scholarly journals Analysis of an M/PH/1 Retrial Queueing-Inventory System with Level Dependent Retrial Rate

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Zaiming Liu ◽  
Xuxiang Luo ◽  
Jinbiao Wu
Keyword(s):  
Life Time ◽  
Maximum Size ◽  
Inventory System ◽  
Phase Type Distribution ◽  
Steady State Probability ◽  
Numerical Illustration ◽  
Retrial Queueing ◽  
Common Life Time ◽  
The Stability ◽  
Number Of Customers

We analyze a queueing-inventory system which can model airline and railway reservation systems. An arriving customer to an idle server joins for service immediately with exactly one item from inventory at the moment of service completion if there are some on-hand inventory, or else he accesses to a buffer of varying size (the buffer capacity varies and equals to the number of the items in the inventory with maximum size S). When the buffer overflows, the customer joins an orbit of infinite capacity with probability p or is lost forever with probability 1−p. Arrivals form a Poisson process, and service time has phase type distribution. The time between any two successive retrials of the orbiting customer is exponentially distributed with parameter depending on the number of customers in the orbit. In addition, the items have a common life time with exponentially distributed. Cancellation of orders is possible before their expiry and intercancellation times are assumed to be exponentially distributed. The stability condition and steady-state probability vector have been studied by Neuts–Rao truncation method using the theory of Level Dependent Quasi-Birth-Death (LDQBD) processes. Several stationary performance measures are also computed. Furthermore, we provide numerical illustration of the system performance with variation in values of underlying parameters and analyze an optimization problem.

scholarly journals A discrete single server queue with Markovian arrivals and phase type group services

1995 ◽  
Vol 8 (2) ◽  
pp. 151-176 ◽  
Author(s):  
Attahiru Sule Alfa ◽  
K. Laurie Dolhun ◽  
S. Chakravarthy
Keyword(s):  
Steady State ◽  
Queueing System ◽  
State Probability ◽  
Arrival Process ◽  
Single Server ◽  
Probability Vector ◽  
Phase Type Distribution ◽  
Steady State Probability ◽  
Phase Type ◽  
Number Of Customers

We consider a single-server discrete queueing system in which arrivals occur according to a Markovian arrival process. Service is provided in groups of size no more than M customers. The service times are assumed to follow a discrete phase type distribution, whose representation may depend on the group size. Under a probabilistic service rule, which depends on the number of customers waiting in the queue, this system is studied as a Markov process. This type of queueing system is encountered in the operations of an automatic storage retrieval system. The steady-state probability vector is shown to be of (modified) matrix-geometric type. Efficient algorithmic procedures for the computation of the rate matrix, steady-state probability vector, and some important system performance measures are developed. The steady-state waiting time distribution is derived explicitly. Some numerical examples are presented.

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.

A revisit to queueing-inventory system with reservation, cancellation and common life time

OPSEARCH ◽  
2016 ◽  
Vol 54 (2) ◽  
pp. 336-350 ◽  
Author(s):  
A. Krishnamoorthy ◽  
Binitha Benny ◽  
Dhanya Shajin
Keyword(s):  
Life Time ◽  
Inventory System ◽  
Common Life ◽  
Common Life Time

The M/G/1 Retrial Queue With Retrial Rate Control Policy

1993 ◽  
Vol 7 (1) ◽  
pp. 29-46 ◽  
Author(s):  
Bong Dae Choi ◽  
Kyung Hyune Rhee ◽  
Kwang Kyu Park
Keyword(s):  
Retrial Queue ◽  
Control Policy ◽  
Single Server ◽  
Retrial Queueing System ◽  
Retrial Queueing ◽  
Virtual Waiting Time ◽  
The Laplace Transform ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Number Of Customers

We consider a single-server retrial queueing system where retrial time is inversely proportional to the number of customers in the system. A necessary and sufficient condition for the stability of the system is found. We obtain the Laplace transform of virtual waiting time and busy period. The transient distribution of the number of customers in the system is also obtained.

Stochastic decomposition in retrial queueing-inventory system

2020 ◽  
Vol 54 (1) ◽  
pp. 81-99 ◽  
Author(s):  
Dhanya Shajin ◽  
A. Krishnamoorthy
Keyword(s):  
Joint Distribution ◽  
Storage System ◽  
Retrial Queue ◽  
Form Solution ◽  
Inventory System ◽  
Product Form ◽  
Stochastic Decomposition ◽  
Retrial Queueing ◽  
Long Run ◽  
Number Of Customers

The purpose of this paper is to obtain product form solution for retrial – queueing – inventory system. We study an M/M/1 retrial queue with a storage system driven by an (s,S) policy. When server is idle, external arrivals enter directly to an orbit. Inventory replenishment lead time is exponentially distributed. The interval between two successive retrials is exponentially distributed and only the customer at the head of the orbit is permitted to access the server. No customer is allowed to join the orbit when the storage system is empty and also when the serer is busy. We first derive the stationary joint distribution of the queue length and the on-hand inventory in explicit product form. Using the joint distribution, we investigate long-run performance measures such as distribution of number of customers served, number of arrivals, number of customers lost during an interval of random duration and a cost function. The optimal pair (s,S) is numerically investigated.

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.

On a Markovian queue with weakly correlated interarrival times

1981 ◽  
Vol 18 (01) ◽  
pp. 190-203 ◽  
Author(s):  
Guy Latouche
Keyword(s):  
Time Distribution ◽  
Queueing System ◽  
Interarrival Time ◽  
Probability Vector ◽  
Common Value ◽  
Markovian Queue ◽  
The Stability ◽  
Number Of Customers ◽  
Stationary Probabilities ◽  
Stationary Probability Vector

A queueing system with exponential service and correlated arrivals is analysed. Each interarrival time is exponentially distributed. The parameter of the interarrival time distribution depends on the parameter for the preceding arrival, according to a Markov chain. The parameters of the interarrival time distributions are chosen to be equal to a common value plus a factor ofε, where ε is a small number. Successive arrivals are then weakly correlated. The stability condition is found and it is shown that the system has a stationary probability vector of matrix-geometric form. Furthermore, it is shown that the stationary probabilities for the number of customers in the system, are analytic functions ofε, for sufficiently smallε, and depend more on the variability in the interarrival time distribution, than on the correlations.

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