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.

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.

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.

A BMAP/PH/N SYSTEM WITH IMPATIENT REPEATED CALLS

2007 ◽  
Vol 24 (03) ◽  
pp. 293-312 ◽  
Author(s):  
VALENTINA I. KLIMENOK ◽  
DMITRY S. ORLOVSKY ◽  
ALEXANDER N. DUDIN
Keyword(s):  
Markov Chain ◽  
Time Distribution ◽  
Arrival Process ◽  
Service Time Distribution ◽  
State Distribution ◽  
Unsuccessful Attempt ◽  
Batch Markovian Arrival Process ◽  
Repeated Calls ◽  
Phase Type ◽  
Multi Server

A multi-server queueing model with a Batch Markovian Arrival Process, phase-type service time distribution and impatient repeated customers is analyzed. After any unsuccessful attempt, the repeated customer leaves the system with the fixed probability. The behavior of the system is described in terms of continuous time multi-dimensional Markov chain. Stability condition and an algorithm for calculating the stationary state distribution of this Markov chain are obtained. Main performance measures of the system are calculated. Numerical results are presented.

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.

Insensitivity in queueing systems

1981 ◽  
Vol 13 (04) ◽  
pp. 846-859 ◽  
Author(s):  
David Y. Burman
Keyword(s):  
Stationary Distribution ◽  
Service Time ◽  
Stochastic Models ◽  
Time Distribution ◽  
Queueing Systems ◽  
Sufficient Conditions ◽  
Service Time Distribution ◽  
Flow Equations

It is well known that the stationary distribution of the number of busy servers in the Erlang blocking system (M/G/c/c) depends on the service-time distribution only through its mean. This insensitivity property is shared by several other queueing systems. In this paper, we give simple sufficient conditions for determining if this insensitivity property holds for general queueing systems and related stochastic models. The conditions involve determining whether the solution of the stationary Markovian flow equations also solves certain restricted flow equations. The proof that these conditions are sufficient is direct and elementary.

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

2004 ◽  
Vol 41 (2) ◽  
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.

Comparison results for M/G/1 queues with waiting and sojourn time deadlines

2019 ◽  
Vol 56 (2) ◽  
pp. 524-532 ◽  
Author(s):  
Yoshiaki Inoue
Keyword(s):  
Performance Measures ◽  
Time Distribution ◽  
Waiting Times ◽  
Loss Probability ◽  
The Other ◽  
Service Time Distribution ◽  
Impatient Customers ◽  
Comparison Results ◽  
New Better Than Used ◽  
Better Than

AbstractThis paper considers two variants of M/G/1 queues with impatient customers, which are denoted by M/G/1+Gw and M/G/1+Gs. In the M/G/1+Gw queue customers have deadlines for their waiting times, and they leave the system immediately if their services do not start before the expiration of their deadlines. On the other hand, in the M/G/1+Gs queue customers have deadlines for their sojourn times, where customers in service also immediately leave the system when their deadlines expire. In this paper we derive comparison results for performance measures of these models. In particular, we show that if the service time distribution is new better than used in expectation, then the loss probability in the M/G/1+Gs queue is greater than that in the M/G/1+Gw queue.

Insensitivity in queueing systems

1981 ◽  
Vol 13 (4) ◽  
pp. 846-859 ◽  
Author(s):  
David Y. Burman
Keyword(s):  
Stationary Distribution ◽  
Service Time ◽  
Stochastic Models ◽  
Time Distribution ◽  
Queueing Systems ◽  
Sufficient Conditions ◽  
Service Time Distribution ◽  
Flow Equations

It is well known that the stationary distribution of the number of busy servers in the Erlang blocking system (M/G/c/c) depends on the service-time distribution only through its mean. This insensitivity property is shared by several other queueing systems. In this paper, we give simple sufficient conditions for determining if this insensitivity property holds for general queueing systems and related stochastic models. The conditions involve determining whether the solution of the stationary Markovian flow equations also solves certain restricted flow equations. The proof that these conditions are sufficient is direct and elementary.

scholarly journals Queueing system with passive servers

1996 ◽  
Vol 9 (2) ◽  
pp. 185-204 ◽  
Author(s):  
Alexander N. Dudin ◽  
Valentina I. Klimenok
Keyword(s):  
Functional Equation ◽  
Markov Chain ◽  
Response Time ◽  
Stationary Distribution ◽  
Random Environment ◽  
Queueing System ◽  
Sufficient Conditions ◽  
Special Kind ◽  
Embedded Markov Chain ◽  
Matrix Analog

In this paper the authors introduce systems in which customers are served by one active server and a group of passive servers. The calculation of response time for such systems is rendered by analyzing a special kind of queueing system in a synchronized random environment. For an embedded Markov chain, sufficient conditions for the existence of a stationary distribution are proved. A formula for the corresponding vector generating function is obtained. It is a matrix analog of the Pollaczek-Khinchin formula and is simultaneously a matrix functional equation. A method for solving this equation is proposed.

Sign in / Sign up

Export Citation Format

Share Document