Abstract
The large deviation principle (LDP) which has been effectively used in queueing analysis is the sample path LDP, the LDP in a function space endowed with the uniform topology. Chang [5] has shown that in the discrete-time G/D/1 queueing system under the FIFO discipline, the departure process satisfies the sample path LDP if so does the arrival process. In this paper, we consider arrival processes satisfying the LDP in a space of measures endowed with the weak* topology (Lynch and Sethuraman [12]) which holds under a weaker condition. It is shown that in the queueing system mentioned above, the departure processes still satisfies the sample path LDP. Our result thus covers arrival processes which can be ruled out in the work of Chang [5]. The result is then applied to obtain the exponential decay rate of the queue length probability in an intree network as was obtained by Chang [5], who considered the arrival process satisfying the sample path LDP.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Ahn and J. Jeon, Analysis of the M/D/1-type queue based on an integer-valued first-order autoregressive process, Oper. Res. Lett. 27 (2000) 235-241.
M.A. Al-Osh and A.A. Alzaid, First-order integer-valued autoregressive (INAR(1)) process, J. Time Series Anal. 3 (1987) 261-275.
A.A. Borovkov, Stochastic Processes in Queueing Theory (Springer, New York, 1976).
C.S. Chang, Stability, queue length and delay of deterministic and stochastic queueing networks, IEEE Trans. Automat. Contol 39 (1994) 913-931.
C.S. Chang, Sample path large deviation and intree network, Queuing Systems 20 (1995) 7-36.
C.S. Chang, P. Heidelberger, S. Juneja and P. Shahabuddin, Effective bandwidth and fast simulation of ATM intree networks, Performance Evaluation 20 (1994) 45-65.
D.A. Dawson and J. Gärtner, Large deviations from the McKean-Vlasov limit for weakly interacting diffusions, Stochastics 20 (1987) 247-308.
A. Dembo and T. Zajic, Large deviations: from empirical mean and measure to partial sums process, Stochastic Process. Appl. 57 (1995) 191-224.
A. Dembo and O. Zeitouni, Large Deviation Technique and Its Applications (Jones and Bartlett, Boston, 1993).
W. Glynn and W. Whitt, Logarithmic asymptotics for steady state tail probabilities in a single server queue, J. Appl. Probab. A 31 (1993) 131-159.
G. Keisidis, J. Walrand and C.S. Chang, Effective bandwiths for multiclass Markov fluids and other ATM sources, IEEE/ACM Trans. Networking 1 (1993) 424-428.
J. Lynch and J. Sethuraman, Large deviations for processes with independent increments, Ann. Probab. 15 (1987) 610-627.
A.A. Mogulskii, Large deviations for trajectories of multi-dimensional random walks, Theory Probab. Appl. 21 (1976) 300-315.
R.T. Rockafellar, Convex Analysis (Princeton Univ. Press, Princeton, 1970).
A. Shwartz and A. Weiss, Large Deviations for Performance Analysis, Queues, Communications and Computing (Chapmann and Hall, London, 1995).
S.R.S. Varadhan, Asymptotic probabilities and differential equations, Comm. Pure Appl. Math. 19 (1966) 261-286.
A.D. Ventsel, Rough limit theorems on large deviations for Markov stochastic processes, I, II, Theory Probab. Appl. 21 (1976) 227-242, 499-512.
W. Whitt, Tail probabilities with statistical multiplexing and effective bandwidths in multi-class queues, Telecommunication Systems 2 (1993) 71-107.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ahn, S., Jeon, J. Analysis of G/D/1 Queueing Systems with Inputs Satisfying Large Deviation Principle under Weak* Topology. Queueing Systems 40, 295–311 (2002). https://doi.org/10.1023/A:1014763513810
Issue Date:
DOI: https://doi.org/10.1023/A:1014763513810