Literature Review on the Queueing Models and Inventory

Posted: August 26th, 2021

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Literature Review on the Queueing Models and Inventory

Introduction                                                     

Inventory controls are some of the critical areas of the management of an organization. Typically, there are usually no defined ways or solutions since every firm or organization is different, being influenced by a variety of features and limiting factors. The problem is related to occurring mathematical models, which have been developed to help formulate strategies that facilitate the achievement of optimal inventory (Bhat 134; Melikov, Agassi, & Ponomarenko 117). The models are uniquely characterized, such that the optimal solutions are easily implemented to adapt to the rapidly changing situation. For instance, the conditions defining the models change daily to match with the dynamism in the organization’s inventory environment (Deep et al. 66; Sericola et al. 33). At the same time, new and useful models need to be developed in readiness for uncertainty. There are always uncertainties regardless of the control measures or objects since it is often difficult to obtain information about some measures or control objects. Such dynamic and complex situations can be quickly addressed using systems analysis and the development of systematic strategies to handle inventory management problems (Porteus 216; Myerson, Roger & Zambrano). The paper conducts a literature review on inventory and queuing models using a qualitative meta-synthesis method using peer review articles, online sources, journals, and books about the topicto assess how the authors view different models in the management of inventory systems. The first section looks at inventory models with impatient customers, the second review examines on the studies about queueing models with impatient customers and the final part looks at general inventory and queueing models before making concussions about the paper.

Keywords: Inventory model, queueing model, inventory control, impatient customer, substitution flexibility, Poisson distribution

Literature Review on Inventory and Queueing Models

Inventory Models with the Impatient Customer

An M/M/1 production system based on (s, S) policy with impatient customers is used to explain this type of inventory model. The model assumes that the arrival process for customers adopts a Poison distribution. Processing and production time are exponentially distributed. The assumptions used in the model are that each production time is one unit and that are no new demands in the queue when no items are available; hence, customers for new demands are considered lost (Sreelekshmi & Shajin 923, Azadeh, Naghavi lhoseiny & Salehi 552). More so, the time taken by an item to move into the queue fromthe production system is inconsequential. That is,it has no impact on overall performance. However, within the model, customers are described to be impatient when there is no item forthcoming in the production site (Melikov, Agassi, & Ponomarenko 86). Hence, the focus of the model is to maximize inventory levels, S, and the production switching levels s while minimizing costs within a steady-state.

            Practically, inventory problems are usually in two categories. The first category is that no circumstances will force customers to leave the system. In this case, the patience factor is infinite, even when the inventory level is at zero. The second problem is that the customer will leave once they become impatient once the level of inventory goes to zero (Sreelekshmi & Shajin 924; Bhat 121). Similarly, Benjaafar & Gayon (3) studied on the optimizing production under the inventory system with impatient customers. In their review, they described the optimal policy using the level of production base stock and admission threshold that limits the extent of accepting orders (Myerson et al. 37). In their assessment, the authors utilized the Markov decision process to help address the problems while comparing performance policies against other policies.More so, while investigating on retrial inventory system with impatient customers, Kathiresan and Krishnan (33)suggested a continuous review (s, Q) inventory model that comprises customers searching the orbit. In their assessment, customers were assumed to arrive according to Poisson distribution such that a customer leaves after receiving the service. Both the lead and service time wassupposed to be independently exponential. Customers that find that the server is busy on arrival are relocated into the orbit space (Beyer 67, Azadeh, Naghavi lhoseiny & Salehi 558). While in orbit, customers will be searched through the server after the current service is completed. The server uses the probability p>0 to explore the orbit for the next customer. Afterward, it remains idle with probability 1 – p once the search is done. The study assumes that the time take to search through the server is negligible. However, problems of reneging and balking are evident amongst impatient customers. In this case, reneging time is exponentially distributed (Beyer 93). Therefore, the joint probability for inventory level, the status of the server, and the number of demands inside the orbit are attained based on the existing steady-state conditions.

            Cases exist when the customers renege the queue, especially when their demands are not met after a stock out. Begain etal. (4) examine this problem using a stochastic (Q, r) inventory model with customer reneging. According to Park, unsatisfied demands are usually backordered to meet high customer demands (65). However, the move eventually causes some customers unsatisfied, therefore, opting out of the queue. The rate of leaving the queue is estimated as p, which represents the reneged customers that will seek satisfaction somewhere (76). In this case, the model assumes that customers exhibit different reactions during stock out such impatient customers will move to find stocks from outside the stations. Those who are patient will wait. According to the model, backorder costs are proportional to the time spent waiting for order replenishment. At the same time, there is a penalty allotted for each unit lost due to unfulfilled demands (88). Based on this background, therefore, Park developed a mathematical model that represents average annual costs incurred when operating an inventory system based on the heuristic treatment of the lot-size reorder-point policy to address the situation. Hence, through the model, it is possible to manage annual costs to optimize overall operations. 

Kathiresan, Anbazhagan & Jeganathan (2) conducted a study on an inventory system with retrial demand and working vacation. Their research focused on factors such as initial inter-arrival periods, service times, retrial times, and the working vacation periods. These factors were considered as random variables that were independent and exponentially distributed. The customer’s arrival was based on the Poisson process (17). The inventory replenishment was following the (s, Q) policy, while the lead times were assumed to follow an exponential distribution. The researchers displayed their results numerically in an attempt to derive the steady-state of the system by employing Matrix analytic approaches, among other performance measures, to illustrate the system behavior. The retrial system worked in such a way that demands that occurred during the periods when the server is busy, or stock out were allowed into the infinite size orbit (19). Equally, the server enters a working vacation once there are no demands, or there are zero levels of inventory. The working vacation is also exponentially distributed. At such periods, during working vacation or zero inventory levels, the orbiting customers become impatient following an exponential distribution.  As such, the joint probability distribution of inventory levels, server status, and demand quantities are obtained during a steady-state (22). Therefore, their system performance measures were derived at this point, including the long-run aggregation of expected cost rates.

            Further, Chen, Jian & Huang studies on compensation mechanisms in the inventory system with impatient customers. A two-backorder model is used where the customer behavior of choice is in a continuous review under which demand and production processes adopt a stochastic approach. In their model, during stock out, the firm employs particular compensation policies to control backorders (25). On their part, customers use decision-making strategies that ensure they maximize their utility through a reduction in item prices and waiting time, which is dependent on the impatience factor (24). The study assumes private information as the impatience factor, which is known to customers alone. In the model, two compensation strategies are designed for controlling backorders (27). These include priority auction and uniform compensation backed with admission charges. In a uniform compensation, the firm provides similar discount levels to all customers. The case is different for auction compensation. In this case, priority is given based on the bid prices that customers have made. Under each of these mechanisms, the researchers strived to obtain the base stock levels and the optimal stock out prices (31). Subsequently, the policies of the respective mechanisms are analyzed. Besides, the model assumes linearized waiting costs under which the impatience factor is uniformly distributed. The study finds that the auction strategy maintained a low base stock-level, giving greater, resulting in higher returns. It was also established that customers with higher or lower impatience factors benefited more than customers with moderate impatience factors (31). Thus, in their conclusion, they ascertain that the two compensation strategies were suitable for the products with high unit profitability, high cost of holding, and high penalty costs on lost sales.

Queueing Models with Impatient Customer

Wang, Kangzhou & Li (1) reviews the queueing model of impatient customers based on their different dimensions. The authors first introduce anxious behaviors such as reneging, balking, and related rules before providing numerical and analytical solutions as well as simulations to demonstrate the model. According to Wang, Kangzhou & Li (2), impatience among customers arises from the need to experience the service. Yet, they must queue, which makes such customers anxious and uncomfortable waiting for long (Begain etal. 148). The authors define impatient behaviors in queueing using two terminologies, that is, balking and reneging. Balking occurs when the customer decides ultimately not to join the line while reneging is about joining the line, but at some point, the customer leaves without being served. Equally, some customers may renege or balk out of the line but later rejoin to be served. The action is referred to as retrial (Wang, Kangzhou & Li 3). A queueing model is essential as it helps address problems faced by industries in the management of its customers. Hence, it has increasingly become a necessary aspect of research for improving customer management policies.

There are three main problems that the authors attempt to address, namely, balking reneging and retrial. A balking customer has two decided to make, that is, whether to join or not join the queue once the service providers are not idle once he is at the service station. In this case, there rules proposed by other researchers (Artalejo & Corral 103). The critical factor that influences the customer’s decision to join the queue is service time. Since service-waiting time is usually invincible, most customers make a decision based on the length of the queue when they arrived. Thus, if a customer comes and finds the queue is beyond the anticipated threshold, he will decide to go back, but he may join otherwise. 

            Subsequently, the authors raise reneging as part of problems that are directly related to balking. Unlike the balking case, reneging occurs once the customer becomes impatient and withdraws from the queue after some time. Also, some customers in the queue may decide to give up space if they realize that the service time exceeds their waiting time. Decisions to get out of the queue or remain until served are defined variedly in literature. According to Wang, Kangzhou & Li (4), the main reneging rule is the following, which is denoted as Type III rule. Other authors explain that the maximum waiting time, T for customers, is exponentially distributed based on the number of customers in the queue. Besides, slow service rate and breakdown of services have the potential to result in impatience amongst customers.  At the same time, there are cases when customers completely abandon the queue, especially when an efficient service arrives. Lastly, the authors also examine retrial as part of the impatient customers in a queue (Artalejo & Corral 123; Duc et al. 83; Wuyi et al. 213). Often, customers who balk or renege out of a queue for different reasons have the potential to rejoin the queue and repeat their particular requests at random times.Therefore,the chance for a balking or reneging customers rejoins the queue if dependent on the customers that are currently on a service queue. Cases of repeated queueing are familiar with communication, computer, and telephone service systems due to their nature of the operation (Artalejo & Corral 133; Baccelli & Pierre Brémaud 88). The authors offer solutions to the queueing issues arising due to customer impatience. These solutions are surveyed based on analytic formulae and formulations against numerical solutions and analytic formulations.Moreover, Baccelli & Pierre Brémaud (88) identifies various factors that influence customers’ impatience. These include queue length, waiting time, and busy periods. Thus, depending on the existing elements, customers are likely to become impatient choose to either balking or reneging the queue.

While expounding on analytical solutions and numerical methods of assessing customer impatience, Wang, Kangzhou & Li (77) acknowledges the need to invoke the use of approximation techniques to solve the queueing problems of impatient customers. This is especially when it is not possible to adopt analytical solutions that tend to rely on actual values and models to prove the existence of queuing problems. The method utilizes the increasing numerical method’s scope following the growing computer power to address the real-life situations involving queueing. According to Wang, Kangzhou & Li (84), the complex scenarios arising from queueing problems and impatient customers are difficult to handle. Thus, it is necessary to exhaust various techniques in addressing such complexities.

Furthermore, Kim, Klimenk & Dudin studied the tandem queue that consists of two multi-server stations under which retrials and impatient customers are given consideration. In the model, customers arrive at each station following the Markovian Arrival Process (17). Although the model employs an intermediate buffer, the first station lacks a barrier. In this case, the customers that fail to access the services immediately once they arrive at the station are allowed to retry accessing the service a given random period. Once the service is completed in the first station, the customer permanently leaves the system; otherwise, it moves to get another service from the second station (18). Some customers that proceed into the second station have priority over others. The priority is availed through reservation for a server section, particularly for priority customers alone. Notably, the model does not allow non-priority customers to occupy the privateparts of the server (22). In an unavoidable situation, non-priority customers would be selected to access the service if the intermediate buffer queues for non-priority customers become overwhelmed beyond the pre-assigned limit. Yet, there are free servers (27). Those customers staying in the buffer servers are impatient ones. Equally, both non-priority and priority customers have particular patient time, and it varies significantly. Hence, they could quit the queue or get back to the first station after the expiry of patience time.

The tandem queueing model adopts a steady state under which it is analyzed. Kim, Klimenk & Dudin (34) derived an existence condition for the stationery regime, including the distribution of constant state and the calculation of various measures of performance. Thus, in their assessment, they concluded that the tandem queueing system is suitable, especially in call modeling using Interactive Voice Response Machines (IVRM).

Garnet & Mandelbaum (77) studied on the appropriate model to support customers at the call centers. In their assessment, they suggested an M/M/N/B model as the most common models. The individual components in the model, notably M/M/N is responsible for modeling busy signals, and the M/M/N/N is accountable for disallowing waiting. However, all these models lack the common feature,that is, a consideration that impatient customers could opt-out of the queue; abandon the system before their service time. As such, the study focused on analyzing the simple abandonment model. In this model, the patience of the customers follows an exponential distribution, and the system has an unlimited waiting capacity, that is, M/M/N+M. Assertions by Garnet & Mandelbaum, in this case, is that the model of this type analyzable and sufficiently rich to offer detailed information that is critical for managers of call-centers. The study first outlined the method for performing an accurate analysis of the M/M/N+M model. Although this model is numerically attractive, it is less insightful. Following this, the study proceeded to outline the specific analysis for M/M/N+M model basing the report within a regime that is suitable for call centers with many agents, high service levels, and efficiency. Thus, based on this model, the study concludes that M/M/N+M model is suitable for large call centers. The model can enhance management of call queueing besides limiting cases of retrials.

Queueing and Inventory Models

Studies about queueing models with inventory control have been vibrant amongst researchers for the past decades. The model features the systematic arrival of customers at the facility to be served. For one to complete serving a customer, an item should come from the inventory. Once the customer is served, he immediately leaves the service point, an action that immediately lowers the inventory size (Altman, Eitan & Yechiali 77). To sustain the level of inventory, there is the sourcing of items, which are supplied from outside the facility. This kind of model is called a queueing-inventory model. The attached inventories on this model directly influence the service, which makes the model different from traditional inventory management models (Neelamegam 189). Another uniqueness is that the consumption of inventories is based on serving rates rather than customer arrival rates when they queue for the service.

            Queueing model with inventory problems are utilized differently depending on the existing problems and environment. As such, they appear in different categories such as single server queuing with inventory, queueing inventory system with stochastic environment, and queueing inventory system with substitution flexibility, among others. Altman, Eitan & Yechiali (49), proposed the single server queueing with an inventory. According to the author, the model relies on the stationary distribution of a joint queue length and an inventory process. In this model, the inventory is continuously reviewed, and new inventory management policies established (Wuyi et al. 77; Choi 42). The demand for this model is based on Poisson distribution, lead, and service time’s exponential distribution. It generally adopts an M/M/1 – system. While reviewing the same model, Altman, Eitan & Yechiali(55) et al. used an M/M/1 queueing-inventory model under (r, Q) policy with lost sales. The demand adopts a Poisson distribution, lead, and service times, which are exponentially distributed. According to the model, once the stock is out, all customers are lost.

Minner (275) conducted a review of reverse logistics inventory models. In the study, he asserts that the increasing consciousness about the customers and respective product environment provides an opportunity to attain customer satisfaction from recovered used products. In this case, it implies that instead of seeking to manufacture new products to meet customer demands, manufacturers need to recycle returned products as a way to managing problems of stock out and taking care of the environment. In this case, there are particular inventory models for product recycling (275). These models share most features, and the common one is the two-supplier inventory model. The two-supplier model consists of second supply mode besides the alternative production. Such a method can help reduce production costs, especially when the total disposal and production unit expenses exceed the costs of remanufacturing. The second measure is about recovery lead-time and manufacturing. That is, it may be less costly to remanufacture than manufacturing. For example, in the case of spare parts, it is possible to remanufacture using waste materials from returns. The review appreciates that it could be more complex to embrace recovery than new manufacturing compared to the situation presented by the inventory models (275).  Some factors that could contribute to such complexity include lack of control over the product returns, and most returns are generally a random variable. Further, most reverse inventory models are characterized by the assumption of stationary and constant demand. However, cycle demand patterns, the lifecycles of the product, and return dependencies, as well as past sales, serve to motivate the choice of a dynamic model (276). Therefore, although it could be efficient to reverse production, there are notable factors that can influence the overall implementation of the strategy.

            Artalejo & Falin (110) performed a comparative review on both standard and retrial queueing models. In their case, they established that there are models that employ generalized retrial guidelines when being implemented. They used the telephony applications and analyzed the situation of a call that is receiving a busy signal (110). Their findings show that whenever a busy signal is initiated, it does not wait for the termination of the existing working conditions. Under these circumstances, the blocked calls are involved in the current generation of independently repeated requests from the source of the calls currently orbiting(Azadeh, Naghavi lhoseiny & Salehi 549). In this case, traditional retrial principles assume a probability of repeated trials during a given time interval with a specific number of calls in the orbiting. However, some applications in the communication networks and computer uses an electronic device to control the time interval between two-successive repeated attempts (110). Hence, the probability at this period is ascertained under the conditions that the orbit is empty. Thus, according to Artalejo & Falin, the two models are treated as a unified approach for defining linear trial principles. 

            Additionally, most of the queuing models with retrial are influenced by telecommunication and computer applications in which repeated attempts appear because of constant system blocking with inadequate service capacity. Besides, retrials can emerge due to other factors. According to the study, Fayolle & Brun (27) retrials also occurs in a single server system consisting of repeated attempts arising from queueing customers with impatience. Another possibility is in the case of mixed models with orbit and waiting for lines (25). In these models, therefore, customers who find long queues may opt to engage in a secondary activity to return in the queue later with the hope that it will have shortened.

Besides, Allen (63) proposed the queueing inventory model with a stochastic environment. The model combines queueing theory and considers lead and demand time as stochastic parameters. While contributing to the same model, Allen (25) suggested the demand process behave according to Poisson distribution and production times to change exponentially for a single item that makes it into the stock system. The model adopts an M/M/1/S queueing system. Equally, the inventory control, in this case, adopts a multi-supplier strategy through two-levels of the supply chain (Tulsian 4). Customers are considered to arrive randomly per two arrivals. Subsequently, Seyedhosenist etal. (76), while examining the inventory and queueing models, attempted to apply the substitution flexibility into the queueing theory and inventory systems. Substitution flexibility helps improve the profitability of the system for multi-product inventory models (Sericola 73). Therefore, many cases of queueing and inventory models exist. Equally, there are many varieties of real-time problems that are solved using suitable methods. As a result, queueing and inventory systems are increasingly being developed to help solve problems in airlines, railways and road transport, among other systems.

Azadeh, Naghavi lhoseiny & Salehi(535-562) deliberated on tandem G/G/K queues operating under disaster situations. In their study, they assert that during the disaster period, the chance that there will be a failure in the system is determined by the quality and capacity of existing equipment. The equipment will make the servers idle, thereby resulting in all customers being removed from the queues and areas of service. As such, queueing customers are forced to orbit within retrial queues once the system fails. In this circumstance, they may choose to go or return to the queue (539). Hence, it is costly to reduce the probability of failure in the system. For instance, this can be done by increasing the number of servers and reducing the number of customers that are lost. Besides, customers must be served without interruption and reduce the system time. As a result, the study compared all these scenarios in five ways, namely; the expenses incurred when customers are lost, costs of the operator, system time, the prices of reducing the rate of disaster arrival, and the number of uninterrupted service customers, among others (546). The computer simulation was used to model the scenarios. Afterward, an optima scenario was selected by using the data envelopment (542). The study concluded that the optimal situationis suitable for maximizing the efficiency of the system. This is mainly on the number of customers that are satisfied, the cost incurred when customers are lost, and the arrival rates of the disasters (558). The main problem is modeled in such a way that the arrival of the catastrophe takes a Poisson distribution. Thus, the study is among the first ones to optimize the G/G/K tandem queueing model under the case of system failure and retrials in Interactive Voice Response Models.

Conclusion

The paper has examined the literature review on various queueing and inventory models as established by different scholars on the topic. As assessed by the study, there are diverse inventory and queueing models. However, they all have various features that distinguish each other. The main distinguishing factors are key assumptions that are developed about main variables such as demand, physical characteristics, and the cost structure of the system. These assumptions are specific to a particular environment and condition due to the effects of variability and uncertainty. At the same time, the literature reveals that the demand to address growing challenges in inventory and customer queueing management has agitated increased studies on queueing and inventory systems. Equally, the literature review identified different inventory models such as the M/M/1 production system based on (s, S) policy with impatient customers, inventory system with retrial demand, and working vacation. Also, it examined compensation mechanisms in the inventory system with impatient customers that tested a two-backorder compensation model. Across these models, it was established that they all appreciate the concept of retrials, primary inter-arrival periods, and service times as well the working vacation periods. Subsequently, the queueing models reviewed based on various studies included the optimal tandem G/G/K queues operating under disaster situations, among others. The main concepts addressed in the queueing models include balking, reneging, and retrial while focusing on impatient customers. Thus, the study reveals that there is a wide range of surveys that appreciate and attempt to address the challenges in inventory and queue management.

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