scholarly journals Queuing-Inventory Models with MAP Demands and Random Replenishment Opportunities

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1092
Author(s):  
Srinivas R. Chakravarthy ◽  
B. Madhu Rao
Keyword(s):  
Point Process ◽  
Inventory Systems ◽  
Arrival Process ◽  
Inventory Level ◽  
Matrix Analytic Methods ◽  
Phase Type Distribution ◽  
Independent Point ◽  
Phase Type ◽  
Probabilistic Rule ◽  
Future Work

Combining the study of queuing with inventory is very common and such systems are referred to as queuing-inventory systems in the literature. These systems occur naturally in practice and have been studied extensively in the literature. The inventory systems considered in the literature generally include (s,S)-type. However, in this paper we look at opportunistic-type inventory replenishment in which there is an independent point process that is used to model events that are called opportunistic for replenishing inventory. When an opportunity (to replenish) occurs, a probabilistic rule that depends on the inventory level is used to determine whether to avail it or not. Assuming that the customers arrive according to a Markovian arrival process, the demands for inventory occur in batches of varying size, the demands require random service times that are modeled using a continuous-time phase-type distribution, and the point process for the opportunistic replenishment is a Poisson process, we apply matrix-analytic methods to study two of such models. In one of the models, the customers are lost when at arrivals there is no inventory and in the other model, the customers can enter into the system even if the inventory is zero but the server has to be busy at that moment. However, the customers are lost at arrivals when the server is idle with zero inventory or at service completion epochs that leave the inventory to be zero. Illustrative numerical examples are presented, and some possible future work is highlighted.

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 Descriptors for point processes based on runs: the Markovian arrival process

1996 ◽  
Vol 9 (4) ◽  
pp. 469-488 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Point Process ◽  
Point Processes ◽  
General Purpose ◽  
Markovian Arrival Process ◽  
The Other ◽  
Arrival Process ◽  
Qualitative Behavior ◽  
Regular Grid ◽  
Matrix Analytic Methods ◽  
Process Map

This paper is part of a broader investigation of properties of a point process that can be identified by imagining that the process is involved in a competition for the generation of runs of events. The general purpose of that methodology is to quantify the prevalence of gaps and bursts in realizations of the process. The Markovian arrival process (MAP) is highly versatile in qualitative behavior and its analysis is numerically tractable by matrix-analytic methods. It can therefore serve well as a benchmark process in that investigation. In this paper, we consider the MAP and a regular grid competing for runs of lengths at least r1 and r2, respectively. A run of length r in one of the processes is defined as a string of r successive events occurring without an intervening event in the other process.This article is dedicated to the memory of Roland L. Dobrushin.

scholarly journals Optimization of Queueing Model with Server Heating and Cooling

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 768 ◽  
Author(s):  
Olga Dudina ◽  
Alexander Dudin
Keyword(s):  
Data Centers ◽  
Threshold Value ◽  
Optimal Choice ◽  
Loss Probability ◽  
Arrival Process ◽  
World Systems ◽  
Phase Type Distribution ◽  
Type Distribution ◽  
Heating And Cooling ◽  
Phase Type

The operation of many real-world systems, e.g., servers of data centers, is accompanied by the heating of a server. Correspondingly, certain cooling mechanisms are used. If the server becomes overheated, it interrupts processing of customers and needs to be cooled. A customer is lost when its service is interrupted. To prevent overheating and reduce the customer loss probability, we suggest temporal termination of service of new customers when the temperature of the server reaches the predefined threshold value. Service is resumed after the temperature drops below another threshold value. The problem of optimal choice of the thresholds (with respect to the chosen economical criterion) is numerically solved under quite general assumptions about the parameters of the system (Markovian arrival process, phase-type distribution of service time, and accounting for customers impatience). Numerical examples are presented.

scholarly journals Analysis of an MMAP/PH1, PH2/N/∞ queueing system operating in a random environment

2014 ◽  
Vol 24 (3) ◽  
pp. 485-501 ◽  
Author(s):  
Chesoong Kim ◽  
Alexander Dudin ◽  
Sergey Dudin ◽  
Olga Dudina
Keyword(s):  
Random Environment ◽  
Queueing System ◽  
Arrival Process ◽  
Service Time Distribution ◽  
Contact Center ◽  
Phase Type Distribution ◽  
Phase Type ◽  
Multi Server

Abstract A multi-server queueing system with two types of customers and an infinite buffer operating in a random environment as a model of a contact center is investigated. The arrival flow of customers is described by a marked Markovian arrival process. Type 1 customers have a non-preemptive priority over type 2 customers and can leave the buffer due to a lack of service. The service times of different type customers have a phase-type distribution with different parameters. To facilitate the investigation of the system we use a generalized phase-type service time distribution. The criterion of ergodicity for a multi-dimensional Markov chain describing the behavior of the system and the algorithm for computation of its steady-state distribution are outlined. Some key performance measures are calculated. The Laplace-Stieltjes transforms of the sojourn and waiting time distributions of priority and non-priority customers are derived. A numerical example illustrating the importance of taking into account the correlation in the arrival process is presented

A Note on Characterizing Service Interruptions with Phase-Type Distribution

2013 ◽  
Vol 31 (4) ◽  
pp. 671-683 ◽  
Author(s):  
A. Krishnamoorthy ◽  
P. K. Pramod ◽  
S. R. Chakravarthy
Keyword(s):  
Phase Type Distribution ◽  
Type Distribution ◽  
Service Interruptions ◽  
Phase Type

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.

scholarly journals A generalized class of correlated run shock models

2018 ◽  
Vol 6 (1) ◽  
pp. 131-138 ◽  
Author(s):  
Femin Yalcin ◽  
Serkan Eryilmaz ◽  
Ali Riza Bozbulut
Keyword(s):  
Stopping Time ◽  
Random Variables ◽  
Random Variable ◽  
Phase Type Distribution ◽  
Type Distribution ◽  
Shock Models ◽  
Phase Type ◽  
Over Time

AbstractIn this paper, a generalized class of run shock models associated with a bivariate sequence {(Xi, Yi)}i≥1 of correlated random variables is defined and studied. For a system that is subject to shocks of random magnitudes X1, X2, ... over time, let the random variables Y1, Y2, ... denote times between arrivals of successive shocks. The lifetime of the system under this class is defined through a compound random variable T = ∑Nt=1 Yt , where N is a stopping time for the sequence {Xi}i≤1 and represents the number of shocks that causes failure of the system. Another random variable of interest is the maximum shock size up to N, i.e. M = max {Xi, 1≤i≤ N}. Distributions of T and M are investigated when N has a phase-type distribution.

A closure characterisation of phase-type distributions

1992 ◽  
Vol 29 (01) ◽  
pp. 92-103 ◽  
Author(s):  
Robert S. Maier ◽  
Colm Art O'Cinneide
Keyword(s):  
Analogous Result ◽  
Finite Automata ◽  
Automata Theory ◽  
Phase Type Distribution ◽  
Exponential Distributions ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Discrete Phase ◽  
Rational Sequence ◽  
Family Of Distributions

We characterise the classes of continuous and discrete phase-type distributions in the following way. They are known to be closed under convolutions, mixtures, and the unary ‘geometric mixture' operation. We show that the continuous class is the smallest family of distributions that is closed under these operations and contains all exponential distributions and the point mass at zero. An analogous result holds for the discrete class. We also show that discrete phase-type distributions can be regarded as ℝ+-rational sequences, in the sense of automata theory. This allows us to view our characterisation of them as a corollary of the Kleene–Schützenberger theorem on the behavior of finite automata. We prove moreover that any summable ℝ+-rational sequence is proportional to a discrete phase-type distribution.

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