no. 1
Computational schemes for two exponential servers where the first has a finite buffer
RAIRO - Operations Research - Recherche Opérationnelle, Tome 45 (2011) no. 1, pp. 17-36.

We consider a system consisting of two not necessarily identical exponential servers having a common Poisson arrival process. Upon arrival, customers inspect the first queue and join it if it is shorter than some threshold n. Otherwise, they join the second queue. This model was dealt with, among others, by Altman et al. [Stochastic Models 20 (2004) 149-172]. We first derive an explicit expression for the Laplace-Stieltjes transform of the distribution underlying the arrival (renewal) process to the second queue. Second, we observe that given that the second server is busy, the two queue lengths are independent. Third, we develop two computational schemes for the stationary distribution of the two-dimensional Markov process underlying this model, one with a complexity of O(nlogδ -1 ), the other with a complexity of O(lognlog 2 δ -1 ), where δ is the tolerance criterion.

DOI : 10.1051/ro/2011101
Classification : 60J22, 60J28
Mots clés : memoryless queues, quasi birth and death processes, matrix geometric 
@article{RO_2011__45_1_17_0,
     author = {Haviv, Moshe and Zlotnikov, Rita},
     title = {Computational schemes for two exponential servers where the first has a finite buffer},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {17--36},
     publisher = {EDP-Sciences},
     volume = {45},
     number = {1},
     year = {2011},
     doi = {10.1051/ro/2011101},
     zbl = {1237.60071},
     mrnumber = {3599268},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ro/2011101/}
}
TY  - JOUR
AU  - Haviv, Moshe
AU  - Zlotnikov, Rita
TI  - Computational schemes for two exponential servers where the first has a finite buffer
JO  - RAIRO - Operations Research - Recherche Opérationnelle
PY  - 2011
SP  - 17
EP  - 36
VL  - 45
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ro/2011101/
DO  - 10.1051/ro/2011101
LA  - en
ID  - RO_2011__45_1_17_0
ER  - 
%0 Journal Article
%A Haviv, Moshe
%A Zlotnikov, Rita
%T Computational schemes for two exponential servers where the first has a finite buffer
%J RAIRO - Operations Research - Recherche Opérationnelle
%D 2011
%P 17-36
%V 45
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ro/2011101/
%R 10.1051/ro/2011101
%G en
%F RO_2011__45_1_17_0
Haviv, Moshe; Zlotnikov, Rita. Computational schemes for two exponential servers where the first has a finite buffer. RAIRO - Operations Research - Recherche Opérationnelle, Tome 45 (2011) no. 1, pp. 17-36. doi : 10.1051/ro/2011101. http://www.numdam.org/articles/10.1051/ro/2011101/

[1] E. Altman, T. Jimenez, R. Nunez Queija and U. Yechiali, Optimal routing among ·/M/1 queues with partial information. Stochastic Models 20 (2004) 149-172 | MR | Zbl

[2] E. Altman, T. Jimenez, R. Nunez Queija and U. Yechiali, A correction to Optimal routing among ·/M/1 queues with partial information. Stochastic Models 21 (2005) 981 | MR | Zbl

[3] F. Avram, Analytic solutions for some QBD models (2010)

[4] R. Hassin, On the advantage of being the first server. Management Sci. 42 (1996) 618-623 | Zbl

[5] M. Haviv and Y. Kerner, The age of the arrival process in the G/M/1 and M/G/1 queues. Math. Methods Oper. Res. 73 (2011) 139-152 | MR | Zbl

[6] A. Kopzon, Y. Nazarathy and G. Weiss, A push-pull network with infinite supply of work. Queueing Systems: Theory and Application 62 (2009) 75-111 | MR | Zbl

[7] S. Karlin and J.L. Mcgregor, The differential equations of birth-and-death processes, and the Stieltjes moment problem. Trans. Am. Math. Soc. 85 (1957) 589-646 | MR | Zbl

[8] W. Keller-Gehring, Fast algorithm for the characteristic polynomial. Theor. Comput. Sci. 36 (1985) 309-317 | MR | Zbl

[9] L. Kleinrock, Queueing Systems 2. John Wiley and Sons, New York (1976) | Zbl

[10] D.P. Kroese, W.R.W. Scheinhardt and P.G. Taylor, Spectral properties of the tandem Jackson network, seen as a quisi-birth-and-death process, Ann. Appl. Prob. 14 (2004) 2057-2089 | MR | Zbl

[11] D. Liu and Y.Q. Zhao, Determination of explict solutions for a general class of Markov processes, in Matrix-Analytic Methods in Stochastic Models, edited by S. Charvarthy and A.S. Alfa, Marcel Dekker (1996) 343-357 | MR | Zbl

[12] M. Neuts Matrix-Geometric Solutions in Stochastic Models. The John Hopkins University Press, Baltimore (1981) | MR | Zbl

[13] V. Ramaswami and G. Latouch, A general class of Markov processes with explicit matrix-geometric solutions. OR Spektrum 8 (1986) 209-218 | MR | Zbl

[14] S.M. Ross Stochastic Processes, 2nd edition, John Wiley and Sons, New York (1996) | MR | Zbl

Cité par Sources :