Skip to main content
SpringerLink
Log in
Book cover

Stochastic Modeling and Analysis of Manufacturing Systems pp 281–323Cite as

Scheduling Networks of Queues: Heavy Traffic Analysis of a Bi-Criteria Problem

  • Chapter

Part of the book series: Springer Series in Operations Research ((ORFE))

Abstract

We consider bi-criteria scheduling problems for three queueing systems (a single queue, a two-station closed network, and a two-station network with controllable inputs) populated by various customer types. The objective is to minimize the long run expected average value of a linear combination of the customer sojourn time and the sojourn time inequity. The inequity at time t is the sum of squares of the pairwise differences of the total number of customers in the system at time t of each type divided by their respective arrival rates. Brownian approximations to these three scheduling problems are solved, and the solutions are interpreted in order to obtain scheduling policies. Simulation results show that the second objective criteria tends to equalize the mean sojourn times of the various customer types, and may lead to a reduction in sojourn time variance. The simulation results also show that in the network settings, in contrast to the single queue case, there are priority sequencing policies that significantly reduce the variance of sojourn times relative to the first-come first-served policy.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Beneš, V. E., Shepp, L. A. and Witsenhausen, H. S. (1980). Some Solvable Stochastic Control Problems. Stochastics 4, 39–83.

    Google Scholar 

  2. Billingsley, P. (1968). Convergence of Probability Measures. John Wiley and Sons, New York.

    Google Scholar 

  3. Chevalier, P. B. and Wein, L. M. (1993). Scheduling Networks of Queues: Heavy Traffic Analysis of a Multistation Closed Network. Operations Research.

    Google Scholar 

  4. Chow, P. L., Menaldi, J. L., and M. Robin (1985). Additive Control of Stochastic Linear Systems With Finite Horizon. SIAM J. Control and Opt. 23, 858–899.

    Article  Google Scholar 

  5. Dorn, W. S. (1960). Duality in Quadratic Programming. Q. Appl. Math. 18, 155–162.

    Google Scholar 

  6. Federgruen, A. and Groenevelt, H. (1988). M/G/c Queueing Systems with Multiple Customer Classes: Characterization and Control of Achievable Performance under Nonpreemptive Priority Rules. Management Science 9, 1121–1138.

    Article  Google Scholar 

  7. Foschini, G. J. (1977). On Heavy Traffic Diffusion Analysis and Dynamic Routing in Packet Switched Networks. In Chandy, K. M. and Reiser, M., editors, Computer Performance, North Holland, Amsterdam.

    Google Scholar 

  8. Foschini, G. J. and Salz, J. (1978). A Basic Dynamic Routing Problem and Diffusion Approximation. IEEE Trans. Comm. COM-26, 320–327.

    Article  Google Scholar 

  9. Groenevelt, H. (1990). The Variances of Waiting Times in Multi-Class Non-Preemptive M/G/1 Queues. Working Paper, Simon Graduate School of Business Administration, U. of Rochester, N.Y.

    Google Scholar 

  10. Harrison, J. M. (1973). A Limit Theorem for Priority Queues in Heavy Traffic. J. Appl. Prob. 10, 907–912.

    Article  Google Scholar 

  11. Harrison, J. M. (1985). Brownian Motion and Stochastic Flow Systems. John Wiley and Sons, New York.

    Google Scholar 

  12. Harrison, J. M. (1988). Brownian Models of Queueing Networks with Heterogeneous Customer Populations. In W. Fleming and P. L. Lions (eds.), Stochastic Differential Systems, Stochastic Control Theory and Applications, IMA Volume 10, Springer-Verlag, New York, 147–186.

    Google Scholar 

  13. Harrison, J. M. and Taksar, M. I. (1983). Instantaneous Control of Brownian Motion. Math. of Oper. Res. 3, 439–453.

    Article  Google Scholar 

  14. Harrison, J. M. and Wein, L. M. (1989). Scheduling Networks of Queues: Heavy Traffic Analysis of a Simple Open Network. Queueing Systems 5, 265–180.

    Article  Google Scholar 

  15. Harrison, J. M. and Wein, L. M. (1990). Scheduling Networks of Queues: Heavy Traffic Analysis of a Two-Station Closed Network. Operations Research 38, 1052–1064.

    Article  Google Scholar 

  16. Jackson, J. R. (1960). Some Problems in Queueing with Dynamic Priorities. Naval Res. Log. Quart. 7, 235–249.

    Article  Google Scholar 

  17. Jackson, J. R. (1961). Queues with Dynamic Priorities. Management Science 1, 18–34.

    Article  Google Scholar 

  18. Jackson, J. R. (1962). Waiting Time Distributions for Queues with Dynamic Priorities. Naval Res. Log. Quart. 9, 31–36.

    Article  Google Scholar 

  19. Johnson, D. P. (1983). Diffusion Approximations for Optimal Filtering of Jump Processes and for Queueing Networks. Unpublished Ph.D. thesis, Dept. of Mathematics, Univ. of Wisconsin, Madison.

    Google Scholar 

  20. Karatzas, I. (1983). A Class of Singular Stochastic Control Problems. Adv. Appl. Prob. 15, 225–254.

    Article  Google Scholar 

  21. Karatzas, I. (1988). Stochastic Control Under Finite-fuel Constraints, in W. Fleming and P. L. Lions (eds.), Stochastic Differential Systems, Stochastic Control Theory and Applications, IMA Volume 10, Springer-Verlag, New York, 225–240.

    Google Scholar 

  22. Karatzas, I. and Shreve, S. E. (1988). Brownian Motion and Stochastic Calculus. Springer-Verlag, New York.

    Google Scholar 

  23. Kelly, F. P. (1979). Reversibihty and Stochastic Networks, John Wiley and Sons, New York.

    Google Scholar 

  24. Kingman, J. F. C. (1962). The Effect of Queue Discipline on Waiting Time Variance. Proc. Camb. Phil. Soc. 58, 163–164.

    Article  Google Scholar 

  25. Kleinrock, L. (1964). A Delay Dependent Queue Discipline. Naval Res. Log. Quart. 11, 329–341.

    Article  Google Scholar 

  26. Kleinrock, L. (1976). Queueing Systems Vol. II: Computer Applications. John Wiley and Sons, New York.

    Google Scholar 

  27. Klimov, G. P. (1974). Time Sharing Service Systems I. Th. Prob. Appl. 19, 532–551.

    Article  Google Scholar 

  28. Kushner, H. J. (1977). Probability Methods for Approximations in Stochastic Control and for Elliptic Equations. Academic Press, New York.

    Google Scholar 

  29. Kushner, H. J., and L. F. Martins, (1993). Limit Theorems for Path-wise Average Cost Per Unit Time Problems for Controlled Queues in Heavy Traffic. Stochastics and Stochastic Reports 42, 25–51.

    Google Scholar 

  30. Kushner, H. J., and Ramachandran, K. M. (1989). Optimal and Approximately Optimal Control Policies for Queues in Heavy Traffic. SIAM J. Control and Optimization 27, 1293–1318.

    Article  Google Scholar 

  31. Laws, C. N. and Louth, G. M. (1990). Dynamic Scheduling of a Four Station Network. Probability in the Engineering and Information Sciences 4, 131–156.

    Article  Google Scholar 

  32. Little, J. D. C. (1961). A Proof of the Queueing Formula L = λW. Operations Research 9, 383–387.

    Article  Google Scholar 

  33. Martins, L. F., and Kushner, H. J. (1989). Routing and Singular Control for Queueing Networks in Heavy Traffic. SIAM J. Control and Optimization 28, 1209–1233.

    Article  Google Scholar 

  34. Ou, J. and Wein, L. M. (1991). Dynamic Scheduling of a Production/Inventory System with By-Products and Random Yield. To appear in Management Science.

    Google Scholar 

  35. Peterson, W. P. (1991). A Heavy Traffic Limit Theorem for Networks of Queues with Multiple Customer Types. Math. Operations Research 16, 90–118.

    Article  Google Scholar 

  36. Reiman, M. I. (1983). Some Diffusion Approximations with State Space Collapse. Proc. Intl. Seminar on Modeling and Performance Evaluation Methodology, Springer-Verlag, Berlin.

    Google Scholar 

  37. Schonberger, R. J. (1982). Japanese Manufacturing Techniques. Free Press.

    Google Scholar 

  38. Shanthikumar, J. G. (1982). On Reducing Time Spent in M/G/1 Systems. European J. Operational Research 9, 286–294.

    Article  Google Scholar 

  39. Shreve, S. E., Lehoczky, J. P. and Gaver, D. P. (1984). Optimal Consumption for General Diffusions with Absorbing and Reflecting Barriers. SIAM J. Control and Opt. 22, 55–75.

    Article  Google Scholar 

  40. Taksar, M. I. (1985). Average Optimal Singular Control and a Related Stopping Problem. Math. Operations Research 10, 63–81.

    Article  Google Scholar 

  41. Walrand, J. (1988). An Introduction to Queueing Networks. Prentice-Hall, Englewood Cliffs, New Jersey.

    Google Scholar 

  42. Wein, L. M. (1990a). Optimal Control of a Two-Station Brownian Network. Math. of Operations Research 15, 215–242.

    Google Scholar 

  43. Wein, L. M. (1990b). Scheduling Networks of Queues: Heavy Traffic Analysis of a Two-Station Network with Controllable Inputs. Operations Research 38, 1065–1078.

    Article  Google Scholar 

  44. Wein, L. M. (1991). Brownian Networks with Discretionary Routing. Operations Research 39, 322–340.

    Article  Google Scholar 

  45. Wein, L. M. (1992b). Scheduling Networks of Queues: Heavy Traffic Analysis of a Multistation Network with Controllable Inputs. Operations Research 40, S312–312S334.

    Article  Google Scholar 

  46. Wein, L. M. (1992b). Dynamic Scheduling of a Multiclass Make-to-Stock Queue. Operations Research 40, 724–735.

    Article  Google Scholar 

  47. Wein, L. M. and Chevalier, P. B. (1992). A Broader View of the Job-Shop Scheduling Problem. Management Science 38, 1018–1033.

    Article  Google Scholar 

  48. Wein, L. M. and Ou, J. (1991). The Impact of Processing Time Knowledge on Dynamic Job-Shop Scheduling. Management Science 37, 1002–1014.

    Article  Google Scholar 

  49. Whitt, W. (1971). Weak Convergence Theorems for Priority Queues: Preemptive-Resume Discipline. J. Appl. Prob. 8, 74–94.

    Article  Google Scholar 

  50. Wood, D. and Sargeant, R. (1984). The Synthesis of Multiclass Single Server Queueing Systems. Unpublished manuscript.

    Google Scholar 

  51. P. Yang (1988). Pathwise Solutions for a Class of Linear Stochastic Systems. Unpublished Ph. D. Thesis, Dept. of Operations Research, Stanford University, Stanford, CA.

    Google Scholar 

Download references

Authors
  1. Lawrence M. Wein
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Industrial Engineering and Operations Research, Columbia University, New York, NY, 10027-6699, USA

    David D. Yao

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Springer-Verlag New York Inc

About this chapter

Cite this chapter

Wein, L.M. (1994). Scheduling Networks of Queues: Heavy Traffic Analysis of a Bi-Criteria Problem. In: Yao, D.D. (eds) Stochastic Modeling and Analysis of Manufacturing Systems. Springer Series in Operations Research. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2670-3_7

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-2670-3_7

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7628-9

  • Online ISBN: 978-1-4612-2670-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation