Fluid Queue Driven by an M/M/1 Queue Subject to Catastrophes
In this paper, we present the stationary analysis of a fluid queueing model modulated by an \( M/M/1 \) queue subject to catastrophes. The explicit expressions for the joint probability of the state of the system and the content of the buffer under steady
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Abstract In this paper, we present the stationary analysis of a fluid queueing model modulated by an M=M=1 queue subject to catastrophes. The explicit expressions for the joint probability of the state of the system and the content of the buffer under steady state are obtained in terms of modified Bessel function of first kind using continued fraction methodology.
Keywords Buffer content distribution Continued fractions Laplace transform Modified Bessel function of first kind
1 Introduction In recent years, fluid queues have been widely accepted as appropriate models for modern telecommunication [4] and manufacturing systems [7]. This modelling approach ignores the discrete nature of the real information flow and treats it as a continuous stream. In addition, fluid models are often useful as approximate models for certain queueing and inventory systems where the flow consists of discrete entities, but the behaviour of individuals is not important to identify the performance analysis [3]. Steady-state behaviour of Markov-driven fluid queues has been extensively studied in the literature. Parthasarathy et al. [8] present an explicit expression for the buffer content distribution in terms of modified Bessel function of first kind using Laplace transforms and continued fractions. Silver Soares and Latouche [11] K. V. Vijayashree (&) A. Anjuka Department of Mathematics, Anna University, Chennai, India e-mail: [email protected] A. Anjuka e-mail: [email protected]
G. S. S. Krishnan et al. (eds.), Computational Intelligence, Cyber Security and Computational Models, Advances in Intelligent Systems and Computing 246, DOI: 10.1007/978-81-322-1680-3_31, Springer India 2014
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expressed the stationary distribution of a fluid queue with finite buffer as a linear combination of matrix exponential terms using matrix analytic methods. Besides, fluid queues also have successful applications in the field of congestion control [14] and risk processes [10]. Fluid models driven by an M=M=1=N queue with single and multiple exponential vacations were recently studied by Mao et al. [5, 6] using spectral method. In this paper, we analyse fluid queues driven by an M=M=1 queue subject to catastrophes. The effect of catastrophes in queueing models induces the system to be instantly reset to zero at some random times. Such models find a wide range of applications in diverse fields [12]. More specifically, birth–death stochastic models subject to catastrophes are popularly used in the study of population dynamics [1, 13], biological process [2], etc. With the arrival of a negative customer into the system (referred to as catastrophe), it induces the positive customers, if any, to immediately leave the system. For example, in computer systems which are not supported by power backup, the voltage fluctuations will lead to the system shut down momentarily and in that process all the work in progress and the jobs waiting to be completed will be lost and the system begins afresh. Such
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