Complutense University Library

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

Artalejo, Jesús R. and Lopez-Herrero, M. J. (2001) Analysis of the busy period for the M/M/c queue: An algorithmic approach. Journal of Applied Probability , 38 (1). pp. 209-222. ISSN 0021-9002

[img] PDF
Restricted to Repository staff only until 31 December 2020.

1MB

Official URL: http://www.jstor.org/stable/3215752

View download statistics for this eprint

==>>> Export to other formats

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.


Item Type:Article
Additional Information:

The authors thank the referee for his detailed comments on an earlier version of this paper. This work was supported by the European Commission through INTAS 96-0828, by the
DGES through project 98-0837 and by the Complutense University of Madrid through project PR64/99-8501.

Uncontrolled Keywords:Multiserve queues; busy period; number of customers served; maximum; entropy principle
Subjects:Sciences > Mathematics > Operations research
ID Code:15826
References:

Arora, K. L. (1964). Two-server bulk-service queuing process. Operat. Res. 12, 286–294.

Artalejo, J. R. (1999). Accessible bibliography on retrial queues. Math. Comput. Model. 30, 1–6.

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.

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.

Daley, D. J. and Servi, L. D. (1998). Idle and busy periods in stable M/M/k queues. J. Appl. Prob. 35, 950–962.

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.

Falin, G. I. and Templeton, J. G. C. (1997). Retrial Queues. Chapman and Hall, London.

Ghahramani, S. (1986). Finiteness of moments of partial busy periods for M/G/c queues. J. Appl. Prob. 23, 261–264. 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.

Karlin, S. and McGregor, J. (1958). Many server queueing processes with Poisson input and exponential service times. Pacific J. Math. 8, 87–118.

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.

Kleinrock, L. (1975). Queueing Systems, Vol. I: Theory. John Wiley, New York.

Kouvatsos, D. D. (1994). Entropy maximization and queueing networks models. Ann. Operat. Res. 48, 63–126.

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.

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 U. N. Bhat and I. V. Basawa, Clarendon Press, Oxford, pp. 240–261.

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.

Natvig, B. (1975b). A Contribution to the Theory of Birth-and-Death Queueing Models. Doctoral Thesis, University of Sheffield.

Sharma, O. P. (1990). Markovian Queues. Ellis Horwood, New York.

Shore, J. E. and Johnson, R. W. (1981). Properties of cross-entropy minimization. IEEE Trans. Inf. Theory 27, 472–482. Syski, R. (1986). Introduction to Congestion Theory in Telephone Systems. Elsevier/North-Holland, Amsterdam.

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.

Wagner, U. and Geyer, A. L. (1995). A maximum entropy method for inverting Laplace transforms of probability density functions. Biometrika 82, 887–892.

Wiens, D. P. (1989). On the busy period distribution of the M/G/2 queueing system. J. Appl. Prob. 27, 858–865.

Deposited On:04 Jul 2012 09:50
Last Modified:13 May 2014 16:54

Repository Staff Only: item control page