Abstract
The areas under the workload process and under the queueing process in a single-server queue over the busy period have many applications not only in queueing theory but also in risk theory or percolation theory. We focus here on the tail behaviour of distribution of these two integrals. We present various open problems and conjectures, which are supported by partial results for some special cases.
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Albrecher, H., Thonhauser, S.: Optimality results for dividend problems in insurance. RACSAM Rev. R. Acad. Cien. Ser. A. Mat. 103(2), 295–320 (2009)
Antunes, N., Fricker, C., Guillemin, F., Robert, P.: Perturbation analysis of the area swept under the queue length process of a variable M/M/1 queue. Performance (2005)
Avram, F., Palmowski, Z., Pistorius, M.: On the optimal dividend problem for a spectrally negative Lévy process. Ann. Appl. Probab. 17, 156–180 (2007)
Avram, F., Palmowski, Z., Pistorius, M.: Exit problem of a two-dimensional risk process from a cone: exact and asymptotic results. Ann. Appl. Probab. 18(6), 2421–2449 (2008)
Baltrunas, A., Daley, D.J., Klüppelberg, C.: Tail behaviour of the busy period of a GI/GI/1 queue with subexponential service times. Stoch. Process. Appl. 111, 237–258 (2004)
Bertoin, J., Yor, M.: On the entire moments of self-similar Markov processes and exponential functionals of Lévy processes. Ann. Fac. Sci. Toulouse Math. (6) 11(1), 33–45 (2002)
Bertoin, J., Biane, P., Yor, M.: Poissonian exponential functionals, q-series, q-integrals, and the moment problem for log-normal distributions. In: Seminar on Stochastic Analysis, Random Fields and Applications IV. Progr. Probab., vol. 58, pp. 45–56. Birkhäuser, Basel (2004)
Biard, R., Loisel, S., Maccib, C., Veraverbeke, N.: Asymptotic behavior of the finite-time expected time-integrated negative part of some risk processes and optimal reserve allocation. J. Math. Anal. Appl. 367(2), 535–549 (2010)
Borovkov, A.A., Boxma, O.J., Palmowski, Z.: On the integral of the workload process of the single server queue. J. Appl. Probab. 40, 200–225 (2003)
Carmona, P., Petit, F., Yor, M.: On the distribution and asymptotic results for exponential functionals of Lévy processes. In: Exponential Functionals and Principal Values Related to Brownian Motion. Bibl. Rev. Mat. Iberoamericana, pp. 73–130 (1997)
Duffy, K.R., Meyn, S.P.: Most likely paths to error when estimating the mean of a reflected random walk. Perform. Eval. 67(12), 1290–1303 (2010)
Goldie, C.M.: Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1(1), 126–166 (1991)
Guillemin, F., Pinchon, D.: On the area swept under the occupation process of an M/M/1 queue in a busy period. Queueing Syst. 29, 383–398 (1998)
Hult, H., Linskog, F.: Extremal behavior of stochastic integrals driven by regularly varying Lévy processes. Ann. Probab. 35, 309–339 (2007)
Jelenković, P.R., Momčilović, P.: Large deviations of square root insensitive random sums. Math. Oper. Res. 29, 398–406 (2004)
Kearney, M.J.: On a random area variable arising in discrete-time queues and compact directed percolation. J. Phys. A, Math. Gen. 37, 8421–8431 (2004)
Kesten, H.: Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248 (1973)
Kyprianou, E.K.: On the quasi-stationary distribution of the virtual waiting time in queues with Poisson arrivals. J. Appl. Probab. 8, 494–507 (1971)
Kyprianou, A., Loeffen, R.: Refracted Lévy processes. Ann. Inst. H. Poincaré Probab. Stat. 46, 24–44 (2010)
Kyprianou, A., Palmowski, Z.: Distributional study of De Finetti’s dividend problem for a general Lévy insurance risk process. J. Appl. Probab. 44(2), 428–443 (2007)
Kulik, R., Palmowski, Z.: Tail behaviour of the area under queue length process of a single server queue with regularly varying service times. Queueing Syst. 50, 299–323 (2005)
Lieshout, P., Mandjes, M.: Asymptotic analysis of Levy-driven tandem queues. Queueing Syst. 60, 203–226 (2008)
Lieshout, P., Mandjes, M.: Brownian tandem queues. Math. Methods Oper. Res. 66, 275–298 (2007)
Maulik, K., Zwart, B.: Tail asymptotics for exponential functionals of Lévy processes. Stoch. Process. Appl. 116, 156–177 (2006)
Meyn, S.P.: Large deviation asymptotics and control variates for simulating large functions. Ann. Appl. Probab. 16(1), 310–339 (2006)
Meyn, S.P.: Control Techniques for Complex Networks. Cambridge University Press, Cambridge (2007)
Schmidli, H.: Stochastic Control in Insurance. Springer, New York (2008)
Zwart, A.P.: Tail asymptotics for the busy period in the GI/G/1 queue. Math. Oper. Res. 26, 485–493 (2001)
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Kulik, R., Palmowski, Z. Tail behaviour of the area under a random process, with applications to queueing systems, insurance and percolations. Queueing Syst 68, 275–284 (2011). https://doi.org/10.1007/s11134-011-9242-1
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DOI: https://doi.org/10.1007/s11134-011-9242-1