ANALYSIS OF FINITE CAPACITY QUEUE WITH NEGATIVE CUSTOMERS AND BUNKER FOR OUSTED CUSTOMERS USING CHEBYSHEV AND GEGENBAUER POLYNOMIALS

2014 ◽  
Vol 31 (04) ◽  
pp. 1450029 ◽  
Author(s):  
ROSTISLAV V. RAZUMCHIK
Keyword(s):  
Finite Size ◽  
Joint Probability ◽  
Gegenbauer Polynomials ◽  
Joint Probability Distribution ◽  
Finite Capacity ◽  
Negative Customers ◽  
Blocking Probabilities ◽  
Regular Customer ◽  
Number Of Customers ◽  
Finite Capacity Queue

Consideration is given to the queueing system with incoming Poisson flows of regular and negative customers. Regular customers await service in buffer of finite size r. Each negative customer upon arrival pushes a regular customer out of the queue in buffer (if it is not empty) and moves it to another queue of finite capacity r (bunker). Customers from both queues are served according to exponential distribution with parameter μ, first-come, first-served discipline, but customers in bunker are served with relative priority. Using method based on Chebyshev and Gegenbauer polynomials the algorithm for computation of stationary blocking probabilities and joint probability distribution of the number of customers in buffer and bunker is obtained. Numerical example is provided.

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.

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 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 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 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.

SCALINGS OF A MODIFIED MANNA MODEL WITH BULK DISSIPATION

2009 ◽  
Vol 20 (02) ◽  
pp. 273-284 ◽  
Author(s):  
CHIEN-FU CHEN ◽  
CHAI-YU LIN
Keyword(s):  
Probability Distribution ◽  
Finite Size ◽  
Joint Probability ◽  
Square Lattice ◽  
Joint Probability Distribution ◽  
Finite Size Scaling ◽  
Scaling Relation ◽  
Critical Scaling ◽  
Crossover Behavior ◽  
Boundary Dissipation

This study incorporates bulk dissipation described by a losing probability f into a modified Manna model on an L × L square lattice. The crossover behavior between bulk and boundary dissipation is investigated using the characteristic lengths produced by bulk dissipation. The toppling number Nn and area Na are studied. For a probability distribution of Ns, [Formula: see text] where s = n or a, the scaling form including the finite-size scaling (f = 0) and the critical scaling (L → ∞) are determined. Subsequently, this paper investigates the joint probability distribution [Formula: see text] and provides the scaling relation between the toppling number and area.

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.

Analysis of Two-Echelon Inventory System with Direct and Retrial Demands

2016 ◽  
pp. 165-182
Author(s):  
Krishnan K.
Keyword(s):  
Poisson Process ◽  
Finite Size ◽  
Joint Probability ◽  
Inventory System ◽  
Arrival Process ◽  
Inventory Level ◽  
Joint Probability Distribution ◽  
Supply Source ◽  
Distribution Centre ◽  
Abundant Supply

In this Chapter, the author considers a continuous review inventory system (both perishable and non-perishable) with Markovian demand. The operating policies are (s, S) and (0,M) policy, that is the maximum inventory level at lower echelon is Sand whenever the inventory drops to s, an order for Q(= S - s) units is placed at the same time in the higher echelon, the maximum inventory level is fixed as M(= nQ: n = 1, 2, ….) and has an instantaneous replenishment facility from an abundant supply source. The demands that occur directly to the distribution centre are called direct demands. The arrival process for the direct demand follows Poisson process. The demand process to the retailer node is independent to the direct demand process and follows Poisson process. The demands that occur during stock out period are enter into the orbit of finite size. The joint probability distribution of the inventory level at lower echelon, higher echelon and the number of customer in the orbit is obtained in the steady state case.

Post-Earthquake Restoration Planning for Los Angeles Electric Power

2006 ◽  
Vol 22 (3) ◽  
pp. 589-608 ◽  
Author(s):  
Zehra Çağnan ◽  
Rachel A. Davidson ◽  
Seth D. Guikema
Keyword(s):  
Los Angeles ◽  
Electric Power ◽  
Cost Benefit Analysis ◽  
Discrete Event ◽  
Joint Probability ◽  
Joint Probability Distribution ◽  
Power Outages ◽  
Restoration Process ◽  
Restoration Planning ◽  
Number Of Customers

This paper describes the application of a new discrete-event-simulation model of the post-earthquake electric power restoration process in Los Angeles. The findings are that (1) Los Angeles residents may experience power outages lasting up to 10 days; (2) what we call the power rapidity risk (the joint probability distribution of restoration of a specified number of customers in a specified amount of time) varies throughout the area; (3) there is a relatively high likelihood that more repair materials than are currently available will be required if a large earthquake occurs; and (4) there are ways to reduce the expected duration of earthquake-initiated power outages and they should be subjected to cost-benefit analysis. These results should be useful to utilities and emergency planners in Los Angeles. The new simulation modeling approach could be used in other seismically active cities to gain insights into the restoration process that other modeling approaches cannot provide.

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