Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-27T18:57:47.412Z Has data issue: false hasContentIssue false
  • English
  • Français

INFINITE-SERVER QUEUES WITH BATCH ARRIVALS AND DEPENDENT SERVICE TIMES

Published online by Cambridge University Press:  27 April 2012

Guodong Pang  and
Ward Whitt
Guodong Pang
Affiliation:
Harold and Inge Marcus Department of Industrial and Manufacturing Engineering, Pennsylvania State University, University Park, PA 16802 E-mail: [email protected]
Ward Whitt
Affiliation:
Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027 E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Motivated by large-scale service systems, we consider an infinite-server queue with batch arrivals, where the service times are dependent within each batch. We allow the arrival rate of batches to be time varying as well as constant. As regularity conditions, we require that the batch sizes be i.i.d. and independent of the arrival process of batches, and we require that the service times within different batches be independent. We exploit a recently established heavy-traffic limit for the number of busy servers to determine the performance impact of the dependence among the service times. The number of busy servers is approximately a Gaussian process. The dependence among the service times does not affect the mean number of busy servers, but it does affect the variance of the number of busy servers. Our approximations quantify the performance impact upon the variance. We conduct simulations to evaluate the heavy-traffic approximations for the stationary model and the model with a time-varying arrival rate. In the simulation experiments, we use the Marshall–Olkin multivariate exponential distribution to model dependent exponential service times within a batch. We also introduce a class of Marshall–Olkin multivariate hyperexponential distributions to model dependent hyper-exponential service times within a batch.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 26 , Issue 2 , April 2012 , pp. 197 - 220
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1.Berkes, I., Hörmann, S., & Schauer, J. (2009). Asymptotic results for the empirical process of stationary sequences. Stochastic Process & Their Applications 119: 1298–1324.CrossRefGoogle Scholar
2.Berkes, I. & Philipp, W. (1977). An almost sure invariance principle for empirical distribution function of mixing random variables. Zeitschrift Wahrscheinlichkeitstheorie und Verwanate Gebiete 41: 115–137.CrossRefGoogle Scholar
3.Bladt, M. & Nielsen, B. F. (2010). On the construction of bivariate exponential distributions with an arbitrary correlation coefficient. Stochastic Models 26: 295–308.CrossRefGoogle Scholar
4.Eckerg, A.E. (1983). Generalized peakedness of teletraffic processes. Proceedings of 10th International Teletraffic Congress. Montreal, Canada.Google Scholar
5.Eick, S.G., Massey, W.A., & Whitt, W. (1993). The physics of The M t/G/∞ queue. Operations Research 41: 731–742.CrossRefGoogle Scholar
6.Eick, S.G., Massey, W.A., & Whitt, W. (1993). M t/G/∞ queues with sinusoidal arrival rates. Management Science 39: 241–252.CrossRefGoogle Scholar
7.Falin, G. (1994). The M k/G/∞ batch arrival queue with heterogeneous dependent demands. Journal of Applied Probability 31: 841–846.CrossRefGoogle Scholar
8.Jacobs, P.A. & Lewis, P.A.W. (1977). A mixed autoregressive-moving average exponential sequence and point process (EARMA 1,1). Advances in Applied Probability 9: 87–104.CrossRefGoogle Scholar
9.Liu, L., Kashyap, B.R.K., & Templeton, J.G.C. (1991). On the GI X/G/∞ system. Journal of Applied Probability 27: 671–683.CrossRefGoogle Scholar
10.Liu, L. & Templeton, J.G.C. (1993). Autocorrelations in infinite server batch arrival queues. Queueing Systems 14: 313–337.CrossRefGoogle Scholar
11.Mark, B.L., Jagerman, D.L., & Ramamurthy, G. (1997). Peakedness measures for traffic characterization in high-speed networks. INFOCOM ’97. Sixteenth Annual Joint Conference of the IEEE Computer and Communications Societies. Vol. 2, 427–435.Google Scholar
12.Marshall, A.W. & Olkin, I. (1967). A multivariate exponential distribution. Journal of the American Statistical Association 62(317): 30–44.CrossRefGoogle Scholar
13.Massey, W.A. & Whitt, W. (1996). Stationary-process approximations for the nonstationary Erlang loss model. Operations Research 44(6): 976–983.CrossRefGoogle Scholar
14.Pang, G., Talreja, R., & Whitt, W. (2007). Martingale proofs of many-server heavy-traffic limits for Markovian queues. Probability Surveys 4, 193–267.CrossRefGoogle Scholar
15.Pang, G. & Whitt, W. (2010). Two-parameter heavy-traffic limits for infinite-server queues. Queueing Systems 65: 325–364.CrossRefGoogle Scholar
16.Pang, G. & Whitt, W. (2011). The impact of dependent service times on large-scale service times. Manufacturing & Service Operations Management http://dx.doi.org/10.1287/msom.1110.0363.Google Scholar
17.Shanbhag, D.N. (1966). On infinite server queues with batch arrivals. Journal of Applied Probability 9: 208–213.Google Scholar
18.Whitt, W. (1983). Comparing batch delays and customer delays. Bell System Technical Journal 62 (7): 2001–2009.CrossRefGoogle Scholar
19.Whitt, W. (1984). Heavy traffic approximations for service systems with blocking. AT&T Bell Laboratories Technical Journal 63, 689–708.CrossRefGoogle Scholar
20.Whitt, W. (2002). Stochastic-process limits. New York: Springer.CrossRefGoogle Scholar