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

Jesus R. Artalejo; Maria J. Lopez-Herrero

RAIRO - Operations Research - Recherche Opérationnelle (2004)

  • Volume: 38, Issue: 3, page 195-213
  • ISSN: 0399-0559

Abstract

top
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).

How to cite

top

Artalejo, Jesus R., and Lopez-Herrero, Maria J.. "Entropy maximization and the busy period of some single-server vacation models." RAIRO - Operations Research - Recherche Opérationnelle 38.3 (2004): 195-213. <http://eudml.org/doc/245246>.

@article{Artalejo2004,
abstract = {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).},
author = {Artalejo, Jesus R., Lopez-Herrero, Maria J.},
journal = {RAIRO - Operations Research - Recherche Opérationnelle},
keywords = {busy period analysis; maximum entropy methodology; $M/G/1$ vacation models; numerical inversion; Busy period analysis; vacation models},
language = {eng},
number = {3},
pages = {195-213},
publisher = {EDP-Sciences},
title = {Entropy maximization and the busy period of some single-server vacation models},
url = {http://eudml.org/doc/245246},
volume = {38},
year = {2004},
}

TY - JOUR
AU - Artalejo, Jesus R.
AU - Lopez-Herrero, Maria J.
TI - Entropy maximization and the busy period of some single-server vacation models
JO - RAIRO - Operations Research - Recherche Opérationnelle
PY - 2004
PB - EDP-Sciences
VL - 38
IS - 3
SP - 195
EP - 213
AB - 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).
LA - eng
KW - busy period analysis; maximum entropy methodology; $M/G/1$ vacation models; numerical inversion; Busy period analysis; vacation models
UR - http://eudml.org/doc/245246
ER -

References

top
  1. [1] J. Abate and W. Whitt, Numerical inversion of Laplace transforms of probability distributions. ORSA J. Comput. 7 (1995) 36–43. Zbl0821.65085
  2. [2] J.R. Artalejo, G-networks: A versatile approach for work removal in queueing networks. Eur. J. Oper. Res. 126 (2000) 233–249. Zbl0971.90007
  3. [3] J.R. Artalejo, On the M/G/1 queue with D-policy. Appl. Math. Modelling 25 (2001) 1055–1069. Zbl0991.60087
  4. [4] K.R. Balachandran and H. Tijms, On the D-policy for the M/G/1 queue. Manage. Sci. 21 (1975) 1073–1076. Zbl0322.60083
  5. [5] B.D. Bunday, Basic Optimization Methods. Edward Arnold, London (1984). Zbl0618.90052
  6. [6] B.T. Doshi, Queueing systems with vacations - A survey. Queue. Syst. 1 (1986) 29–66. Zbl0655.60089
  7. [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. Zbl0494.60095
  8. [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. Zbl0818.68038
  9. [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. Zbl0819.60085
  10. [10] E. Gelenbe, Product-form queueing networks with negative and positive customers. J. Appl. Prob. 28 (1991) 656–663. Zbl0741.60091
  11. [11] E. Gelenbe, G-networks: A unifying model for neural and queueing networks. Ann. Oper. Res. 48 (1994) 433–461. Zbl0803.90058
  12. [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. Zbl0433.60086
  13. [13] S. Guiasu, Maximum entropy condition in queueing theory. J. Opl. Res. Soc. 37 (1986) 293–301. Zbl0582.60090
  14. [14] D. Heyman, The T-policy for the M/G/1 queue. Manage. Sci. 23 (1977) 775–778. Zbl0357.60022
  15. [15] L. Kleinrock, Queueing Systems, Volume 1: Theory. John Wiley & Sons, Inc., New York (1975). Zbl0334.60045
  16. [16] D.D. Kouvatsos, Entropy maximisation and queueing networks models. Ann. Oper. Res. 48 (1994) 63–126. Zbl0789.90032
  17. [17] Y. Levy and U. Yechiali, Utilization of idle time in an M/G/1 queueing system. Manage. Sci. 22 (1975) 202–211. Zbl0313.60067
  18. [18] J.A. Nelder and R. Mead, A simplex method for function minimization. Comput. J. 7 (1964) 308–313. Zbl0229.65053
  19. [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). Zbl0778.65002MR1196230
  20. [20] J.E. Shore, Information theoretic approximations for M/G/1 and G/G/1 queuing systems. Acta Inform. 17 (1982) 43–61. Zbl0456.68038
  21. [21] L. Tadj and A. Hamdi, Maximum entropy solution to a quorum queueing system. Math. Comput. Modelling 34 (2001) 19–27. Zbl0989.60095
  22. [22] H. Takagi, Queueing Analysis. Vol. 1–3, North-Holland, Amsterdam (1991). Zbl0744.60114
  23. [23] J. Teghem Jr., Control of the service process in a queueing system. Eur. J. Oper. Res. 23 (1986) 141–158. Zbl0583.60092
  24. [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. Zbl0861.62007
  25. [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. Zbl1175.90115
  26. [26] M. Yadin and P. Naor, Queueing systems with a removable server station. Oper. Res. Quar. 14 (1963) 393–405. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.