Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-25T11:13:43.623Z Has data issue: false hasContentIssue false
  • English
  • Français

A REVERSIBLE ERLANG LOSS SYSTEM WITH MULTITYPE CUSTOMERS AND MULTITYPE SERVERS

Published online by Cambridge University Press:  14 September 2010

Ivo Adan ,
Cor Hurkens  and
Gideon Weiss
Ivo Adan
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands Department of Quantitative Economics, University of Amsterdam, 1000 GG Amsterdam, The Netherlands E-mail: iadan@win.tue.nl
Cor Hurkens
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600 MB, Eindhoven, The Netherlands E-mail: wscor@win.tue.nl
Gideon Weiss
Affiliation:
Department of Statistics, The University of Haifa, Mount Carmel 31905, Israel E-mail: gweiss@stat.haifa.ac.il
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a memoryless Erlang loss system with servers = {1, …, J}, and with customer types = {1, …, I}. Servers are multitype, so that server j can serve a subset of customer types C(j). We show that the probabilities of assigning arriving customers to idle servers can be chosen in such a way that the Markov process describing the system is reversible, with a simple product form stationary distribution. Furthermore, the system is insensitive; these properties are preserved for general service time distributions.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 24 , Issue 4 , October 2010 , pp. 535 - 548
Copyright
Copyright © Cambridge University Press 2010

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

REFERENCES

1.Adan, I., Foley, R., & McDonald, D. (2009). Exact asymptotics of the stationary distribution of a Markov Chain: A production model. Queuing Systems 62: 311–344.CrossRefGoogle Scholar
2.Adan, I.J.B.F. & Weiss, G. (2010). Exact FCFS matching rates for two infinite multi-type sequences. Eurandom Preprint Series No. 2010-025, Eindhoven: Eurandom.Google Scholar
3.Adan, I.J.B.F., Wessels, J., & Zijm, W.H.M. (1989). Queuing analysis in a flexible assembly system with a job-dependent parallel structure. In Pressmar, D., Jäger, K.E., Krallmann, H., Schellhaas, H., Streitferd, L. (eds). Operations Research Proceedings 1988. Berlin: Springer-Verlag, pp. 551–558.CrossRefGoogle Scholar
4.Adan, I.J.B.F., Wessels, J., & Zijm, W.H.M. (1991). Flexible assembly and shortest queue problems. In Fandel, G. & Zaepfel, G. (eds.), Modern production concepts, theory and applications. Berlin: Springer-Verlag, pp. 644–659.CrossRefGoogle Scholar
5.Aerts, J., Korst, J., & Verhaegh, W. (2007). Load balancing for redundant storage strategies: Multiprocessor scheduling with machine eligibility. Journal of Scheduling 4: 245–257.CrossRefGoogle Scholar
6.Aksin, O.Z., Armony, M., & Mehrotra, V. (2007). The modern call-center: A multi-disciplinary perspective on operations management research. Production and Operations Management 16: 665–688.CrossRefGoogle Scholar
7.Caldentey, R., Kaplan, E.H., & Weiss, G. (2009) FCFS infinite bipartite matching of servers and customers. Advances in Applied Probability 41: 695–730.CrossRefGoogle Scholar
8.de Bruin, A.M., Bekker, R., van Zanten, L., & Koole, G.M. (2008). Dimensioning hospital wards using the Erlang loss model. Annals of Operations Research 178: 23–43.CrossRefGoogle Scholar
9.Ford, L.R. Jr. & Fulkerson, D.R. (1962) Flows in networks. Princeton, NJ: Princeton University Press.Google Scholar
10.Garnett, O. & Mandelbaum, A. (2000). An introduction to skill-based routing and its operational complexities. Available from http://iew3.technion.ac.il/serveng/Lectures/SBR.pdfGoogle Scholar
11.Kelly, F.P. (1979). Reversibility and stochastic networks. New York: Wiley.Google Scholar
12.McManus, M.L., Long, M.C., Copper, A., & Litvak, E. (2004). Queuing theory accurately models the need for critical care resources. Anesthesiology 100: 1271–1276.CrossRefGoogle ScholarPubMed
13.Sevastyanov, B.A. (1957). An ergodic theorem for Markov processes and its application to telephone systems with refusals. Theory of Probability and its Applications 2: 104–112.CrossRefGoogle Scholar
14.Talreja, R. & Whitt, W. (2007). Fluid models for overloaded multi-class many-service queueing systems with FCFS routing. Management Science 54: 1513–1527.CrossRefGoogle Scholar
15.Visschers, J.W.C.H. (2000). Random walks with geometric jumps. Ph.D. thesis, Eindhoven University of Technology.Google Scholar
16.Visschers, J.W.C.H., Adan, I.J.B.F., & Weiss, G. (2009). A multi-class multi-server system with customer-class service restrictions. Preprint.Google Scholar
17.Zijm, W.H.M. & Laarhoven, P.J.M. (1993). Production preparation and numerical control in PCB assembly. International Journal of Flexible Manufacturing Systems 5(3): 187–207.Google Scholar