## 1. INTRODUCTION

Research on queueing models with inventory control has captured much attention of researchers over the last decades. In this system, customers arrive at the service facility one by one and they require service. In order to complete customer service, an item from the inventory is needed. A served customer departs immediately from the system and the on-hand inventory decreases by one at the moment of service completion. The decreased inventory is supplied by an outside supplier. This system is called *queueing inventory systems* (QIS) (Schwarz *et al*., 2006). The queueing inventory systems are described in Figure 1.

Meanwhile phase-type distributions are defined the time to absorption of a finite Markov chain in continuous time, when there is a single absorbing state and the stochastic process starts in a transient state (Latouche and Ramaswzmi, 1999). Since their introduction by Neuts in 1975, phase-type distributions have been used in a wide range of stochastic modeling applications in areas as diverse as telecommunications, finance, reliability theory, queueing theory, biostatistics and inventory theory. Phasetype distributions have enjoyed such popularity because they constitute a very versatile class of distributions defined on the nonnegative real numbers that lead to models which are algorithmically tractable. Their formulation also allows the Markov structure of stochastic models to be retained when they replace the familiar exponential distribution.

Prior to Neuts’s 1975 work much of the research in stochastic modeling and queueing theory relied on random variables of interest and service time was modeled by the exponential or Erlang distributions, and point and interarrival processes by the Poisson process or Erlang renewal processes. Phase-type distributions constitute a much more useful class of distributions for several reasons. First, they form a versatile class of distributions that are dense in the class of distributions defined on the nonnegative real numbers. That is, they can approximate any nonnegative distribution arbitrarily closely although the number of states needed may be large (Asmussen, 2003). Second, since they have a simple probabilistic interpretation in terms of continuous-time Markov chain, they exhibit a Markov structure which enables an easier analysis of models that use them instead of general distributions (Neuts, 1981). Third, the use of phase-type distributions in stochastic models often enables algorithmically tractable solutions to be found. If phase-type distributions are used, many quantities of interest that are used in algorithms to compute performance measures can be expressed simply in terms of the inverse and exponential of matrices that contain only real entries (Neuts, 1981). Therefore, we can easily calculate because of this recent advance in computational aspect of Linear Algebra.

There have been studied a variety of literature reviews for inventory models. Lost-sales inventory theory was produced by Bijvank and Vis (2011), Barron and Hermel (2017). Bakker *et al*. (2012) and Li *et al*. (2010) reviewed inventory systems with deterioration. Krishnamoorthy *et al* .(2011) investigated inventory model with positive service time. Minner (2003) studied multiplesupplier inventory models in supply chain management. Nahmias (1982) and Saranya and Lawrence (2019) reviewed perishable inventory models. Paterson *et al*. (2011) investigated inventory models with lateral transshipments. While, very few results were suggested for the review papers on phase-type distributions. Mogens (2005) reviewed phase-type distributions in risk theory. Shama (2014) studied the work done by the prominent researchers on the phase service queues and their applications in several realistic queueing situations.

This review contains an overview of the work done in queueing inventory systems with phase-type service distributions. We have included all those papers which can give insights to solve various inventory models. Finally, the framework of literature review in this paper clearly provides an overview of the queueing inventory systems study, which can be used as a starting point for further study. Figure 2

This paper is organized as follows; In section 2, we categorize inventory models by features in queueing model. In section 3, we categorize features in inventory model which are important control variables. Finally, conclusion and future work is drawn in Section 4.

## 2. CATEGORIZATION BY FEATURES IN QUEUEING MODEL

### 2.1 Customer's Arrival

One useful way to organize the researches done on the queueing inventory model is to look at the core of each model in queueing terms, i.e. the basic assumptions concerning the demand process, the lead time distribution, and the number of servers present. However, we assume that service time is phase-type distributions in our research. The most commonly used distribution for either the product lifetime or demand process has been exponential (symbol M as in Markov), as in (Altiok, 1989b;Brown and Zipkin, 1991;Schwarz *et al*., 2006;Saranya and Lawrence, 2019;Svoronos and Zipkin, 1991;Xu and Wang, 2019), as this greatly simplifies mathematical analysis. Some researchers have broadened the applicability of their work by generalizing this to the deterministic (D), compounded Poisson process (CP), general (G) distribution or Markovian Arrival Process (MAP) case. Table 1 shows a basic queueing inventory system structures by customer’s arrival.

It is important how demand or order process assumes in an inventory model. It is possible to represent whether the demand or order process is deterministic or stochastic process in QIS. Boute *et al*. (2007), Balciogiu and Gureler (2011) and Choi *et al*. (2015) are applied to the inventory model that demand process is deterministic. However, most of the researches on inventory model are assumed that the demand or order process is stochastic process. It is interesting that Boute *et al*. (2007), Van Houdt and Pérez (2012) and Choi *et al*. (2015) analyzed the inventory model with *endogenous lead time* by matrix analytic method. In addition, Choi and Yoon (2013) analyzed it using roots method which accurately calculates the unknown constants and then uses these constants to get probability distributions of the number in the system at pre-arrival, arbitrary, and post-departure epochs as well as the sojourn-time distribution and other performance measures. The replenishment lead times in production/ inventory (P/I) and inspection/inventory (I/I) systems are endogenously determined by the production and inspection system, respectively. These *endogenous* replenishment lead times are load-dependent and affected by the current size if the order queues in the production and inspection system. Since the rate of interarrival time or service time changes dependent on what lead time is considered, it is important to distinguish between *endogenous lead time* and *exogenous lead time*. Since it can be different performance measures according to the type of lead time, the applications for queueing model are very essential in QIS.

In addition, any general and finite distribution can be approximated to a phase-type distribution (Boute, 2007). Instead of using general distributions, it is preferable to use phase-type distributions in the analysis (Chang and Lu, 2011). Wang (2012), Wang and Su (2007) approximated M/G/1 queueing inventory model into M/PH/1, Buzacott and Shanthikumar (1993) applied to the PH/PH/1 queueing inventory model instead of GI/G/1 model. Phase-type distribution is able to approximate a discrete queueing model as well as a continuous queueing model (Lee and Zipkin, 1992, 1995). For more details on the approximation by phase-type distribution, see Altiok (1985b), Asmussen *et al*. (1996), Thummler *et al*. (2006).

### 2.2 Vacation

A vacation, in queueing system, is the period when the server does not attend to a particularly targeted queue. The server may be under repair, attending to other queues, or simple forced to stop serving. Vacation models may be classified into two categories. In a gated system, once the server returns from a vacation, it places a gate behind the last waiting customer. It then begins to serve only the customers who are within the gate, based on some rules of how many or for how long it could serve. While, in an ungated system, the server only applies the rule of how many or for how long it could serve (Alfa, 2003). Under each of these categories, we may have further features, such as: single or multiple vacations; time-limited service- preemptive and non-preemptive; number limited service; random interruptions for vacation; and others.

Krishnamoorthy and Narayanan (2011) considered when either there is no customer in the system or no inventory or both, the server goes on a vacation. When he return in the system, if he finds one or both of these still not available then he goes for another vacation. They investigated the system stability. Under the condition of stability, they investigated the system state distribution and then several performance measures are computed. Narayanan *et al*. (2008) considered the server goes on a vacation of random duration at a service completion epoch if no customer is waiting or the inventory level is zero by matrix analytic method. They all considered the vacation time is phase-type distributions. Xu and Wang (2019) studied fluid model driven by the queue length process of a working vacation queue.

### 2.3 Customer Classification

Customer classification is made up of single class in many cases. However, if arriving customers have different characteristics, customer classification needs to be divided by their characteristics. Manuel *et al*. (2008) considered two types of customers, ordinary and negative, arrive according to a Markovian Arrival Process (MAP). An ordinary customer joins the queue and a negative customer removes one ordinary customer from the queue instead of joining the queue. They obtained the joint probability distribution of the number of customers in the waiting room, number of customers in the orbit and the inventory level in the steady-state case. Choi *et al*. (2015) developed a method to analyze the delay of order fulfillment using the counting process of D-BMAP with 2-class, order failure and inspection failure.

### 2.4 Customer Behavior

It is necessary to know the reaction of a customer upon entering the system. A customer may decide to wait no matter how long the queue becomes, or, on the other hand, if the queue is too long, the customer may decide not to enter the system. If a customer decides not to enter the queue upon arrival, the customer is said to have *balked*. A customer may enter the queue, but after a time lose patience and decide to leave. In this case, the customer is said to have *reneged*. In the event that there are two or more parallel waiting lines, customers may switch from one to another, that is, *jockey* for position. These three situations are all examples of queues with *impatient customers* (Gross *et al*., 2008). The impatient customers can be a *retrial* to take service (Falin, 1990). Krishnamoorthy *et al*. (2008) considered customers renege from the system at a constant rate in a PH/PH/1 inventory system with positive service time and shortage by matrix analytic method which obtaining the rate matrix R using the logarithmic reduction algorithm due to Latouche and Ramswami (1999). Narayanan *et al*. (2008) studied customer arrival process is subject to balking and customers become impatient and leave the system (renege) for waiting for service in MAP/PH/1 model with (*s*, *S*) inventory policy. Rejitha and Jose (2018) considered a MAP/PH/1 queueing inventory model which arriving customer who determinds the inventory level zero or the server busy enters into an orbit of infinite capacity and will retry for service time from there.

### 2.5 Service Discipline

When services are completed or keeping, we should decide customers who get service next time by any rules. It is called *service discipline*. There are many service disciplines such as FCFS (First Come First Served), LCFS (Last Come First Served), RSS (Random Selection for Service), PR (Priority), SJF (Shortest Job First), LJF (Longest Job First), SRPT (Shortest Remaining Processing Time). In most cases, we found inventory models with FCFS service discipline. Altiok (1987) studied FCFS service discipline inventory models with compound poisson arrivals and phase-type service distributions.

## 3. CATEGORIZATION BY FEATURES IN INVENTORY MODEL

### 3.1 Inventory Review Policy

There are two-type policies of reviewing the inventory by time, *continuous review* and *periodic review*. The continuous review policy is that the inventory position is continuously monitored and new orders are triggered by some events. On the other hand, the periodic review policy is that the inventory position is reviewed only once every *d _{i}* period. The length of

*d*is always integral multiple of the base period. Generally, continuous review policy applies to the continuous-time queueing inventory model (Baron

_{i}*et al*., 2010;Brown and Zipkin, 1991) and periodic review policy can be expressed by the discretetime queueing inventory model (Boute

*et al*., 2007;Lian

*et al*., 2005).

### 3.2 Replenishment Rule

Historically, the issue of replenishment rule involved answering only two fundamental questions: How much to reorder, and when to reorder (Duri *et al*., 2000). The simplest way to analyze the performance measures of replenishment rule for inventory model is to fix one of two factors: (re)order quantity and (re)order interval. The fixed order quantity model involves placing an order for a fixed amount of product each time reordering takes places. The exact amount of product to be ordered depends upon the cost and demand characteristics of the product itself and upon the relevant costs of carrying inventory and reordering. We generally need to develop a minimum stock level to determine when the fixed quantity is to be reordered. This is usually called the reorder point. When the number of items in inventory reaches the predetermined level, the fixed order quantity (also called the economic order quantity, EOQ) is automatically ordered. In a sense, the predetermined level for ordering acts as a trigger for setting the wheels in motion for the next order, and hence the reference made previously to the triggering activity.

The second form of the basic approach is the fixed order interval approach to replenish inventory. This is also referred to many times as the fixed period or fixed review period approach. In essence, this technique involves the ordering of inventory at fixed or regular intervals in time, and generally the ordered amount will vary depending on how much is in stock and available at the time of review. It is customary to make a count of inventory near the end of the interval, and to reorder based on what is on hand at the time.

After two replenishment rules were presented, there have been sustainable developments to achieve the economic and sturdy inventory management. The (*s*, *Q*) rule and (*s*, *S*) rule are two such inventory rules which are defined by two parameters. The first parameter is called the reorder point (or level) s. The second parameter is the quantity to be ordered (*Q*) for (*s*, *Q*) policy and order up to level *S* for (*s*, *S*) policy. In (*s*, *Q*) rule, each time the inventory falls below the reorder point, a new order of quantity *Q* is placed. Similarly, in (*s*, *S*) the order quantity is so as to make the total inventory level to *S*. Therefore, we calculate that *Q* denoted by order quantity equals *S* minus s in this policy. Base-stock rule, in which customer demands must be satisfied immediately or it will be considered lost (Chang and Lu, 2011;Duri *et al*., 2000;Ko *et al*., 2016;Song and Zipkin, 1992). In general, the basestock rule is an optimal inventory policy in systems where there is no fixed ordering cost and both holding and shortage costs are proportional to the volume of on-hand inventory or shortage (Sivakumar *et al*., 2012). However, if a fixed order cost is present, clearly an (*s*, *S*) policy is optimal. Ordering in every period would be uneconomical, and batching of orders would occur (Boute, 2007).

(*s*, *r*, *S*) rule is basically the same concept with (*s*, *S*) rule. However, (*s*, *r*, *S*) rule is considered a cancellation threshold *r* (Berman *et al*., 2008). It can derive explicit formulas for the cost minimization and as functions of the decision variables *s*, *r* and *S*.

(*R*, *r*) inventory model with phase-type service distributions is suggested by Altiok and Shiue (1995), Gurgur and Altiok (2004). It is once the inventory level drops to *r*, production or transportation continues until the inventory level reaches *R*. Boute *et al*. (2007), Van Houdt and Pérez (2012) used (*r*, *S*) policy, which the inventory position is raised after each order to some fixed position *S*, called the base-stock level and what at the regular time instants *r*. Brown and Zipkin (1991) are dealt with (*r*, *q*) inventory model: when the inventory position (stock on hand plus stock on order minus backorders) reaches the order point *r*, an order is placed for the fixed amount *q*, the batch size.

### 3.3 Stock-out Assumption

The stock-out generally may be modeled in one of three ways or all: backorder (backlog), lost sales and combinations of the two cases, called a partial backorder. When an entity within the supply chain is unable to cope with customer demand from on-hand inventory, the demand is either back ordered, meaning that the demand will be fulfilled in the future when on-hand inventory becomes available, or the demand is lost, meaning that the demand will not be fulfilled.

Inventory models with backorder assumption have received by far the greatest attention in inventory literature at the beginning of study (Bijvank and Vis, 2011;Chakravarthy, 2015;Lian *et al*., 2005;Song and Zipkin, 1992;Svoronos and Zipkin, 1991). However, the backorder assumption for excess demand is not realistic in many conditions. When the lost-sales system is approximated by a backorder model, the cost deviations can run up to 30% (Zipkin, 2008). Therefore, in general, inventory system has to be modeled with a lost-sales model instead of a backorder model to compute exact performance measures.

In a backorder model, the inventory position (inventory on hand plus inventory on order minus backorders) is used as main indicator of the inventory status. It increases when an order is placed and decreases when a demand occurs. Note that backorders are included in the definition for the inventory position. When the demand is lost instead of backordered, the inventory position does not decrease if the system is out of stock. It is no longer true that the amount of inventory after the lead time equals the inventory position after the order placement minus the demand during the lead time. While in a lost sales model, it is not possible to track the changes in the inventory position independently of the on-hand inventory level. Consequently, lost sales model has to keep track of the available inventory on hand and the quantities of the individual outstanding orders that were placed in the past and have not yet arrived. Figure 3

There were inventory models with lost sales and phase-type service distributions by Chang and Lu (2011), Sivakumar and Arivarignan (2007), Sivarkumar *et al*. (2012). While Lian *et al*. (2005), Chakravarthy and Daniel (2004), Xu (2010), Federgruen and Zipkin (1985) apply the backorder (backlog) to the inventory model with phase-type service distributions. However, very little seems to have been done on queueing inventory systems with partial backorder.

### 3.4 The Others

#### 3.4.1 Supply Disruption

Silver (1981) and Nahmias (1982) first pointed out the need of studying supplier unavailability, disruptions in the context of inventory models were usually considered to be arising due to strikes, shut down plants under preventive maintenance, or vendors committed fully to the contracts of other patrons. Balciogiu and Gureler (2011) studied an inventory problem with deterministic demand where supply can be disrupted at random times for random durations

#### 3.4.2 Information Sharing

The information sharing has substantially lowered the time and cost to process an order, leading to impressive improvements in supply chain performance such as reduction in lead times, inventory cost and inventory level (Cachon and Fisher, 1997;Chakravarthy, 2010;Clark and Hammond, 1997). He *et al*. (2002) analyzed numerically the value of information about the unfilled demands and the replenishment times in inventory system. They observed that queue length of information can significantly reduce the inventory cost and that higher level information is more valuable than lower level information.

#### 3.4.3 Shelf Time (Life Time)

In general, product is perished and system or equipment is deteriorated after a certain amount of time. We called shelf time or life time in inventory model. The shelf time of units is divided into infinite (items do not perish or deteriorate) or finite (items can become obsolete or perish) (Liu and Shi, 1991). Models for finite deteriorating inventory can be broadly categorized according to the lifetime of products and characteristics of demand. The three categories are distinguished based on shelf life characteristics. The first is fixed lifetime, i.e. it predetermined deterministic lifetime, e.g. two days or one season. The second is age dependent deterioration rate which implies a probability distributed life time, e.g. exponential (Chakravarthy and Daniel, 2004). The third is time or inventory (but not age) dependent deterioration rate. The perishable inventory models occur naturally in many practical situations. For example, analyzing inventory systems involving food items such as vegetables, fresh fruits, and meats; drug, photographic materials, and even electronic items such as memory chips, fall under perishable inventory models. The literature reviews on perishable inventory models with phase-type service distributions can be found in Lawrence *et al*. (2013). Tang *et al*. (2010) analyzed phase-type deteriorating reparable system which is regarded as one of the shelf time.

#### 3.4.4 Positive Service Time

Mostly, if the customers are available and server is ready to serve then the service starts. Nevertheless, this is not the case in inventory with positive service time. Because it needs positive amount of time to serve the inventory (Krishnamoorthy *et al*., 2011). Server may be ready to serve and there may be customers waiting to get serve. However, inventory may not be available on stock. Thus queue of customers builds up. Needless to say that in the case of positive lead time the server may remain idle even when customers are waiting for want of items in inventory. In classical inventory models, queue of customers is formed only when the system is out of order and unsatisfied customers are permitted to wait. However, in inventory with positive service time, queue is formed even when inventoried items are available. This is so because while a service is going on new customers are waiting. This is the case since the item to be served is needed in the inventory to provide the service (Krishnamoorthy and Narayanan, 2011). Inventory with positive service time is first investigated by Berman *et al*. (1993) where demands and service formed two distinct deterministic processes. In QIS, Krishnamoorthy and Narayanan (2011), Narayanan *et al*. (2008), Rejitha and Jose (2019) studied inventory model with positive service time, vacation to server and correlated lead time.

#### 3.4.5 Multi Items (Products) and Stages (Echelon, Location)

In many cases, the inventory problems in a real world treat with multi items and stage. Van Houdt and Pérez (2012) investigated inventory system that orders consist of multiple items and the production time of a single item is i.i.d (independent and identically distributed) according to a discrete-time phase-type service distributions. Svoronos and Zipkin (1991) proposed the system consists of several facilities or locations whose supplydemand relationships form a hierarchy. Altiok and Shiue (1995), Rejitha and Jose (2019), Song and Zipkin (1996), Wang (2009), Zhang and Archibald (2011) also studied multi items and/or stages. Phase-type distribution is easy to compute the performance measures for complex circumstances like this multi items or multi stages.

#### 3.4.6 Order Cancellation

Order cancellation can be the one of many ways to cope with rapid demand fluctuations before delivery in case the content has raised to a sufficiently high level during lead time due to its random fluctuations. Berman *et al*. (2008) suggested (*s*, *r*, *S*) inventory model with phase-type service time. They were interested in the cost minimization for an inventory, choosing a content level s at which to order, a replenishment level S and a cancellation threshold r.

#### 3.4.7 Postponed Demands

The demands that occurred during the stock-out period are either lost (lost sales) or satisfied only after the arrival of ordered items (back orders). In the case of back orders, the back ordered demand may have to wait even after the replenishment. This type of inventory problems are called inventory with *postponed demands* (Sivakumar and Arivarignan, 2007). The concept of postponed demands in inventory has been introduced by Berman *et al*. (1993) and postponed customers in queueing model has considered by Deepak *et al*. (2004). Sivakumar and Arivarignan (2007) considered a continuous review perishable inventory system with postponed demands by assuming that the items are perishable, demands occur according to a Markovian arrival process (MAP). Sivakumar *et al*. (2012) dealt with a discrete time inventory system with postponed demands.

## 4. CONCLUSION

In this paper, we have classified the literature addressing queueing inventory systems with the phasetype service distributions and discussed their contributions to inventory management. We classify the models in the literature based on the queueing model and model features such as replenishment policy, stock out assumption, shelf time (life time) and positive service time. Table 2 gives the summary of queueing inventory system with phase-type service distributions. In addition, we suggest future works which has little studied in the past times through literature review. The literature review framework in this paper provides a clear overview of the queueing inventory systems study field, which can be used as a starting point for further study.

As for further research, multi-server queues and the limited capacity can be applied to more queueing inventory systems with phase-type service distribution. In addition, various replenishment rule such as (*s*, *S*), (*s*, *Q*), (*s*, *r*, *S*) and vacation policy can be extended to those with queueing inventory systems in a various analysis way. We definitely can find new and creative research topics on queueing inventory systems with phase-type service distribution through Table 2.

Phase-type distributions have enjoyed such popularity because they constitute a very versatile class of distributions defined on the nonnegative real numbers that lead to models which are algorithmically tractable. Their formulation also allows the Markov structure of stochastic models to be retained when they replace the familiar exponential distribution.