Semi-Markov Model of a Single-Server Queue with Losses and Maintenance of an Unreliable Server

Mostafaei

et al. 2020

Pak.j.stat.oper.res.

Semi-Markov processes can be considered as a generalization of both Markov and renewal processes. One of the principal characteristics of these processes is that in opposition to Markov models, they represent systems whose evolution is dependent not only on their last visited state but on the elapsed time since this state. Semi-Markov processes are replacing the exponential distribution of time intervals with an optional distribution. In this paper, we give a statistical approach to test the semi-Markov hypothesis. Moreover, we describe a Monte Carlo algorithm able to simulate the trajectories of the semi-Markov chain. This simulation method is used to test the semi-Markov model by comparing and analyzing the results with empirical data. We introduce the database of Network traffic which is employed for applying the Monte Carlo algorithm. The statistical characteristics of real and synthetic data from the models are compared. The comparison between the semi-Markov and the Markov models is done by computing the Autocorrelation functions and the probability density functions of the Network traffic real and simulated data as well. All the comparisons admit that the Markovian hypothesis is rejected in favor of the more general semi Markov one. Finally, the interval transition probabilities which show the future predictions of the Network traffic are given.

Section: Introductionmentioning

confidence: 99%

Testing the Semi Markov Model Using Monte Carlo Simulation Method for Predicting the Network Traffic

Mostafaei

et al. 2020

Pak.j.stat.oper.res.

“…It should be emphasized that the most complete results were obtained under the condition that the random factors describing unreliable queueing systems have exponential distributions. In [7,8], models of unreliable single-server queueing systems in the case of general-form distributions were constructed. The publications cited above do not claim to be a complete list of studies on the characterization of queueing systems with possible occurrence of server failures.…”

Section: Introductionmentioning

confidence: 99%

“…In a number of cases, it is possible to increase the efficiency of a queueing system by carrying out maintenance of its server(s). For example, in [9], a strategy to carry out preventive maintenance depending on the service time of a request was studied. In [10][11][12][13], a series of queueing system models with maintenance were constructed as follows: in [10], a model taking into account the server's operating time between failures and delayed maintenance; in [11], a model taking into account the system maintenance in case of hidden server failures; in [12], a model taking into account maintenance depending on the total operating time to failure; in [13], a model taking into account different types of server restoration.…”

Section: Introductionmentioning

confidence: 99%

“…For example, in [9], a strategy to carry out preventive maintenance depending on the service time of a request was studied. In [10][11][12][13], a series of queueing system models with maintenance were constructed as follows: in [10], a model taking into account the server's operating time between failures and delayed maintenance; in [11], a model taking into account the system maintenance in case of hidden server failures; in [12], a model taking into account maintenance depending on the total operating time to failure; in [13], a model taking into account different types of server restoration. In this paper, another maintenance strategy of an unreliable restorable single-server queueing system with losses is explored: the preventive maintenance of the server begins as soon as the accumulated operating time (the service time of all requests after completion of restoration works) exceeds an upper permissible threshold.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Stationary Characteristics of an Unreliable Single-Server Queueing System with Losses and Preventive Maintenance

Peschansky

2020

Autom Remote Control

Self Cite

This paper considers an unreliable restorable single-server queueing system with losses in which server failures may occur during operation. The random variables describing the system have general distributions. For increasing the efficiency of this system, it is proposed to carry out preventive maintenance of the server as soon as the accumulated operating time exceeds an upper permissible threshold. A semi-Markov model of the system's evolution over time is constructed. The explicit-form expressions for the final probabilities and mean sojourn times of the system in different physical states are derived using the stationary distribution of the embedded Markov chain. The frequency of preventive maintenance of the server is optimized via maximizing the average specific profit and minimizing the average specific costs of the system.

Multicriterial optimization of preventive maintenance of informational/technical stochastic system

Kovalenko

Rogachev

2016

ITM Web of Conferences

Abstract. The problem of optimization of reliability and economical indexes is solved for informational/technical system with periodical diagnostics and restoration. The system considered is the one with users accesses and errors occurring. The probability that the error occurs increases in time. Therefore, the longer is the restoration period the more there is a risk of system operation with error. But too frequent restoration can be too expensive. To take into account all these factors two-criteria optimization is made with the restriction of reliability characteristic being of the given level.