Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-28T17:43:54.184Z Has data issue: false hasContentIssue false
  • English
  • Français

Poisson traffic flow in a general feedback queue

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

Published online by Cambridge University Press:  14 July 2016

Erol A. Peköz  and
Nitindra Joglekar
Erol A. Peköz*
Affiliation:
Boston University
Nitindra Joglekar*
Affiliation:
Boston University
*
∗ Postal address: School of Management, Boston University, 595 Commonwealth Ave, Boston, MA 02215, USA.
∗ Postal address: School of Management, Boston University, 595 Commonwealth Ave, Boston, MA 02215, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

Consider a ·/G/k finite-buffer queue with a stationary ergodic arrival process and delayed customer feedback, where customers after service may repeatedly return to the back of the queue after an independent general feedback delay whose distribution has a continuous density function. We use coupling methods to show that, under some mild conditions, the feedback flow of customers returning to the back of the queue converges to a Poisson process as the feedback delay distribution is scaled up. This allows for easy waiting-time approximations in the setting of Poisson arrivals, and also gives a new coupling proof of a classic highway traffic result of Breiman (1963). We also consider the case of nonindependent feedback delays.

Keywords

Feedback queuePoisson convergencecoupling methods

MSC classification

Primary: 60K25: Queueing theory 90B22: Queues and service
Secondary: 60G55: Point processes 90B20: Traffic problems
Type
Short Communications
Information
Journal of Applied Probability , Volume 39 , Issue 3 , September 2002 , pp. 630 - 636
Copyright
Copyright © Applied Probability Trust 2002 

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

Barbour, A. D., and Brown, T. (1996). Approximate versions of Melamed's theorem. J. Appl. Prob. 33, 472–489.CrossRefGoogle Scholar
Bohn, R. (2000). Stop fighting fires. Harvard Business Rev. 78, No. 4, 82–91.Google Scholar
Breiman, L. (1963). The Poisson tendency in traffic distribution. Ann. Math. Statist. 34, 308–311.CrossRefGoogle Scholar
Daley, D. J., and Vere-Jones, D. (1988). An introduction to the theory of point processes. Springer, New York.Google Scholar
D’Avignon, G. R., and Disney, R. L. (1977/78). Queues with instantaneous feedback. Management Sci. 24, 168–180.CrossRefGoogle Scholar
Foley, R. D., and Disney, R. L. (1983). Queues with delayed feedback. Adv. Appl. Prob. 15, 162–182.CrossRefGoogle Scholar
Jackson, J. R. (1957). Networks of waiting lines. Operat. Res. 5, 518–521.CrossRefGoogle Scholar
Kelly, F. P. (1979). Reversibility and Stochastic Networks. John Wiley, New York.Google Scholar
Kumar, P. R. (1993). Re-entrant lines. Queueing Systems 13, 87–110.CrossRefGoogle Scholar
Kuri, J., and Kumar, A. (1997). On the optimal control of arrivals to a single queue with arbitrary feedback delay. Queueing Systems 27, 1–16.CrossRefGoogle Scholar
Lindvall, T. (2000). On simulation of stochastically ordered life-length variables. Prob. Eng. Inf. Sci. 14, 1–7.CrossRefGoogle Scholar
Melamed, B. (1979). Characterizations of Poisson traffic streams in Jackson queueing networks. Adv. Appl. Prob. 11, 422–438.CrossRefGoogle Scholar
Mountford, T., and Prabhakar, B. (1995). On the weak convergence of departures from an infinite series of ·/M/1 queues. Ann. Appl. Prob. 5, 121–127.CrossRefGoogle Scholar
Prabhakar, B., Mountford, T. S., and Bambos, N. (1996). Convergence of departures in tandem networks of ·/GI/∞ queues. Prob. Eng. Inf. Sci. 10, 487–500.CrossRefGoogle Scholar
Takacs, L. (1963). A single-server queue with feedback. Bell System Tech. J. 42, 505–519.CrossRefGoogle Scholar
Wortman, M. A., Disney, R. L., and Kiessler, P. C. (1991). The M/GI/1 Bernoulli feedback queue with vacations. Queueing Systems 9, 353–363.CrossRefGoogle Scholar