Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T18:59:45.607Z Has data issue: false hasContentIssue false
  • English
  • Français

Analysis of the busy period for the M/M/c queue: an algorithmic approach

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

J. R. Artalejo  and
M. J. Lopez-Herrero
J. R. Artalejo*
Affiliation:
Universidad Complutense de Madrid
M. J. Lopez-Herrero*
Affiliation:
Universidad Complutense de Madrid
*
∗ Postal address: Faculty of Mathematics, Department of Statistics and O.R., Complutense University of Madrid, Madrid 28040, Spain. Email address: jesus_artalejo@mat.ucm.es
∗∗ Postal address: School of Statistics, Complutense University of Madrid, Madrid 28040, Spain.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper presents an algorithmic analysis of the busy period for the M/M/c queueing system. By setting the busy period equal to the time interval during which at least one server is busy, we develop a first step analysis which gives the Laplace-Stieltjes transform of the busy period as the solution of a finite system of linear equations. This approach is useful in obtaining a suitable recursive procedure for computing the moments of the length of a busy period and the number of customers served during it. The maximum entropy formalism is then used to analyse what is the influence of a given set of moments on the distribution of the busy period and to estimate the true busy period distribution. Our study supplements a recent work of Daley and Servi (1998) and other studies where the busy period of a multiserver queue follows a different definition, i.e., a busy period is the time interval during which all servers are engaged.

Keywords

multiserver queuesbusy periodnumber of customers servedmaximum entropy principle

MSC classification

Primary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 1 , March 2001 , pp. 209 - 222
Copyright
Copyright © by the Applied Probability Trust 2001 

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

Arora, K. L. (1964). Two-server bulk-service queuing process. Operat. Res. 12, 286–294.Google Scholar
Artalejo, J. R. (1999). Accessible bibliography on retrial queues. Math. Comput. Model. 30, 1–6.CrossRefGoogle Scholar
Artalejo, J. R., and Gomez-Corral, A. (1997). Steady state solution of a single-server queue with linear repeated request. J. Appl. Prob. 34, 223–233.Google Scholar
Artalejo, J. R., and Lopez-Herrero, M. J. (2000). On the busy period of the M/G/1 retrial queue. Naval Res. Logist. 47, 115–127.3.0.CO;2-C>CrossRefGoogle Scholar
Daley, D. J., and Servi, L. D. (1998). Idle and busy periods in stable M/M/k queues. J. Appl. Prob. 35, 950–962.CrossRefGoogle Scholar
Falin, G. I., Martin, M., and Artalejo, J. R. (1994). Information theoretic approximations for the M/G/1 retrial queue. Acta Inf. 31, 559–571.CrossRefGoogle Scholar
Falin, G. I., and Templeton, J. G. C. (1997). Retrial Queues. Chapman and Hall, London.Google Scholar
Ghahramani, S. (1986). Finiteness of moments of partial busy periods for M/G/c queues. J. Appl. Prob. 23, 261–264.CrossRefGoogle Scholar
Ghahramani, S. (1990). On remaining full busy periods of GI/G/c queues and their relation to stationary point processes. J. Appl. Prob. 27, 232–236.Google Scholar
Karlin, S., and McGregor, J. (1958). Many server queueing processes with Poisson input and exponential service times. Pacific J. Math. 8, 87–118.Google Scholar
Kiefer, J., and Wolfowitz, J. (1956). On the characteristics of the general queuing process with applications to random walks. Ann. Math. Statist. 27, 147–161.Google Scholar
Kleinrock, L. (1975). Queueing Systems, Vol. 1: Theory. John Wiley, New York.Google Scholar
Kouvatsos, D. D. (1994). Entropy maximization and queueing networks models. Ann. Operat. Res. 48, 63–126.Google Scholar
Kouvatsos, D. D., and Awan, I. U. (1998). MEM for arbitrary closed queueing networks with RS-blocking and multiple job classes. Ann. Operat. Res. 79, 231–269.Google Scholar
McCormick, W. P., and Park, Y. S. (1992). Approximating the distribution of the maximum queue length for M/M/s queues. In Queuing and Related Models, eds Bhat, U. N. and Basawa, I. V., Clarendon Press, Oxford, pp. 240–261.Google Scholar
Natvig, B. (1975a). On the waiting-time and busy period distributions for a general birth-and-death queueing model. J. Appl. Prob. 12, 524–532.CrossRefGoogle Scholar
Natvig, B. (1975b). A Contribution to the Theory of Birth-and-Death Queueing Models. Doctoral Thesis, University of Sheffield.Google Scholar
Sharma, O. P. (1990). Markovian Queues. Ellis Horwood, New York.Google Scholar
Shore, J. E., and Johnson, R. W. (1981). Properties of cross-entropy minimization. IEEE Trans. Inf. Theory 27, 472–482.CrossRefGoogle Scholar
Syski, R. (1986). Introduction to Congestion Theory in Telephone Systems. Elsevier/North-Holland, Amsterdam.Google Scholar
Omahen, K., and Marathe, V. (1978). Analysis and applications of the delay cycle for the M/M/c queueing system. J. Assoc. Comput. Mach. 25, 283–303.Google Scholar
Wagner, U., and Geyer, A. L. (1995). A maximum entropy method for inverting Laplace transforms of probability density functions. Biometrika 82, 887–892.CrossRefGoogle Scholar
Wiens, D. P. (1989). On the busy period distribution of the M/G/2 queueing system. J. Appl. Prob. 27, 858–865.Google Scholar