Hostname: page-component-cb9f654ff-hqlzj Total loading time: 0 Render date: 2025-09-10T05:03:19.362Z Has data issue: false hasContentIssue false
  • English
  • Français

Queues with marked customers

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Qi-Ming He
Qi-Ming He*
Affiliation:
University of Waterloo
*
∗ Postal address: Department of Management Sciences, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada. Email: qmhe@mansci.uwaterloo.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

Queueing systems with distinguished arrivals are described on the basis of Markov arrival processes with marked transitions. Customers are distinguished by their types of arrival. Usually, the queues observed by customers of different types are different, especially for queueing systems with bursty arrival processes. We study queueing systems from the points of view of customers of different types. A detailed analysis of the fundamental period, queue lengths and waiting times at the epochs of arrivals is given. The results obtained are the generalizations of the results of the MAP/G/1 queue.

Keywords

QUEUEING THEORYMARKOV ARRIVAL PROCESSMAP/G/1 QUEUEMATRIX-ANALYTIC METHOD

MSC classification

Primary: 60K25: Queueing theory

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 28 , Issue 2 , June 1996 , pp. 567 - 587
Copyright
Copyright © Applied Probability Trust 1996 

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.)

Article purchase

Temporarily unavailable

Footnotes

The research was supported by the K. C. Wang Education Foundation and the National Science Foundation through Grant Nr DDM-8915235.

References

[1] Asmussen, S. and Ramaswami, V. (1990) Probabilistic interpretation of some duality results of the matrix paradigms in queueing theory. Stoch. Mod. 6, 715–734.Google Scholar
[2] Chandramouli, Y., Neuts, M. F. and Ramaswami, V. (1989) A queueing model with service interruptions with applications to meteor burst packet communication systems. IEEE Trans. Commun. 37, 1024–1030.Google Scholar
[3] ÇInlar, E. (1969) Markov renewal theory. Adv. Appl. Prob. 1, 123–187.CrossRefGoogle Scholar
[4] Cohen, J. W. (1982) The Single Server Queue. North-Holland, Amsterdam.Google Scholar
[5] He, Qi-Ming and Neuts, M. F. (1995) Markov arrival processes with marked transitions. (Submitted for publication.) Google Scholar
[6] Lindley, D. V. (1952) The theory of queues with a single server. Proc. Camb. Phil. Soc. 48, 227–289.CrossRefGoogle Scholar
[7] Lucantoni, D. M. (1991) New results on the single server queue with a batch Markovian arrival process. Stoch. Mod. 7, 1–46.Google Scholar
[8] Lucantoni, D. M., Meier-Hellstern, K. S. and Neuts, M. F. (1990) A single server queue with server vacations and a class of non-renewal arrival processes. Adv. Appl. Prob. 22, 676–705.Google Scholar
[9] Neuts, M. F. (1979) A versatile Markovian point process. J. Appl. Prob. 16, 764–79.CrossRefGoogle Scholar
[10] Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. The Johns Hopkins University Press, Baltimore.Google Scholar
[11] Neuts, M. F. (1986) Generalizations of the Pollaczek-Khinchin integral equation in the theory of queues. Adv. Appl. Prob. 18, 952–90.Google Scholar
[12] Neuts, M. F. (1989) The fundamental period of the queue with Markov-modulated arrivals In: Probability, Statistics and Mathematics: Papers in Honor of Samuel Karlin. pp 187–200. Academic Press, New York.CrossRefGoogle Scholar
[13] Neuts, M. F. (1989) Structured Stochastic Matrices of M/G/1 type and their Applications. Marcel Dekker, New York.Google Scholar
[14] Ramaswami, V. (1980) The N/G/1 queue and its detailed analysis. Adv. Appl. Prob. 12, 222–61.CrossRefGoogle Scholar
[15] Ramaswami, V. (1990) A duality theorem for the matrix paradigms in queueing theory. Stoch. Mod. 6, 151–163.Google Scholar
[16] Ramaswami, V. (1990) From the matrix-geometric to the matrix-exponential. Queueing Systems. 6, 229–260.Google Scholar
[17] Stanford, D. A. (1991) Interdeparture-time distributions in the non-preemptive priority SiMi/Gi/1 queue. Perf. Eval. 12, 43–60.Google Scholar
[18] Stanford, D. A. and Fischer, W. (1989) The interdeparture time distribution for each class in the SiMi/Gi/1 queue. Queueing Systems 4, 179–191.Google Scholar
[19] Stanford, D. A. and Fischer, W. (1991) Characterising interdeparture times for bursty input streams in the queue with pooled renewal arrivals. Stoch. Mod. 7, 311–320.Google Scholar