Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T04:31:30.392Z Has data issue: false hasContentIssue false
  • English
  • Français

Transient behavior of coverage processes by applications to the infinite-server queue

Part of: Special processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  14 July 2016

Sid Browne  and
J. Michael Steele
Sid Browne*
Affiliation:
Columbia University
J. Michael Steele*
Affiliation:
University of Pennsylvania
*
∗ Postal address: 402 Uris Hall, Graduate School of Business, Columbia University, New York, NY 10027 USA. email: [email protected]
∗∗ Postal address: Department of Statistics, The Wharton School, University of Pennsylvania, Philadelphia, PA 19104, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

We obtain the distribution of the length of a clump in a coverage process where the first line segment of a clump has a distribution that differs from the remaining segments of the clump. This result allows us to provide the distribution of the busy period in an M/G/∞ queueing system with exceptional first service, and applications are considered. The result also provides the tool necessary to analyze the transient behavior of an ordinary coverage process, namely the depletion time of the ordinary M/G/∞ system.

Keywords

CLUMPSBUSY PERIODSDEPLETION TIMESEXCEPTIONAL FIRST SERVICE

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 3 , September 1993 , pp. 589 - 601
Copyright
Copyright © Applied Probability Trust 1993 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Brown, M. and Ross, S. M. (1969) Some results for infinite server Poisson queues. J. Appl. Prob. 6, 604–611.Google Scholar
[2] Brown, M. and Solomon, H. (1974) Some results for secondary processes generated by a Poisson process. Stoch. Proc. Appl. 2, 337–348.Google Scholar
[3] Browne, S. and Yechiali, U. (1989) Dynamic priority rules for cyclic type queues. Adv. Appl. Prob. 21, 432–450.Google Scholar
[4] Cowan, R. D. (1989) Objects arranged randomly in space: an accessible theory. Adv. Appl. Prob. 21, 543–569.Google Scholar
[5] Doshi, B. (1986) Queueing systems with vacations - a survey. QUESTA 1, 29–66.Google Scholar
[6] Glynn, P. W. and Whitt, W. (1988) Ordinary CLT and WLLN versions of L = ?W. Math. Operat. Res. 13, 674–692.Google Scholar
[7] Hall, P. (1988) Introduction to the Theory of Coverage Processes. Wiley, New York.Google Scholar
[8] Keilson, J. and Ramaswamy, R. (1988) The backlog and depletion-time process for M/G/1 vacation models with exhaustive service. J. Appl. Prob. 25, 404–412.Google Scholar
[9] Kulkarni, V. G. and Puryear, L. C. (1992) A queueing model for data base systems. University of North Carolina at Chapel Hill, Technical Report No. UNC/OR/TR/92-3, Feb. 1992.Google Scholar
[10] Ramalhoto, M. F. (1984) Bounds for the variance of the busy period of the M/G/8 queue. Adv. Appl. Prob. 16, 929–932.Google Scholar
[11] Shanbag, D. N. (1966) On infinite server queues with batch arrivals. J. Appl. Prob. 3, 274–279.Google Scholar
[12] Stadje, W. (1985) The busy period of the queueing system M/G/8. J. Appl. Prob. 22, 697–704.Google Scholar
[13] Stadje, W. (1989) Coverage problems for random intervals. SIAM J. Appl. Math. 49, 1538–1551.Google Scholar
[14] Stoyan, D., Kendall, W. S. and Mecke, J. (1987) Stochastic Geometry and its Applications. Wiley, New York.Google Scholar
[15] Takács, L. (1956) On a probability problem arising in the theory of counters. Proc. Camb. Phil. Soc. 52, 488–498.Google Scholar
[16] Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar
[17] Widder, D. V. (1941) The Laplace Transform. Princeton University Press.Google Scholar