Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-05T08:51:05.280Z Has data issue: false hasContentIssue false
  • English
  • Français

Correlation functions in queueing theory

Published online by Cambridge University Press:  14 July 2016

S. K. Srinivasan ,
R. Subramanian  and
R. Vasudevan
S. K. Srinivasan
Affiliation:
Indian Institute of Technology, Madras
R. Subramanian
Affiliation:
Indian Institute of Technology, Madras
R. Vasudevan
Affiliation:
Institute of Mathematical Sciences, Madras
Get access Rights & Permissions [Opens in a new window]

Abstract

The object of this paper is to study the actual waiting time of a customer in a GI/G/1 queue. This is an important criterion from the viewpoint of both the customers and the efficient functioning of the counter. Suitable point processes in the product space of load and time parameters for any general inter-arrival and service time distributions are defined and integral equations governing the correlation functions are set up. Solutions of these equations are obtained and with the help of these, explicit expressions for the first two moments of the number of customers who have waited for a time longer than w in a given time interval (0, T) are calculated.

Keywords

CORRELATION FUNCTIONSQUEUESPOINT PROCESSESLOADBUSY PERIODIDLE PERIODWAITING TIMEPRODUCT DENSITYMARKOV CHAIN
Type
Research Papers
Information
Journal of Applied Probability , Volume 9 , Issue 3 , June 1972 , pp. 604 - 616
Copyright
Copyright © Applied Probability Trust 1972 

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

Bellman, R. and Harris, T. E. (1948) On the theory of age-dependent stochastic branching processes. Proc. Nat. Acad. Sci. (USA) 34, 601–604.Google Scholar
Cohen, J. W. (1969) The Single Server Queue. North-Holland Publishing Company, Amsterdam, Chapter V, p. 278.Google Scholar
Kendall, D. G. (1951) Some problems in the theory of queues. J. R. Statist. Soc. B 13, 151–185.Google Scholar
Kendall, D. G. (1953) Stochastic processes occurring in the theory of queues and their analysis by the method of imbedded Markov chains. Ann. Math. Statist. 24, 338–354.CrossRefGoogle Scholar
Kendall, D. G. (1964) Some recent work and further problems in the theory of queues. Theor. Probability Appl. 9, 1–13.CrossRefGoogle Scholar
Lindley, D. V. (1952) The theory of queues with a single server. Proc. Camb. Phil. Soc. 43, 277–289.CrossRefGoogle Scholar
Ramakrishnan, Alladi (1950) Stochastic processes relating to particles distributed in a continuous infinity of states. Proc. Camb. Phil. Soc. 46, 595–602.CrossRefGoogle Scholar
Smith, W. L. (1953) On the distribution of queueing times. Proc. Camb. Phil. Soc. 49, 449–461.CrossRefGoogle Scholar
Srinivasan, S. K. and Subramanian, R. (1969) Queueing theory and imbedded renewal processes. J. Math. Phys. Sci. 3, 221–244.Google Scholar