scholarly journals Retrial Queues with Unreliable Servers and Delayed Feedback

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2415
Author(s):  
Agassi Melikov ◽  
Sevinj Aliyeva ◽  
Janos Sztrik
Keyword(s):  
Steady State ◽  
Performance Measures ◽  
Numerical Experiments ◽  
Three Dimensional ◽  
Delayed Feedback ◽  
Retrial Queues ◽  
Bernoulli Feedback ◽  
Bernoulli Scheme ◽  
Multi Server ◽  
Steady State Probabilities

In this paper, models of unreliable multi-server retrial queues with delayed feedback are examined. The Bernoulli retrial is allowed upon the arrival of both primary (from outside) and feedback customers (from orbit), as well as the Bernoulli feedback that may occur after each service in this system. Servers can break down both during the service of customers and when they are idle. If a server breaks down during the service of a customer, then the interrupted customer, in accordance with the Bernoulli scheme, decides either to leave the system or join a common orbit of retrial and feedback customers. An approximate method, based on the space merging approach of three-dimensional Markov chains, is proposed for the calculation of the steady-state probabilities, as well as performance measures of the system. The results of the numerical experiments are demonstrated.

scholarly journals Analysis of Queueing System MMPP/M/K/K with Delayed Feedback

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1128 ◽  
Author(s):  
Agassi Melikov ◽  
Sevinj Aliyeva ◽  
Janos Sztrik
Keyword(s):  
Numerical Experiments ◽  
Queueing System ◽  
Three Dimensional ◽  
Delayed Feedback ◽  
State Dependent ◽  
Repeated Calls ◽  
Markov Modulated Poisson Process ◽  
Markov Modulated ◽  
Steady State Probabilities ◽  
Dependent Probability

The model of multi-channel queuing system with Markov modulated Poisson process (MMPP) flow and delayed feedback is considered. After the customer is served completely, they will decide either to join the retrial group again for another service (feedback) with some state-dependent probability or to leave the system forever with complimentary probability. Feedback calls organize an orbit of repeated calls (r-calls). If upon arrival of an r-call all the channels of the system are busy, then it either leaves the system with some state-dependent probability or with a complementary probability returns to orbit. Methods to calculate the steady-state probabilities of the appropriate three-dimensional Markov chain as well as performance measures of investigated system are developed. Results of numerical experiments are demonstrated.

A single server Markovian queuing system with limited buffer and reverse balking

2021 ◽  
Vol 12 (7) ◽  
pp. 1774-1784
Author(s):  
Girin Saikia ◽  
Amit Choudhury
Keyword(s):  
Steady State ◽  
Performance Measures ◽  
Mutual Fund ◽  
System Size ◽  
Queuing System ◽  
Single Server ◽  
Insurance Company ◽  
Number Of Customers ◽  
Steady State Probabilities

The phenomena are balking can be said to have been observed when a customer who has arrived into queuing system decides not to join it. Reverse balking is a particular type of balking wherein the probability that a customer will balk goes down as the system size goes up and vice versa. Such behavior can be observed in investment firms (insurance company, Mutual Fund Company, banks etc.). As the number of customers in the firm goes up, it creates trust among potential investors. Fewer customers would like to balk as the number of customers goes up. In this paper, we develop an M/M/1/k queuing system with reverse balking. The steady-state probabilities of the model are obtained and closed forms of expression of a number of performance measures are derived.

scholarly journals Analysis of M/M/1 Queuing System with Customer Reneging During Server Vacations Subject To Server Breakdown and Delayed Repair

2018 ◽  
Vol 7 (4.10) ◽  
pp. 552
Author(s):  
Ch. Swathi ◽  
V. Vasanta Kumar
Keyword(s):  
Steady State ◽  
Closed Form ◽  
Detailed Analysis ◽  
Performance Measures ◽  
Queuing System ◽  
Multiple Vacation ◽  
Delayed Repair ◽  
Server Breakdown ◽  
Number Of Customers ◽  
Steady State Probabilities

In this paper, we consider an M/M/1 queuing system with customer reneging for an unreliable sever. Customer reneging is assumed to occur due to the absence of the server during vacations.  Detailed analysis for both single and multiple vacation models during different states of the server such as busy, breakdown and delayed repair periods is presented. Steady state probabilities for single and multiple vacation policies are obtained. Closed form expressions for various performance measures such as average number of customers in the system, proportion of customers served and reneged per unit time during single and multiple vacations are obtained.   

The steady state of a multi-server mixed queue

1973 ◽  
Vol 5 (03) ◽  
pp. 614-631 ◽  
Author(s):  
N. B. Slater ◽  
T. C. T. Kotiah
Keyword(s):  
Steady State ◽  
Statistical Mechanics ◽  
Generating Functions ◽  
Queueing System ◽  
Linear Equations ◽  
Weighted Sum ◽  
System Of Linear Equations ◽  
Different Types ◽  
Multi Server ◽  
Steady State Probabilities

In a multi-server queueing system in which the customers are of several different types, it is useful to define states which specify the types of customers being served as well as the total number present. Analogies with some problems in statistical mechanics are found fruitful. Certain generating functions are defined in such a way that they satisfy a system of linear equations. Solution of the associated eigenvector problem shows that the steady-state probabilities for states in which all the servers are busy can be represented by a weighted sum of geometric probabilities.

Approximations for the steady-state probabilities in the M/G/c queue

1981 ◽  
Vol 13 (1) ◽  
pp. 186-206 ◽  
Author(s):  
H. C. Tijms ◽  
M. H. Van Hoorn ◽  
A. Federgruen
Keyword(s):  
Steady State ◽  
Queue Size ◽  
Numerical Experience ◽  
General Service Times ◽  
Server Queue ◽  
Poisson Arrivals ◽  
The Mean ◽  
Multi Server ◽  
General Service ◽  
Steady State Probabilities

For the multi-server queue with Poisson arrivals and general service times we present various approximations for the steady-state probabilities of the queue size. These approximations are computed from numerically stable recursion schemes which can be easily applied in practice. Numerical experience reveals that the approximations are very accurate with errors typically below 5%. For the delay probability the various approximations result either into the widely used Erlang delay probability or into a new approximation which improves in many cases the Erlang delay probability approximation. Also for the mean queue size we find a new approximation that turns out to be a good approximation for all values of the queueing parameters including the coefficient of variation of the service time.

Approximations for the steady-state probabilities in the M/G/c queue

1981 ◽  
Vol 13 (01) ◽  
pp. 186-206 ◽  
Author(s):  
H. C. Tijms ◽  
M. H. Van Hoorn ◽  
A. Federgruen
Keyword(s):  
Steady State ◽  
Queue Size ◽  
Numerical Experience ◽  
General Service Times ◽  
Server Queue ◽  
Poisson Arrivals ◽  
The Mean ◽  
Multi Server ◽  
General Service ◽  
Steady State Probabilities

For the multi-server queue with Poisson arrivals and general service times we present various approximations for the steady-state probabilities of the queue size. These approximations are computed from numerically stable recursion schemes which can be easily applied in practice. Numerical experience reveals that the approximations are very accurate with errors typically below 5%. For the delay probability the various approximations result either into the widely used Erlang delay probability or into a new approximation which improves in many cases the Erlang delay probability approximation. Also for the mean queue size we find a new approximation that turns out to be a good approximation for all values of the queueing parameters including the coefficient of variation of the service time.

scholarly journals Analysis of Instantaneous Feedback Queue with Heterogeneous Servers

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2186
Author(s):  
Agassi Melikov ◽  
Sevinj Aliyeva ◽  
János Sztrik
Keyword(s):  
High Speed ◽  
Three Dimensional ◽  
Threshold Value ◽  
High Accuracy ◽  
Approximate Algorithms ◽  
Heterogeneous Servers ◽  
Bernoulli Scheme ◽  
Markov Modulated ◽  
Steady State Probabilities ◽  
Feedback Queue

A system with heterogeneous servers, Markov Modulated Poisson flow and instantaneous feedback is studied. The primary call is serviced on a high-speed server, and after it is serviced, each call, according to the Bernoulli scheme, either leaves the system or requires re-servicing. After the completion of servicing of a call in a slow server, according to the Bernoulli scheme, it also either leaves the system or requires re-servicing. If upon arrival of a primary call the queue length of such calls exceeds a certain threshold value and the slow server is free, then the incoming primary call, according to the Bernoulli scheme, is either sent to the slow server or joins its own queue. A mathematical model of the studied system is constructed in the form of a three-dimensional Markov chain. Approximate algorithms for calculating the steady-state probabilities of the models with finite and infinite queues are proposed and their high accuracy is shown. The results of numerical experiments are presented.

scholarly journals M/M/1 Queueing system with Customer Balking and Reneging

2019 ◽  
Vol 8 (9) ◽  
pp. 2636-2646
Keyword(s):  
Steady State ◽  
Performance Measures ◽  
Generating Functions ◽  
Queueing System ◽  
Probability Generating Functions ◽  
Multiple Vacation ◽  
Steady State Probabilities ◽  
Explicit Expressions

This paper deals with an M/M/1 queueing system with customer balking and reneging. Balking and reneging of the customers are assumed to occur due to non-availability of the server during vacation and breakdown periods. Steady state probabilities for both the single and multiple vacation scenarios are obtained by employing probability generating functions. We evaluate the explicit expressions for various performance measures of the queueing system.

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