [5] have commenced the idea of postponed demand in the inventory model.
Correlated along with the postponed demand article, Sivakumar and Arivarignan [6] scrutinized the perishable inventory model along the infinite size waiting for a place, customers appear under MAP, and when the customer appears in the stock-out, either they move to a pool or fall out. In addition, Paul Manuel et al. [7] deal with a perishable inventory model having infinite waiting space, two kinds of customers arrive according to MAP, and service time follows PH distribution.
Krishnamoorthy et al. [8] commenced inter-cancellation into the queueing-inventory system. Dhanaya Shajin and Krishnamoorthy [9] scrutinized a queueing-inventory model with the MAP, service time under PH distribution, all the items have a common lifetime, and items are overbooked. The waiting customers are fall out when the system is full, forthwith S items are formed. Dhanya Shajin et al. [10] explored advanced reservation and overbooking with MAP in the queueing-inventory model. There are only a few papers related to inter-cancellation. Some notable work done by Sung-Seok Ko [11], Srinivas R. Chakravarthy [12], and Krishnamoorthy et al. [13] on the Markovian arrival process.
Nair and Jose [14] considered the retrial customers, the customers appear according to MAP, the service time is exponentially distributed, and the process of production complies with PH distribution. Production begins and service at a reduced rate up to the zero level of inventory when the inventory goes to a pre-assigned level s. Incoming customers are aimed at a buffer of finite capacity equal to the existing inventory level. Suppose that buffer is complete, customers move to an orbit of infinite size or fall out. Punalal and Babu [15] contemplated a model in which the customers appear, governed with MAP, and all customers are treated as ordinary at the moment of arrival. During busy periods, incoming customers fill the orbit’s infinite capacity. Every orbit customer, regardless of others, makes a priority with the time that occurs. Once, the customer got precedence rabidly taken service during the free period; otherwise, the customer went right into a waiting space that is reserved only for precedence generated customers.
Ayyappan and Gowthami [16] considered a queueing model along with the arrival types, such as incoming and outgoing calls. Customers who appear on the system under the MAP make incoming calls, and the server makes outgoing calls during idle time. Service times of incoming or outgoing calls under phase-type distribution. Seokjun Lee et al. [17] contemplated the queueing system with a single server and heterogeneous the arrival flow is under the marked Markov arrival process. There is distinct impatience for customers of several kinds. It is assumed that the difficulty of setting non-preemptive priorities for various types of consumers is solved within the assumption that customers can improve their precedence while being in the buffer. The distribution of service time is of the phase type. Valentina Klimenok et al. [18] analysed the queueing system with a single server. They categorized two kinds of customers who appear under a batch-marked Markov arrival process. Customers with low priority are entitled to obtain higher priority after a random period. Non-priority customers are also allowed in the buffer, but fix the timer. If the timer is expired, the customer falls away from the system with some probability. The high priority acquires the complementary probability. Alexander Dudin and Sergei Dudin [19] analysed the queueing model along with a single server and arrival types, such as customers of type 1 can be queued into the buffer with infinite capacity and customers of type 2 have a finite capacity buffer. Customers of both types can be impatient, and the arrival harmonizes with the marked Markovian arrival process. The author introduces a new form of distribution called phase-type with failures (when a failure can occur while a customer is being served in generalizes of phase-type distribution). The distribution of service time is PHF. In a related bibliography [20,21,22,23], the stochastic inventory system with MAP arrivals is shown.
The findings of the previous poll sparked our work, and to our knowledge, there has been little research into impulsive customers with MAP.

## 2. Model Description

This paper investigates the two classes of customers in a queueing-inventory system with a cancellation policy. The two classes of customers are defined as ordinary and impulse customers. In real-life phenomena, we observe two types of customers who may approach the inventory system to buy a product. Many customers visit an inventory system without planing to buy the product. Suppose at the end of the visiting process, a customer decides whether to buy a product or not. This type of customer is called an impulse customer. On the other hand, once the customers enter into the system, they buy the product compulsorily. These kinds of customers are called ordinary customers. In such a way, the proposed model allows these two classes of customers to purchase inventory items. The arrival pattern of both ordinary and impulse customers are assumed to be independent MAP.
The arrival process of ordinary customer representation is $( E 0 , E 1 )$, where $E i ( i = 0 , 1 )$ are square matrices of dimension $k 1$, such that $E 0$ governs transitions corresponding to no arrival and $E 1$ governs transitions corresponding to an arrival. The underlying Markov chain $U 3 ( t )$ of the MAP has the generator E is a square matrix of dimension $k 1$, where $E = E 0 + E 1$.
The stationary rate $λ 1$ of an ordinary customer is defined by $λ 1 = η 1 E 1 e$, where stationary row vector $η 1$ of dimension $1 × k 1$ is to be obtained by using $η 1 E = 0 and η 1 e = 1$.
Similarly, $( F 0 , F 1 )$ represents the arrival pattern of an impulse customer, where $F i ( i = 0 , 1 )$ are the square matrices of size $k 2$, such that $F 0$ governs transitions corresponding to no arrival and $F 1$ governs transitions corresponding to an arrival. The underlying Markov chain $U 4 ( t )$ of the MAP has the generator F is a square matrix of size $k 2$, where $F = F 0 + F 1$.
The stationary rate $λ 2$ of an impulse customer is defined by $λ 2 = η 2 F 1 e$, where stationary row vector $η 2$ of dimension $1 × k 2$ is to be obtained by using $η 2 F = 0 and η 2 e = 1$. The parameters $k 1$ and $k 2$ represents the phase of the arrival process of ordinary and impulse customers, respectively.
The service process of the system is assumed to be instantaneous when the inventory level is positive. Any arriving customer finds that there exists a positive inventory, he/she starts purchasing their product. After the purchase completion of the customer, the inventory will be decreased by one unit of an item. In this system, at the end of the visiting process of impulse customers, they leave the system either under the transition rate $p F 1$ if they buy the product or $q F 1$ if they do not buy the product. Suppose the inventory system is empty, the arriving impulse customers are considered as lost. In contrast, the ordinary customers may join in the finite size, N of pooled place under the Bernoulli schedule. That is, the arriving ordinary customer enters into the pooled place with probability $r 1$ or leave the system with probability $r 2$, where $r 2 = 1 − r 1$. The customers in the pool approach the inventory system whenever they find the positive stock in the inventory with the rate $θ$. The time between two successive approaches of a pooled customers follows an exponential distribution.
Cancellation policy:
The customers return the purchased product due to their dissatisfaction. Suppose the purchased item is damaged and the system has at most S (maximum inventory level) items in the inventory, they can not be allowed to return the product. Whenever there exists a $( S − i )$ item in the inventory, where $i ( 1 ≤ i ≤ S )$ represents the purchased item, the transition rate of the return or cancellation of the product is defined by $i β$. The time between two successive cancellations of the product is assumed to be exponentially distributed.
Further, if the storage of the system falls to s, there must be $Q ( = S − s > s + 1 )$ items immediately replenished with the transition rate $μ ( > 0 )$. The lead time follows an exponential distribution.

## 3. Analysis

In this sector, we construct the transition rate matrix on the queueing-inventory system. The Markov process of the form ${ ( U 1 ( t ) , U 2 ( t ) , U 3 ( t ) , U 4 ( t ) ) , t ≥ 0 }$ with state space
$C = { ( u 1 , u 2 , u 3 , u 4 ) : u 1 = 0 , 1 , ⋯ , N ; u 2 = 0 , 1 , ⋯ , S ; u 3 = 1 , 2 , ⋯ , k 1 ; u 4 = 1 , 2 , ⋯ , k 2 }$
where
$U 1 ( t ) : The number of customers in pool of finite size waiting place at time t . U 2 ( t ) : The number of items in the inventory at time t . U 3 ( t ) : Phase of the ordinary customers arrival process at time t . U 4 ( t ) : Phase of the impulse customers arrival process at time t .$
Transition rates are:
• Transition due to ordinary customers arrival
(a)
$( u , v ) → ( u , v − 1 )$: rate $E 1 ⊗ I k 2$, $u = 0 , 1 , ⋯ , N ; v = 1 , 2 , ⋯ , S$.
(b)
$( u , 0 ) → ( u + 1 , 0 )$: rate $r 1 E 1 ⊗ I k 2$, $u = 0 , 1 , ⋯ , N − 1$.
• Transition due to impulse customers arrival
$( u , v ) → ( u , v − 1 )$: rate $I k 1 ⊗ p F 1$, $u = 0 , 1 , ⋯ , N ; v = 1 , 2 , ⋯ , S$.
• Transition due to cancellation
$( u , v ) → ( u , v + 1 )$: rate $( S − v ) β I k 1 ⊗ I k 2$, $u = 0 , 1 , ⋯ , N ; v = 0 , 1 , ⋯ , S − 1 .$
• Transition due to approach from pooled customers
$( u , v ) → ( u − 1 , v − 1 )$: rate $θ I k 1 ⊗ I k 2$, $u = 1 , 2 , ⋯ , N ; v = 1 , 2 , ⋯ , S .$
• Transition due to replenishment
$( u , v ) → ( u , v + Q )$: rate $μ I k 1 ⊗ I k 2$, $u = 0 , 1 , ⋯ , N ; v = 0 , 1 , ⋯ , s ,$
where $u = u 1 , v = u 2 .$
The process’s infinitesimal generator A is generated by
$A = 0 1 2 3 ⋯ N − 1 N 0 1 2 3 ⋮ N − 1 N ( A 00 A 01 0 0 ⋯ 0 0 A 10 A 11 A 01 0 ⋯ 0 0 0 A 10 A 11 A 01 ⋯ 0 0 0 0 A 10 A 11 ⋱ 0 0 ⋮ ⋮ ⋮ ⋱ ⋱ ⋱ ⋮ 0 0 0 0 ⋱ A 11 A 01 0 0 0 0 ⋯ A 10 A 22 ) ,$
where $[ A 10 ] v v ′ = { θ I k 1 ⊗ I k 2 v ′ = v − 1 , v = 1 , 2 , ⋯ , S , 0 otherwise .$
$[ A 01 ] v v ′ = { r 1 E 1 ⊗ I k 2 v ′ = v , v = 0 , 0 otherwise .$
$[ A 00 ] v v ′ = { E 1 ⊕ p F 1 v ′ = v − 1 , v = 1 , 2 , ⋯ , S , ( S − v ) β I k 1 ⊗ I k 2 v ′ = v + 1 , v = 0 , 1 , ⋯ , S − 1 , μ I k 1 ⊗ I k 2 v ′ = v + Q , v = 0 , 1 , ⋯ , s , ( r 2 E 1 + E 0 ) ⊕ F − ( μ + ( S − v ) β ) I k 1 ⊗ I k 2 v ′ = v , v = 0 , E 0 ⊕ ( F 0 + q F 1 ) − ( μ + ( S − v ) β ) I k 1 ⊗ I k 2 v ′ = v , v = 1 , 2 , ⋯ , s , E 0 ⊕ ( F 0 + q F 1 ) − ( S − v ) β I k 1 ⊗ I k 2 v ′ = v , v = s + 1 , s + 2 , ⋯ , S − 1 , E 0 ⊕ ( F 0 + q F 1 ) v ′ = v , v = S , 0 otherwise .$
$[ A 11 ] v v ′ = { E 1 ⊕ p F 1 v ′ = v − 1 , v = 1 , 2 , ⋯ , S , ( S − v ) β I k 1 ⊗ I k 2 v ′ = v + 1 , v = 0 , 1 , ⋯ , S − 1 , μ I k 1 ⊗ I k 2 v ′ = v + Q , v = 0 , 1 , ⋯ , s , ( r 2 E 1 + E 0 ) ⊕ F − ( μ + ( S − v ) β ) I k 1 ⊗ I k 2 v ′ = v , v = 0 , E 0 ⊕ ( F 0 + q F 1 ) − ( μ + ( S − v ) β + θ ) I k 1 ⊗ I k 2 v ′ = v , v = 1 , 2 , ⋯ , s , E 0 ⊕ ( F 0 + q F 1 ) − ( ( S − v ) β + θ ) I k 1 ⊗ I k 2 v ′ = v , v = s + 1 , ⋯ , S − 1 , E 0 ⊕ ( F 0 + q F 1 ) − θ I k 1 ⊗ I k 2 v ′ = v , v = S , 0 otherwise .$
$[ A 22 ] v v ′ = E 1 ⊕ p F 1 v ′ = v − 1 , v = 1 , 2 , ⋯ , S , ( S − v ) β I k 1 ⊗ I k 2 v ′ = v + 1 , v = 0 , 1 , ⋯ , S − 1 , μ I k 1 ⊗ I k 2 v ′ = v + Q , v = 0 , 1 , ⋯ , s , E ⊕ F − ( μ + ( S − v ) β ) I k 1 ⊗ I k 2 v ′ = v , v = 0 , E 0 ⊕ ( F 0 + q F 1 ) − ( μ + ( S − v ) β + θ ) I k 1 ⊗ I k 2 v ′ = v , v = 1 , 2 , ⋯ , s , E 0 ⊕ ( F 0 + q F 1 ) − ( ( S − v ) β + θ ) I k 1 ⊗ I k 2 v ′ = v , v = s + 1 , ⋯ , S − 1 , E 0 ⊕ ( F 0 + q F 1 ) − θ I k 1 ⊗ I k 2 v ′ = v , v = S , 0 otherwise .$
It may be noted that the matrices $A 10$, $A 01$, $A 00$, $A 11$, and $A 22$ are all square matrices of dimension $( S + 1 ) k 1 k 2$.

#### 3.1. Steady State Probability Vector

The Markov process ${ ( U 1 ( t ) , U 2 ( t ) , U 3 ( t ) , U 4 ( t ) ) , t ≥ 0 }$ on the state space C and the limiting distribution $Θ ( u 1 , u 2 , u 3 , u 4 ) =$
$lim t → ∞ P r [ U 1 ( t ) = u 1 , U 2 ( t ) = u 2 , U 3 ( t ) = u 3 , U 4 ( t ) = u 4 | U 1 ( 0 ) , U 2 ( 0 ) , U 3 ( 0 ) , U 4 ( 0 ) ]$
exists and is independent of the initial state.
Take
$Θ = ( Θ 0 , Θ 1 , Θ 2 , … , Θ N − 1 , Θ N ) ,$
where
• $Θ u 1 = ( Θ ( u 1 , 0 ) , Θ ( u 1 , 1 ) , Θ ( u 1 , 2 ) , ⋯ , Θ ( u 1 , S ) ) , u 1 = 0 , 1 , ⋯ , N$
• $Θ ( u 1 , u 2 ) = ( Θ ( u 1 , u 2 , 1 ) , Θ ( u 1 , u 2 , 2 ) , ⋯ , Θ ( u 1 , u 2 , k 1 ) ) , u 1 = 0 , 1 , ⋯ , N ; u 2 = 0 , 1 , ⋯ , S$
• $Θ ( u 1 , u 2 , u 3 ) = ( Θ ( u 1 , u 2 , u 3 , 1 ) , Θ ( u 1 , u 2 , u 3 , 2 ) , ⋯ , Θ ( u 1 , u 2 , u 3 , k 2 ) ) , u 1 = 0 , 1 , ⋯ , N ; u 2 = 0 , 1 , ⋯ , S ; u 3 = 1 , 2 , ⋯ , k 1$.
Our general matrix A has same structure in Gaver [24], so we make use of the same arguments to determine the limiting probability vectors.
We present the Gaver algorithm here.
Gaver Algorithm:
• Determine the matrices $Z n$ recursively by initializing,
$Z 0 = A 00$
$Z n = A 11 + A 10 ( − Z n − 1 − 1 ) A 01 , 1 ≤ n ≤ N − 1$
$Z N = A 22 + A 10 ( − Z N − 1 − 1 ) A 01 .$
• Compute the limiting probability vectors $Θ n$ using,
$Θ n = Θ ( n + 1 ) A 10 ( − Z n − 1 )$, for $n = 0 , ⋯ , N − 1 .$
• Determine the system of equations
$Θ N Z N = 0$;
$∑ n = 0 N Θ n e = 1 .$
From the above system of equations $Θ N Z N = 0$, vector $Θ N$ could be determine distinctively, up to a multiplicative constant. The constant is resolved by $Θ n = Θ ( n + 1 ) A 10 ( − Z n − 1 )$, $n = 0 , ⋯ , N − 1$ and $∑ n = 0 N Θ n e = 1 .$

#### 3.2. Few Significant of the System Peculiarities

In this segment, we acquire a few significant peculiarities measures.
• Mean inventory level
Let $η I$ is mean inventory level in the steady state. Since $Θ ( i 1 , i 2 )$ denote the limiting probability vector with the inventory level represents as $i 1$ and the number of customers in the pool represents as $i 2$. This is given by
$η I = ∑ i 1 = 0 N ∑ i 2 = 1 S i 2 Θ ( i 1 , i 2 ) e .$
• Mean reorder rate
Let $η R$ denote the mean reorder rate in the steady-state. When the inventory level reduces to s from $s + 1$ due to any of the following situations, a reorder is triggered:
(a)
The purchase of an ordinary customer.
(b)
Any one of pooled customers approaches.
(c)
The purchase of an impulse customer.
$η R = ∑ i 1 = 0 N Θ ( i 1 , s + 1 ) ( E 1 ⊗ I k 2 ) e + ∑ i 1 = 1 N Θ ( i 1 , s + 1 ) θ I e + ∑ i 1 = 0 N Θ ( i 1 , s + 1 ) ( I k 1 ⊗ p F 1 ) e .$
• Mean number of customers in the pool
Let $η P C$ denote the mean number of customers in the pool. Since $Θ i 1$ denote the stationary probability vector with the inventory level $i 1$. Hence, the mean number of customers in the pool is given by
$η P C = ∑ i 1 = 1 N i 1 Θ i 1 e .$ Mean rate of arrival of impulse customers
Let $η I C$ denote the mean rate of arrival of impulse customers in the steady state. Then, $η I C$ is given by
$η I C = ∑ i 1 = 0 N ∑ i 2 = 0 S Θ ( i 1 , i 2 ) ( I k 1 ⊗ F 1 ) e .$
• Mean number of lost customers in the system
Let $η L$ denote the mean number of lost customers in the system. This is given by
$η L = ∑ i 1 = 0 N − 1 Θ ( i 1 , 0 ) ( r 2 E 1 ⊗ I k 2 ) e + ∑ i 2 = 0 S Θ ( N , i 2 ) ( E 1 ⊗ I k 2 ) e + ∑ i 1 = 0 N Θ ( i 1 , 0 ) ( I k 1 ⊗ F 1 ) e + ∑ i 1 = 0 N ∑ i 2 = 1 S Θ ( i 1 , i 2 ) ( I k 1 ⊗ q F 1 ) e .$
• Mean cancellation rate of return product
Let $η C$ denote the mean cancellation rate of return product in the steady state. Then, $η C$ is given by
$η C = ∑ i 1 = 0 N ∑ i 2 = 0 S − 1 Θ ( i 1 , i 2 ) ( S − i 2 ) β e .$

#### 3.3. Construction of the Cost Feature

The expected total cost function per unit time is constructed by
$C ( S , s ) = c h η I + c w η P C + c s η R + c c l η L + c i η I C ,$
where
$c h : The inventory carrying cos t per unit time . c w : Waiting cos t of a customer in the pool per unit time . c s : Setup cos t per order . c c l : Cos t of a customer lost due to the zero stock per unit time . c i : The cos t due to the arrival of impulse customer per unit time .$

## 4. Numerical Illustration

We give a few descriptive numerical examples that expose the convexity of the expected cost rate and the MAP for ordinary and impulse customers’ appearance. We consider, $E 0 = F 0$ and $E 1 = F 1$.
• Hyper-exponential (HEX):
$E 0 = F 0 = − 15 0 0 − 5 ; E 1 = F 1 = 13.5 1.5 4.5 0.5$
• Erlang (ER):
$E 0 = F 0 = − 3 3 0 0 − 3 3 0 0 − 3 ; E 1 = F 1 = 0 0 0 0 0 0 3 0 0$
• Negative Correlation (NC):
$E 0 = F 0 = − 2.35 2.35 0 0 − 2.35 0 0 0 − 3.5$; $E 1 = F 1 = 0 0 0 0.0235 0 2.3265 3.465 0 0.035$
• Positive Correlation (PC):
$E 0 = F 0 = − 2.35 2.35 0 0 − 2.35 0 0 0 − 3.5 ; E 1 = F 1 = 0 0 0 2.3265 0 0.0235 0.035 0 3.465$
The ordinary customer process has negative(positive) correlated arrival with coefficient of variance $c v a r = 2 λ 1 η 1 ( − E 0 ) − 1 e − 1 = 0 . 9342 ( 0 . 9342 )$ and coefficient of correlation $c c o r = ( λ 1 η 1 ( − E 0 ) − 1 E 1 ( − E 0 ) − 1 ) e − 1 ) / c v a r = − 0 . 2595 ( 0 . 2595 )$ with arrival rate $λ 1 = 1.7594$. By our consideration, the values of $c v a r$ and $c c o r$ for the impulse customer are the same as the values of $c v a r$ and $c c o r$ for ordinary customers.
Table 1 gives the behaviour of the cost function of two variables $C ( S , s )$ for the case of hyper-exponential distribution. The values are divulged bold in each column indicate the minimum cost rate whereas, the least cost rate is specified in each row by underlining the values. Thus a value (bold and underlined) spectacles the local minimum of the function $C ( S , s )$. The optimal cost value $C * ( S , s )$ = 34.3798 is achieve at $S *$ = 23, $s *$ = 7 with the values $r 1 = 0.6 , r 2 = 1 − r 1 , p = 0.6 , q = 1 − p , θ = 6.5 , λ 1 = 12.5 , λ 2 = 12.5 , μ = 0.5 , β = 1 , N = 7 , c h = 0.9 , c w = 3 , c s = 10$, $c c l = 0.1 , c i = 0.3$. Table 1 and Figure 1 shows that the function $C ( S , s )$ is convex.
Table 2 scrutinized the ramifications of lead time rate $μ$ and a customer’s approach rate from the pool $θ$, the total cost rate $C ( S , s ) *$ and analogous optimal value $( S * , s * )$ with values $r 1 = 0.6 , r 2 = 1 − r 1 , p = 0.6 , q = 1 − p , λ 1 = 12.5 , λ 2 = 12.5 , β = 1 , N = 7 , c h = 0.9$, $c w = 3 , c s = 10 , c c l = 0.1 , c i = 0.3$. We observe, the total expected cost rate decreases whenever $θ$ and $μ$ increases. Table 3 and Table 4 are bestow the total expected cost rate is increase when $c w$, $c h$ and $c s$ increase but total expected cost rate is decrease when $c c l$ increases.
Table 5, Table 6 and Table 7 give the mean inventory level. The arrival rate of impulse customers, reorder rate, the number of lost customer, and the mean cancellation rate decreases, and the mean number of pooled customers increase whenever $θ$, $μ$, and $β$ increase under hyper-exponential, erlang, negative correlation and positive correlation.
Table 8 indicates the ordinary and impulse customer appears under erlang, a negative and positive correlation on the optimal total cost rate values at optimal value $S *$ and $s *$. Table 9 bestow the ordinary customer appears under erlang, and impulse customer appears with various distributions (erlang, negative and positive correlation) on the various measures and also negative and positive correlation. Table 10 bestow the effect of $λ 1$ for various MAP(ER, NC and PC) with $λ 2 = 0.1$. The value of $λ 1$ increase as $C ( S , s )$ increase with different MAP(ER, NC, and PC) and the values $r 1 = 0.4 , r 2 = 1 − r 1 , p = 0.6 , q = 1 − p , θ = 10.5 , μ = 2.5 , β = 2 , N = 7 , c h = 9 , c w = 3 , c s = 10$, $c c l = 0.1 c i = 0.3$. Table 11 bestow the effect of $λ 2$ for various MAP(ER, NC and PC) with $λ 1 = 1$. The value of $λ 2$ increase as $C ( S , s )$ increase with different MAP(ER,NC and PC) and the values $r 1 = 0.4 , r 2 = 1 − r 1 , p = 0.6 , q = 1 − p , θ = 10.5 , μ = 0.5 , β = 02 , N = 7 , c h = 9 , c w = 3 , c s = 10 , c c l = 0.1 , c i = 0.3$. Figure 2 shows the correlation values of ordinary customers are plotted against the $C ( S , s )$ values. It is possible to see that the $C ( S , s )$ is increase as correlation coefficient values increases and impulse customer pursuant to MAP(ER, NC and PC) with the values $r 1 = 0.5 , r 2 = 1 − r 1 , p = 0.5 , q = 1 − p , λ 1 = 12.5 , λ 2 = 12.5 , β = 1.5 , N = 7 , c h = 0.9 , c w = 3 , c s = 9 , c c l = 0.1 , c i = 0.3$. Figure 3 shows the correlation values of impulse customer are plotted against the $C ( S , s )$. It is possible to see that the $C ( S , s )$ values non-decrease as the correlation coefficient values increases and ordinary customer under MAP(ER, NC, and PC).

## 5. Conclusions

In this article, we introduced the impulse customer with cancellation policy in the queueing-inventory system. This work helps improve the QoS of an inventory management system. We presented the behaviour of the total cost function with variables s and S under hyper exponential distribution. We analysed the total cost function with various distributions like erlang, negative correlation, and positive correlation for ordinary and impulsed customers’ arrival streams. We analysed the effects of the pooled customer approach, lead time, and cancellation rates with various arrival streams like hyper-exponential, erlang, negative correlation, and positive correlation. The effects of an average number of impulse customer arrival rates and the average loss rate are indicated. Finally, we showed the effects of ordinary and impulsed customers’ arrival correlation with total cost rate.
In the future, we will be interested in extending this model with multi-server, and service time follows PH distribution.

## Author Contributions

Conceptualization, V.V.; data curation, V.V. and K.J.; formal analysis, V.V.; funding acquisition, G.P.J. and S.S.; investigation, W.C.; methodology, N.A. and K.J.; project administration, G.P.J.; resources, S.S.; software, S.A.; supervision, G.P.J., W.C. and S.S.; validation, S.S.; visualization, N.A. and S.A.; writing—original draft, V.V.; writing—review and editing, G.P.J. All authors have read and agreed to the published version of the manuscript.

## Funding

This research received no external funding.

Not applicable.

Not applicable.

Not applicable.

## Acknowledgments

Anbazhagan and Amutha would like to thank RUSA Phase 2.0 (F 24-51/2014-U), DST-FIST (SR/FIST/MS-I/2018/17), DST-PURSE 2nd Phase programme (SR/PURSE Phase 2/38), Govt. of India.

## Conflicts of Interest

The authors declare no conflict of interest.

## Abbreviations

The following Notations and Abbreviations are used in this manuscript:
 $[ A ] i j$ The element submatrix at $( i , j )$ the position of A $0$ Zero matrix $e$ A column vector of 1’s appropriate dimension S Maximum inventory level $A ⊗ B$ Kronecker product of matrices A and B $A ⊕ B$ Kronecker sum of matrices A and B MAP Markovian Arrival Process PH Phase-type PHF Phase-type with Failure.

## References

1. Jeganathan, K.; Harikrishnan, T.; Selvakumar, S.; Anbazhagan, N.; Amutha, S.; Acharya, S.; Dhakal, R.; Joshi, G.P. Analysis of Interconnected Arrivals on Queueing-Inventory System with Two Multi-Server Service Channels and One Retrial Facility. Electronics 2021, 9, 576. [Google Scholar] [CrossRef]
2. Sigman, K.; Simchi-Levi, D. Light traffic heuristic for an M/G/1 queue with limited inventory. Ann. Oper. Res. 1992, 40, 371–380. [Google Scholar] [CrossRef]
3. Melikov, A.Z.; Molchanov, A.A. Stock optimization in transportation/storage systems. Cybern. Syst. Anal. 1992, 28, 484–487. [Google Scholar] [CrossRef]
4. Neuts, M.F. Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach; Courier Corporation: North Chelmsford, MA, USA, 1994. [Google Scholar]
5. Berman, O.; Kaplan, E.H.; Shevishak, D.G. Deterministic approximations for inventory management at service facilities. IIE Trans. 1993, 25, 98–104. [Google Scholar] [CrossRef]
6. Sivakumar, B.; Arivarignan, G. A stochastic inventory system with postponed demands. Perform. Eval. 2009, 66, 47–58. [Google Scholar] [CrossRef]
7. Manuel, P.; Sivakumar, B.; Arivarignan, G. A perishable inventory system with service facilities, MAP arrivals and PH Service times. J. Syst. Sci. Syst. Eng. 2007, 16, 62–73. [Google Scholar] [CrossRef]
8. Krishnamoorthy, A.; Vladimir, V.; Manjunath, A.S.; Dhanya, S. Single server with several services. Reliab. Theory Appl. 2017, 12, 14–30. [Google Scholar]
9. Shajin, D.; Krishnamoorthy, A. On a queueing-inventory system with impatient customers, advanced reservation, cancellation, overbooking and common life time. Oper. Res. 2021, 21, 1229–1253. [Google Scholar] [CrossRef]
10. Shajin, D.; Krishnamoorthy, A.; Dudin, A.; Joshua, V.; Jacob, V. On a queueing-inventory system with advanced reservation and cancellation for the next K time frames ahead: The case of overbooking. Queueing Syst. 2020, 94, 3–37. [Google Scholar] [CrossRef]
11. Ko, S.-S. A Nonhomogeneous Quasi-birth-death process approach for an (s,S) policy for a perishable inventory system with retrial demands. J. Ind. Manag. Optim. 2020, 37, 1415–1433. [Google Scholar]
12. Chakravarthy, S.R. Analysis of MAP/PH1,PH2/1 queue with vacations and optional secondary services. Appl. Math. Model. 2013, 37, 8886–8902. [Google Scholar] [CrossRef]
13. Krishnamoorthy, A.; Shajin, D.; Lakshmy, B. GI/M/1 type queueing-inventory systems with postponed work, reservation, cancellation and common life time. Indian J. Pure Appl. Math. 2016, 47, 357–388. [Google Scholar] [CrossRef]
14. Nair, S.S.; Jose, K.P. A PH Distributed Production Inventory Model with Different Modes of Service and MAP Arrivals. In Applied Probability and Stochastic Processes; Infosys Science Foundation Series; Springer: Singapore, 2020. [Google Scholar]
15. Punalal, J.; Babu, S. The MAP/(PH,PH,PH)/1 Model with Self-Generation of Priorities, Customer Induced Interruption and Retrial of Customers. In Applied Probability and Stochastic Processes; Infosys Science Foundation Series; Springer: Singapore, 2020. [Google Scholar]
16. Ayyappan, G.; Gowthami, R. Analysis of MAP, PH2OA/PH1I, PH2O/1 retrial queue with vacation, feedback, two-way communication and impatient customers. Soft Comput. 2020, 25, 9811–9838. [Google Scholar] [CrossRef]
17. Lee, S.; Dudin, S.; Dudina, O.; Kim, C.; Klimenok, V. A Priority Queue with Many Customer Types, Correlated Arrivals and Changing Priorities. Mathematics 2020, 8, 1292. [Google Scholar] [CrossRef]
18. Klimenok, V.; Dudin, A.; Dudina, O.; Kochetkova, I. Queuing System with Two Types of Customers and Dynamic Change of a Priority. Mathematics 2020, 8, 824. [Google Scholar] [CrossRef]
19. Dudin, A.; Dudin, S. Analysis of a Priority Queue with Phase-Type Service and Failures. Int. J. Stoch. Anal. 2016, 2016, 9152701. [Google Scholar] [CrossRef] [Green Version]
20. He, Q.-M. The Versatility of MMAP[K] and the MMAP[K]/G[K]/1 Queue. Queueing Syst. 2001, 38, 397–418. [Google Scholar] [CrossRef]
21. Salini, S.; Nair, K.P.; Jose, A. MAP/PH/1 Production Inventory Model with Varying Service Rates. Int. J. Pure Appl. Math. 2017, 117, 373–381. [Google Scholar]
22. Radhamani, V.; Devi, P.C.; Sivakumar, B. A Stochastic Inventory System with Postponed Demands and Infinite Pool in Discrete-Time Setup. J. Oper. Res. Soc. China 2014, 2, 455–480. [Google Scholar] [CrossRef] [Green Version]
23. Barron, Y.; Hermel, D. Shortage decision policies for a fluid production model with MAP arrivals. Int. J. Prod. Res. 2017, 55, 3946–3969. [Google Scholar] [CrossRef]
24. Gaver, D.P.; Jacobs, P.A.; Latouche, G. Finite Birth- And- Death Models in Randomly Changing Environments. Adv. Appl. Probab. 1984, 16, 715–731. [Google Scholar] [CrossRef]
Figure 1. Convexity of the total cost with two variables S and s.
Figure 1. Convexity of the total cost with two variables S and s.
Figure 2. Ordinary customer arrival correlation vs. total cost rate.
Figure 2. Ordinary customer arrival correlation vs. total cost rate.
Figure 3. Impulse customer arrival correlation vs. total cost rate.
Figure 3. Impulse customer arrival correlation vs. total cost rate.
Table 1. The function of total cost rate with two variables S and s.
Table 1. The function of total cost rate with two variables S and s.
$S / s$5678
2173.909250.131635.950851.6445
2270.149848.717535.186350.1225
2366.653047.328134.379848.6287
2464.288846.863434.436948.0564
2564.396046.990835.033648.0747
Table 2. Ramification of the rate of lead time $( μ )$ and approach from the pool customer $( θ )$ on the optimal values.
Table 2. Ramification of the rate of lead time $( μ )$ and approach from the pool customer $( θ )$ on the optimal values.
$μ / θ$0.51.01.51.52.0
6237237237237237
35.205334.902534.593834.281033.9654
8247247247237237
32.016731.741831.463431.182530.8967
10247247247247247
29.105428.853228.598128.341328.0834
12247247247247247
26.521026.290926.058625.825025.5906
14247247247247247
24.230324.021423.810823.599223.3871
Table 3. Ramification of the cost rate of a lost customer during the stock out period $( c c l )$ and cost rate of waiting customer in the pool $( c w )$ on the optimal values.
Table 3. Ramification of the cost rate of a lost customer during the stock out period $( c c l )$ and cost rate of waiting customer in the pool $( c w )$ on the optimal values.
$c cl / c w$12345
0.1247247247247247
25.323726.377527.431228.485029.5388
0.2237237237237237
20.620821.706322.791823.877324.9628
0.3237237237237237
15.871616.957218.042719.128220.2137
0.4237237237237237
11.122512.208013.293514.379015.4645
0.5227237237237237
6.36117.45898.54449.629910.7154
Table 4. Ramification of the cost of setup cost $( c s )$ and holding cost $( c h )$ on the optimal values.
Table 4. Ramification of the cost of setup cost $( c s )$ and holding cost $( c h )$ on the optimal values.
$c s / c h$0.70.80.91.01.1
8237237237237237
24.807225.173125.538925.904726.2705
9247247247237237
25.747526.134026.520526.905727.2716
10247247247247247
26.658227.044727.431227.817828.2043
11247247247247247
27.568927.955528.342028.728529.1150
12247247247247247
28.479728.866229.252729.639230.0258
Table 5. Ramification of MAP with ER, NC and PC for ($S , s , μ , β$) = (21, 7, 2.5, 0.1).
Table 5. Ramification of MAP with ER, NC and PC for ($S , s , μ , β$) = (21, 7, 2.5, 0.1).
Customer Arrivals$θ$$η I$$η R$$η PC$$η IC$$η L$$η C$
MAP with HEX67.14895.55420.94920.19550.96554.9701
86.18455.24181.82820.18410.71654.8983
105.54455.12972.12210.17550.5984.8031
125.29715.11852.32090.15870.26554.7455
144.18425.03952.79860.14840.04954.2965
MAP with ER610.47161.56054.77010.19070.27183.9203
810.36391.10164.80310.16180.17113.7462
1010.21590.78284.83550.14210.10713.3194
129.95230.55794.86720.12710.06693.2674
149.60100.39774.89830.11480.04193.0355
MAP with NC611.93610.81363.61260.50720.18696.5235
811.80470.65693.65020.46940.14156.1452
1011.71420.53003.68720.43750.10765.4512
1210.13720.42783.72350.40990.08215.2201
149.26490.34553.75920.38570.06295.1258
MAP with PC611.81790.00553.00610.49160.15558.2835
811.96990.00633.01130.47380.16558.1521
1011.97040.00693.01600.45900.17317.9203
1211.42440.00763.02030.44640.17907.5603
1410.14320.00823.02440.43520.18357.3595
Table 6. Ramification of MAP with ER, NC and PC for ($S , s , θ , β$) = (21, 7, 10.5, 0.1).
Table 6. Ramification of MAP with ER, NC and PC for ($S , s , θ , β$) = (21, 7, 10.5, 0.1).
Customer Arrivals$μ$$η I$$η R$$η PC$$η IC$$η L$$η C$
MAP with HEX0.59.25111.25432.35330.75300.95326.2472
1.05.45241.02162.37510.46520.75926.1552
1.55.24551.01782.37510.37280.53135.2428
2.03.94251.01492.37510.20860.30975.1588
2.53.70581.01262.37510.05320.09275.0214
MAP with ER0.59.53251.20293.83550.64170.80137.2472
1.09.92821.12253.83550.51240.70326.8435
1.58.81401.02123.83560.45320.62336.5472
2.08.22871.01293.83560.36810.52376.3758
2.58.02661.00253.83570.25220.42516.2457
MAP with NC0.56.24592.00543.68720.74900.15957.8643
1.06.12052.00563.68730.63780.16457.5184
1.513.04762.00553.68730.54310.16357.4259
2.06.01022.00533.68740.46290.15857.3445
2.55.31392.00513.68740.39520.15106.2454
MAP with PC0.55.81842.00514.52121.70600.96546.9637
1.03.99542.00514.61221.59710.76526.7235
1.52.37292.00494.62221.50890.64496.6282
2.02.95222.00464.75221.43710.53726.4645
2.51.71772.00434.76221.37820.32866.1278
Table 7. Ramification of MAP with ER, NC and PC for ($S , s , θ , μ$) = (21, 7, 10.5, 2.5).
Table 7. Ramification of MAP with ER, NC and PC for ($S , s , θ , μ$) = (21, 7, 10.5, 2.5).
Customer Arrivals$β$$η I$$η R$$η PC$$η IC$$η L$$η C$
MAP with HEX0.54.39154.78841.055438.83449.95144.2575
1.04.38054.47501.032030.73078.99134.1702
1.54.26814.31671.112330.32238.99134.1434
2.04.18684.06681.992130.27152.01511.5565
2.53.63453.28792.047530.12811.94401.3230
MAP with ER0.56.57332.55661.02141.16025.10510.7440
1.02.06541.42141.26131.03154.05830.6516
1.50.88321.37711.33150.72741.42300.5629
2.00.40740.30741.37010.31220.53640.2604
2.50.19880.00651.41280.16820.23500.1353
MAP with NC0.57.76804.15440.92910.03020.11300.0098
1.02.07473.54910.93140.01390.01390.0024
1.51.56721.56720.94370.00020.00060.0001
2.01.75860.57860.95600.00010.00050.0001
2.50.21070.37540.96840.00010.00040.0001
MAP with PC0.50.32340.16161.18712.26193.40101.0797
1.00.31600.07881.12481.01171.72190.5441
1.50.05900.02061.21370.15210.11980.1073
2.00.00870.00191.35210.01940.01610.0158
2.50.00200.00031.42870.00400.00340.0035
Table 8. Ramification of ordinary and impulse arrival on optimal value.
Table 8. Ramification of ordinary and impulse arrival on optimal value.
Ordinary/Impulse ArrivalsMAP with ERMAP with NCMAP with PC
MAP with ER237238247
23.111221.697212.0351
MAP with NC237238247
22.808221.455621.1303
MAP with PC237238247
21.569823.804522.7606
Table 9. Ramification of MAP of ordinary and impulse customer arrivals on various measures.
Table 9. Ramification of MAP of ordinary and impulse customer arrivals on various measures.
Impulse Customer Arrivals$η I$$η R$$η PC$$η IC$$η L$$η C$
MAP of ordinary arrival with ER
MAP with ER4.07490.30741.03700.31220.53640.2604
MAP with NC2.83860.29350.94440.27830.08780.0417
MAP with PC1.83690.00381.01980.01890.05620.0154
MAP of ordinary arrival with NC
MAP with ER1.75860.00550.57860.54310.16350.1720
MAP with NC0.67410.01610.00160.92800.47870.0128
MAP with PC0.01640.00860.00290.85910.25630.0031
MAP of ordinary arrival with PC
MAP with ER8.07590.00190.01940.01610.01582.1521
MAP with NC2.25890.01710.00150.85940.50841.2315
MAP with PC0.09090.88070.22350.13000.14861.1763
Table 10. Effects of $λ 1$ with various MAP.
Table 10. Effects of $λ 1$ with various MAP.
$λ 1$Ordinary/Impulse ArrivalsMAP with ERMAP with NCMAP with PC
1.0MAP with ER309308319
3.23122.83331.9895
MAP with NC337337319
2.75142.81492.8366
MAP with PC318348337
3.07062.82752.8560
1.5MAP with ER337357358
2.90482.92772.8302
MAP with NC359338338
3.78052.85422.8954
MAP with PC318348337
24.72672.389020.5588
2.0MAP with ER337357358
2.89222.89212.8967
MAP with NC359338338
2.92652.81772.8953
MAP with PC318348337
2.88552.88512.8920
Table 11. Effects of $λ 2$ with various MAP.
Table 11. Effects of $λ 2$ with various MAP.
$λ 2$Impulse/Ordinary ArrivalsMAP with ERMAP with NCMAP with PC
0.2MAP with ER309308319
4.21721.38510.1314
MAP with NC337337319
1.31001.05231.0050
MAP with PC318348337
1.12530.52227.0213
0.4MAP with ER383383383
3.02332.22232.3503
MAP with NC359338338
1.01232.22112.4125
MAP with PC318348337
3.45612.18472.4875
0.6MAP with ER359338338
2.56543.48421.1551
MAP with NC3393111358
3.24841.54220.4412
MAP with PC318348337
3.16542.16853.1517
 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Share and Cite

MDPI and ACS Style

Vinitha, V.; Anbazhagan, N.; Amutha, S.; Jeganathan, K.; Joshi, G.P.; Cho, W.; Seo, S. Steady State Analysis of Impulse Customers and Cancellation Policy in Queueing-Inventory System. Processes 2021, 9, 2146. https://doi.org/10.3390/pr9122146

AMA Style

Vinitha V, Anbazhagan N, Amutha S, Jeganathan K, Joshi GP, Cho W, Seo S. Steady State Analysis of Impulse Customers and Cancellation Policy in Queueing-Inventory System. Processes. 2021; 9(12):2146. https://doi.org/10.3390/pr9122146

Chicago/Turabian Style

Vinitha, V., N. Anbazhagan, S. Amutha, K. Jeganathan, Gyanendra Prasad Joshi, Woong Cho, and Suseok Seo. 2021. "Steady State Analysis of Impulse Customers and Cancellation Policy in Queueing-Inventory System" Processes 9, no. 12: 2146. https://doi.org/10.3390/pr9122146

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.