Positive Service Time Research Articles

Two commodity queueing inventory system with random common lifetime, two demand classes and pool of customers

This paper analyzes a two-commodity perishable queueing inventory system with random common lifetimes commodity-wise, positive service time, and individual ordering policy. We have considered two types of customers: priority customers and non-priority customers. There is a buffer of finite capacity for the priority customers and a pool with infinite capacity made available for the non-priority customers. The arrival of these customers is occurred by two independent Poisson processes. The lifetimes of each of the commodities and service times follow independent exponential distributions. The matrix geometric approach is used to analyze the model's steady-state behavior and we computed various system performance measures. This article examines the effects of the common lifetime parameters on the measures of performance, such as average cycle time, the average number of reorders due to the realization of the common lifetime of commodities, etc. Also, a sensitivity analysis of the system parameters on the performance measures is carried out. Further, a cost function is constructed based on the performance measures, and its convexity is examined numerically.

Stochastic optimization of level-dependent perishable inventory system by Jackson network

In this research work, we developed a model to investigate the long-run behavior of the solutions of continuous review ( s , S ) inventory policy for level-dependent perishable products with positive service time on Jackson queueing network. The service facility is assumed that the waiting capacities of each queue are different. In this model, we assume two independent single server nodes in a M / M / 1 queueing system with an associated inventory, where each node represents a queue. In a Poisson process, customers enter the system. The service times ( μ i > 0 ) and lead times ( θ i > 0 ) are independent and also spread exponentially at the node i for 1 ⩽ i ⩽ 2 . If the inventory levels of goods from the warehouse i are below s i at any moment, an order for Q i = ( ( S i - s i ) > 0 ; ∀ i ) units is made, where S i is the highest capacity of storage for each warehouse in the system. In the steady state case, the matrix analytic technique is used to derive the combined probability distribution for the number of customers and inventory level. The overall anticipated cost rate is determined using some essential system performance measures. Numerical and graphical examples are presented to analyze the effect of variation of parameters on the system.

Comparison of discrete time inventory systems with positive service time and lead time

This paper investigates two discrete time queueing inventory models with positive service time and lead time. Customers arrive according to a Bernoulli process and service time and lead time follow geometric distributions. The first model under discussion based on replenishment of order upto $S$ policy where as the second model is based on order placement by a fixed quantity $Q$, where $Q=S-s$, whenever the inventory level falls to $s$. We analyse this queueing systems using the matrix geometric method and derive an explicit expression for the stability condition. We obtain the steady-state behaviour of these systems and several system performance measures. The influence of various parameters on the systems performance measures and comparison on the cost analysis are also discussed through numerical example.

Analysis of self service (<i>s</i>, <i>S</i>) inventory with general lead time and positive service time

Analysis of self service (<i>s</i>, <i>S</i>) inventory with general lead time and positive service time

Analysis of self service (s,S) inventory with general lead time and positive service time

This paper deals with the analysis of (s, S) inventory with general lead time and positive service time in a self service environment. The time for serving the inventory is considered as the servic...

Reducing the variability of inter-departure times of a single-server queueing system – The effects of skewness

A critical performance measure in serial production lines is the variability of inter-departure times of a single-server queue. Increasing upstream inter-departure time variability generates greater downstream variability, diminishing overall line performance. Theory suggests that the variability of inter-departure times of a single-server queue is reduced by decreasing the variance of inter-arrival and service times. This study investigates the effects of the skewness of inter-arrival and service time distributions on the variability of inter-departure times. Contrary to previous results suggesting that mean waiting times of a GI/G/1 queue can be reduced by increasing inter-arrival time skewness, this experimental study of a GI/G/1 queue with triangular inter-arrival and service times shows that the inter-departure time coefficient of variation is reduced through a combination of negative inter-arrival time skewness and positive service time skewness. These results also suggest that the absolute value of the negative autocorrelation between consecutive departures is reduced by the same combination of negative inter-arrival time skewness and positive service time skewness for low values of server’s utilization, while positive skewness for both inter-arrival and service times reduces this value for high values of server’s utilization. Finally, it was found that queue capacity constraints increase the coefficient of variation of inter-departure times, as has been previously suggested, as well as the skewness and the absolute correlation values of the inter-departure time distribution.

Markov models of inventory management systems with positive service time applications

Markov models of inventory management systems with positive service time applications

Discrete Product Inventory Control System with Positive Service Time and Two Operation Modes

Consideration is given to the Markov inventory control system of a discrete product of maximum volume S under the strategies (s,Q) and (s, S) with a positive service time. Upon arrival, the order is queued if the inventory level is positive or, otherwise, leaves the system unserviced. One server handles the queued orders one-by-one in the sequence of their arrival. If the inventory level exceeds s, then the service time has the exponential distribution of intensity μ; otherwise, of intensity αμ, 0 < α ⩽ 1. The product in the inventory is consumed only at the instant when the service (of the order) ends. Inventory deficit is not allowed. When the inventory is empty new orders are not admitted into the system, and the service process of the queued orders (if any) is stopped. The lead time is assumed to be exponentially distributed. Analytical relations are established for the basic stationary performance characteristics of the system.

Markov Models of Inventory Management Systems with a Positive Service Time

Markov models of perishable inventory systems where some customers are withdrawn after completion of the service without buying the stock are studied. It is assumed that the service time and filling supply orders are positive random variables. The inventory replenishment size is a variable and depends on the current inventory level. Exact and approximate methods for calculating the joint distribution of the inventory level and the number of orders in the system are developed. Formulas for calculating the main characteristics of the studied systems are proposed and the problem of their optimization is solved. The high level of accuracy of the proposed formulas is confirmed by the numerical experiments.

Discrete product inventory control system with positive service time and two operation modes

Discrete product inventory control system with positive service time and two operation modes

A stochastic inventory system with two modes of service and retrial of customers

This paper analyses an inventory model with positive service time and retrial of customers. The customers arrive according to a Markovian Arrival Process with representation ( $$D_{0} ,D_{1}$$ ). The service times are assumed to be of Phase type distribution with representation $$\left( {\eta ,U} \right)$$ . When the inventory level depletes to the reorder point s, the service is given at a reduced rate and the service time distribution has the representation $$\left( {\eta ,\alpha U} \right)$$ , where $$0 < \alpha < 1$$ . An arriving customer, who finds the inventory level zero or the server busy, enters into an orbit of infinite capacity and will retry for service from there. The lead time follows an exponential distribution with rate β. We analyze the system using Matrix Analytic Method. Some important performance measures in the steady state are obtained. A suitable cost function for the expected total cost is constructed and analyzed numerically and graphically.

Joint inventory control and pricing in a service-inventory system

This study addresses joint inventory control and pricing decisions for a service-inventory system. In such a system both an on-hand inventory item and a positive service time are required to fulfill customer demands. The service-inventory system also captures main features of the classical inventory systems with a positive processing time, e.g., make-to-order systems. In this study the service-inventory system is modeled as an M/M/1 queue in which the customer arrival rate is price dependent. The inventory of an individual item is continuously reviewed under an (r,Q) policy. The replenishment lead times of the inventory are exponentially distributed. Furthermore, customers arriving during stock-out periods are lost. The stochastic customer inter-arrival times, service times, and inventory replenishment lead times cause the high complexity of the problem and the difficulty in solving it. The aim of this study is to formulate the problem and solve it to optimality. We make three main contributions: (1) We integrate inventory control and pricing in the service-inventory system. The problem is formulated and analyzed as a fractional programming problem, and structural properties are explored for the model. (2) Two solution algorithms are proposed. The first one provides optimal solutions, while the second one is more efficient. (3) The impact of the integrated inventory control and pricing decisions on the overall system performance is investigated. We compare the solutions of the models both with and without fill-rate and service-reliability constraints and report the main interesting managerial insights.

Analysis of a Multiserver Queueing-Inventory System

We attempt to derive the steady-state distribution of theM/M/cqueueing-inventory system with positive service time. First we analyze the case ofc=2servers which are assumed to be homogeneous and that the service time follows exponential distribution. The inventory replenishment follows the(s,Q)policy. We obtain a product form solution of the steady-state distribution under the assumption that customers do not join the system when the inventory level is zero. An average system cost function is constructed and the optimal pair(s,Q)and the corresponding expected minimum cost are computed. As in the case ofM/M/cretrial queue withc≥3, we conjecture thatM/M/cforc≥3, queueing-inventory problems, do not have analytical solution. So we proceed to analyze such cases using algorithmic approach. We derive an explicit expression for the stability condition of the system. Conditional distribution of the inventory level, conditioned on the number of customers in the system, and conditional distribution of the number of customers, conditioned on the inventory level, are derived. The distribution of two consecutivestostransitions of the inventory level (i.e., the first return time tos) is computed. We also obtain several system performance measures.

S,S Inventory with Positive Service Time and Retrial of Demands: An Approach through Multiserver Queues

We analyze an s,S inventory with positive service time and retrial of demands by considering the inventory as servers of a multiserver queuing system. Demands arrive according to a Poisson process and service time distribution is exponential. On each service completion, the number of demands in the system as well as the number of inventories (servers) is reduced by one. When all servers are busy, new arrivals join an orbit from which they try to access the service at an exponential rate. Using matrix geometric methods the steady state joint distribution of the demands and inventory has been analyzed and a numerical illustration is given.

Production inventory with service time and interruptions

This paper considers an (s,S) production inventory system with positive service time, with time for producing each item following Erlang distribution. Customers arrive according to a Poisson process. A customer who arrives when there is no inventory in the system is considered lost. On the other hand, a customer who finds a busy server with at least one inventory in the system joins a queue of infinite capacity. When the inventory level falls to s, production process is switched on, and it is switched off when the inventory level reaches back to S. Service time to each customer also follows an Erlang distribution. The service of a customer may be interrupted, where the time for such a phenomenon follows an exponential distribution, whenever it occurs. An interrupted service, after repair, resumes from where it was stopped. The correction/repair time follows an exponential distribution. We assume that the service of a single customer may encounter any number of interruptions and that the customer being served waits there until his service is completed. Moreover, at a time the server is subject to at most one interruption. We also assume that no inventory is lost due to a service interruption. Like the service process, the production process also is subject to interruptions, where the duration to an interruption follows an exponential distribution. However, in contrast to the service interruption, in the case of interruption to production process, we assume that the item being processed is lost because of interruption. That is, the production process, on being interrupted, restarts from the beginning, after repair. The repair time of an interrupted production process follows exponential distribution. Few of the last service phases are assumed to be protected in the sense that the service will not be interrupted while being in these phases. The same is assumed for the production process also.The model is analysed as a level-independent quasi-birth–death process. We apply a novel method to obtain an explicit expression for the necessary and sufficient condition for the stability of the system under study. This method works even if we assume general phase-type distributions for the production as well as the service processes, and hence can be used to characterise the stability of inventory systems where the assumption of disallowing the customers to join the system, when there is a shortage of inventory has been made. Under stability, we apply matrix analytic methods to compute the system state distribution. In consequence to that, several system performance measures have been derived, and their dependence on the system parameters has been studied numerically.

A revisit to queueing-inventory system with positive service time

A queueing-inventory system, with the item given with probability γ to a customer at his service completion epoch, is considered in this paper. Two control policies, (s,Q) and (s,S) are discussed. In both cases we obtain the joint distribution of the number of customers and the number of items in the inventory as the product of their marginals 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. Further we investigate numerically an expression for per unit time cost as a function of γ. This function exhibit convexity property. A comparison with Schwarz et al. (Queueing Syst. 54:55–78, 2006) is provided. The case of arbitrarily distributed service time is briefly indicated.

Two Server (s, S) Inventory System with Positive Service Time, Positive Lead Time, Retrial Customers and Negative Arrivals

This paper deals with a two server (s,S) inventory system with positive service time, positive lead time, retrial of customers and negative arrivals. In this system, 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. 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. Assume that the capacity of the orbit is infinite. The negative arrival plays an important role in this paper and it controls the congestion in the orbit by removing one customer from the orbit and further it is assumed that it removes the customer from the orbit only if inventory level is zero or both servers are busy. It is also assumed that the access from orbit to the service facility is governed by the classical retrial policy. This model is solved by using Direct Truncation Method. Numerical and graphical studies have been done for analysis of mean number of customers in the orbit, average inventory level, Truncation level, mean number of busy servers and system performance measures. A suitable cost function is defined. General Terms Stochastic Process, Queueing theory, Retrial queues, Inventory system, Negative arrivals.

Stochastic decomposition in production inventory with service time (abstract only)

We study an (s, S) inventory system with positive service time (for an overview of the work reported so far in inventory with positive service time one may refer to Krishnamoorthy, Lakshmi and Manikandan: A survey on inventory models with positive service time, OPSEARCH, DOI 10.1007/s12597-010-0032-z). This leads to a queue of demands being formed. The process of demand arrival constitutes a Poisson process. The duration of each service is exponentially distributed. Our model is a supply chain where items are added to the inventory through a production process. This starts each time the inventory level goes down to s and continues to be on until inventory level reaches S with the time required to add one unit of the item into the inventory when the production is on, are independent, identically distributed exponential random variables. Further all distributions involved in this paper are assumed to be mutually independent. We assume that no customer joins the queue when the inventory level is 0. This assumption leads us to an explicit product form solution for the steady state probability vector, using a simple approach. This is despite the fact that there is a strong correlation between lead time (the time required to add an item into the inventory) and the number of customers joining the queue during the lead time (except when the inventory level is zero during which time no customer joins the queue). The technique is to combine the steady state probability vector of the classical M/M/1 queue and that of the production inventory system where each service requires negligible time and no backlogs are allowed. Using a similar technique, the expected length of a production cycle is also obtained explicitly. The optimality of the highest inventory level S and the production switching on level s has been studied using a cost function constructed using the steady state system performance measures. Since we have obtained explicit expressions for these measures, analytic expressions have been derived for the optimal values of S and s. To show that our method can be applied to other similar problems, we analyze in detail a variant of the above problem (discussed in Schwarz M, Sauer. C, Daduna, H., Kulik, R and Szekli, R: M/M/1 Queueing systems with inventory, Queueing Systems, 54,55-78, 2006). For that model, we assume that in a production run, production occurs only once in a cycle and the amount produced is sufficient to take the inventory level back to S. A brief discussion on the application of our method to inventory system with lead time for replenishment has also been provided.

A survey on inventory models with positive service time

A detailed review of inventory models involving positive service time is given. These include classical and retrial cases. Also contributions to production inventory with service time is indicated towards the end. In addition directions for future work are indicated.

Production inventory with service time and vacation to the server

This paper considers a production inventory with positive service time. The time for producing each item is assumed to follow a ‘Markovian production scheme’. Because of this, correlation automatically gets into the production process. The customer arrival process follows a Markovian arrival process. When inventory falls to s, the production process is switched on, and is switched off when the on-hand inventory reaches S. The service time to each customer has a phase-type distribution. We investigate the system stability. Under the condition of stability, we investigate the system state distribution. Next, several performance measures are computed, such as the fraction of time server is on vacation; fraction of time the inventory level is zero; fraction of time the production process is On; expected number of customers in the system while server is on vacation and also while it is busy. Numerical results indicate, among other things, the effect of the control variables s and S on the fraction of time the system goes out of inventory and on expected loss rate of customers.