A Perishable Inventory Model with Bonus Service for Certain Customers, Balking and N + 1 Policy

2014 ◽  
Vol 2 (3-4) ◽  
pp. 83-104
Author(s):  
Kathirvel Jeganathan
Keyword(s):  
Steady State ◽  
Joint Probability ◽  
Life Time ◽  
Inventory Level ◽  
Joint Probability Distribution ◽  
Perishable Inventory ◽  
Cost Rate ◽  
Waiting Area ◽  
Waiting Line ◽  
Number Of Customers

AbstractWe consider a perishable inventory system with bonus service for certain customers. The maximum inventory level is S. The ordering policy is (0, S) policy and the lead time is zero. The life time of each items is assumed to be exponential. Arrivals of customers follow a Poisson process with parameter λ. Arriving customers form a single waiting line based on the order of their arrivals. The capacity of the waiting line is restricted to M including the one being served. The first N ≥ 1 customers who arrive after the system becomes empty must wait for service to begin. We have assumed that these N customers received an additional service, called bonus service, in addition to the essential service rendered to all customers. The service times are exponentially distributed. An arriving customer who finds the server busy or waiting area size ≥ N, proceeds to the waiting area with probability p and is lost forever with probability (1 − p). The Laplace–Stieltjes transform of the waiting time of the tagged customer is derived. The joint probability distribution of the number of customers in the waiting area and the inventory level is obtained for the steady state case. Some important system performance measures and the long-run total expected cost rate are derived in the steady state.

scholarly journals A Perishable Inventory System with Postponed Demands and Multiple Server Vacations

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
R. Jayaraman ◽  
B. Sivakumar ◽  
G. Arivarignan
Keyword(s):  
Steady State ◽  
Joint Probability ◽  
Life Time ◽  
Inventory System ◽  
Inventory Level ◽  
Joint Probability Distribution ◽  
Single Server ◽  
Cost Rate ◽  
Stochastic Inventory ◽  
Number Of Customers

A mathematical modelling of a continuous review stochastic inventory system with a single server is carried out in this work. We assume that demand time points form a Poisson process. The life time of each item is assumed to have exponential distribution. We assume(s,S)ordering policy to replenish stock with random lead time. The server goes for a vacation of an exponentially distributed duration at the time of stock depletion and may take subsequent vacation depending on the stock position. The customer who arrives during the stock-out period or during the server vacation is offered a choice of joining a pool which is of finite capacity or leaving the system. The demands in the pool are selected one by one by the server only when the inventory level is aboves, with interval time between any two successive selections distributed as exponential with parameter depending on the number of customers in the pool. The joint probability distribution of the inventory level and the number of customers in the pool is obtained in the steady-state case. Various system performance measures in the steady state are derived, and the long-run total expected cost rate is calculated.

scholarly journals Markovian inventory model with two parallel queues, jockeying and impatient customers

2016 ◽  
Vol 26 (4) ◽  
pp. 467-506 ◽  
Author(s):  
K. Jeganathan ◽  
J. Sumathi ◽  
G. Mahalakshmi
Keyword(s):  
Steady State ◽  
Finite Size ◽  
Joint Probability ◽  
Life Time ◽  
The Other ◽  
Inventory Level ◽  
Joint Probability Distribution ◽  
Cost Rate ◽  
Parallel Queues ◽  
Number Of Customers

This article presents a perishable stochastic inventory system under continuous review at a service facility consisting of two parallel queues with jockeying. Each server has its own queue, and jockeying among the queues is permitted. The capacity of each queue is of finite size L. The inventory is replenished according to an (s; S) inventory policy and the replenishing times are assumed to be exponentially distributed. The individual customer is issued a demanded item after a random service time, which is distributed as negative exponential. The life time of each item is assumed to be exponential. Customers arrive according to a Poisson process and on arrival; they join the shortest feasible queue. Moreover, if the inventory level is more than one and one queue is empty while in the other queue, more than one customer are waiting, then the customer who has to be received after the customer being served in that queue is transferred to the empty queue. This will prevent one server from being idle while the customers are waiting in the other queue. The waiting customer independently reneges the system after an exponentially distributed amount of time. The joint probability distribution of the inventory level, the number of customers in both queues, and the status of the server are obtained in the steady state. Some important system performance measures in the steady state are derived, so as the long-run total expected cost rate.

scholarly journals A Perishable Inventory System with Postponed Demands and Finite

2016 ◽  
Vol 12 (8) ◽  
pp. 6500-6515
Author(s):  
R Jayaraman
Keyword(s):  
Steady State ◽  
Joint Probability ◽  
Life Time ◽  
Inventory System ◽  
Inventory Level ◽  
Joint Probability Distribution ◽  
Single Item ◽  
Perishable Inventory ◽  
Perishable Inventory System ◽  
Number Of Customers

In this article, we consider a continuous review perishable inventory system with a finite number of homogeneous sources generating demands. The demand time points form quasi random process and demand is for single item. The maximum storage capacity is assumed to be The life time of each item is assumed to have exponential distribution. The order policy is policy, that is, whenever the inventory level drops to a prefixed level an order for items is placed. The ordered items are received after a random time which is distributed as exponential. We assume that the demands that occur during the stock out periods either enter a pool or leave the system which is according to a Bernoulli trial. The demands in the pool are selected one by one, while the stock is above the level with interval time between any two successive selections is distributed as exponential. The joint probability distribution of the number of customers in the pool and the inventory level is obtained in the steady state case. Various system performance measures are derived to compute the total expected cost per unit time in the steady state. The optimal cost function and the optimal are studied numerically.

scholarly journals Perishable Inventory System with Postponed Demands and Negative Customers

2007 ◽  
Vol 2007 ◽  
pp. 1-12 ◽  
Author(s):  
Paul Manuel ◽  
B. Sivakumar ◽  
G. Arivarignan
Keyword(s):  
Steady State ◽  
Joint Probability ◽  
Inventory System ◽  
Arrival Process ◽  
Inventory Level ◽  
Joint Probability Distribution ◽  
Time Interval ◽  
Negative Customers ◽  
Perishable Inventory System ◽  
Number Of Customers

This article considers a continuous review perishable (s,S) inventory system in which the demands arrive according to a Markovian arrival process (MAP). The lifetime of items in the stock and the lead time of reorder are assumed to be independently distributed as exponential. Demands that occur during the stock-out periods either enter a pool which has capacity N(<∞) or are lost. Any demand that takes place when the pool is full and the inventory level is zero is assumed to be lost. The demands in the pool are selected one by one, if the replenished stock is above s, with time interval between any two successive selections distributed as exponential with parameter depending on the number of customers in the pool. The waiting demands in the pool independently may renege the system after an exponentially distributed amount of time. In addition to the regular demands, a second flow of negative demands following MAP is also considered which will remove one of the demands waiting in the pool. The joint probability distribution of the number of customers in the pool and the inventory level is obtained in the steady state case. The measures of system performance in the steady state are calculated and the total expected cost per unit time is also considered. The results are illustrated numerically.

Perishable Inventory System with Server Interruptions, Multiple Server Vacations, and N Policy

2015 ◽  
Vol 6 (2) ◽  
pp. 32-52 ◽  
Author(s):  
K. Jeganathan ◽  
N. Anbazhagan ◽  
B. Vigneshwaran
Keyword(s):  
Steady State ◽  
Poisson Process ◽  
Finite Size ◽  
Joint Probability ◽  
Inventory System ◽  
Joint Probability Distribution ◽  
Perishable Inventory ◽  
Waiting Area ◽  
Perishable Inventory System ◽  
Negative Exponential

This article presents a perishable inventory system under continuous review at a service facility in which a waiting area for customers is of finite size . The authors assume that the replenishment of inventory is instantaneous. The items of inventory have exponential life times. It is assumed that demand for the commodity is of unit size. The service starts only when the customer level reaches a prefixed level , starting from the epoch at which no customer is left behind in the system. The arrivals of customers to the service station form a Poisson process. The server goes for a vacation of an exponentially distributed duration whenever the waiting area is zero. If the server finds the customer level is less than when he returns to the system, he immediately takes another vacation. The individual customer is issued a demanded item after a random service time, which is distributed as negative exponential. The service process is subject to interruptions, which occurs according to a Poisson process. The interrupted server is repaired at an exponential rate. Also the waiting customer independently reneges the system after an exponentially distributed amount of time. The joint probability distribution of the number customers in the system and the inventory levels is obtained in steady state case. Some measures of system performance in the steady state are derived and the total expected cost is also considered. The results are illustrated with numerical examples.

A Stochastic Inventory Model with Multiple Vacations and N - Policy

2016 ◽  
pp. 106-123
Author(s):  
Jeganathan Kathirvel
Keyword(s):  
Poisson Process ◽  
Finite Size ◽  
Joint Probability ◽  
Joint Probability Distribution ◽  
Stochastic Inventory ◽  
Left Behind ◽  
Stochastic Inventory Model ◽  
Waiting Area ◽  
Perishable Inventory System ◽  
Number Of Customers

We consider a perishable inventory system under continuous review at a service facility in which a waiting area for customers is of finite size. We assume that the replenishment of inventory is instantaneous. The items of inventory have exponential life times. The service starts only when the customer level reaches a prefixed level, starting from the epoch at which no customer is left behind in the system. The arrivals of customers to the service station form a Poisson process. The server goes for a vacation of an exponentially distributed duration whenever the waiting area is zero. The service process is subject to interruptions, which occurs according to a Poisson process. The interrupted server is repaired at an exponential rate. Also the waiting customer independently reneges the system after an exponentially distributed amount of time. The joint probability distribution of the number of customers in the system and the inventory levels is obtained in steady state case. The results are illustrated with numerical examples.

scholarly journals Perishable Inventory System with N-Policy, MAP Arrivals, and Impatient Customers

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1514
Author(s):  
R. Suganya ◽  
Lewis Nkenyereye ◽  
N. Anbazhagan ◽  
S. Amutha ◽  
M. Kameswari ◽  
...  
Keyword(s):  
Inventory System ◽  
Joint Probability Distribution ◽  
Single Server ◽  
Impatient Customers ◽  
Expected Cost ◽  
Perishable Inventory ◽  
Cost Rate ◽  
Perishable Inventory System ◽  
The Impact ◽  
Number Of Customers

In this study, we consider a perishable inventory system that has an (s, Q) ordering policy, along with a finite waiting hall. The single server, which provides an item to the customer after completing the required service performance for that item, only begins serving after N customers have arrived. Impatient demand is assumed in that the customers waiting to be served lose patience and leave the system if the server’s idle time overextends or if the arriving customers find the system to be full and will not enter the system. This article analyzes the impatient demands caused by the N-policy server to an inventory system. In the steadystate, we obtain the joint probability distribution of the level of inventory and the number of customers in the system. We analyze some measures of system performance and get the total expected cost rate in the steadystate. We present a beneficial cost function and confer the numerical illustration that describes the impact of impatient customers caused by N-policy on the inventory system’s total expected cost rate.

Markov processes whose steady state distribution is matrix-exponential with an application to the GI/PH/1 queue

1989 ◽  
Vol 21 (01) ◽  
pp. 159-180 ◽  
Author(s):  
Bhaskar Sengupta
Keyword(s):  
Steady State ◽  
Markov Process ◽  
Supply And Demand ◽  
Joint Probability ◽  
Matrix Exponential ◽  
Linear Matrix ◽  
Joint Probability Distribution ◽  
Rate Matrix ◽  
Steady State Distribution

This paper is concerned with a bivariate Markov process {Xt, Nt ; t ≧ 0} with a special structure. The process Xt may either increase linearly or have jump (downward) discontinuities. The process Xt takes values in [0,∞) and Nt takes a finite number of values. With these and additional assumptions, we show that the steady state joint probability distribution of {Xt, Nt ; t ≧ 0} has a matrix-exponential form. A rate matrix T (which is crucial in determining the joint distribution) is the solution of a non-linear matrix integral equation. The work in this paper is a continuous analog of matrix-geometric methods, which have gained widespread use of late. Using this theory, we present a new and considerably simplified characterization of the waiting time and queue length distributions in a GI/PH/1 queue. Finally, we show that the Markov process can be used to study an inventory system subject to seasonal fluctuations in supply and demand.

Perishable Inventory System with Bi-Level Service System

2016 ◽  
pp. 200-217
Author(s):  
D. Gomathi
Keyword(s):  
Steady State ◽  
Performance Measures ◽  
Service Time ◽  
System Performance ◽  
Service System ◽  
Joint Probability ◽  
Inventory System ◽  
Service Time Distribution ◽  
Perishable Inventory ◽  
Perishable Inventory System

In this chapter we consider a perishable inventory system under continuous review at a bi-level service system with finite waiting hall of size N. The maximum storage capacity of the inventory is S units. We assumed that a demand for the commodity is of unit size. The arrival time points of customers form a Poisson process. The individual customer is issued a demanded item after a random service time, which is distributed as negative exponential. The effect of the two modes of operations on the system performance measures is also discussed. It is also assumed that lead time for the reorders is distributed as exponential and is independent of the service time distribution. The items are perishable in nature and the life time of each item is assumed to be exponentially distributed. The demands that occur during stock out periods are lost. The joint probability distribution of the number of customers is obtained in the steady-state case. Various system performance measures in the steady state are derived. The results are illustrated numerically.

scholarly journals On the Nonsymmetric Longer Queue Model: Joint Distribution, Asymptotic Properties, and Heavy Traffic Limits

2013 ◽  
Vol 2013 ◽  
pp. 1-21 ◽  
Author(s):  
Charles Knessl ◽  
Haishen Yao
Keyword(s):  
Heavy Traffic ◽  
Asymptotic Properties ◽  
Joint Probability ◽  
Integral Representations ◽  
Joint Probability Distribution ◽  
Single Server ◽  
Poisson Arrival ◽  
Large Numbers ◽  
Node Network ◽  
Number Of Customers

We consider two parallel queues, each with independent Poisson arrival rates, that are tended by a single server. The exponential server devotes all of its capacity to the longer of the queues. If both queues are of equal length, the server devotesνof its capacity to the first queue and the remaining1−νto the second. We obtain exact integral representations for the joint probability distribution of the number of customers in this two-node network. Then we evaluate this distribution in various asymptotic limits, such as large numbers of customers in either/both of the queues, light traffic where arrivals are infrequent, and heavy traffic where the system is nearly unstable.

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