Abstract
In this chapter we characterize (in terms of transforms) the distribution of the transient workload, that is, the workload Q t at some time t ≥ 0, conditional on the initial workload Q 0 = x ≥ 0. Again we distinguish between the spectrally one-sided case and the two-sided case. Rather explicit expressions are presented for the transform of Q t in the spectrally positive case, and for the density of Q t (in terms of scale functions) in the spectrally negative case. The chapter is concluded by an analysis of the two-sided case, relying, as in Chapter 3, on Wiener–Hopf-type arguments.
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References
Asmussen, S.: Applied Probability and Queues, 2nd edn. Springer, New York (2003)
Beneš, V.: On queues with Poisson arrivals. Ann. Math. Stat. 3, 670–677 (1957)
Bertoin, J.: Regularity of the half-line for Lévy processes. Bull. Sci. Math. 121, 345–354 (1997)
Doney, R.: Some excursion calculations for spectrally one-sided Lévy processes. Séminaire de Probabilités XXXVIII, 5–15 (2005)
Harrison, J.: Brownian Motion and Stochastic Flow Systems. Wiley, New York (1985)
Kella, O., Boxma, O., Mandjes, M.: A Lévy process reflected at a Poisson age process. J. Appl. Probab. 43, 221–230 (2006)
Kella, O., Mandjes, M.: Transient analysis of reflected Lévy processes. Stat. Probab Lett. 83, 2308–2315 (2013)
Kyprianou, A.: Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin (2006)
Pistorius, M.: On exit and ergodicity of the spectrally one-sided Lévy process reflected at its infimum. J. Theor. Probab. 17, 183–220 (2004)
Rogers, L.: Evaluating first-passage probabilities for spectrally one-sided Lévy processes. J. Appl. Probab. 37, 1173–1180 (2000)
Suprun, V.: Ruin problem and the resolvent of a terminating process with independent increments. Ukr. Mat. Zh. 28, 53–61 (1976)
Surya, B.: Evaluating scale functions of spectrally negative Lévy processes. J. Appl. Probab. 45, 135–149 (2008)
Takács, L.: Investigations of waiting time problems by reduction to Markov processes. Acta Math. Acad. Sci. Hung. 6, 101–129 (1955)
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Dębicki, K., Mandjes, M. (2015). Transient Workload. In: Queues and Lévy Fluctuation Theory. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-20693-6_4
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DOI: https://doi.org/10.1007/978-3-319-20693-6_4
Publisher Name: Springer, Cham
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