Stochastic monotonicity approach for a non-Markovian priority retrial queue

2021 ◽  
pp. 2150156
Author(s):  
Mohamed Boualem ◽  
Nassim Touche
Keyword(s):  
Markov Chain ◽  
General Theory ◽  
Joint Distribution ◽  
Retrial Queue ◽  
Stochastic Orders ◽  
Embedded Markov Chain ◽  
Stochastic Monotonicity

This paper considers a non-Markovian priority retrial queue which serves two types of customers. Customers in the regular queue have priority over the customers in the orbit. This means that the customer in orbit can only start retrying when the regular queue becomes empty. If another customer arrives during a retrial time, this customer is served and the retrial has to start over when the regular queue becomes empty again. In this study, a particular interest is devoted to the stochastic monotonicity approach based on the general theory of stochastic orders. Particularly, we derive insensitive bounds for the stationary joint distribution of the embedded Markov chain of the considered system.

scholarly journals Insensitive Bounds for the Stationary Distribution of a Single Server Retrial Queue with Server Subject to Active Breakdowns

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Mohamed Boualem
Keyword(s):  
Markov Chain ◽  
Stationary Distribution ◽  
Retrial Queue ◽  
Embedded Markov Chain ◽  
Single Server ◽  
Waiting Room ◽  
Monotonicity Properties

The paper addresses monotonicity properties of the single server retrial queue with no waiting room and server subject to active breakdowns. The obtained results allow us to place in a prominent position the insensitive bounds for the stationary distribution of the embedded Markov chain related to the model in the study. Numerical illustrations are provided to support the results.

scholarly journals Stochastic analysis of a single server unreliable queue with balking and general retrial time

2020 ◽  
Vol 28 (4) ◽  
pp. 319-326
Author(s):  
Mohamed Boualem
Keyword(s):  
Markov Chain ◽  
Cognitive Radio ◽  
Stochastic Analysis ◽  
Manufacturing Systems ◽  
Cognitive Radio Network ◽  
Stochastic Orders ◽  
Embedded Markov Chain ◽  
Single Server ◽  
Radio Network ◽  
If Current

In this investigation, we consider an M/G/1 queue with general retrial times allowing balking and server subject to breakdowns and repairs. In addition, the customer whose service is interrupted can stay at the server waiting for repair or leave and return while the server is being repaired. The server is not allowed to begin service on other customers until the current customer has completed service, even if current customer is temporarily absent. This model has a potential application in various fields, such as in the cognitive radio network and the manufacturing systems, etc. The methodology is strongly based on the general theory of stochastic orders. Particularly, we derive insensitive bounds for the stationary distribution of the embedded Markov chain of the considered system.

scholarly journals Bounds of the stationary distribution in M/G/1 retrial queue with two-way communication and n types of outgoing calls

2019 ◽  
Vol 29 (3) ◽  
pp. 375-391
Author(s):  
Lala Alem ◽  
Mohamed Boualem ◽  
Djamil Aissani
Keyword(s):  
Markov Chain ◽  
Stationary Distribution ◽  
Performance Measures ◽  
Time Distribution ◽  
Retrial Queue ◽  
Main Idea ◽  
Embedded Markov Chain ◽  
Stochastic Comparison ◽  
Service Time Distribution ◽  
Comparison Approach

In this article we analyze the M=G=1 retrial queue with two-way communication and n types of outgoing calls from a stochastic comparison viewpoint. The main idea is that given a complex Markov chain that cannot be analyzed numerically, we propose to bound it by a new Markov chain, which is easier to solve by using a stochastic comparison approach. Particularly, we study the monotonicity of the transition operator of the embedded Markov chain relative to the stochastic and convex orderings. Bounds are also obtained for the stationary distribution of the embedded Markov chain at departure epochs. Additionally, the performance measures of the considered system can be estimated by those of an M=M=1 retrial queue with two-way communication and n types of outgoing calls when the service time distribution is NBUE (respectively, NWUE). Finally, we test numerically the accuracy of the proposed bounds.

Loss Probability of a Burst Arrival Finite Queue with Synchronized Service

1992 ◽  
Vol 6 (2) ◽  
pp. 201-216 ◽  
Author(s):  
Masakiyo Miyazawa
Keyword(s):  
Markov Chain ◽  
Loss Probability ◽  
Buffer Size ◽  
Embedded Markov Chain ◽  
Main Concern ◽  
Single Server ◽  
Finite Buffer ◽  
Single Server Queue ◽  
Server Queue ◽  
Computation Algorithm

We are concerned with a burst arrival single-server queue, where arrivals of cells in a burst are synchronized with a constant service time. The main concern is with the loss probability of cells for the queue with a finite buffer. We analyze an embedded Markov chain at departure instants of cells and get a kind of lumpability for its state space. Based on these results, this paper proposes a computation algorithm for its stationary distribution and the loss probability. Closed formulas are obtained for the first two moments of the numbers of cells and active bursts when the buffer size is infinite.

Analysis of an M/G/1 queue with two types of impatient units

1995 ◽  
Vol 27 (03) ◽  
pp. 840-861 ◽  
Author(s):  
M. Martin ◽  
J. R. Artalejo
Keyword(s):  
Markov Chain ◽  
Performance Measures ◽  
Mathematical Analysis ◽  
Service System ◽  
Probability Distributions ◽  
Embedded Markov Chain ◽  
Renewal Processes ◽  
Stationary Probability ◽  
Markov Renewal Processes ◽  
Specific Performance

This paper deals with a service system in which the processor must serve two types of impatient units. In the case of blocking, the first type units leave the system whereas the second type units enter a pool and wait to be processed later. We develop an exhaustive analysis of the system including embedded Markov chain, fundamental period and various classical stationary probability distributions. More specific performance measures, such as the number of lost customers and other quantities, are also considered. The mathematical analysis of the model is based on the theory of Markov renewal processes, in Markov chains of M/G/l type and in expressions of ‘Takács' equation' type.

A stable algorithm for stationary distribution calculation for a BMAP/SM/1 queueing system with Markovian arrival input of disasters

2004 ◽  
Vol 41 (02) ◽  
pp. 547-556 ◽  
Author(s):  
Alexander Dudin ◽  
Olga Semenova
Keyword(s):  
Markov Chain ◽  
Stationary State ◽  
Stationary Distribution ◽  
Queueing System ◽  
Embedded Markov Chain ◽  
State Distribution ◽  
Stable Algorithm ◽  
Distribution Calculation

Disaster arrival into a queueing system causes all customers to leave the system instantaneously. We present a numerically stable algorithm for calculating the stationary state distribution of an embedded Markov chain for the BMAP/SM/1 queue with a MAP input of disasters.

Ergodicity of age structure in populations with Markovian vital rates. II. General states

1977 ◽  
Vol 9 (01) ◽  
pp. 18-37 ◽  
Author(s):  
Joel E. Cohen
Keyword(s):  
Markov Chain ◽  
Age Structure ◽  
Joint Distribution ◽  
Sample Path ◽  
Initial Distribution ◽  
Transition Function ◽  
Leslie Matrix ◽  
Vital Rates ◽  
Closed Population ◽  
Leslie Matrices

The age structure of a large, unisexual, closed population is described here by a vector of the proportions in each age class. Non-negative matrices of age-specific birth and death rates, called Leslie matrices, map the age structure at one point in discrete time into the age structure at the next. If the sequence of Leslie matrices applied to a population is a sample path of an ergodic Markov chain, then: (i) the joint process consisting of the age structure vector and the Leslie matrix which produced that age structure is a Markov chain with explicit transition function; (ii) the joint distribution of age structure and Leslie matrix becomes independent of initial age structure and of the initial distribution of the Leslie matrix after a long time; (iii) when the Markov chain governing the Leslie matrix is homogeneous, the joint distribution in (ii) approaches a limit which may be easily calculated as the solution of a renewal equation. A numerical example will be given in Cohen (1977).

Analysis of an M/G/1 queue with two types of impatient units

1995 ◽  
Vol 27 (3) ◽  
pp. 840-861 ◽  
Author(s):  
M. Martin ◽  
J. R. Artalejo
Keyword(s):  
Markov Chain ◽  
Performance Measures ◽  
Mathematical Analysis ◽  
Service System ◽  
Probability Distributions ◽  
Embedded Markov Chain ◽  
Renewal Processes ◽  
Stationary Probability ◽  
Markov Renewal Processes ◽  
Specific Performance

This paper deals with a service system in which the processor must serve two types of impatient units. In the case of blocking, the first type units leave the system whereas the second type units enter a pool and wait to be processed later.We develop an exhaustive analysis of the system including embedded Markov chain, fundamental period and various classical stationary probability distributions. More specific performance measures, such as the number of lost customers and other quantities, are also considered. The mathematical analysis of the model is based on the theory of Markov renewal processes, in Markov chains of M/G/l type and in expressions of ‘Takács' equation' type.

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