Abstract
In this paper, we provide an exact analysis of a discrete-time queueing system driven by a discrete autoregressive model of order 1 (DAR(1)) characterized by an arbitrary marginal batch size distribution and a correlation coefficient. Closed-form expressions for the probability generating function and mean queue length are derived. It is shown that the system performance is quite sensitive to the correlation of the arrival process. In addition, a comparison with traditional Markovian processes shows that arrival processes of DAR(1) type exhibit larger queue length as compared with the traditional Markovian processes when the marginal densities and correlation coefficients are matched.
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References
G.E.P. Box and G.M. Jenkins, Time Series Analysis: Forecasting and Control (Holden-Day, 1970).
P.C. Consul, A generalization of Poisson distribution, Technometrics 15(4) (1973).
D.R. Cox, Some statistical methods connected with series of events, J. Roy. Statist. Soc. Ser. B 17 (1955) 129–164.
D.R. Cox and P.A.W. Lewis, Statistical Analysis of Series of Events (Chapman and Hall, London, 1966).
Z. Drezner and N. Farnum, A correlated Poisson distribution for correlated events, Comm. Statist. Theory Methods 23(3) (1994) 841–857.
A. Elwalid, D. Heyman, T.V. Lakshman, D. Mitra and A. Weiss, Fundamental bounds and approximations for ATM multiplexers with applications to video teleconferencing, IEEE J. Selected Areas Commun. 13(6) (1995) 1004–1016.
P.D. Finch, The single server queueing system with non-recurrent input-process and Erlang service time, J. Australian Math. Soc. 3 (1963) 220–236.
P.D. Finch and C. Pearce, A second look at a queueing system with moving average input process, J. Australian Math. Soc. 5 (1965) 100–106.
P. Franken, D. Konig, U. Arndt and V. Schmidt, Queues and Point Processes (Akademie-Verlag, Berlin, 1982).
D.P. Gaver and P.A.W. Lewis, First-order autoregressive gamma sequences and point processes, Adv. Appl. Probab. 12 (1980) 727–745.
B. Hajek and L. He, On variations of queue response for inputs with the same mean and autocorrelation function, IEEE/ACM Trans. Networking 6(5) (1998) 588–598.
J. He and K. Sohraby, A new analysis framework for discrete time queueing systems with general stochastic sources, in:IEEE INFOCOM '2001, Anchorage, Alaska, April 2001.
C.L. Hwang and S.Q. Li, On the convergence of traffic measurement and queueing analysis: A statistical-match queueing (SMAQ) tool, in: Proc. IEEE INFOCOM '95, 1995, pp. 602–613.
P.A. Jacob and P.A.W. Lewis, Time series generated by mixtures, J. Time Series Anal. 4(1) (1983) 19–36.
A.J. Lawrence and P.A.W. Lewis, An exponential moving average sequence and point process EMA1 process, J. Appl. Probab. 14 (1977) 98–113.
R. Szekli, R.L. Disney and S. Hur, A MR/GI/1 queues with positively correlated arrival stream, J. Appl. Probab. 31 (1994) 497–514.
S.K. Srinivasan,Stochastic Point Processes and Their Applications (Charles Griffin, London, 1974).
P. Tin, A queueing system with Markov-dependent arrivals, J. Appl. Probab. 22 (1985) 668–677.
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Hwang, G.U., Sohraby, K. On the Exact Analysis of a Discrete-Time Queueing System with Autoregressive Inputs. Queueing Systems 43, 29–41 (2003). https://doi.org/10.1023/A:1021848330183
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DOI: https://doi.org/10.1023/A:1021848330183