Abstract
This paper studies an important aspect of queueing theory, autocorrelation properties of system processes. A general infinite server queue with batch arrivals is considered. There areM different types of customers and their arrivals are regulated by a Markov renewal input process. Batch sizes and service times depend on the relevant customer types. With a conditional approach, closed form expressions are obtained for the autocovariance of the continuous time and prearrival system sizes. Some special models are also discussed, giving insights into steady state system behaviour. Autocorrelation functions have a wide range of applications. We highlight one area of application by using autocovariances to derive variances of sample means for a number of special models.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Abate and W. Whitt, The correlation functions of RBM andM/M/1, Stochastic Models 4 (1988) 315–359.
V.E. Beneš,Mathematical Theory of Connecting Networks and Telephone Traffic (Academic Press, New York, 1965).
A.A. Borovkov, On limit laws for service processes in multi-channel systems, Siberian Math. J. 8 (1967) 746–763.
E. Çinlar,Introduction to Stochastic Processes (Prentice-Hall, Englewood Cliffs, NJ, 1975).
D.R. Cox and H.D. Miller,Stochastic Processes (Chapman and Hall, London, 1972).
P.W. Glynn, Some asymptotical formulas for Markov chains with applications to simulation, J. Statist. Comput. Simul. 19 (1984) 97–112.
P.W. Glynn and W. Whitt, A new view of the heavy-traffic limit theorem for infinite-server queues, Adv. Appl. Prob. 23 (1991) 188–209.
W.K. Grassmann, Means and variances of time averages in Markovian environments, Eur. J. Oper. Res. 31 (1987) 132–139.
S. Halfin and W. Whitt, Heavy-traffic limits for queues with many exponential servers, Oper. Res. 29 (1981) 567–588.
S. Halfin, Linear estimators for a class of stationary queueing processes, Oper. Res. 30 (1982) 515–529.
G.B. Hazen and A.A.B. Pritsker, Formulas for the variance of sample mean in finite state Markov processes, J. Statist. Comput. Simul. (1980) 25–40.
J.M. Harrison and A.J. Lemoine, A note on networks of infinite-server queues, J. Appl. Prob. 18 (1981) 561–567.
D.F. Holman, M.L. Chaudhry and B.R.K. Kashyap, On the number in the systemGI X/M/∞, Sankhyā Ser. A44, Pr. 1 (1982) 294–297.
D.L. Iglehart, Limit diffusion approximations for the many server queue and the repairman problem, J. Appl. Prob. 2 (1965) 429–441.
N. Kaplan, Limit theorems for aGI/G/∞ queue, Ann. Prob. 3 (1975) 780–789.
L. Liu, Infinite server queues with batch arrivals, Dissertation, Department of Industrial Engineering, University of Toronto (1990).
L. Liu, B.R.K. Kashyap and J.G.C. Templeton, On theGI X/G/∞ system, J. Appl. Prob. 27 (1990) 671–683.
L. Liu and J.G.C. Templeton, The\(GR^{X_n } /G_n /\infty \), system: system size, Queueing Systems 8 (1991) 323–356.
L. Liu and J.G.C. Templeton, The departure process in\(GR^{X_n } /G_n /\infty \), in preparation (1992).
W.A. Massey and W. Whitt, Networks of infinite-server queues with nonstationary Poisson input, Queueing Systems 13 (1993) 183.
M.F. Neuts and S.E. Chen, The infinite server queue with semi-Markovian arrivals and negative exponential services, J. Appl. Prob. 9 (1972) 178–184.
T.J. Ott, The stableM/G/1 queue in heavy traffic and its covariance function, Adv. Appl. Prob. 9 (1977) 169–186.
E. Parzen,Stochastic Processes (Holden-Day, San Francisco, 1962).
J.S. Rao, An application of stationary point processes to queueing theory and textile research, J. Appl. Prob. 3 (1966) 231–246.
J.F. Reynolds, N linearly regressive processes, J. Appl. Prob. 9 (1972) 208–213.
J.F. Reynolds, The covariance structure of queues and related processes — A survey of recent work, Adv. Appl. Prob. 7 (1975) 383–415.
D.N. Shanbhag, On infinite server queues with batch arrivals, J. Appl. Prob. 3 (1966) 274–279.
L. Takács, On a coincidence problem in telephone traffic, Acta Math. Acad. Sci. Hung. 9 (1958) 45–80.
W. Whitt, On the heavy-traffic limit theorem forGI/G/∞ queues, Adv. Appl. Prob. 14 (1982) 171–190.
W. Whitt, Asymptotic formulas for Markov processes with application to simulation, Oper. Res. 40 (1992) 279–291.
C.M. Woodside, B. Pagurek and G.F. Newell, A diffusion approximation for correlation in queues, J. Appl. Prob. 17 (1980) 1033–1047.
Author information
Authors and Affiliations
Additional information
This work has been supported by the Natural Sciences and Engineering Council of Canada through Grant A5639 and by the National Natural Science Foundation of China through Grant 19001015.
Rights and permissions
About this article
Cite this article
Liu, L., Templeton, J.G.C. Autocorrelations in infinite server batch arrival queues. Queueing Syst 14, 313–337 (1993). https://doi.org/10.1007/BF01158871
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01158871