Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-16T15:07:40.615Z Has data issue: false hasContentIssue false
  • English
  • Français

Waiting Time in M/G/1 Queues with Impolite Arrival Disciplines

Published online by Cambridge University Press:  27 July 2009

Süleyman Òzekici ,
Jingwen Li  and
Fee Seng Chou
Süleyman Òzekici
Affiliation:
Department of Industrial Engineering, Boǵaziçi University, 80815 Bebek, Istanbul, Turkey
Jingwen Li
Affiliation:
Department of Decision Sciences, National University of Singapore, 10 Kent Ridge Crescent, Singapore, 0511
Fee Seng Chou
Affiliation:
Department of Decision Sciences, National University of Singapore, 10 Kent Ridge Crescent, Singapore, 0511
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a queueing system where arriving customers join the queue at some random position. This constitutes an impolite arrival discipline because customers do not necessarily go to the end of the line upon arrival. Although mean performance measures like the average waiting time and average number of customers in the queue are the same for all such disciplines, we show that the variance of the waiting time increases as the arrival discipline becomes more impolite, in the sense that a customer is more likely to choose a position closer to the server. For the M/G/1 model, we also provide an iterative procedure for computing the moments of the waiting time distribution. Explicit computational formulas are derived for an interesting special model where a customer joins the queue either at the head or at the end of the line.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 9 , Issue 2 , April 1995 , pp. 255 - 267
Copyright
Copyright © Cambridge University Press 1995

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.Abate, J. & Whitt, W. (1992). The Fourier-series transforms of probability distributions. Queueing Systems 10: 5–88.CrossRefGoogle Scholar
2.Burke, P.J. (1959). Equilibrium delay distribution for one channel with constant holding time. Poisson input and random service. Bell System Technical Journal 38: 1021–1031.CrossRefGoogle Scholar
3.Carter, G.M. & Cooper, R.B. (1972). Queues with service in random order. Operations Research 20: 389–405.CrossRefGoogle Scholar
4.Fuhrmann, S.W. (1991). Second moment relationships for waiting times in queueing systems with Poisson input. Queueing Systems 8: 397–406.CrossRefGoogle Scholar
5.Kingman, J.F.C. (1962). On queues in which customers are served in random order. Proceedings of the Cambridge Philosophical Society 58: 79–91.CrossRefGoogle Scholar
6.Kingman, J.F.C. (1962). The effect of queue discipline on waiting time variance. Proceedings of the Cambridge Philosophical Society 58: 163–164.CrossRefGoogle Scholar
7.Larson, R.C. (1987). Perspectives on queues: Social justice and the psychology of queueing. Operations Research 35: 895–905.CrossRefGoogle Scholar
8.Niu, S.C. & Cooper, R.B. (1993). Transform-free analysis of M/G/1/K and related queues. Mathematics of Operations Research 18: 486–510.Google Scholar
9.Özekici, S., Li, J., & Chou, F.S. (1994). Queues with impolite customers. Queueing Systems 15: 261–277.CrossRefGoogle Scholar
10.Rosenblum, D.M. (1992). Allocation of waiting time by trading in position on a C/M/s queue. Operations Research 40: 338–341.CrossRefGoogle Scholar
11.Shanthikumar, J.G. & Sumita, U. (1987). Convex ordering of sojourn times in single-server queues: Extremal properties of FIFO and LIFO service disciplines. Journal of Applied Probability 24: 734–748.CrossRefGoogle Scholar
12.Takács, L. (1963). Delay distributions for one line with Poisson input, general holding times, and various orders of service. Bell System Technical Journal 43: 487–503.CrossRefGoogle Scholar
13.Tambouratzis, D.G. (1968). On a property of the variance of the waiting time of a queue. Journal of Applied Probability 5: 702–703.CrossRefGoogle Scholar
14.Vasicek, O.A. (1977). An inequality for the variance of waiting time under a general queueing discipline. Operations Research 25: 879–844.CrossRefGoogle Scholar
15.Whitt, W. (1984). The amount of overtaking in a network of queues. Networks 14: 411–426.CrossRefGoogle Scholar
16.Wishart, D.M. (1960). Queueing systems in which the discipline is ‘last-come, first-served.’ Operations Research 8: 591–599.CrossRefGoogle Scholar