Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-05-26T00:00:18.660Z Has data issue: false hasContentIssue false
  • English
  • Français

Concavity of queueing systems with NBU service times

Part of: Computer system organization Operations research and management science Special processes

Published online by Cambridge University Press:  01 July 2016

Rajendran Rajan  and
Rajeev Agrawal
Rajendran Rajan*
Affiliation:
IBM
Rajeev Agrawal*
Affiliation:
University of Wisconsin-Madison
*
∗ Postal address: IBM T.J. Watson Research Center, P.O. Box 704, Yorktown Heights, NY 10598, USA. Email address: raju@watson.ibm.com
∗∗ Postal address: Department of Electrical and Computer Engineering, University of Wisconsin-Madison, Madison, WI 53706-1691, USA. Email address: agrawal@engr.wisc.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper establishes structural properties for the throughput of a large class of queueing networks with i.i.d. new-better-than-used service times. The main result obtained in this paper is applied to a wide range of networks, including tandems, cycles and fork-join networks with general blocking and starvation (as well as certain networks with splitting and merging of traffic streams), to deduce the concavity of their throughput as a function of system parameters, such as buffer and initial job configurations, and blocking and starvation parameters. These results have important implications for the optimal design and control of such queueing networks by providing exact solutions, reducing the search space over which optimization need be performed, or establishing the convergence of optimization algorithms. In order to obtain results for such disparate networks in a unified manner, we introduce the framework of constrained discrete event systems (CDES), which enables us to characterize any permutable and non-interruptive queueing network through its constraint set. The main result of this paper establishes comparison properties of the event occurrence processes of CDES as a function of the constraint sets, which are then translated into the above-mentioned concavity of the throughput as a function of system parameters in the context of queueing networks.

Keywords

Queueing networksnew-better-than-usedthroughputmonotonicityconcavitydiscrete parameter optimizationwork-in-progressbuffer allocationtimed discrete event systemsgeneralized semi-Markov processes

MSC classification

Primary: 60K25: Queueing theory
Secondary: 68M10: Network design and communication 68M20: Performance evaluation; queueing; scheduling 90B22: Queues and service
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 2 , June 1998 , pp. 551 - 567
Copyright
Copyright © Applied Probability Trust 1998 

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 supported in part by NSF Grant No. NCR-9305018.

References

[1] Anantharam, V. and Tsoucas, P. (1990). Stochastic concavity of throughput in series of queues with finite buffers. Adv. Appl. Prob. 22, 761–763.Google Scholar
[2] Baccelli, F. (1992). Ergodic theory of stochastic Petri networks. Ann. Prob. 20, 375–396.Google Scholar
[3] Baccelli, F. and Liu, Z. (1992). Comparison properties of stochastic decision free Petri nets. IEEE Trans. Automatic Control 37, 1905–1920.Google Scholar
[4] Buzacott, J. A. and Shanthikumar, J. G. (1992). Design of manufacturing systems using queueing models. Queueing Systems 12, 135–214.CrossRefGoogle Scholar
[5] Cassandras, C. G. (1993). Discrete Event Systems: Modelling and Performance Analysis. Aksen Irwin.Google Scholar
[6] Cheng, D. W. and Yao, D. D. (1993). Tandem queues with general blocking: a unified model and comparison results. DEDS: Theory and Applications 2, 207–234.Google Scholar
[7] Dallery, Y., Liu, Z. and Towsley, D. (1992). Properties of fork/join networks with blocking under various operating mechanisms. Technical Report COINS 92–39. Department of Computer Science, University of Massachusetts at Amherst.Google Scholar
[8] Dallery, Y., Liu, Z. and Towsley, D. (1994). Equivalence, reversibility, symmetry and concavity properties in fork/join queueing networks with blocking. J. ACM 41, 903–942.Google Scholar
[9] Dallery, Y. and Towsley, D. (1991). Symmetry property of the throughput in closed tandem queueing networks with finite buffers. Operat. Res. Lett. 10, 541–547.Google Scholar
[10] Glasserman, P. and Yao, D. D. (1992). Monotonicity in generalized semi-Markov processes. Math. Operat. Res. 17, 1–21.Google Scholar
[11] Glasserman, P. and Yao, D. D. (1992). Second-order properties of generalized semi-Markov processes. Math. Operat. Res. 17, 444–469.Google Scholar
[12] Glasserman, P. and Yao, D. D. (1994). A GSMP framework for the analysis of production lines. In Stochastic Modeling and Analysis of Manufacturing Systems, ed. Yao, D. D.. Springer.Google Scholar
[13] Glasserman, P. and Yao, D. D. (1994). Monotone Structure in Discrete-Event Systems. John Wiley, New York.Google Scholar
[14] Glasserman, P. and Yao, D. D. (1995). Subadditivity and stability of a class of discrete-event systems. IEEE Trans. Automatic Control 40, 1514–1527.Google Scholar
[15] Glasserman, P. and Yao, D. D. (1996). Structured buffer allocation problems. DEDS: Theory and Applications 6, 9–42.Google Scholar
[16] Hillier, F. S., So, K. C. and Boling, R. W. (1993). Notes: Toward characterizing the optimal allocation of storage space in production line systems with variable processing times. Management Sci. 39, 1, 126–133.Google Scholar
[17] Ho, Y. C., Eyler, M. A. and Cien, T. T. (1979). A gradient technique for general buffer storage design in a production line. Int. J. Production Res. 17, 557–580.Google Scholar
[18] Ichikawa, A. and Hiraishi, K. (1988). Analysis and control of discrete event systems represented by Petri nets. In Discrete Event Systems: Models and Applications (Sopron, Hungary). IIASA, Springer, New York.Google Scholar
[19] Kamae, T., Krengel, U. and O'Brien, G. L. (1977). Stochastic inequalities on partially ordered spaces. Ann. Prob. 5, 899–912.Google Scholar
[20] Lazar, A. A. (1983). Optimal flow control of a class of queueing networks in equilibrium. IEEE Trans. Automatic Control AC-28, 1001–1007.Google Scholar
[21] Meester, L. and Shanthikumar, J. G. (1990). Concavity of throughput and optimal buffer space allocation for tandem queueing systems with finite buffer storage space. Adv. Appl. Prob. 22, 764–767.Google Scholar
[22] Mitra, D. and Mitrani, I. (1990). Analysis of a kanban discipline for cell coordination in production lines. Management Sci. 36, 1548–1566.Google Scholar
[23] Rajan, R. (1995). General fluid models for queueing networks. Ph.D. dissertation. University of Wisconsin–Madison.Google Scholar
[24] Rajan, R. and Agrawal, R. (1994). Cyclic networks with general blocking and starvation. Queueing Systems 15, 99–136.Google Scholar
[25] Rajan, R. and Agrawal, R. (1995). Second-order properties of families of discrete event systems. IEEE Trans. Automatic Control 40, 261–272.Google Scholar
[26] Shanthikumar, J. G. and Yao, D. D. (1988). Second-order properties of the throughput of a closed queueing network. Math. Operat. Res. 13, 524–534.Google Scholar
[27] Shanthikumar, J. G. and Yao, D. D. (1989). Monotonicity and concavity properties in cyclic queueing networks with finite buffers. In Queueing networks with blocking: Proc. of first international workshop, Raleigh, North Carolina, May 20–21, 1988, ed. Perros, H. G. and Altiok, T.. North-Holland, pp. 325–344.Google Scholar
[28] Shanthikumar, J. G. and Yao, D. D. (1989). Second-order stochastic properties in queueing systems. Proc. IEEE 77, 162–170.Google Scholar
[29] Stoyan, D. (1983). Comparison Methods for Queues and Other Stochastic Models. Wiley, New York.Google Scholar
[30] Tayur, S. R. (1993). Structural properties and a heuristic for kanban-controlled serial lines. Management Sci. 39, 1347–1368.Google Scholar
[31] Varaiya, P. (1988). Finitely recursive processes. In Discrete Event Systems: Models and Applications (Sopron, Hungary). IIASA, Springer, New York.Google Scholar
[32] Yamazaki, G. and Sakasegawa, H. (1975). Properties of duality in tandem queueing systems. Ann. Inst. Stat. Math. 27, 201–212.Google Scholar