A FINITE CAPACITY RESEQUENCING MODEL WITH MARKOVIAN ARRIVALS

2005 ◽  
Vol 22 (03) ◽  
pp. 409-443 ◽  
Author(s):  
SRINIVAS R. CHAKRAVARTHY ◽  
STEFANKA CHUKOVA
Keyword(s):  
Steady State ◽  
Optimization Problems ◽  
Arrival Process ◽  
Finite Buffer ◽  
Finite Capacity ◽  
Departure Process ◽  
Queuing Model ◽  
Process Map ◽  
Steady State Analysis ◽  
Numerical Examples

In this paper, we consider a two-server finite capacity queuing model in which messages should leave the system in the order in which they entered the system. Messages arrive according to a Markovian arrival process (MAP) and any message finding the buffer full is considered lost. Out-of-sequence messages are stored in a (finite) buffer and may lead to blocking when a processed message cannot be placed in the buffer. The steady state analysis of the model is performed by exploiting the structure of the coefficient matrices. The departure process is characterized and two interesting optimization problems along with illustrative numerical examples are discussed.

scholarly journals Two parallel finite queues with simultaneous services and Markovian arrivals

1997 ◽  
Vol 10 (4) ◽  
pp. 383-405 ◽  
Author(s):  
S. R. Chakravarthy ◽  
S. Thiagarajan
Keyword(s):  
Markov Process ◽  
Performance Measures ◽  
System Performance ◽  
Arrival Process ◽  
Single Server ◽  
Queueing Model ◽  
Finite Capacity ◽  
Numerical Examples ◽  
Finite Queues ◽  
Large State Space

In this paper, we consider a finite capacity single server queueing model with two buffers, A and B, of sizes K and N respectively. Messages arrive one at a time according to a Markovian arrival process. Messages that arrive at buffer A are of a different type from the messages that arrive at buffer B. Messages are processed according to the following rules: 1. When buffer A(B) has a message and buffer B(A) is empty, then one message from A(B) is processed by the server. 2. When both buffers, A and B, have messages, then two messages, one from A and one from B, are processed simultaneously by the server. The service times are assumed to be exponentially distributed with parameters that may depend on the type of service. This queueing model is studied as a Markov process with a large state space and efficient algorithmic procedures for computing various system performance measures are given. Some numerical examples are discussed.

Steady State Analysis of Impulse Customers and Cancellation Policy in Queueing-Inventory System

Processes ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 2146
Author(s):  
V. Vinitha ◽  
N. Anbazhagan ◽  
S. Amutha ◽  
K. Jeganathan ◽  
Gyanendra Prasad Joshi ◽  
...  
Keyword(s):  
Steady State ◽  
Lead Time ◽  
Arrival Process ◽  
Inventory Level ◽  
Stationary Probability ◽  
Probability Vector ◽  
Steady State Analysis ◽  
Proposed Model ◽  
Stationary Probability Vector ◽  
Bernoulli Schedule

This article discusses the queueing-inventory model with a cancellation policy and two classes of customers. The two classes of customers are named ordinary and impulse customers. A customer who does not plan to buy the product when entering the system is called an impulse customer. Suppose the customer enters into the system to buy the product with a plan is called ordinary customer. The system consists of a pool of finite waiting areas of size N and maximum S items in the inventory. The ordinary customer can move to the pooled place if they find that the inventory is empty under the Bernoulli schedule. In such a situation, impulse customers are not allowed to enter into the pooled place. Additionally, the pooled customers buy the product whenever they find positive inventory. If the inventory level falls to s, the replenishment of Q items is to be replaced immediately under the (s, Q) ordering principle. Both arrival streams occur according to the independent Markovian arrival process (MAP), and lead time follows an exponential distribution. In addition, the system allows the cancellation of the purchased item only when there exist fewer than S items in the inventory. Here, the time between two successive cancellations of the purchased item is assumed to be exponentially distributed. The Gaver algorithm is used to obtain the stationary probability vector of the system in the steady-state. Further, the necessary numerical interpretations are investigated to enhance the proposed model.

scholarly journals Strong, Strongly Universal and Weak Interval Eigenvectors in Max-Plus Algebra

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1348
Author(s):  
Martin Gavalec ◽  
Ján Plavka ◽  
Daniela Ponce
Keyword(s):  
Steady State ◽  
Computer Network ◽  
Optimization Problems ◽  
Linear Equations ◽  
Recognition Algorithm ◽  
Interval Matrix ◽  
General Character ◽  
Discrete Events ◽  
Linear Combinations ◽  
Numerical Examples

The optimization problems, such as scheduling or project management, in which the objective function depends on the operations maximum and plus, can be naturally formulated and solved in max-plus algebra. A system of discrete events, e.g., activations of processors in parallel computing, or activations of some other cooperating machines, is described by a systems of max-plus linear equations. In particular, if the system is in a steady state, such as a synchronized computer network in data processing, then the state vector is an eigenvector of the system. In reality, the entries of matrices and vectors are considered as intervals. The properties and recognition algorithms for several types of interval eigenvectors are studied in this paper. For a given interval matrix and interval vector, a set of generators is defined. Then, the strong and the strongly universal eigenvectors are studied and described as max-plus linear combinations of generators. Moreover, a polynomial recognition algorithm is suggested and its correctness is proved. Similar results are presented for the weak eigenvectors. The results are illustrated by numerical examples. The results have a general character and can be applied in every max-plus algebra and every instance of the interval eigenproblem.

scholarly journals A Queueing System with Heterogeneous Impatient Customers and Consumable Additional Items

2017 ◽  
Vol 27 (2) ◽  
pp. 367-384 ◽  
Author(s):  
Janghyun Baek ◽  
Olga Dudina ◽  
Chesoong Kim
Keyword(s):  
Waiting Time ◽  
Queueing System ◽  
Optimization Problems ◽  
Markovian Arrival Process ◽  
Arrival Process ◽  
Single Server ◽  
Finite Buffer ◽  
Threshold Strategy

Abstract A single-server queueing system with a marked Markovian arrival process of heterogeneous customers is considered. Type-1 customers have limited preemptive priority over type-2 customers. There is an infinite buffer for type-2 customers and no buffer for type-1 customers. There is also a finite buffer (stock) for consumable additional items (semi-products, half-stocks, etc.) which arrive according to the Markovian arrival process. Service of a customer requires a fixed number of consumable additional items depending on the type of the customer. The service time has a phase-type distribution depending on the type of the customer. Customers in the buffer are impatient and may leave the system without service after an exponentially distributed amount of waiting time. Aiming to minimize the loss probability of type-1 customers and maximize throughput of the system, a threshold strategy of admission to service of type-2 customers is offered. Service of type-2 customer can start only if the server is idle and the number of consumable additional items in the stock exceeds the fixed threshold. Stationary distributions of the system states and the waiting time are computed. In the numerical example, we show some interesting effects and illustrate a possibility of application of the presented results for solution of optimization problems.

scholarly journals A finite capacity queue with Markovian arrivals and two servers with group services

1994 ◽  
Vol 7 (2) ◽  
pp. 161-178 ◽  
Author(s):  
S. Chakravarthy ◽  
Attahiru Sule Alfa
Keyword(s):  
Steady State ◽  
Waiting Time ◽  
Queue Length ◽  
Time Distribution ◽  
Arrival Process ◽  
Finite Capacity ◽  
Waiting Time Distribution ◽  
Phase Type ◽  
Stationary Waiting Time ◽  
Number Of Customers

In this paper we consider a finite capacity queuing system in which arrivals are governed by a Markovian arrival process. The system is attended by two exponential servers, who offer services in groups of varying sizes. The service rates may depend on the number of customers in service. Using Markov theory, we study this finite capacity queuing model in detail by obtaining numerically stable expressions for (a) the steady-state queue length densities at arrivals and at arbitrary time points; (b) the Laplace-Stieltjes transform of the stationary waiting time distribution of an admitted customer at points of arrivals. The stationary waiting time distribution is shown to be of phase type when the interarrival times are of phase type. Efficient algorithmic procedures for computing the steady-state queue length densities and other system performance measures are discussed. A conjecture on the nature of the mean waiting time is proposed. Some illustrative numerical examples are presented.

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