A Lévy-Driven Stochastic Queueing System with Server Breakdowns and Vacations

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.

Equilibrium analysis of cloud user request based on the Markov queue with variable vacation and vacation interruption

“…Wang et al [21] analyzed the equilibrium strategies of customers in terms of the overall welfare in an M/M/k queue with asynchronous or synchronous multiple vacations. Motivated by modeling the data transmission in telecommunication networks, Peng and Wu [14] studied a Lévy-driven stochastic queueing system where the server may be subject to breakdowns and repairs. Because introducing a vacation policy into the inventory system has more realistic significance, Padmavathi et al [13] considered two different vacation policies in the inventory queueing system, where the number of demands in the orbit and the joint probability distribution of the inventory level are obtained by using the method of matrix-geometric solution.…”

On the stability of Lévy driven stochastic jump systems. towards an unified approach for modeling and stabilizing impulsive and switched systems

“…Nevertheless, such systems can effectively be described based on their statistical properties. Examples of such applications include: biological phenomenon [42,12], queuing systems [36], financial/economic market dynamics [13,14]. Interestingly, also for the modeling and analysis of reinforcement learning algorithms [44], where the online learning of the value function can experience switching and impulsive dynamics.…”

Preface to the Special Issue “Mathematical Modeling in Industrial Engineering and Electrical Engineering”—Special Issue Book

“…The paper by Y. Peng and J. Wu [11] studies a stochastic fluid queuing system led by Lévy where the server is subject to failure and/or repair. By exploiting the joint use of the Lévy and Kella-Whitt processes, the limiting distribution of the workload process was achieved, also investigating the period occupied and the correlation structure.…”