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ON THE VARIANCES OF SYSTEM SIZE AND SOJOURN TIME IN A DISCRETE-TIME DAR(1)/D/1 QUEUE

Published online by Cambridge University Press:  21 July 2011

Daniel Wei-Chung Miao  and
Hung Chen
Daniel Wei-Chung Miao
Affiliation:
National Taiwan University of Science and Technology, Taipei, 106, Taiwan E-mail: miao@mail.ntust.edu.tw
Hung Chen
Affiliation:
National Taiwan University, Taipei, 106, Taiwan E-mail: hchen@math.ntu.edu.tw
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Abstract

We consider a discrete-time DAR(1)/D/1 queue and provide an analysis on the variances of both its system size and sojourn time. Our approach is simple, but the results are nice, as these variances are found in closed form. We first establish the relation between these variances, based on which we then use the conditioning technique to analyze the expected cross terms that come from its system recurrence relation. The closed-form results allow us to explicitly examine the effect from the batch size distribution and the autocorrelation parameter p. It is observed that as p grows toward 1, the standard deviations of the two performance measures will blow up in same asymptotic order of O(1/(1−p)) as their means. These are demonstrated through numerical examples.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 25 , Issue 4 , October 2011 , pp. 519 - 535
Copyright
Copyright © Cambridge University Press 2011

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References

1.Bertsimas, D. & Nakazato, D. (1995). The distributional Little's law and its applications. Operations Research 43: 298–310.CrossRefGoogle Scholar
2.Brumelle, S.L. (1972). A generalization of L=λ W to moments of queue length and waiting times. Operations Research 20: 1127–1136.CrossRefGoogle Scholar
3.Chaudhry, M.L., Yoon, B.K. & Chae, K.C. (2002). Waiting-time distribution of a discrete-time multiserver queue with correlated arrivals and deterministic service times: D-MAP/D/k system. Operations Research Letters 30: 174–180.CrossRefGoogle Scholar
4.Choi, B.D., Kim, B., Hwang, G.U. & Kim, J. (2004). The analysis of a multiserver queue fed by a discrete autoregressive process of order 1. Operations Research Letters 32: 85–93.CrossRefGoogle Scholar
5.Fiems, D. & Bruneel, H. (2002). A note on the discretization of Little's result. Operations Research Letters 30: 17–18.CrossRefGoogle Scholar
6.Heffes, H. & Lucantoni, D.M. (1986). A Markov modulated characterization of packetized voice and data traffic and related statistical multiplexer performance. IEEE Journal Selected Areas on Communications 4: 856–868.CrossRefGoogle Scholar
7.Herrmann, C. (2001). The complete analysis of the discrete time finite DBMAP/G/1/N queue. Performance Evaluation 43: 95–121.CrossRefGoogle Scholar
8.Hwang, G.U., Choi, B.D. & Kim, J. (2002). The waiting time analysis of a discrete-time queue with arrivals as a discrete autoregressive process of order 1. Journal of Applied Probability 39: 619–629.CrossRefGoogle Scholar
9.Hwang, G.U. & Sohraby, K. (2003). On the exact analysis of a discrete-time queueing system with autogressive inputs. Queueing Systems 43: 29–41.CrossRefGoogle Scholar
10.Kang, S.H., Sung, D.K. & Choi, B. D. (1998). An empirical real-time approximation of waiting time distribution in MMPP(2)/D/1. IEEE Communications Letters 2: 17–19.CrossRefGoogle Scholar
11.Kim, B. & Sohraby, K. (2006). Tail behavior of the queue size and waiting time in a queue with discrete autogressive arrivals. Advances in Applied Probability 38: 1116–1131.CrossRefGoogle Scholar
12.Kim, B., Chang, Y., Kim, Y.C. & Choi, B.D. (2007). A queueing system with discrete autoregressive arrivals. Performance Evaluation 64: 148–161.CrossRefGoogle Scholar
13.Kim, N.K. & Chaudhry, M. L. (2008). The use of the distributional Little's law in the computational analysis of discrete-time GI/G/1 and GI/G/c queues. Performance Evaluation 65: 3–9.CrossRefGoogle Scholar
14.Kim, J., Kim, B. & Sohraby, K. (2008). Mean queue size in a queue with discrete autogressive arrivals of order p. Annals of Operations Research 162: 69–83.CrossRefGoogle Scholar
15.Lucantoni, D.M. (1991). New results on the single server queue with a batch Markovian arrival process. Stochastic Models 7: 1–46.CrossRefGoogle Scholar
16.Ozekici, S. (1997). Markov modulated Bernoulli process. Mathematical Methods of Operations Research 45: 331–324.CrossRefGoogle Scholar
17.Takahashi, Y. & Miyazawa, M. (1995). Relationship between queue length and waiting time in a stationary discrete-time queue. Stochastic Models 11: 249–271.CrossRefGoogle Scholar
18.Whitt, W. (1991). A review of L=λ W and extensions. Queueing Systems 9: 235–268.CrossRefGoogle Scholar
19.Xiong, Y., Bruneel, H. & Steyaert, B. (1996). Deriving delay characteristics from queue length statistics in discrete-time queues with multiple servers. Performance Evaluation 24: 189–204.CrossRefGoogle Scholar