Abstract
We consider asymptotic expansions for defective and excessive renewal equations that are close to being proper. These expansions are applied to the analysis of processor sharing queues and perturbed risk models, and yield approximations that can be useful in applications where moments are computable, but the distribution is not.
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References
Abate J, Whitt W (1994) A Heavy-traffic expansion for the asymptotic decay rates of tail probabilities in multi-channel queues. Oper Res Lett 15: 223–230
Blanchet J, Glynn P (2007) Uniform renewal theory with applications to expansions of geometric sums. Adv Appl Probab 39: 1070–1097
Borovkov A, Foss S (2000) Estimates for overshooting and arbitrary boundary by a random walk and their applications. Theor Probab Appl 44: 231–253
Dufresne F, Gerber H (1991) Risk theory for the compound Poisson process that is perturbed by diffusion. Insur Math Econom 10: 51–59
Egorova R, Zwart AP (2007) Tail behavior of conditional sojourn times in processor-sharing queues. Queueing Syst 55: 107–121
Feller W (1968) An introduction to probability theory and its applications, vol 2. Wiley, New York
Fuh C (2004) Uniform renewal theorems and ruin probabilities in Markov random walks. Ann Appl Prob 14: 1202–1241
Grishechkin S (1992) On a relationship between processor-sharing queues and Crump-Mode-Jagers branching processes. Adv Appl Probab 24: 653–698
Gyllenberg M, Silvestrov D (2000) Nonlinearly perturbed regenerative processes and pseudo-stationary phenomena for stochastic systems. Stoch Proc Appl 86: 1–27
Gyllenberg M, Silvestrov D (2000) Cramér–Lundberg approximation for nonlinearly perturbed risk processes. Insurance: Mathematics and Economics 26: 75–90
Gyllenberg M, Silvestrov D (2008) Quasi-stationary phenomena in nonlinearly perturbed stochastic systems. De Gruyter, Berlin
Lin S, Willmot G (2000) Lundberg approximations for compound distributions with insurance applications. Springer, New York
Ott TJ (1984) The sojourn-time distribution in the M/G/1 queue with processor sharing. J Appl Probab 21: 360–378
Resnick S (1992) Adventures in stochastic processes. McGraw-Hill, New York
Siegmund D (1979) Corrected diffusion approximations in certain random walk problems. Adv Appl Probab 11: 701–719
Yashkov SF (1983) A derivation of response time distribution for a M/G/1 processor-sharing queue. Probl Control Inf Theor 12: 133–148
Yashkov SF (1993) On a heavy traffic limit theorem for the M/G/1 processor-sharing queue. Stoch Model 9: 467–471
Zhen Q, Knessl C (2010) Asymptotic expansions for the sojourn time distribution in the M/G/1-PS queue. Math Methods Oper Res 71: 201–244
Zwart AP, Boxma OJ (2000) Sojourn time asymptotics in the M/G/1 processor sharing queue. Queueing Syst 35: 141–166
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Blanchet, J., Zwart, B. Asymptotic expansions of defective renewal equations with applications to perturbed risk models and processor sharing queues. Math Meth Oper Res 72, 311–326 (2010). https://doi.org/10.1007/s00186-010-0321-6
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DOI: https://doi.org/10.1007/s00186-010-0321-6