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Integration of Discrete-Time Correlated Markov Processes in a TDM System

Published online by Cambridge University Press:  27 July 2009

Cheng-Shang Chang ,
Xiuli Chao  and
Michael Pinedo
Cheng-Shang Chang
Affiliation:
IBM Research Division T.J. Watson Research Center P.O. Box 704, Yorktown Heights, New York 10598
Xiuli Chao
Affiliation:
IBM Research Division T.J. Watson Research Center P.O. Box 704, Yorktown Heights, New York 10598
Michael Pinedo
Affiliation:
Center for Telecommunications Research Department of Industrial Engineering and Operations Research Columbia University, New York, New York 10027
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In this paper, we consider a discrete-time queueing model for a Time Division Multiplexing (TDM) system with integration of voice and data (a model introduced by Li and Mark [16]). The voice traffic is a superposition of N Markov chains, which alternate between two states: the talkspurt state and the silence state. The data traffic is Poisson and independent of the voice sources. We show that the average queue size is increasing in certain correlation coefficients of the voice sources, increasing convex in the proportion of time the voice sources are in talkspurts, increasing convex in the number of voice sources, and increasing convex in the data traffic intensity. However, it is decreasing convex in the number of channels. These structural results yield various bounds. To take video traffic into account as well, we adapt a model of Maglaris et al. [18]. In their model, video traffic is generated by a continuous-state autoregressive Markov process that matches the average rate and the autocovariance of the output of a video coder. We show that if we replace their autoregressive model by a two-state Markov chain model with the same rate and correlation coefficient, we obtain an upper bound for the queue size. This replacement enables us to treat the video traffic as a voice source and use the techniques developed for dealing with voice/data integration to obtain bounds and estimates.

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 4 , Issue 1 , January 1990 , pp. 29 - 56
Copyright
Copyright © Cambridge University Press 1990

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References

Asmussen, S. & Thorisson, H. (1987). A Markov-chain approach to periodic queues. Journal of Applied Probability 24: 215–225.CrossRefGoogle Scholar
Borovkov, A.A. (1976). Stochastic processes in queueing theory. New York: Springer-Verlag.CrossRefGoogle Scholar
Burman, D.Y. & Smith, D.R. (1986). An asymptotic analysis of a queueing system with Markov-modulated arrivals. Operations Research 34 (1): 105–119.CrossRefGoogle Scholar
Chang, C.S. (1989). Comparison theorems for queueing systems and their applications to ISDN. Ph.D. dissertation. Department of Electrical Engineering, Columbia University, New York.Google Scholar
Chang, C.S., Chao, X.L., & Pinedo, M. (1988). Monotonicity results for queues with doubly stochastic Poisson arrivals: Ross's Conjecture. Submitted to Advances in Applied Probability.Google Scholar
Chang, C.S. & Pinedo, M. (1988). Bounds and inequalities for single server loss systems. Submitted to Queueing Systems.Google Scholar
Chow, Y.-S. & Teicher, H. (1988). Probability theory, independence, interchangeability, martingales. New York: Springer-Verlag.Google Scholar
De, Serres Y. & Mason, L.G. (1988). A multiserver queue with narrow- and wide-band customers and wide-band restricted access. IEEE Transactions on Communication 36(6): 675–684.Google Scholar
Fond, S. & Ross, S.M. (1978). A heterogeneous arrival and service queueing loss model. Naval Research Logistics Quarterly 25: 483–488.CrossRefGoogle Scholar
Gaver, D.P. & Lehoczky, J.P. (1982). Channels that cooperatively service a data stream and voice messages. IEEE Transactions on Communication. COM-30: 1153–1162.CrossRefGoogle Scholar
Heffes, H. & Lucantoni, D.M. (1986). A Markov-modulated characterization of packetized voice and data traffic and related statistical multiplexer performance. IEEE Journal of Selected Areas in Communication SAC-4(6): 856–867.CrossRefGoogle Scholar
Heyman, D.P. (1982). On Ross's conjecture about queues with nonstationary Poisson arrivals. Journal of Applied Probability 19: 245–249.CrossRefGoogle Scholar
Kraimeche, B. & Schwartz, M. (1986). Bandwidth allocation strategies in wide-band integrated networks. IEEE Journal of Selected Areas in Communication SAC-4: 869–878.CrossRefGoogle Scholar
Lehoczky, J.P. & Gaver, D.P. (1981). Diffusion approximations for the cooperative service and voice and data messages. Journal of Applied Probability 18: 660–671.CrossRefGoogle Scholar
Li, S.-Q. & Mark, J.W. (1985). Performance of voice/data integration on a TDM system. IEEE Transactions on Communication COM-33(12): 1265–1273.Google Scholar
Li, S.-Q. & Mark, J.W. (1988). Performance trade-offs in an integrated voice/data service TDM systems. Performance Evaluation 9: 51–64.CrossRefGoogle Scholar
Li, S.-Q. & Mark, J.W. (1988). Traffic characterization for integrated services networks. Resubmitted to IEEE Transactions on Communication.Google Scholar
Maglaris, B., Anastassiou, D., Sen, P., Karlsson, G., & Robbins, J. (1988). Performance models of statistical multiplexing in packet video communications. IEEE Transactions on Communication 36(7): 834–844.CrossRefGoogle Scholar
Marshall, A.W. & Olkin, I. (1979). Inequalities: Theory of majorization and its applications. New York: Academic Press.Google Scholar
Mitrani, I.L. & Avi-Itzhak, B. (1968). A many-server queue with service interruptions. Operations Research 16(3): 628–638.CrossRefGoogle Scholar
Neuts, M.F. (1978). Further results in the M/M/l queue with randomly varying rates. Opsearch 15(14): 158–168.Google Scholar
Neuts, M.F. & Lucantoni, D.M. (1979). A Markov chain with N servers subject to breakdowns and repairs. Management Science 25(9): 849–861.CrossRefGoogle Scholar
Niu, S.-C. (1980). A single server queueing loss model with heterogeneous arrival and service. Operations Research 28: 584–593.CrossRefGoogle Scholar
Papoulis, A. (1984). Probability, random variables, and stochastic processes. New York: McGraw-Hill.Google Scholar
Pfanzagl, J. (1974). Convexity and conditional expectation. Annals of Probability 2: 490–494.CrossRefGoogle Scholar
Regterschot, G.J.K. & de, Smit J.H.A. (1986). The queue M/G/1 with Markov-modulated arrivals and services. Mathematics of Operations Research 11(3): 465–483.CrossRefGoogle Scholar
Rolski, T. (1981). Queues with nonstationary input stream: Ross's conjecture. Advances in Applied Probability 13: 603–618.CrossRefGoogle Scholar
Rolski, T. (1983). Comparison theorems for queues with dependent interarrival times, modeling, and performance evaluation. Proceeding of the International Seminar, Paris, France, 01 24–26, pp. 42–67.Google Scholar
Rolski, T. (1986) Upper bounds for single server queues with doubly stochastic Poisson arrivals. Mathematics of Operations Research. 11(3): 442–450.CrossRefGoogle Scholar
Ross, S.M. (1978). Average delay in queues with nonstationary Poisson arrivals. Journal of Applied Probability 15: 602–609.CrossRefGoogle Scholar
Ross, S.M. (1983). Stochastic processes. J. Wiley & Sons.Google Scholar
Shaked, M. & Shanthikumar, J.G. (1988). Temporal stochastic convexity and concavity. Stochastic Process and their Applications 27: 1–20.CrossRefGoogle Scholar
Sigman, K. (1988). Queues as Harris recurrent Markov chains. Queueing Systems 3: 179–198.CrossRefGoogle Scholar
Sriram, K., Varshney, P.K., & Shanthikumar, J.G. (1983). Discrete-time analysis of integrated voice-data multiplexer with and without speech activity detectors. IEEE Journal of Selected Areas in Communication (Special Issue on Packet Switched Voice and Data Communications) SAC-1: 1124–32.CrossRefGoogle Scholar
Stoyan, D. (1983). Comparison methods for queues and other stochastic models. New York: J. Wiley & Sons.Google Scholar
Svoronos, A. & Green, L. (1987). The N-seasons S-servers loss system. Naval Research Logistics 34: 579–591.3.0.CO;2-K>CrossRefGoogle Scholar
Svoronos, A. & Green, L. (1988). A convexity result for single server exponential loss systems with nonstationary arrivals. Journal of Applied Probability 25: 224–227.CrossRefGoogle Scholar
Tchen, A.H. (1980). Inequalities for distributions with given marginals. Annals of Probability 8(4): 814–827.CrossRefGoogle Scholar
Wolff, R.W. (1982). Poisson arrivals see time averages. Operations Research 30(2): 223–231.CrossRefGoogle Scholar