Advances in Applied Probability

FCFS infinite bipartite matching of servers and customers

Caldentey René, Kaplan Edward H., and Weiss Gideon

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We consider an infinite sequence of customers of types 𝓒={1,2,...,I} and an infinite sequence of servers of types 𝓢={1,2,...,J}, where a server of type j can serve a subset of customer types C(j) and where a customer of type~$i$ can be served by a subset of server types S(i). We assume that the types of customers and servers in the infinite sequences are random, independent, and identically distributed, and that customers and servers are matched according to their order in the sequence, on a first-come--first-served (FCFS) basis. We investigate this process of infinite bipartite matching. In particular, we are interested in the rate ri,j that customers of type i are assigned to servers of type j. We present a countable state Markov chain to describe this process, and for some previously unsolved instances, we prove ergodicity and existence of limiting rates, and calculate ri,j.

Article information

Adv. in Appl. Probab. Volume 41, Number 3 (2009), 695-730.

First available in Project Euclid: 18 September 2009

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Zentralblatt MATH identifier

Primary: 90B22: Queues and service [See also 60K25, 68M20]
Secondary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40] 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx]

Service systems first-come--first-served infinite bipartite matching Markov chain


René, Caldentey; Edward H., Kaplan; Gideon, Weiss. FCFS infinite bipartite matching of servers and customers. Adv. in Appl. Probab. 41 (2009), no. 3, 695--730. doi:10.1239/aap/1253281061.

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  • Adan, I., Foley, R. D. and McDonald, D. R. (2007). Exact asymptotics of the stationary distribution of a Markov chain: a production model. EURANDOM Report 2008-036, Eindhoven, Netherlands. Available at
  • Aksin, Z., Armony, M. and Mehrotra, V. (2007). The modern call-center: a multi-disciplinary perspective on operations management research. Production Operat. Manag. 16, 665--688.
  • Brémaud, P. (1999). Markov Chains. Springer, New York.
  • Birch, M. W. (1963). Maximum likelihood in three-way contingency tables. J. R. Statist. Soc. B 25, 220--233.
  • Bishop, Y. M. M. and Fienberg, S. E. (1969). Incomplete two-dimensional contingency tables. Biometrics 25, 119--128.
  • Caldentey, R. A. and Kaplan, E. H. (2002). A heavy traffic approximation for queues with restricted customer-service matchings. Unpublished manuscript.
  • Dao-Thi, T.-H. and Mairesse, J. (2006). Zero-automatic networks. In Proc. VALUETOOLS (Pisa, Italy), ACM, New York.
  • Dao-Thi, T.-H. and Mairesse, J. (2007). Zero-automatic queues and product form. Adv. Appl. Prob. 39, 502--536.
  • Durrett, R. (1995). Probability: Theory and Examples, 2nd edn. Duxbury Press, Belmont, CA.
  • Fayolle, G., Malyshev, V. A. and Menshikov, M. V. (1995). Topics in the Constructive Theory of Countable Markov Chains. Cambridge University Press.
  • Fienberg, S. E. (1970). Quasi-independence and maximum likelihood estimation in incomplete contingency tables. J Amer. Statist. Assoc. 65, 1610--1615.
  • Ford, L. R., Jr. and Fulkerson, D. R. (1962). Flows in Networks. Princeton University Press.
  • Garnett, O. and Mandelbaum, A. (2000). An introduction to skill-based routing and its operational complexities. Unpublished manuscript. Available at
  • Goodman, L. (1968). The analysis of cross classified data: independence, quasi-independence, and interactions in contingency tables with and without missing entries. J. Amer. Statist. Assoc. 63, 1091--1131.
  • Hwang, N. H. C. (1981). Fundamentals of Hydraulic Engineering Systems. Prentice Hall, Englewood Cliffs, NJ.
  • Kaplan, E. H. (1984). Managing the demand for public housing. ORC Tech. Rep. 183, MIT.
  • Kaplan, E. H. (1988). A public housing queue with reneging and task-specific servers. Decision Sci. 19, 383\nobreakdash--391.
  • Mairesse, J. (2005). Random walks on groups and monoids with a Markovian harmonic measure. Electron. J. Prob. 10, 1417--1441.
  • Mairesse, J. and Mathéus, F. (2007). Random walks on free products of cyclic groups. J. London Math. Soc. 75, 47--66. Appendix available at
  • Talreja, R. and Whitt, W. (2007). Fluid models for overloaded multiclass many-service queueing systems with FCFS routeing. Manag. Sci. 54, 1513--1527.
  • Zenios, S. A. (1999). Modeling the transplant waiting list: a queueing model with reneging. Queueing Systems 31, 239--251.