Abstract
Next to the distribution of the (stationary and transient) workload, in queueing theory much attention is paid to the analysis of the distribution of the busy period. The question addressed in this chapter is, given the workload is in stationarity at time 0, how long does it take for the queue to idle? Explicit results in terms of Laplace transforms are presented. The last part of this chapter addresses the distribution of the minimal value attained by the workload process in an interval of given length.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Alili, L., Kyprianou, A.: Some remarks on first passage of Lévy processes, the American put and smooth pasting. Ann. Appl. Probab. 15, 2062–2080 (2004)
Asmussen, S.: Applied Probability and Queues, 2nd edn. Springer, New York (2003)
Darling, D., Liggett, T., Taylor, H.: Optimal stopping for partial sums. Ann. Math. Stat. 43, 1363–1368 (1972)
Dębicki, K., Kosiński, K., Mandjes, M.: On the infimum attained by a reflected Lévy process. Queueing Syst. 70, 23–35 (2012)
Kyprianou, A.: Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin (2006)
Mandjes, M., Palmowski, Z., Rolski, T.: Quasi-stationary workload of a Lévy-driven queue. Stoch. Mod. 28, 413–432 (2012)
Pecherskii, E., Rogozin, B.: On the joint distribution of random variables associated with fluctuations of a process with independent increments. Theory Probab. Appl. 14, 410–423 (1969)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Dębicki, K., Mandjes, M. (2015). Busy Period. In: Queues and Lévy Fluctuation Theory. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-20693-6_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-20693-6_6
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-20692-9
Online ISBN: 978-3-319-20693-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)