Probability Towards 2000 pp 340-356 | Cite as

# Some Recent Developments for Queueing Networks

Chapter

## Abstract

Early investigations in queueing theory provided detailed analysis of the behavior of a single queue and of networks that in a sense could be decomposed into a product of single queues. Whilst insights from these early investigations are still used, more recent investigations have focussed on understanding how network components interact.

## Keywords

Brownian Motion Heavy Traffic Service Discipline Queueing Network Tandem Queue
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## Reference

- [1]Bertsekas, D., and Gallagher, R. (1992).
*Data Networks*. Prentice-Hall, Englewood Cliffs, N.J.MATHGoogle Scholar - [2]Bertsimas, D., Paschalidis, I. Ch., and Tsitsiklis, J. N. (1994). Optimization of multiclass queueing networks: Polyhedral and nonlinear characterizations of achievable performance.
*Annals of Applied Probability*, 4, 43–75.MathSciNetMATHCrossRefGoogle Scholar - [3]Bramson, M. (1994). Instability of FIFO queueing networks.
*Annals of Applied Probability*,**4**, 414 - 431.MathSciNetMATHCrossRefGoogle Scholar - [4]Bramson, M. (1994). Instability of FIFO queueing networks with quick service times.
*Annals of Applied Probability*,**4**, 693–718.MathSciNetMATHCrossRefGoogle Scholar - [5]Bramson, M. (1995). Convergence to equilibria for fluid models of FIFO queueing networks.
*Queueing Systems: Theory and Applications, to appear*.Google Scholar - [6]Bramson, M. (1995). Convergence to equilibria for fluid models of processor sharing queueing networks. Preprint.Google Scholar
- [7]Chen, H. (1995). A sufficient condition for the positive recurrence of a semimartin- gale reflecting Brownian motion in an orthant. Preprint.Google Scholar
- [8]Chen, H., and Mandelbaum, A. (1991). Stochastic discrete flow networks: diffusion approximations and bottlenecks.
*Annals of Probability*, 4, 1463–1519.MathSciNetCrossRefGoogle Scholar - [9]Chen, H., and Zhang, H. (1995). Stability of multiclass queueing networks under FIFO service discipline. Preprint.Google Scholar
- [10]Dai, J. G. (1995). On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid limit models.
*Annals of Applied Probability*,**5**, 49–77.MathSciNetMATHCrossRefGoogle Scholar - [11]Dai, J. G. (1995). Stability of open multiclass queueing networks via fluid models. In
*Stochastic Networks*, IMA Volumes in Mathematics and Its Applications, F. P. Kelly and R. J. Williams (eds.), 71, Springer-Verlag, New York, 71–90.Google Scholar - [12]Dai, J. G., and Harrison, J. M. (1992). Reflected Brownian motion in an orthant: numerical methods for steady-state analysis.
*Annals of Applied Probability*,**2**, 65–86.MathSciNetMATHCrossRefGoogle Scholar - [13]Dai, J. G., and Kurtz, T. G. (1995). A multiclass station with Markovian feedback in heavy traffic.
*Mathematics of Operations Research*,**20**, 721 - 742.MathSciNetMATHCrossRefGoogle Scholar - [14]Dai, J. G., and Nguyen, V. (1994). On the convergence of multiclass queueing networks in heavy traffic.
*Annals of Applied Probability*,**4**, 26 - 42.MathSciNetMATHCrossRefGoogle Scholar - [15]Dai, J. G., and VandeVate, J. The stability of two-station queueing networks. In preparation.Google Scholar
- [16]Dai, J. G., and Wang, Y. (1993). Nonexistence of Brownian models of certain multiclass queueing networks.
*Queueing Systems: Theory and Applications*,**13**, 41–46.MathSciNetMATHCrossRefGoogle Scholar - [17]Dai, J. G., and Weiss, G. (1996) Stability and instability of fluid models for certain re-entrant lines.
*Mathematics of Operations Research*, to appear.Google Scholar - [18]Dai, J. G., and Williams, R. J. (1995). Existence and uniqueness of semimartin- gale reflecting Brownian motions in convex polyhedrons.
*Theory of Probability and Its Applications*,**40**, 3–53 (in Russian), to appear in the SIAM translation journal of the same name.MathSciNetMATHCrossRefGoogle Scholar - [19]Dupuis, P., and Williams, R. J. (1994). Lyapunov functions for semimartingale reflecting Brownian motions.
*Annals of Probability*,**22**, 680–702.MathSciNetMATHCrossRefGoogle Scholar - [20]Ethier, S. N., and Kurtz, T. G. (1986).
*Markov Processes: Characterization and Convergence*. Wiley, New York.MATHGoogle Scholar - [21]Harrison, J. M. (1978). The diffusion approximation for tandem queues in heavy traffic.
*Adv. Appl Prob*, 10, 886–905.MATHCrossRefGoogle Scholar - [22]Harrison, J. M. (1988). Brownian models of queueing networks with heteroge-neous customer populations. In Stochastic Differential Systems,
*Stochastic Control Theory and Applications*, IMA Volumes in Mathematics and Its Applications, W. Fleming and P.-L. Lions (eds.), Springer-Verlag, New York, 147–186.Google Scholar - [23]Harrison, J. M. (1995). Balanced fluid models of multiclass queueing networks: a heavy traffic conjecture. In
*Stochastic Networks*, IMA Volumes in Mathematics and Its Applications, F. P. Kelly and R. J. Williams (eds.),**71**, Springer-Verlag, New York, 1–20.Google Scholar - [24]Harrison, J. M., and Nguyen, V. (1990). The QNET method for two-moment analysis of open queueing networks.
*Queueing Systems: Theory and Applications*,**6**, 1–32.MathSciNetMATHCrossRefGoogle Scholar - [25]Harrison, J. M., and Nguyen, V. (1995). Brownian models of multiclass queueing networks: current status and open problems.
*Queueing Systems: Theory and Applications*,**13**, 5–40.MathSciNetCrossRefGoogle Scholar - [26]Harrison, J. M., and Pich, M. T. (1993). Two-moment analysis of open queueing networks with general workstation capabilities. Operations Research, to appear.Google Scholar
- [27]Harrison, J. M., and Reiman, M. I. (1981). Reflected Brownian motion on an orthant.
*Annals of Probability*, 9, 302–308.MathSciNetMATHCrossRefGoogle Scholar - [28]Harrison, J. M., and Wein, L. M. (1989). Scheduling networks of queues: heavy traffic analysis of a simple open network.
*Queueing Systems: Theory and Applications*,**5**, 265–280.MathSciNetMATHCrossRefGoogle Scholar - [29]Harrison, J. M., and Williams, R. J. (1987). Brownian models of open queueing networks with homogeneous customer populations. Stochastics, 22, 77–115.MathSciNetMATHGoogle Scholar
- [30]Harrison, J. M., and Williams, R. J. (1987). Multidimensional reflected Brownian motions having exponential stationary distributions.
*Annals of Probability*,**15**, 115–137.MathSciNetMATHCrossRefGoogle Scholar - [31]Harrison, J. M., and Williams, R. J. (1992). Brownian models of feedforward queueing networks: quasireversibility and product form solutions.
*Annals of Applied Probability*,**2**, 263–293.MathSciNetMATHCrossRefGoogle Scholar - [32]Harrison, J. W., and Williams, R. J. (1995). A multiclass closed queueing network with unconventional heavy traffic behavior.
*Annals of Applied Probability*, to appear.Google Scholar - [33]Harrison, J. M., Williams, R. J., and Chen, H. (1990). Brownian models of closed queueing networks with homogeneous customer populations.
*Stochastics and Stochastics Reports*,**29**, 37–74.MATHGoogle Scholar - [34]Hobson, D. G., and Rogers, L. C. G. (1993). Recurrence and transience of reflecting Brownian motion in the quadrant.
*Math. Proc. Cambridge Philosophical Society*,**113**, 387–399.MathSciNetMATHCrossRefGoogle Scholar - [35]Iglehart, D. L., and Whitt, W. (1970). Multiple channel queues in heavy trafficGoogle Scholar
- [36]Iglehart, D. L., and Whitt, W. (1970). Multiple channel queues in heavy trafficGoogle Scholar
- [37]Johnson, D. P. (1983). Diffusion Approximations for Optimal Filtering of Jump Processes and for Queueing Networks. Ph.D. dissertation, Department of Mathematics, University of Wisconsin, Madison, WI.Google Scholar
- [38]Kaspi, H., and Mandelbaum, A. (1992). Regenerative closed queueing networks.
*Stochastics and Stochastics Reports*,**39**, 239–258.MathSciNetMATHGoogle Scholar - [39]Kelly, F. P., and Laws, C. N. (1993). Dynamic routing in open queueing networks: Brownian models, cut constraints and resource pooling.
*Queueing Systems: Theory and Applications*,**13**, 47–86.MathSciNetMATHCrossRefGoogle Scholar - [40]Krichagina, E. V., Liptser, R. S., and Puhalsky, A. A. (1988). Diffusion approximation for the system with arrival process depending on queue and arbitrary service distribution.
*Theory of Probability and Its Applications*,**33**, 124–135.MathSciNetMATHCrossRefGoogle Scholar - [41]Kumar, P. R., and Meyn, S. P. (1995). Stability of queueing networks and scheduling policies.
*IEEE Transactions on Automatic Control*,**40**, 251–260.MathSciNetMATHCrossRefGoogle Scholar - [42]Kumar, P. R., and Seidman, T. I. (1990). Dynamic instabilities and stabilization methods in distributed real-time scheduling of manufacturing systems.
*IEEE Transactions on Automatic Control*,**35**, 289–298.MathSciNetMATHCrossRefGoogle Scholar - [43]Kushner, H. J. (1995). A control problem for a new type of public transportation system, via heavy traffic analysis. In
*Stochastic Networks*, IMA Volumes in Mathematics and Its Applications, F. P. Kelly and R. J. Williams (eds.), 71, Springer-Verlag, New York, 139–167.Google Scholar - [44]Lu, S. H., and Kumar, P. R. (1991). Distributed scheduling based on due dates and buffer priorities.
*IEEE Transactions on Automatic Control*,**36**, 1406–1416.CrossRefGoogle Scholar - [45]Mandelbaum, A., and Massey, W. A. (1995). Strong approximations for time- dependent queues.
*Mathematics of Operations Research*,**20**, 33–64.MathSciNetMATHCrossRefGoogle Scholar - [46]Massey, W. A. (1981).
*Nonstationary Queueing Networks*. Ph.D. dissertation, Department of Mathematics, Stanford University, Stanford, CA.Google Scholar - [47]Meyn, S. P., and Down, D. (1994). Stability of generalized Jackson networks.
*Annals of Applied Probability*,**4**, 124–148.MathSciNetMATHCrossRefGoogle Scholar - [48]Pats, G. (1995). State Dependent Queueing Networks:
*Approximations and Applications*. Ph.D. dissertation, Department of Industrial Engineering and Management, Technion, Haifa, Israel.Google Scholar - [49]Peterson, W. P. (1991). Diffusion approximations for networks of queues with multiple customer types.
*Mathematics of Operations Research*,**9**, 90–118.CrossRefGoogle Scholar - [50]Reiman, M. I. (1984). Open queueing networks in heavy traffic.
*Mathematics of Operations Research*,**9**, 441–458.MathSciNetMATHCrossRefGoogle Scholar - [51]Reiman, M. I. (1988). A multiclass feedback queue in heavy traffic.
*Mathematics of Operations Research*,**20**, 179–207.MathSciNetMATHGoogle Scholar - [52]Reiman, M. I., and Williams, R. J. (1988–89). A boundary property of semi- martingale reflecting Brownian motions.
*Probability Theory and Related Fields*,**77**, 87–97, and**80**, 633.MathSciNetMATHCrossRefGoogle Scholar - [53]Rybko, A. N., and Stolyar, A. L. (1991). Ergodicity of stochastic processes describing the operation of an open queueing network.
*Problemy Peredachi Informatsil*,**28**, 2–26.Google Scholar - [54]Sauer, C. H., and Chandy, K. M. (1981).
*Computer Systems Performance Modeling*. Prentice-Hall, Englewood Cliffs, N.J.Google Scholar - [55]Seidman, T. I. (1994). ‘First come, first served’ can be unstable!
*IEEE Transactions on Automatic Control*,**39**, 2166–2171.MathSciNetMATHCrossRefGoogle Scholar - [56]Skorokhod, A. V. (1956). Limit Theorems for Stochastic Processes.
*Theory of Probability and Its Applications*,**1**, 261 - 290.CrossRefGoogle Scholar - [57]Taylor, L. M., and Williams, R. J. (1993). Existence and uniqueness of semimartingale reflecting Brownian motions in an orthant. Probability Theory and Related Fields,
**96**, 283–317.MathSciNetMATHCrossRefGoogle Scholar - [58]Wein, L. M. (1990). Scheduling networks of queues: heavy traffic analysis of a two-station network with controllable inputs. Operations Research,
**38**, 1065–1078.MathSciNetMATHCrossRefGoogle Scholar - [59]Wein, L. M. (1992). Scheduling networks of queues: heavy traffic analysis of a multistation network with controllable inputs. Operations Research,
**40**(suppl.), S312–S334.MathSciNetCrossRefGoogle Scholar - [60]Whitt, W. (1971). Weak convergence theorems for priority queues: preemptive resume discipline.
*J. Applied Probability*,**8**, 74–94.MathSciNetMATHCrossRefGoogle Scholar - [61]Whitt, W. (1993). Large fluctuations in a deterministic multiclass network of queues.
*Management Science*,**39**, 1020–1028.MATHCrossRefGoogle Scholar - [62]Williams, R. J. (1985). Recurrence classification and invariant measure for reflected Brownian motion in a wedge.
*Annals of Probability*,**13**, 758–778.MathSciNetMATHCrossRefGoogle Scholar - [63]Williams, R. J. (1987). Reflected Brownian motion with skew symmetric data in a polyhedral domain. Probability Theory and Related Fields,
**75**, 459–485.MathSciNetMATHCrossRefGoogle Scholar - [64]Williams, R. J. (1995). Semimartingale reflecting Brownian motions in an or-thant. In
*Stochastic Networks*, IMA Volumes in Mathematics and Its Applications, F. P. Kelly and R. J. Williams (eds.),**71**, Springer-Verlag, New York, 125–137.Google Scholar - [65]Williams, R. J. (1996). On the approximation of queueing networks in heavy traffic. In
*Stochastic Networks*: Theory and Applications, F. P. Kelly, S. Zachary and I. Ziedins (eds.), Oxford University Press, Oxford.Google Scholar - [66]Yao, D. D. (ed.) (1994).
*Stochastic Modeling and Analysis of Manufacturing Systems*. Springer-Verlag, New York.MATHGoogle Scholar

## Copyright information

© Springer-Verlag New York, Inc. 1998