Hostname: page-component-cb9f654ff-rkzlw Total loading time: 0 Render date: 2025-08-21T16:38:44.684Z Has data issue: false hasContentIssue false
  • English
  • Français

How Nearly do Arriving Customers See Time-Average Behavior?

Part of: Operations research and management science Stochastic processes Special processes

Published online by Cambridge University Press:  14 July 2016

Erol A. Peköz ,
Sheldon M. Ross  and
Sridhar Seshadri
Erol A. Peköz*
Affiliation:
Boston University
Sheldon M. Ross*
Affiliation:
University of Southern California
Sridhar Seshadri*
Affiliation:
New York University
*
∗ Postal address: Department of Operations and Technology Management, Boston University, 595 Commonwealth Avenue, Boston, MA 02215, USA. Email address: pekoz@bu.edu
∗∗ Postal address: Department of Industrial and System Engineering, University of Southern California, Los Angeles, CA 90089, USA.
∗∗∗ Current address: Department of Information, Risk, and Operations Management, McCombs School of Business, University of Texas at Austin, 1 University Station, B6500, Austin, TX 78712, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Customers arriving at a queue do not usually see its time-average behavior unless arrivals occur according to a Poisson process. In this article we study how nearly customers see time-average behavior. We give total variation error bounds for comparing the distance between the time- and customer-average distributions of a queueing system in terms of properties of the interarrival distribution. Some refinements are given for special cases and numerical computations are used to demonstrate the performance of the inequalities.

Keywords

PASTAarrivals that see time averagesarrival theorem

MSC classification

Secondary: 60K25: Queueing theory 90B22: Queues and service 60K05: Renewal theory 60G10: Stationary processes

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 4 , December 2008 , pp. 963 - 971
Copyright
Copyright © Applied Probability Trust 2008 

References

Baccelli, F. and Brèmaud, P. (1987). Palm Probabilities and Stationary Queues (Lecture Notes Statist. 41). Springer, Berlin.Google Scholar
Brèmaud, P. (1993). A Swiss Army formula of Palm calculus. J. Appl. Prob. 30, 40–51.Google Scholar
Brèmaud, P., Kannurpatti, R. and Mazumdar, R. (1992). Event and time averages: a review. Adv. Appl. Prob. 24, 377–411.Google Scholar
Heyman, D. P. and Stidham, S. Jr. (1980). The relation between customer and time averages in queues. Operat. Res. 28, 983–994.Google Scholar
Gross, D. and Harris, C. M. (1998). Fundamentals of Queueing Theory, 3rd edn. John Wiley, New York.Google Scholar
Köning, D. and Schmidt, V. (1980). Stochastic inequalities between customer-stationary and time-stationary characteristics of queueing systems with point processes. J. Appl. Prob. 17, 768–777.CrossRefGoogle Scholar
Köning, D. and Schmidt, V. (1981). Relationships between time- and customer-stationary characteristics of service systems. In Point Processes and Queueing Problems, eds Bartfai, P. and Tomko, J., North-Holland, Amsterdam, pp. 181–225.Google Scholar
Melamed, B. and Whitt, W. (1990a). On arrivals that see time averages. Operat. Res. 38, 156–172.Google Scholar
Melamed, B. and Whitt, W. (1990b). On arrivals that see time averages: a martingale approach. J. Appl. Prob. 27, 376–384.Google Scholar
Melamed, B. and Yao, D. D. (1995). The ASTA property. In Advances in Queueing, ed. Dshalalow, J. H., CRC, Boca Raton, FL, pp. 195–224.Google Scholar
Niu, S. (1984). Inequalities between arrival averages and time averages in stochastic processes arising from queueing theory. Operat. Res. 32, 785–795.Google Scholar
Peköz, E. A. and Sheldon, R. (2008). Relating time and customer averages for queues using ‘forward’ coupling from the past. J. Appl. Prob. 45, 568–574.Google Scholar
Ross, S. and Peköz, E. A. (2007). A Second Course in Probability. ProbabilityBookstore.com, Boston, MA.Google Scholar
Shanthikumar, G. and Zazanis, M. (1999). Inequalities between event and time averages. Prob. Eng. Inf. Sci. 13, 293–308.CrossRefGoogle Scholar
Wolff, R. (1982). Poisson arrivals see time averages. Operat. Res. 30, 223–231.Google Scholar