Abstract
Motivated by modelling the data transmission in computer communication networks, we study a Lévy-driven stochastic fluid queueing system where the server may subject to breakdowns and repairs. In addition, the server will leave for a vacation each time when the system is empty. We cast the workload process as a Lévy process modified to have random jumps at two classes of stopping times. By using the properties of Lévy processes and Kella–Whitt martingale method, we derive the limiting distribution of the workload process. Moreover, we investigate the busy period and the correlation structure. Finally, we prove that the stochastic decomposition properties also hold for fluid queues with Lévy input.
Highlights
Lévy processes are very important stochastic processes with stationary and independent increments
We give a detailed analysis of the system busy period and the queue’s correlation structure
We end this section by presenting the famous Kella–Whitt martingale associated with a spectrally positive Lévy process X with exponent φ(θ ) that will be very useful for the analysis the stationary distribution of the workload process of our fluid queue
Summary
Lévy processes are very important stochastic processes with stationary and independent increments. Kella and Whitt [11] first considered Lévy processes with secondary jump input which were applied to analyze queues with server vacations. Boxma and Kella [17] generalized known workload decomposition results for Lévy queues with secondary jump inputs and queues with server vacations or service interruptions. We consider a single-server Lévy-driven fluid queue with multiple vacations (exhaustive service) and a server subject to breakdowns and repairs. The principal purpose of the present paper is to apply the martingale results which were derived in Kella and Whitt [24] to investigate the stochastic dynamics of the system and realize an extensive analysis of the system from the transient virtual waiting time process to the steady-state distribution of the waiting time process. A summary of the results is presented in Conclusion
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