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On the serial correlation for number in system in the stationary GI/M/1 bulk arrival and GI/Em/1 queues

Part of: Operations research and management science Computer system organization Special processes Multivariate analysis

Published online by Cambridge University Press:  14 July 2016

D. A. Stanford  and
B. Pagurek
D. A. Stanford*
Affiliation:
University of Western Ontario
B. Pagurek*
Affiliation:
Carleton University
*
∗Postal address: Department of Statistical and Actuarial Sciences, The University of Western Ontario, London, Canada N6A 5B9.
∗∗Postal address: Department of Systems and Computer Engineering, Carleton University, Ottawa, Canada.
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Abstract

The generating functions for the serial covariances for number in system in the stationary GI/M/1 bulk arrival queue with fixed bulk sizes, and the GI/Em/1 queue, are derived. Expressions for the infinite sum of the serial correlation coefficients are also presented, as well as the first serial correlation coefficient in the case of the bulk arrival queue. Several numerical examples are considered.

Keywords

MODULO PROBABILITIESCORRELATION COEFFICIENT SUMSCORRELATION COEFFICIENT GENERATING FUNCTIONS

MSC classification

Primary: 60K25: Queueing theory
Secondary: 62H20: Measures of association (correlation, canonical correlation, etc.) 90B22: Queues and service 68U20: Simulation 68M20: Performance evaluation; queueing; scheduling
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 2 , June 1992 , pp. 404 - 417
Copyright
Copyright © Applied Probability Trust 1992 

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References

[1] Blomqvist, N. (1967) The covariance function of the M/G/1 queueing system. Skand. Akt. 50, 157–174.Google Scholar
[2] Boxma, O. J. (1984) Joint distribution of sojourn time, waiting time, and queue length in the M/G/1 queue with (in)finite capacity. Eur. J. Operat. Res. 16, 246–256.CrossRefGoogle Scholar
[3] Daley, D. J. (1968) Monte Carlo estimation of the mean queue size in a stationary GI/M/1 queue. Operat. Res. 16, 1002–1005.CrossRefGoogle Scholar
[4] Daley, D. J. (1968) The serial correlation coefficients of waiting times in a stationary single server queue. J. Austral. Math. Soc. 8, 683–699.CrossRefGoogle Scholar
[5] Danielyan, E. A. and Zemlyanoy, N. C. (1978) Correlation coefficients of lengths of queues in a single-channel queueing system with relative priority. Eng. Cybernet. 16, 81–87.Google Scholar
[6] Grimmett, G. R. and Stirzaker, D. R. (1982) Probability and Random Processes. Oxford University Press, Oxford.Google Scholar
[7] Ott, T. J. (1977) The covariance function of the virtual waiting time process in an M/G/1 queue. Adv. Appl. Prob. 9, 158–168.Google Scholar
[8] Ott, T. J. (1977) The stable M/G/1 queue in heavy traffic and its covariance function. Adv. Appl. Prob. 9, 169–186.CrossRefGoogle Scholar
[9] Pagurek, B. and Woodside, C. M. (1979) The sum of serial correlations of waiting and system times in GI/G/1 queues. Operat. Res. 27, 755–766.CrossRefGoogle Scholar
[10] Pakes, A. G. (1971) The correlation coefficients of the queue lengths of some stationary single server queues. J. Austral. Math. Soc. 12, 35–46.CrossRefGoogle Scholar
[11] Pakes, A. G. (1971) The serial correlation coefficients of waiting times in the stationary GI/M/1 queue. Ann. Math. Statist. 42, 1727–1734.Google Scholar
[12] Reynolds, J. F. (1975) The covariance structure of queues and related processes - a survey of recent work. Adv. Appl. Prob. 7, 383–415.CrossRefGoogle Scholar
[13] Srinivasan, S. K. and Chudalaimuthu Pillai, C. (1980) Correlation functions in the G/M/1 system. Adv. Appl. Prob. 12, 530–540.Google Scholar
[14] Stanford, D. A., Pagurek, B. and Woodside, C. M. (1983) Optimal prediction of times and queue lengths in the GI/M/1 queue. Operat. Res. 31, 322–337.Google Scholar
[15] Stanford, D. A., Pagurek, B. and Woodside, C. M. (1988) The serial correlation coefficients for waiting times in the stationary GI/M/m queue. QUESTA 2, 373–380.Google Scholar
[16] Takács, L. J. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar
[17] Woodside, C. M. and Pagurek, B. (1979) An algorithm for computing serial correlations of times in GI/G/1 queues with rational arrival processes. Management Sci. 25, 54–63.CrossRefGoogle Scholar
[18] Woodside, C. M., Pagurek, B. and Newell, G. F. (1980) A diffusion approximation for correlation in queues. J. Appl. Prob. 17, 1033–1047.Google Scholar