Distributions of suprema of Lévy processes via the Heavy Traffic Invariance Principle

• Autorzy: Szczotka W. , Woyczyński W. A.
• Języki publikacji: EN

Abstrakty

EN

We study the relationship between the distribution of the supremum functional M_X = sup_{0 ≤ t < ∞} (X(t) − βt) for a process X with stationary, but not necessarily independent increments, and the limiting distribution of an appropriately normalized stationary waiting time for G/G/l queues in heavy traffic. As a by-product we obtain explicit expressions for the distribution of M_X in several special cases of Lévy processes.

Słowa kluczowe

EN

Lévy process; supremum; heavy traffic; queueing systems

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2003
• Tom: Vol. 23, Fasc. 2
• Strony: 251--272
• Opis fizyczny: Bibliogr. 17 poz.

Twórcy

autor

Szczotka W.

• Institute of Mathematics, University of Wrocław, 50-384 Wrocław, Poland

autor

Woyczyński W. A.

• Department of Statistics, and the Center for Stochastic and Chaotic Processes in Science and Technology, Case Western Reserve University, Cleveland, OH 44106, U.S.A

Bibliografia

• [1] G. Baxter and M. D. Donsker, On the distribution and the supremum functional for processes with stationary independent increments, Trans. Amer. Math. Soc. 85 (1957), pp. 73-87.
• [2] P. Billingsley, Convergence of Probability Measures, Wiley, New York 1968.
• [3] N. H. Bingham, Fluctuation theory in continuous time, Adv. in Appl. Probab. 7 (1975), pp. 705-766.
• [4] O. J. Boxma and J. W. Cohen, Heavy-trafic analysis for the GI/G/1 queue with heavy-tailed distributions, Queueing Systems 33 (1999), pp. 177-204.
• [5] J. W. Cohen, The Single Server Queue, North Holland/Elsevier, 1969.
• [6] J. M. Harrison, The supremum distribution of the Lévy process with no negative jumps, Adv. in Appl. Probab. 9 (1977), pp. 417-422.
• [7] S. Karlin and H. M. Taylor, A First Course in Stochastic Processes, Academic Press INC (London), 1975.
• [8] O. Kella and W. Witt, Queues with server vacations and Lévy processes with secondary jump input, Ann. Appl. Probab. 1 (1991), pp. 104-117.
• [9] S. Kwapień and W. A. Woyczyński, Random Series and Stochastic Integrals: Single and Multiple, Birkhäuser, Boston 1992.
• [10] T. Lindvall, Weak convergence of probability measures and random functions in the function space D [0, ∞), J. Appl. Probab. 10 (1973), pp. 109-112.
• [11] Yu. V. Prokhorov, Convergence of Random Processes and Limit Theorems in Probability Theory, Theory Probab. Appl., Vol. 1 (1956), pp. 157-214.
• [12] W. Szczotka, Exponential approximation of waiting time and queue size for queues in heavy traffic, Adv. in Appl. Probab. 22 (1990), pp. 230-240.
• [13] W. Szczotka, Tightness of the stationary waiting time in heavy traffic, Adv. in Appl. Probab. 31 (1999), pp. 788-794.
• [14] L. Takács, On the distribution of the supremum of stochastic processes with exchangeable increments, Trans. Amer. Math. Soc. 119 (1965), pp. 367-379.
• [15] W. Whitt, Heavy traffic limit theorems for queues: a survey, in: Mathematical Methods in Queueing Theory, Proceedings 1973, A. B. Clarke (Ed.), Lecture Notes in Economics and Mathematical Systems 98, Springer, Berlin 1973, pp. 307-350.
• [16] W. Whitt, Stochastic-Process Limits. An Introduction to Stochastic-Process Limits and Their Application to Queues, Springer, New York 2002.
• [17] V. M. Zolotarev, The first passage time to a level and the behavior at infinity of processes with independent increments, Theory Probab. Appl. 9 (1964), pp. 653-661.