Mean Waiting Time Approximations in the G/G/1 Queue

It is known that correlations in an arrival stream offered to a single-server queue profoundly affect mean waiting times as compared to a corresponding renewal stream offered to the same server. Nonetheless, this paper uses appropriately constructed GI/G/1 models to create viable approximations for queues with correlated arrivals. The constructed renewal arrival process, called PMRS (Peakedness Matched Renewal Stream), preserves the peakedness of the original stream and its arrival rate; furthermore, the squared coefficient of variation of the constructed PMRS equals the index of dispersion of the original stream. Accordingly, the GI/G/1 approximation is termed PMRQ (Peakedness Matched Renewal Queue). To test the efficacy of the PMRQ approximation, we employed a simple variant of the TES+ process as the autocorrelated arrival stream, and simulated the corresponding TES+/G/1 queue for several service distributions and traffic intensities. Extensive experimentation showed that the proposed PMRQ approximations produced mean waiting times that compared favorably with simulation results of the original systems. Markov-modulated Poisson process (MMPP) is also discussed as a special case.

[1]  Benjamin Melamed,et al.  ON MARKOVIAN TRAFFIC WITH APPLICATIONS TO TES PROCESSES , 1994 .

[2]  Benjamin Melamed,et al.  An Overview of Tes Processes and Modeling Methodology , 1993, Performance/SIGMETRICS Tutorials.

[3]  Miron Livny,et al.  The Impact of Autocorrelation on Queuing Systems , 1993 .

[4]  Bruce E. Hajek,et al.  On variations of queue response for inputs with the same mean and autocorrelation function , 1998, TNET.

[5]  R. Marie.  Modélisation par réseaux de files d'attente , 1978 .

[6]  Ronald W. Wolff,et al.  Stochastic Modeling and the Theory of Queues , 1989 .

[7]  B. Melamed,et al.  The transition and autocorrelation structure of tes processes: Part II: Special Cases , 1992 .

[8]  B. Eddy Patuwo,et al.  The effect of correlated arrivals in queues , 1993 .

[9]  Tayfur Altiok,et al.  The Case for Modeling Correlation in Manufacturing Systems , 2001 .

[10]  Paul Bratley,et al.  A guide to simulation (2nd ed.) , 1986 .

[11]  Leonard Kleinrock,et al.  Queueing Systems - Vol. 1: Theory , 1975 .

[12]  Leonard Kleinrock,et al.  Theory, Volume 1, Queueing Systems , 1975 .

[13]  D. Cox,et al.  The statistical analysis of series of events , 1966 .

[14]  B. Melamed,et al.  The transition and autocorrelation structure of tes processes , 1992 .

[15]  Ward Whitt,et al.  Dependence in packet queues , 1989, IEEE Trans. Commun..

[16]  David R. Cox,et al.  The statistical analysis of series of events , 1966 .

[17]  Benjamin Melamed,et al.  TES: A Class of Methods for Generating Autocorrelated Uniform Variates , 1991, INFORMS J. Comput..

[18]  Linus Schrage,et al.  A guide to simulation , 1983 .

[19]  David L. Jagerman.  Approximations for waiting time in GI/G/1 systems , 1987, Queueing Syst. Theory Appl..

[20]  Wolfgang Fischer,et al.  The Markov-Modulated Poisson Process (MMPP) Cookbook , 1993, Perform. Evaluation.

[21]  Riccardo Gusella,et al.  Characterizing the Variability of Arrival Processes with Indexes of Dispersion , 1991, IEEE J. Sel. Areas Commun..

[22]  Benjamin Melamed,et al.  BURSTINESS DESCRIPTORS OF TRAFFIC STREAMS: INDICES OF DISPERSION AND PEAKEDNESS , 1994 .

[23]  Paul Bratley,et al.  A guide to simulation , 1983 .

[24]  D. L. Jagerman.  Methods in traffic calculations , 1984, AT&T Bell Laboratories Technical Journal.

[25]  J. A. Buzacott,et al.  On the approximations to the single server queue , 1980 .