1. INTRODUCTION:

Research on queueing systems 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 require service. In order to complete the 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 inventory is supplied by an outside supplier. This system is called a queueing-inventory system (Schwarz et al., 2006). The queueing-inventory system is different from the traditional queueing system because the attached inventory influences the service. If there is no inventory on hand, the service will be interrupted. Also, it is different from the traditional inventory management because the inventory is consumed at the serving rate rather than the customers’ arrival rate when there are customers queued up for service.

2. QUEUEING SYSTEMS WITH INVENTORY:

(i) Single Server Queueing Systems with Inventory

Maike Schwarz et. al. [49] proposed the model M/M/1 Queueing systems with inventory. They derived stationary distributions of joint queue length and inventory processes in explicit product form for various M/M/1-systems with inventory under continuous review and different inventory management policies, and with lost sales. Here demand follows Poisson distribution, service times and lead times exponential distribution. Mohammad Saffari [55] et. al. considered an M/M/1 queueing system with inventory under the (r, Q) policy and with lost sales, in which demands occur according to a Poisson process and service times are exponentially distributed. All arriving customers during stock out are lost. They derived the stationary distributions of the joint queue length and on hand inventory when lead times are random variables and can take various distributions.ToktasPalut and U ¨lengin [93] coordinated the inventory policies in a two-stage decentralized supply chain, where each supplier has been considered as an M/M/1 queue and the manufacture has been assumed as a GI/M/1 queue. Alimardani et al.[1] applied continuous review (S1,S) policy for inventory control and supposed a bi-product three-echelon supply chain which is modelled as an (M/M/ 1) queue model for each type of products offered through the developed network. In addition, to show the performance of the proposed bi-product supply chain, they also considered a network including two (M/M/1) queue for each type of products.

Saffari and Haji[68] considered a two-echelon supply chain in which the supplier is modelled by an M/M/1 queue with inventory, exponential lead time distribution, and lost sale. They computed performance measures of the system to find optimal supplier’s order size.Schwarz et.al.[74] derived stationary distributions of joint queue length and inventory processes in explicit product form for M/M/1 queuing-inventory system with lost sales under various inventory management policies such as (r,Q) policy and (r,S) policy. The M/M/1 queueing-inventory system with backordering was investigated by Schwarz and Daduna[73]. The authors derived the system steady state behaviour under reorder policy which is (0, Q) policy with an additional threshold 1 for the queue length as a decision variable.

(ii) Queueing Inventory System with Stochastic Environment

Parlar [63] presented an inventory model combined with queueing theory and considered demand and lead time as stochastic parameters. Ha[25] proposed Poisson distribution for demands and exponential for production times in a single item make-to stock production system and M/M/1/S queuing system for modelling. Arda and Hennet[4] addressed inventory control of a multi-supplier strategy in a two-level supply chain. They considered random arrivals for customers and random delivery time for suppliers and represented the system as a queuing network. Yu and Dong[99] considered a two-stage production lot sizing problem which used an inventory system with random demand arrival and proposed a numerical approach to solve the problem. Baek and Moon[8] considered a lost sales production-inventory system in an uncertain environment. They used queuing theory to present a stochastic model for the system. Karimi-Nasab and Seyedhoseini[37] assumed that the lead time is negligible or it can be ignored in practice when it is short in contrast to other time factors for some production-inventory paper. Most of such research papers did not consider lost sales. Many of them, however, considered back orders as a rational managerial policy. Chang and Lu[17] considered a serial production system controlled by the base-stock policy and presented a phase-type approximation for a controlled base-stock serial production system and proposed a cost model to determine the optimal base-stock level.Babai et al.[7] considered demand and lead time stochastic and analysed a single-echelon single-item inventory system by means of queuing theory. Ravichandran[65] developed an inventory system with Poisson demand and Erlangian life time where lead time is assumed to be positive. Arivarignan[5] considered a perishable (−r, S) inventory system with renewal demand and exponentially distributed life time of each item.Pal and Ghosh[62] considered a perishable (−s, S) inventory system with Poisson demand, exponentially distributed life time of each item and instantaneous supply. William Karush[95] discussed an explicit mathematical solution which is obtained by methods of general interest for a probabilistic model that arose in connection with consulting work for an industrial client. Here the customer demand for a given commodity is a Poisson process with mean rate λ, and replenishment time for restocking is random. Stephen C.Graves [90] developed two distinct models for studying inventory systems with continuous production and perishable items. The perishable items have a deterministic usable life after which they must be outdated. For each of the models, analytical expressions derived from queueing theory, are found for the steady-state distribution of system inventory. Knowledge of this steady-state behaviour may be used for evaluation of system performance, and for consideration of alternatives for improving system performance. The analysis for both models exploits the similarity of the inventory system with a single-server queueing system.Liming Liu, Xiaoming Liu, David Yago[47] first developed a multistage inventory-queue model and a job-queue decomposition approach that evaluates the performance of serial manufacturing and supply systems with inventory control at every stage. Then they presented an efficient procedure to minimize the overall inventory in the system while meeting the required service level.

(iii).Queueing Inventory System with Substitution Flexibility

S.M. Seyedhoseiniet. al.[76] introduced an application of queueing theory in inventory systems with substitution ﬂexibility which can improve proﬁts in many multi-product inventory systems. They prepared a comprehensive substitution inventory model, where an inventory system with two substitute products with ignorable lead time has been considered, and effects of simultaneous odering have been examined and demands of customers for both of the products have been regarded as stochastic parameters, and queuing theory has been used to construct a mathematical model. Bayindiret. al. [9] considered a one-way substitution system with two products which uses (S-1, S) policy. They use a two-dimensional Markov process to develop the model, where the objective of their research was to ﬁnd the optimal order up to levels. Liu and Lee[48] proposed three different policies to use one-way substitution, and developed an inventory system with backlogs. Knessl.C[39] et.al. introduced a continuous-level production rate with a base-stock level inventory policy subject to fluctuating demand. The inventory level at time t is denoted by I(t) and is the time required to produce the items currently in stock given a production rate of unity. Negative inventory levels I(t) < 0, reflect orders that cannot be satisfied by current inventory and are backlogged assumed that the inventory is produced at a rate which depends on the current inventory level and may differ from unity. Thus the production rate is controlled by the producer, which introduces a feed- back mechanism for inventory control. They define the base-stock level M as the level of inventory at which production stops. Production only restarts when I(t) < M. Orders or demands arrive randomly with an inter-arrival time that is exponentially distributed with a parameter that depends on the current inventory level. The size of the orders is exponentially distributed. The dependence of the arrival rate on the current inventory level can have several interpretations. The model is appropriate if production is highly automated and easily controlled.

Ning Zhos [59] et.al. introduced a queueing inventory system with two classes of customers. Customers arrive at a service facility according to Poisson process. Service time follows exponential distribution. Each service uses one item in the attached inventory supplied by an outside supplied by an outside supplier with exponentially distributed lead time. They find a priority service rule to minimize the long run expected waiting cost of dynamic programming method and obtain the necessary and sufficient condition for the priority queueing inventory system being stable. Using Bright-Tailor algorithm, they computed the steady state probability distribution by formulating the model as a level-dependent-quasi-birth-death-process.. Berman and Sapna[13] studied queueing-inventory systems with Poisson arrivals, arbitrary distribution service times and zero lead times. The optimal value of the maximum allowable inventory which minimizes the long-run expected cost rate has been obtained. Berman and Sapna[14] discussed a finite capacity system with Poisson arrivals, exponential distributed lead times and service times. The existence of a stationary optimal service policy has been proved. Berman and Kim[12] addressed an infinite capacity queuing-inventory system with Poisson arrivals, exponential service times and exponential lead times. The authors identified a replenishment policy which maximized the system profit. Berman and Kim[11] studied internet based supply chains with Poisson arrivals, exponential service times and the Erlang lead times and found that the optimal ordering policy has a monotonic threshold structure. Krishnamoorthy et.al.[44] discussed an (s,S) inventory system with service time where the server keeps processing the items even in the absence of customers. Krishnamoorthy et. al. [45] introduced an additional control policy (N-policy) into (s,S) inventory system with positive service time. In Manuel, et.al.[50]the perishable queueing-inventory systems with Markovian arrival process (MAP) were studied. The joint probability distributions of the number of customers in the system and the inventory level were obtained for the steady state case. The stationary system performance measures and the total expected cost rate were both calculated.

(iv).Queueing System with Production-Inventory

Some related works in the production industry are He and Jewkes[27] and He et. al.[28]. He et. al.[30] studied the inventory replenishment policy of an M/M/1 make-to-order inventory-production system with zero lead times. They explored the structure of the optimal replenishment policy which minimizes the average total cost per product. For the M/PH/1 make-to-order inventory-production system with Erlang distributed lead times, Heet. al.[29] quantified the value of information used in inventory control. A logically related model has been studied by He et al.[31] who analyzed a Markovian inventory production system, where customer demand occurs at a workshop comprising a single machine in a batch of size one.Jung Woo Baekand Seung Ki Moon[36] proposed the new model which is “The M/M/1 queue with a production-inventory system and lost sales”. The authors considered an extension of the queueing system with inventory in which the stocks are delivered both by an outside supplier and an internal production and called the proposed queueing system as an M/M/1 queue with an attached production-inventory system. Customers arrive in the system according to a Poisson process, and a single server serves the customers. The customers leave the system with exactly one item from the inventory at his service completion epoch. If there is no inventory item, all arriving customers are lost. The stocks are replenished by (1) an external order under (r, Q)-policy, or (2) an internal production. The internal production process is assumed to be a Poisson process. They derived the stationary joint distribution of the queue length and the on-hand inventory in product form. Using the joint distribution, they introduced long-run performance measures and a cost model. Then, they established numerical examples, which minimize the long-run cost per unit time. Reza Rashid et.al.[66] proposed an application of queuing theory in production-inventory optimization. They developed a mathematical model for an inventory control system in which customer’s demands and supplier’s service times are considered as stochastic parameters. They solved the problem through queuing theory for a single item. Then the model is extended to the case of multi-item inventory systems and developed a new heuristic algorithm to deal with the complexity of this problem. He et.al. discussed some related ideas in the production industry. Sajeev[71] discussed Production inventory with service time and protection to a few of the ﬁnal phases of production and service. Hennet and Arda[32] proposed inventory control of a multi-supplier strategy in a two-level supply chain. They considered random arrivals for customers and random delivery time for suppliers and represented the system as a queuing network. The objective of inventory management is to balance conﬂicting goals such as stock costs and shortage costs.

(v).Queueing System with Service Inventory

In inventory management point of view, the assembly-like queue can be applied to a service-inventory system in which the customers can be served only when the level of the attached inventory is positive. The simplest example is a retail market where customers spend time to pay (the service time) for the item (the inventory) that they want to purchase. M. Geetha Rani and C. Elango [24] developed a supply network model for a service facility system with perishable inventory by considering a two dimensional stochastic process of the form (L, X) = {(L(t), X(t)), t ≥ 0 )} , where L (t) is the level of the on hand inventory and X (t) is the number of customers at time t. The inter-arrival time to the service station is assumed to be exponentially distributed with mean 1/λ and the service time for each customer is exponentially distributed with mean 1/ µ. The maximum inventory level is S and the maximum capacity of the waiting space is N. The replenishment process is assumed to be (S1, S) with a replenishment of only one unit at any level of the inventory. Lead time is exponentially distributed with parameter β. The items are replenished at a rate of β whose mean replenishment time is 1/β. Item in inventory is perishable when it’s utility drops to zero or the inventory item become worthless while in storage. Perishable of any item occurs at a rate of γ. Once entered a queue, the customer may choose to leave the queue at a rate of α if they have not been served after a certain time. They derived the steady state probability distributions for the system states.

Berman and Kim[10] analysed the situation in a stochastic environment where customers arrive at service facilities in a Poisson process and service times are exponentially distributed with mean inter-arrival times greater than the mean service time and each service require one item from inventory. A logically related model was studied by, He, Q-M, E.M. Jewes and Buzzacott, who analysed a complete Markov production – inventory system, where demands arrive at a workshop and are processed one by one in order. Mohebbi and Posner[56] considered a continuous-review inventory system with compound Poisson demand, Erlang as well as hyper-exponentially distributed lead time and lost sales. They derived the stationary distribution of inventory level for the purpose of formulating long-run average cost functions with/without a service level constraint. Mohebbi and Hao [57] considered inventory system with compound Poisson demand, Erlang-distributed lead times, random supply interruptions and derived the stationary distribution of the inventory level under an (r, Q)-type control policy. Sigman. K [77] et. al. given a contribution where alight trafﬁc heuristic was derived for an M/G/1 queue with an attached inventory. A few analytical models in this ﬁeld have been developed up to now. Bermanand Kim[12] considered a service system with an attached inventory, with Poisson customer arrival process, exponential service times and Erlang distribution of replenishment. Their formulation was a Markov decision problem to characterize an optimal inventory policy as a monotonic threshold structure which minimizes system costs. Berman and Kim presented an extension where revenue is generated upon the service. They found an optimal policy which maximizes the proﬁt.

Schwarz and Daduna [73] considered M/M/1/∞ queue with inventory under continuous-review with backordering when lead times are exponentially distributed. They computed performance measures and derived optimality conditions under different order policies. For evaluating performance measures and steady state probabilities they presented an approximation scheme. Krishnamoorthy et. al. [46] introduced a queueing-inventory system, with the item given with probability γ to a customer at his service completion epoch, is considered in the article. Two control policies, (s,Q) and (s,S) are discussed. In both cases they obtain the joint distribution of the number of customers and the number of items in the inventory as the product of their marginal under the assumption that customers do not join when inventory level is zero. Optimization problems associated with both models are investigated and the optimal pairs (s,S) and (s,Q) and the corresponding expected minimum costs are obtained. Saffariet. al.[72] consider an M/M/1 queue with inventoried items for service. The control policy followed is (s,Q) and lead time is mixed exponential distribution. When inventory is out of stock, fresh arrivals are lost to the system. This leads to a product form solution for the system state probability. Schwarz et al.[75] consider queueing networks with attached inventory. At each service station an order for replenishment is made when the inventory level at that station drops to its reorder level. They consider rerouting of customers served out from a particular station, when the immediately following station has zero inventory. Thus no customer is lost to the system. The authors derive joint stationary distribution of queue length and inventory level in explicit product form. Using dynamic programing Zhao and Lian[100] obtain necessary and sufﬁcient condition for a priority queueing inventory system to be stable. In a very recent paper of signiﬁcance Saffariet. al. [69] analyse an inventory model with positive service time and arbitrarily distributed lead time. They assumed that no customer joins the system when the inventory level is zero. A product form solution for system state is obtained here as well. Another very recent contribution of interest to inventory with positive service time in a random environment is by Ruslan and Dadun[67]. They prove a necessary and sufﬁcient condition for a product form steady state distribution of the joint queueing-environment process and contributed inventory with positive service time in a random environment. They prove a necessary and sufﬁcient condition for a product form steady state distribution of the joint queueing-environment process.Anoop N. Nair[3] et. al. considered a multi-server Markovian queuing model where each server provides service only to one customer. Arrival of customers is according to a Poisson process and whenever a customer leaves the system after getting service, that server is also removed from the system. Here the servers are considered as an inventory that will be replenished according to the standard (s,S) policy. Behaviour of this system is studied using a two dimensional QBD process. The condition for checking ergodicity, the steady state solutions and average inventory cycle time are obtained using matrix analytic methods. Also they have studied an optimization problem that minimizes the total cost induced by the waiting cost of arrivals, holding cost of the inventory of servers and ordering cost. Berman et al.[16] considered an inventory system at a service facility with a deterministic service time and Poisson demands. Krishnamoorthy and Narayanan[41]analysed a production inventory with positive service time wherein no customer joins when inventory level is zero and obtain a stochastic decomposition of the system state thereby subsuming the results given in Schwarz et.al.[73]. Saffari.et.al.[54] considered an arbitrary distribution for the lead time with customers arriving during stock out period lost to the system. They derived the system state distribution as the product of the marginal distributions of the components. Following these pioneering works, there had been numerous other studies on inventory with positive service time. For further details we refer to a recent survey article by Krishnamoorthy et al.[40]. It can be observed that all models discussed above considered queues where the single server services an inventory according to different inventory policies. Multi server Markovian queues (M/M/C) with attached inventory have also been studied recently. Padmavathi·B.et.al.[60] considers a continuous review stochastic (s,S) inventory system with Poisson demand and exponentially distributed delivery time. The demands that occur during the stock out period or during the server vacation period enter into an orbit of inﬁnite size. These orbiting demands retry to get satisﬁed by sending out signals so that the time durations are exponentially distributed. They consider two models which differ in the way that server go for vacation. The joint probability distribution of the inventory level, the number of demands in the orbit and the server status is obtained in the steady state case. Various system performance measures in the steady state is derived and the long-run total expected cost rate is calculated. Several numerical examples are presented, which provide insight in to the behaviour of the cost function.Hsin Rau et. al. [34] introduced the classical inventory replenishment problem with a linear function in demand uses a single-segment linear function as its demand and modelled by a simple algorithm. They extended it to provide a heuristic solution for the inventory replenishment model with a two-segment linear function in demand called the two-segment piecewise linear demand model. Also the authors proposed a general procedure for solving both models.

(vi).Continuous Review Inventory Systems with Server Vacation

Daniel and Ramanarayanan [19] introduced the concept of server vacation in the inventory system with two servers. In Daniel and Ramanarayanan[20], they had studied an (s,S) inventory system in which the server took a rest when the level of the inventory became zero. They assumed that the demands that occurred during stock-out period or the server rest periods were lost. The inter-occurrence times between successive demands, the lead times, and the rest times were assumed to follow general distributions. Using renewal and convolution techniques, they obtained expressions involving the steady state transition probabilities. Narayanan et. al.[58] considered an inventory system with random positive service time. Customers arrived at the service station according to a Markovian arrival process and the service time for each customer had phase-type distribution. They assumed correlated lead time for the orders and an inﬁnite waiting hall for the customers. The customers who wait for service may renege after a random time. The server took multiple vacations whenever there was no customer waiting in the system or the inventory level was zero. Under the above assumptions, they analysed the level dependent quasi birth-death process. Sivakumar[81] introduced the concept of server vacation in retrial inventory system. He assumed exponential distribution for inter-demand times, lead times, inter-retrial times and server vacation times. He also assumed that all these events are mutually independent. He adopted a multiple vacation policy. The concept of retrial demands in inventory was introduced by Artalejo et.al.[6] by assuming Poisson demands, exponentially distributed lead time and exponentially distributed inter-retrial time.

(vii).Queueing Inventory System with Postponed Demands/ Customers.

Padmavathi. I et. al .[61] study a finite source (s,S) inventory system with postponed demands and server vacation. Their modified M vacation policy is defined as: Whenever the inventory level reaches zero, the server goes to inactive period which comprises the inactive-idle and vacation period. If replenishment occurs during the inactive-idle period, the server becomes active immediately, or otherwise he goes for a vacation period. The server can take at most M inactive periods repeatedly until replenishment takes place. This inactive-idle time, the vacation time and lead time follow independent PH distributions. After the M^th inactive period, the server remains dormant in the system irrespective of the replenishment of order. Demands that occur during stock out or inactive periods, enter the pool and these demands are selected if the inventory level is above. The inter-selection time follows exponential distribution. The joint distribution of the mode of the server, server status, the inventory level and the number of demands in the pool is obtained in the steady state. They have derived several system performance measures and total expected cost function. Berman et al. introduced the concept of postponed demands/customers in inventory system and then by Deepak et al. introduced the same concept in queueing theory. Krishnamoorthy and Islam [42] considered (s,S) inventory system with postponed demands in which arrival follows a Poisson process, exponential lead time. They also assumed that the arriving demands who find the inventory level zero join the pool of finite size and these pooled demands are selected one by one according to an exponential distributed time lag as long as the inventory level is greater than the reorder point. Sivakumar and Arivarignan[78] studied a perishable inventory system with postponed demands in which the demands that occur during stock out period enter the pool with independent Bernoulli trial. They assumed that arrival process is Markovian arrival process, the lead time is distributed as phase type and that the life time and the inter-selection time are assumed to have independent exponential distribution. Sivakumar et.al.[79]considered an inventory with postponed demands in discrete time in which arrival follows a discrete Markovian arrival process with independent phase type distribution for lead time and selection time. Sivakumar and Arivarignan [80] select the demand from the pool of infinite size until the inventory level drops to prefixed level N(1 ≤ N ≤ s). They assumed that primary and negative demand arrives according to independent Markovian arrival process and the life time, the lead time and the inter-selection time are independently distributed as exponential. Jayaraman et. al. [35] dealt with multiple vacation to the server in an inventory system with retrial demands and postponed demands respectively.

(viii).Queueing Systems with New Inventory Models

Narayanan et. al.[58] studied a Markovian inventory system with positive service time and Krishnamoorthy et. al. [41] considered a production inventory with service time. In both model, server took multiple vacation due to lack of inventory or the lack of customer or both. Queueing systems with server vacation have been widely studied in the literature. Tian and Zhang[92] have studied various vacation policies such as single and multiple vacation policy for single server and also synchronous and asynchronous with single and multiple vacation for multi-server queueing system. Ke[38] have studied M[X]/G/1 queue with J vacation policy in which after the exhaustive service, the server takes at most J vacation of constant length repeatedly until at least one customer present in the system. Amirthakodi.M.et.al.[2] considered a continuous review perishable inventory system with service facility consisting of ﬁnite waiting hall and a single server. The primary customers arrive according to a Markovian arrival process. An arriving customer, who ﬁnds the waiting hall is full, is considered to be lost. The individual customer’s unit demand is satisﬁed after a random time of service which is distributed as exponential. The life time of each item is assumed to be exponentially distributed. The items are replenished based on variable ordering policy. The lead time is assumed to have phase type distribution. After the service completion, the primary customer may decide either to join the secondary (feedback) queue, which is of inﬁnite size, or leave the system according to a Bernoulli trial and the server decides to serve either for primary or feedback customer according to a Bernoulli trail. The primary and secondary services are at different counters. The service time for feedback customers is assumed to be independent exponential distribution. After the service completion for feedback customer, the server starts immediately for primary customer’s service, whenever the inventory level and the primary customer level is positive, otherwise the server becomes idle for an exponential duration. If the primary customer level and inventory level becomes positive during the server idle period then he starts service for primary customer immediately. After completing his idle period, the server goes to secondary counter to serve for feedback customer, if any. The joint probability distribution of the system is obtained in the steady state. Important system performance measures are derived and the long-run total expected cost rate is also calculated. Sahin [70] considered (s, S) inventory system and general random demand process with ﬁxed lead time and backordering. He derived time-dependent and stationary distribution to derive approximations for optimal control policy. Wu and Dong [96] developed a new methodology by combining multi-class queueing networks and inventory models for the performance analysis of multiproduct manufacturing logistic chains. Their approximation method was based on dividing the whole network into multiple single sites and approximating queue length of each site in the steady state. Deepak et al. [22] treated an inventory system with service times of customers waiting in two different queues where customers could be transferred from the longer to the shorter queue and each queue had its own continuous review policy and exponential lead time distribution. They also considered inventory transferring from a queue to the other which has waiting customer(s) but zero on-hand inventory. They analysed stability of the system and computed some performance measures. Schwarz et al. [73] developed a model for customers who arrive during stock out and considered different inventory management policies with lost sale. They derived stationary distributions of joint queue length and inventory processes in explicit product form. Stationary distributions are then used to calculate performance measures of the systems. Berman and Sapna[15] considered optimization of service time of perishable goods. They considered a finite capacity system where arrivals form a Poisson process and replenishment is instantaneous. They determine the service rates to be employed at each instant of time so that the long-run expected cost rate is minimized for fixed maximum inventory level and capacity. The problem is modeled as a semi-Markov decision problem. Also they established the existence of a stationary optimal policy and is derived by reducing it to a linear programing problem.

Maike Schwarz and Hans Daduna [49] investigated M/M/1/∞-systems with inventory management, continuous review, exponentially distributed lead times and backordering. They compute performance measures and derive optimality conditions under different order policies. For performance measures, which are not explicitly at hand, we present an approximation scheme for all possible parameter combinations. Although they cannot completely determine analytically the steady state probabilities for the system, they are able to derive functional relations between interesting probabilities and show surprising insensitivity properties of several performance measures. Chakravarthy. S.R et. al.[18] considered a single-server queueing model in which the customers are served in batches of varying size depending on predetermined thresholds as well as available inventory. An (s, S)-type inventory system is used for the models considered in this article and the model is studied in detail using the matrix-analytic method by assuming all the underlying random variables to be exponentially distributed. Due to complexity of the model when more general assumptions are made on the underlying random variables, simulation is opted after a satisfactory validation with the analytic counter part of the exponential model. Vidyadhar Kulkarni. Keqi Yan [94] considered a production-inventory system where the production and demand rates are modulated by a ﬁnite state Continuous Time Markov Chain (CTMC). When the inventory position falls to a reorder point r, they place an order of size q from an external supplier. They considered the case of stochastic lead times, where the lead times are i.i.d. exponential(μ) random variables, and orders may or may not be allowed to cross and derived the distribution of the inventory level, and analyse the long run holding, backlogging and ordering cost rate per unit time. Sivakumar[84] introduced the concept of server vacation in retrial inventory system. He assumed exponential distribution for inter-demand times, lead times, inter-retrial times and server vacation times. He also assumed that all these events are mutually independent. He adopted a multiple vacation policy. Sivakumar [85] considered a two-commodity substitutable retrial inventory system with joint ordering policy. Sivakumar [86] considered a perishable inventory system with retrial demands. The author has assumed that demands are generated by a ﬁnite source, the exponentially distributed life times for the stored items, exponentially distributed lead time for order and exponentially distributed inter-retrial times. Sivakumar. B and Arivarignan. G [87] considered a continuous review (s, S) inventory system in which the arriving customers belong to any one of the two types (type-1 or type-2). The customers are not distinguished as to their type and their demanded items are delivered immediately to them when the inventory level is above s. When the inventory level drops to s(≥ 0), an order for Q items is placed and thereafter the demands of type-2 customers alone are satisfied. The type-1 customers are sent to a place called orbit which is of infinite size. These orbiting demands retry for their demand after a random time which is assumed to have exponential distribution. The arrivals of customers are assumed to follow a Markovian arrival process and the lead time is assumed to have phase-type distribution. Sivakumar.B and Arivarignan. G [88] considered a continuous review perishable (s, S) inventory system with a service facility consisting of a waiting line of finite capacity and a single server. 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 some ordinary customers from the queue. They considered the following removal rule: a negative customer at an arrival epoch removes one or more ordinary waiting customers and the number of removals is a random variable depending on the number of waiting customers in the system. The life time of each item, the service time and the lead time of the reorders are all assumed to have independent exponential distributions. Various stationary system performance measures are computed and the total expected cost rate is calculated. Sivakumar. B Elango. C, Arivarignan. G [89] considered a continuous review (s, S) inventory system at a service facility in which the waiting hall for customers is of fixed size. They assumed batch arrivals of customers. The time points of arrivals of each batch are assumed to form a Poisson process and the number of customers in each batch is a random variable. Each customer undergoes a random time of service at the end of which he/she is issued their order. The service time follows a state-dependent negative exponential distribution. The items of inventory are perishable in nature with exponential life time. It is also assumed that lead time for the reorders is assumed to be distributed as exponential independent of the service time distribution.

Maike Schwarz, Cornelia Sauer, Hans Daduna, Rafal Kulik, Ryszard Szekli [53]derived stationary distributions of joint queue length and inventory processes in explicit product form for various M/M/1-systems with inventory under continuous review and different inventory management policies, and with lost sales. Here demand is Poisson, service times and lead times are exponentially distributed. These distributions are used to calculate performance measures of the respective systems. In case of infinite waiting room the key result is that the limiting distributions of the queue length processes are the same as in the classical M/M/1/∞-system. Ushakumari[91] considered a retrial inventory system with classical retrial policy. Krishnamoorthy and Jose [43] analysed three different retrial inventory systems with positive service time and positive lead-time. The authors assumed Poisson arrivals, exponentially distributed lead time and exponentially distributed inter-retrial time. Sivakumar[82] considered a two-commodity substitutable retrial inventory system with joint ordering policy. Sivakumar[83] considered a perishable inventory system with retrial demands. The author has assumed that demands are generated by a ﬁnite source, the exponentially distributed life times for the stored items, exponentially distributed lead time for order and exponentially distributed inter-retrial times. Manuel et al. [51] studied a perishable inventory system with service facilities. They assumed that demand points form Markovian arrival process, service time is a phase type distribution, exponentially distributed life times for the items, exponentially distributed lead time and exponentially distributed inter-retrial time. Manuel et al. [52] studied a perishable inventory system with service facilities. They assumed that demand points form Markovian arrival process, service time is a phase type distribution, exponentially distributed life times for the items, exponentially distributed lead time and exponentially distributed inter-retrial time. Yadavalli et al.[97] studied a multi-server inventory system with service facility to which apart from the usual demands, another stream of demands, called negative demands arrive. Yong joo Lee, Paul Zinkin[98] explored a natural generalization of the classic tandem-queue model, designed specifically to represent make- to-stock production processes. In such systems, intermediate and finished goods can be produced and stored in advance of demand. They considered the simplest version of the model, where demand is a Poisson process, and the unit production times are exponentially distributed. Also they proposed and tested a tractable approximation scheme. Hannah Revathy. P Bharathiar [26] developed the model with a two server (s,S) inventory system with positive service time, positive lead time, retrial of customers and negative arrivals. Here arrival of customers form a Poisson process, lead time and service time are exponentially distributed. The system starts with S units of inventory on hand and each arriving customer is served a single unit of the item by any one of the servers. When the inventory level reaches s, an order is placed for (S-s) units. If the inventory level is zero or both servers are busy, then the arriving customer goes to orbit and becomes a source of repeated calls and this model is solved by using Direct Truncation Method. Rashid, R., Hoseini, S.F., Gholamian, M.R. et. al.[64], presented a mathematical model for an inventory control system in which customers’ demands and suppliers’ service time are considered as stochastic parameters. The proposed problem is solved through queuing theory for a single item. In this case, transitional probabilities are calculated in steady state. Afterward, the model is extended to the case of multi-item inventory systems. Then, to deal with the complexity of this problem, a new heuristic algorithm is developed. Finally, they presented bi-level inventory-queuing model is implemented as a case study in Electroestil Company. Ebrahim Teimoury, Ali Mazlomi, Raheleh Nadafioun, Iman G. Khondabi, and Mehdi Fathi[23] considered a supply chain that includes a manufacture which produced more than one product that are demanded by several retailers. After production, each all type of products is hold in separated warehouses. Each warehouse has different holding cost and each product has a different backorder cost. They formulated a linear cost function to aggregate all the costs of the holding, back ordering and ordering. The manufactured incurs a setup time whenever it switches from producing one product type to another and it has a finite production rate and stochastic production times. The manufacturer modelled as a FCFS, GI/G/1 queue. They extended the proposed model in order to analyse the logistics process to three-echelon inventory model. Hill et. al.[33]proposed a single-item, two-echelon, continuous-review inventory model. Most of the retailers have their stock replenished from a central warehouse and the warehouse in turn replenishes stock from an external supplier. The demand processes on the retailers are independent Poisson. The performance measures of interest are the average total stock in the system and the fraction of demand met in the retailers. Davidperry and Benny Levikson[21] considered a two stage production systems in which items are produced continuously over time with fixed rate. The inventory level distributions and other important functional associated with these storage systems are derived and it is accomplished by an analogy existing between the storage systems and certain queuing systems and a finite Dam model.

3. CONCLUSION:

In this review article we have discussed queuing inventory systems with various cases. There are many real time problems arising in the worldwide involving inventory with queueing discipline solved by using the suitable methods. More recently we have queueing-inventory system that has applications in railway, airline and Bus reservation systems. Advance reservation/purchase of inventory for future use is a common phenomenon. Sometimes items reserved are subject to cancellation such type of problems are solved by numerically.

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Received on 16.07.2016 Modified on 08.08.2016

Research J. Pharm. and Tech 2016; 9(11):2056-2066.

DOI: 10.5958/0974-360X.2016.00421.2