Entropy maximization and the busy period of some single-server vacation models

Artalejo, Jesus R.; Lopez-Herrero, Maria J.

doi:10.1051/ro:2004020

Artalejo, Jesus R. ; Lopez-Herrero, Maria J.

RAIRO - Operations Research - Recherche Opérationnelle, Tome 38 (2004) no. 3, pp. 195-213.

Résumé

In this paper, information theoretic methodology for system modeling is applied to investigate the probability density function of the busy period in $M / G / 1$ vacation models operating under the $N$ -, $T$ - and $D$ -policies. The information about the density function is limited to a few mean value constraints (usually the first moments). By using the maximum entropy methodology one obtains the least biased probability density function satisfying the system’s constraints. The analysis of the three controllable $M / G / 1$ queueing models provides a parallel numerical study of the solution obtained via the maximum entropy approach versus “classical” solutions. The maximum entropy analysis of a continuous system descriptor (like the busy period) enriches the current body of literature which, in most cases, reduces to discrete queueing measures (such as the number of customers in the system).

MR 2091752 | Zbl pre02110162

DOI : https://doi.org/10.1051/ro:2004020
Mots clés : busy period analysis, maximum entropy methodology,

M / G / 1

vacation models, numerical inversion

@article{RO_2004__38_3_195_0,
     author = {Artalejo, Jesus R. and Lopez-Herrero, Maria J.},
     title = {Entropy maximization and the busy period of some single-server vacation models},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {195--213},
     publisher = {EDP-Sciences},
     volume = {38},
     number = {3},
     year = {2004},
     doi = {10.1051/ro:2004020},
     zbl = {pre02110162},
     mrnumber = {2091752},
     language = {en},
     url = {www.numdam.org/item/RO_2004__38_3_195_0/}
}

Artalejo, Jesus R.; Lopez-Herrero, Maria J. Entropy maximization and the busy period of some single-server vacation models. RAIRO - Operations Research - Recherche Opérationnelle, Tome 38 (2004) no. 3, pp. 195-213. doi : 10.1051/ro:2004020. http://www.numdam.org/item/RO_2004__38_3_195_0/

Bibliographie

[1] J. Abate and W. Whitt, Numerical inversion of Laplace transforms of probability distributions. ORSA J. Comput. 7 (1995) 36-43. | Zbl 0821.65085

[2] J.R. Artalejo, G-networks: A versatile approach for work removal in queueing networks. Eur. J. Oper. Res. 126 (2000) 233-249. | Zbl 0971.90007

[3] J.R. Artalejo, On the M/G/1 queue with D-policy. Appl. Math. Modelling 25 (2001) 1055-1069. | Zbl 0991.60087

[4] K.R. Balachandran and H. Tijms, On the D-policy for the M/G/1 queue. Manage. Sci. 21 (1975) 1073-1076. | Zbl 0322.60083

[5] B.D. Bunday, Basic Optimization Methods. Edward Arnold, London (1984). | Zbl 0618.90052

[6] B.T. Doshi, Queueing systems with vacations - A survey. Queue. Syst. 1 (1986) 29-66. | Zbl 0655.60089

[7] M.A. El-Affendi and D.D. Kouvatsos, A maximum entropy analysis of the M/G/1 and G/M/1 queueing systems at equilibrium. Acta Inform. 19 (1983) 339-355. | Zbl 0494.60095

[8] G.I. Falin, M. Martin and J.R. Artalejo, Information theoretic approximations for the M/G/1 retrial queue. Acta Inform. 31 (1994) 559-571. | Zbl 0818.68038

[9] K.G. Gakis, H.K. Rhee and B.D. Sivazlian, Distributions and first moments of the busy period and idle periods in controllable M/G/1 queueing models with simple and dyadic policies. Stoch. Anal. Appl. 13 (1995) 47-81. | Zbl 0819.60085

[10] E. Gelenbe, Product-form queueing networks with negative and positive customers. J. Appl. Prob. 28 (1991) 656-663. | Zbl 0741.60091

[11] E. Gelenbe, G-networks: A unifying model for neural and queueing networks. Ann. Oper. Res. 48 (1994) 433-461. | Zbl 0803.90058

[12] E. Gelenbe and R. Iasnogorodski, A queue with server of walking type (autonomous service). Annales de l'Institut Henry Poincaré, Series B 16 (1980) 63-73. | Numdam | Zbl 0433.60086

[13] S. Guiasu, Maximum entropy condition in queueing theory. J. Opl. Res. Soc. 37 (1986) 293-301. | Zbl 0582.60090

[14] D. Heyman, The T-policy for the M/G/1 queue. Manage. Sci. 23 (1977) 775-778. | Zbl 0357.60022

[15] L. Kleinrock, Queueing Systems, Volume 1: Theory. John Wiley & Sons, Inc., New York (1975). | Zbl 0334.60045

[16] D.D. Kouvatsos, Entropy maximisation and queueing networks models. Ann. Oper. Res. 48 (1994) 63-126. | Zbl 0789.90032

[17] Y. Levy and U. Yechiali, Utilization of idle time in an M/G/1 queueing system. Manage. Sci. 22 (1975) 202-211. | Zbl 0313.60067

[18] J.A. Nelder and R. Mead, A simplex method for function minimization. Comput. J. 7 (1964) 308-313. | Zbl 0229.65053

[19] W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes in Fortran, The Art of Scientific Computing. Cambridge University Press (1992). | MR 1196230 | Zbl 0778.65002

[20] J.E. Shore, Information theoretic approximations for M/G/1 and G/G/1 queuing systems. Acta Inform. 17 (1982) 43-61. | Zbl 0456.68038

[21] L. Tadj and A. Hamdi, Maximum entropy solution to a quorum queueing system. Math. Comput. Modelling 34 (2001) 19-27. | Zbl 0989.60095

[22] H. Takagi, Queueing Analysis. Vol. 1-3, North-Holland, Amsterdam (1991).

[23] J. Teghem Jr., Control of the service process in a queueing system. Eur. J. Oper. Res. 23 (1986) 141-158. | Zbl 0583.60092

[24] U. Wagner and A.L.J. Geyer, A maximum entropy method for inverting Laplace transforms of probability density functions. Biometrika 82 (1995) 887-892. | Zbl 0861.62007

[25] K.H. Wang, S.L. Chuang and W.L. Pearn, Maximum entropy analysis to the N policy M/G/1 queueing systems with a removable server. Appl. Math. Modelling 26 (2002) 1151-1162. | Zbl pre01878812

[26] M. Yadin and P. Naor, Queueing systems with a removable server station. Oper. Res. Quar. 14 (1963) 393-405.