Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T19:45:37.405Z Has data issue: false hasContentIssue false
  • English
  • Français

Some analyses on the control of queues using level crossings of regenerative processes

Published online by Cambridge University Press:  14 July 2016

J. G. Shanthikumar
J. G. Shanthikumar*
Affiliation:
University of Toronto
Get access Rights & Permissions [Opens in a new window]

Abstract

Some properties of the number of up- and downcrossings over level u, in a special case of regenerative processes are discussed. Two basic relations between the density functions and the expected number of upcrossings of this process are derived. Using these reults, two examples of controlled M/G/1 queueing systems are solved. Simple relations are derived for the waiting time distribution conditioned on the phase of control encountered by an arriving customer. The Laplace-Stieltjes transform of the distribution function of the waiting time of an arbitrary customer is also derived for each of these two examples.

Keywords

QUEUEING SYSTEMSCONTROL OF QUEUESUP- AND DOWNCROSSINGSREGENERATIVE PROCESSES
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 3 , September 1980 , pp. 814 - 821
Copyright
Copyright © Applied Probability Trust 1980 

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] Brill, P. H. (1975) System Point Theory in Exponential Queues. Ph. D. Thesis, Department of Industrial Engineering, University of Toronto.Google Scholar
[2] Brill, P. H. and Posner, M. J. M. (1974) On the the equilibrium waiting time distribution for a class of exponential queues. WP 74–012, Department of Industrial Engineering, University of Toronto.Google Scholar
[3] Brill, P. H. and Posner, M. J. M. (1976) A characterization of exponential queues with heterogeneous servers. WP 76–021, Department of Industrial Engineering, University of Toronto.Google Scholar
[4] Brill, P. H. and Posner, M. J. M. (1977) Level crossings in point process applied to queues; single server case. Operat. Res. 25, 662–674.Google Scholar
[5] Cohen, J. W. (1976) On Regenerative Processes in Queueing Theory. Lecture Notes in Economics and Mathematical Systems 121, Springer-Verlag, Berlin.Google Scholar
[6] Cohen, J. W. (1977) On up- and downcrossings. J. Appl. Prob. 14, 405–410.CrossRefGoogle Scholar
[7] Heyman, D. P. (1968) Optimal operating policies for M/G/l queueing systems. Operat. Res. 16, 362–382.CrossRefGoogle Scholar
[8] Levy, Y. and Yechiali, U. (1975) Utilization of idle time in an M/G/l queueing system. Management Sci. 22, 202–211.CrossRefGoogle Scholar
[9] Shanthikumar, J. G. (1979) Approximate Queueing Models of Dynamic Job Shops. Ph. D. Thesis, Department of Industrial Engineering, University of Toronto.Google Scholar
[10] Shanthikumar, J. G. (1979) Some analyses on the control of queues using level crossings of regenerative processes. WP 79–002, Department of Industrial Engineering, University of Toronto.Google Scholar
[11] Shanthikumar, J. G. (1979) Some properties of the alternating regenerative processes. WP 79–018, Department of Industrial Engineering, University of Toronto.Google Scholar
[12] Smith, W. L. (1955) Regenerative stochastic processes. Proc. R. Soc. London A 232, 6–31.Google Scholar
[13] Smith, W. L. (1958) Renewal theory and its ramifications. J. R. Statist. Soc. B 20, 243–302.Google Scholar
[14] Stidham, S. (1972) Regenerative processes in the theory of queues, with application to the alternating-priority queue. Adv. Appl. Prob. 4, 542–577.Google Scholar
[15] Tijms, H. C. (1977) Stationary distributions for control policies in an M/G/l queue with removable server. Report BW 79/77. Mathematisch Centrum, Amsterdam.Google Scholar
[16] Van Der Duyn Schouten, F. (1978) An M/G/l queueing model with vacation times. Z. Operat. Res. 22, 95–105.Google Scholar