TY - JOUR
T1 - Computational schemes for two exponential servers where the first has a finite buffer
AU - Haviv, Moshe
AU - Zlotnikov, Rita
PY - 2011/1
Y1 - 2011/1
N2 - 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 $O(n \log\delta-1)$, the other with a complexity of $O(\log n\log2\δ -1)$, where δ is the tolerance criterion.
AB - 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 $O(n \log\delta-1)$, the other with a complexity of $O(\log n\log2\δ -1)$, where δ is the tolerance criterion.
KW - Matrix geometric
KW - Memoryless queues
KW - Quasi birth and death processes
UR - http://www.scopus.com/inward/record.url?scp=79959647280&partnerID=8YFLogxK
U2 - 10.1051/ro/2011101
DO - 10.1051/ro/2011101
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AN - SCOPUS:79959647280
SN - 0399-0559
VL - 45
SP - 17
EP - 36
JO - RAIRO - Operations Research
JF - RAIRO - Operations Research
IS - 1
ER -