The Annals of Applied Probability

Large deviations of a modified Jackson network: Stability and rough asymptotics

Robert D. Foley and David R. McDonald

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Consider a modified, stable, two node Jackson network where server 2 helps server 1 when server 2 is idle. The probability of a large deviation of the number of customers at node one can be calculated using the flat boundary theory of Schwartz and Weiss [Large Deviations Performance Analysis (1994), Chapman and Hall, New York]. Surprisingly, however, these calculations show that the proportion of time spent on the boundary, where server 2 is idle, may be zero. This is in sharp contrast to the unmodified Jackson network which spends a nonzero proportion of time on this boundary.

Article information

Ann. Appl. Probab., Volume 15, Number 1B (2005), 519-541.

First available in Project Euclid: 1 February 2005

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Zentralblatt MATH identifier

Primary: 60K25: Queueing theory [See also 68M20, 90B22]
Secondary: 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) [See also 90Bxx]

Rare events change of measure h transform quasi-stationarity queueing networks


Foley, Robert D.; McDonald, David R. Large deviations of a modified Jackson network: Stability and rough asymptotics. Ann. Appl. Probab. 15 (2005), no. 1B, 519--541. doi:10.1214/105051604000000666.

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  • Alanyali, M. and Hajek, B. (1998). On large deviations of Markov processes with discontinuous statistics. Ann. Appl. Probab. 8 45--66.
  • Bartle, R. G. (1976). The Elements of Real Analysis, 2nd ed. Wiley, New York.
  • Bertsekas, D. (1995). Nonlinear Programming. Athena Scientific, Belmont, MA.
  • Blinovskii, V. M. and Dobrushin, R. L. (1994). Process level large deviations for a class of piecewise homogeneous random walks. In The Dynkin Festschrift: Markov Processes and Their Applications 1--59. Birkhäuser, Boston.
  • Dupuis, P. and Ellis, R. S. (1995). The large deviation principle for a general class of queueing systems I. TAMS 8 2689--2751.
  • Fayolle, G. and Iasnogorodski, R. (1979). Two coupled processors: The reduction to a Riemann--Hilbert problem. Z. Wahrsch. Verw. Gebiete 47 325--351.
  • Flatto, L. and Hahn, S. (1984). Two parallel queues created by arrivals with two demands I. SIAM J. Appl. Math. 44 1041--1053.
  • Foley, R. and McDonald, D. (2001). Join the shortest queue: Stability and exact asymptotics. Ann. Appl. Probab. 11 569--607.
  • Foley, R. and McDonald, D. (2005). Bridges and networks: Exact asymptotics. Ann. Appl. Probab. 15 542--586.
  • Ignatiouk-Robert, I. (2001). Sample path large deviations and convergence parameters. Ann. Appl. Probab. 11 1292--1329.
  • McDonald, D. (1999). Asymptotics of first passage times for random walk in a quadrant. Ann. Appl. Probab. 9 110--145.
  • Ney, P. and Nummelin, E. (1987). Markov additive processes I. Eigenvalue properties and limit theorems. Ann. Probab. 15 561--592.
  • Rardin, R. L. (1998). Optimization in Operations Research. Prentice-Hall, New York.
  • Rockafellar, R. T. (1970). Convex Analysis. Princeton Univ. Press.
  • Shwartz, A. and Weiss, A. (1994). Large Deviations for Performance Analysis. Chapman and Hall, New York.