Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-27T23:09:42.307Z Has data issue: false hasContentIssue false
  • English
  • Français

Control of arrivals to a stochastic input–output system

Published online by Cambridge University Press:  01 July 2016

Søren Glud Johansen  and
Shaler Stidham Jr
Søren Glud Johansen*
Affiliation:
University of Aarhus
Shaler Stidham Jr*
Affiliation:
North Carolina State University at Raleigh
*
∗Postal address: Department of Operations Research, University of Aarhus, Building 530, Ny Munkegade, 8000 Aarhus C, Denmark.
∗∗Postal address: Department of Industrial Engineering, North Carolina State University, Box 5111, Raleigh, NC 27650, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

The problem of controlling input to a stochastic input-output system by accepting or rejecting arriving customers is analyzed as a semi-Markov decision process. Included as special cases are a GI/G/1 model and models with compound input and/or output processes, as well as several previously studied queueing-control models. We establish monotonicity of socially and individually optimal acceptance policies and the more restrictive nature of the former, with random rewards for acceptance and both customer-oriented and system-oriented non-linear waiting costs. Distinctive features of our analysis are (i) that it allows dependent interarrival times and (ii) that the monotonicity proofs do not rely on the standard concavity-preservation arguments.

Keywords

INPUT–OUTPUTSTORAGE SYSTEMGI/G/1 QUEUECONTROL OF QUEUESSEMI-MARKOV DECISION PROCESSDYNAMIC PROGRAMMINGSOCIALLY AND INDIVIDUALLY OPTIMAL POLICIESNON-LINEAR WAITING COSTS
Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 4 , December 1980 , pp. 972 - 999
Copyright
Copyright © Applied Probability Trust 1980 

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.)

Footnotes

Research partially supported by NATO Research Grant No. SRG. SS. 5, administered by the NATO Special Programme Panel on System Science.

References

[1] Adler, I. and Naor, P. (1969) Social optimization versus self-optimization in waiting lines. Tech. Rpt. No. 4, Department of Operations Research, Stanford University.Google Scholar
[2] Boxma, O. J. (1975) The single-server queue with random service output. J. Appl. Prob. 12, 763–778.CrossRefGoogle Scholar
[3] Cramer, M. (1971) Optimal customer selection in exponential queues. ORC 71-24, Operations Research Center, University of California, Berkeley.Google Scholar
[4] Doshi, B. T. (1977) Continuous-time control of the arrival process in an M/G/l queue. Stoch. Proc. Appl. 5, 265–284.Google Scholar
[5] Hillier, F. S. (1963) Economic models for industrial waiting-line problems. Management Sci. 10, 119–130.CrossRefGoogle Scholar
[6] Knudsen, N. C. (1972) Individual and social optimization in a multi-server queue with a general cost-benefit structure. Econometrica 40, 515–528.Google Scholar
[7] Knudsen, N. C. and Stidham, S. (1976) Individual and social optimization in a birth-death congestion system with a general cost-benefit structure. NCSU-IE Tech. Rpt. No. 76-8, Department of Industrial Engineering, North Carolina State University, Raleigh.Google Scholar
[8] Lippman, S. A. (1975) Applying a new device in the optimization of exponential queueing systems. Operat. Res. 23, 687–710.Google Scholar
[9] Lippman, S. A. and Stidham, S. Jr (1977) Individual versus social optimization in exponential congestion systems. Operat. Res. 25, 233–247.Google Scholar
[10] McGill, J. T. (1969) Optimal control of queueing systems with variable number of exponential servers. Technical Report No. 123, Department of Operations Research and Statistics, Stanford University.Google Scholar
[11] Miller, B. L. (1969) A queueing reward system with several customer classes. Management Sci. 16, 234–245.CrossRefGoogle Scholar
[12] Mitchell, W. (1973) Optimal service-rate selection in an M/G/l queue. SIAM J. Appl. Math. 24, 19–35.Google Scholar
[13] Naor, P. (1969) On the regulation of queue size by levying tolls. Econometrica 37, 15–24.Google Scholar
[14] Ross, S. M. (1970) Applied Probability Models with Optimization Applications. Holden-Day, San Francisco.Google Scholar
[15] Schäl, M. (1975) Conditions for optimality in dynamic programming and for the limit of n-stage optimal policies to be optimal. Z. Wahrscheinlichkeitsth. 32, 179–296.Google Scholar
[16] Serfozo, R. F. (1976) Monotone optimal policies for Markov decision processes. Math. Prog. 6, North-Holland, 202–215.Google Scholar
[17] Sobel, M. J. (1975) The optimality of full-service policies. School of Organization and Management, Yale University.Google Scholar
[18] Stidham, S. Jr (1970) On the optimality of single-server queueing systems. Operat. Res. 18, 708–732.Google Scholar
[19] Stidham, S. Jr (1978) Socially and individually optimal control of arrivals to a GI/M/1 queue. Management Sci. 24, 1598–1610.Google Scholar
[20] Stidham, S. Jr and Prabhu, N. U. (1974) Optimal control of queueing systems. In Mathematical Methods in Queueing Theory. Lecture Notes in Economics and Mathematical Systems 98, Springer-Verlag, Berlin, 263–294.Google Scholar
[21] Stoyan, D. (1977) Bounds and approximations in queueing through monotonicity and continuity. Operat. Res. 25, 851–863.CrossRefGoogle Scholar
[22] Topkis, D. (1978) Minimizing a submodular function on a lattice. Operat. Res. 26, 305–321.Google Scholar
[23] Veinott, A. F. (1965) Optimal policy in a dynamic, single-product, non-stationary inventory model with several demand classes. Operat. Res. 13, 761–778.CrossRefGoogle Scholar
[24] Veinott, A. F. (1967) Unpublished class notes. Department of Operations Research, Stanford University.Google Scholar
[25] Winston, W. (1978) Optimality of monotonic policies for multiple-server exponential queueing systems with state-dependent arrival rates. Operat. Res. 26, 1089–1094.Google Scholar
[26] Yechiali, U. (1971) On optimal balking rules and toll charges in a GI/M/1 queueing process. Operat. Res. 19, 349–370.Google Scholar
[27] Yechiali, U. (1972) Customers' optimal joining rules for the GI/M/s queue. Management Sci. 18, 434–443.Google Scholar