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 $O(nlog{\delta}^{-1})$, the other with a complexity of $O(logn{log}^{2}{\delta}^{-1})$, where δ is the tolerance criterion.

Keywords: 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://archive.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://archive.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://archive.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, Volume 45 (2011) no. 1, pp. 17-36. doi : 10.1051/ro/2011101. http://archive.numdam.org/articles/10.1051/ro/2011101/

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