Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-28T00:01:52.238Z Has data issue: false hasContentIssue false
  • English
  • Français

The geometricity of the limiting distributions in queues and dams

Part of: Special processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

R. M. Phatarfod
R. M. Phatarfod*
Affiliation:
Monash University
*
∗ Postal address: Department of Mathematics, Monash University, Clayton, VIC 3168, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

There are a number of cases in the theories of queues and dams where the limiting distribution of the pertinent processes is geometric with a modified initial term — herein called zero-modified geometric (ZMG). The paper gives a unified treatment of the various cases considered hitherto and some others by using a duality relation between random walks with impenetrable and with absorbing barriers, and deriving the probabilities of absorption by using Waldian identities. Thus the method enables us to distinguish between those cases where the limiting distribution would be ZMG and those where it would not.

Keywords

MOVING AVERAGE INPUTSEMI-MARKOV INPUTBOTTOMLESS DAMGI/M/1 QUEUEDUALITY OF RANDOM WALKSWALDIAN IDENTITIES

MSC classification

Primary: 60K25: Queueing theory
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) 60G40: Stopping times; optimal stopping problems; gambling theory 60G42: Martingales with discrete parameter
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 2 , June 1993 , pp. 438 - 445
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

Finch, P. D. (1963) The single-server queueing system with non-recurrent input-process and Erlang service time. J. Austral. Math. Soc. 3, 220–236.Google Scholar
Finch, P. D. and Pearce, C. (1965) A second look at a queueing system with moving average input process. J. Austral. Math. Soc. 5, 100–106.Google Scholar
Herbert, H. G. (1972) An infinite discrete dam with dependent inputs. J. Appl. Prob. 9, 404–413.Google Scholar
Karlin, S. and Taylor, H. M. (1975) A First Course in Stochastic Processes, 2nd edn. Academic Press, New York.Google Scholar
Lawrance, A. J. and Lewis, P. A. W. (1977) An exponential moving-average sequence and point process (EMA 1). J. Appl. Prob. 14, 98–113.Google Scholar
Pakes, A. G. (1981) Discrete dams with Markovian inputs. Stoch. Proc. Appl. 11, 57–77.Google Scholar
Pakes, A. G. and Phatarfod, R. M. (1979) The limiting distribution for the infinitely deep dam with a Markovian input. Stoch. Proc. Appl. 8, 199–209.Google Scholar
Pearce, C. (1966) A queueing system with general moving average input and negative exponential service time. J. Austral. Math. Soc. 6, 223–236.Google Scholar
Phatarfod, R. M. (1979) The bottomless dam. J. Hydrol. 40, 337–363.Google Scholar
Phatarfod, R. M. (1982) On some applications of Wald's identity to dams. Stoch. Proc. Appl. 13, 279–292.Google Scholar
Phatarfod, R. M., Speed, T. P. and Walker, A. M. (1971) A note on random walks. J. Appl. Prob. 8, 198–201.Google Scholar
Speed, T. P. (1973) A note on random walks. II. J. Appl. Prob. 10, 218–222.Google Scholar
Tin, P. (1985) A queueing system with Markov-dependent arrivals. J. Appl. Prob. 22, 668–677.Google Scholar