Computational schemes for two exponential servers where the first has a finite buffer

Moshe Haviv; Rita Zlotnikov

RAIRO - Operations Research (2011)

  • Volume: 45, Issue: 1, page 17-36
  • ISSN: 0399-0559

Abstract

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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 Models20 (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 δ - 1 ) , the other with a complexity of O ( log n log 2 δ - 1 ) , where δ is the tolerance criterion.

How to cite

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Haviv, Moshe, and Zlotnikov, Rita. "Computational schemes for two exponential servers where the first has a finite buffer." RAIRO - Operations Research 45.1 (2011): 17-36. <http://eudml.org/doc/197791>.

@article{Haviv2011,
abstract = { 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 Models20 (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 $O(\log n \log^2\delta^\{-1\})$, where δ is the tolerance criterion. },
author = {Haviv, Moshe, Zlotnikov, Rita},
journal = {RAIRO - Operations Research},
keywords = {Memoryless queues; quasi birth and death processes; matrix geometric; queueing; performance evaluation; scheduling; queues and service},
language = {eng},
month = {5},
number = {1},
pages = {17-36},
publisher = {EDP Sciences},
title = {Computational schemes for two exponential servers where the first has a finite buffer},
url = {http://eudml.org/doc/197791},
volume = {45},
year = {2011},
}

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
DA - 2011/5//
PB - EDP Sciences
VL - 45
IS - 1
SP - 17
EP - 36
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 Models20 (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 $O(\log n \log^2\delta^{-1})$, where δ is the tolerance criterion.
LA - eng
KW - Memoryless queues; quasi birth and death processes; matrix geometric; queueing; performance evaluation; scheduling; queues and service
UR - http://eudml.org/doc/197791
ER -

References

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  1. E. Altman, T. Jimenez, R. Nunez Queija and U. Yechiali, Optimal routing among · /M/1 queues with partial information. Stochastic Models20 (2004) 149–172 Zbl1060.60087
  2. E. Altman, T. Jimenez, R. Nunez Queija and U. Yechiali, A correction to Optimal routing among · /M/1 queues with partial information. Stochastic Models21 (2005) 981 Zbl1060.60087
  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 Zbl0880.90049
  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 Zbl1209.93141
  6. A. Kopzon, Y. Nazarathy and G. Weiss, A push-pull network with infinite supply of work. Queueing Systems: Theory and Application62 (2009) 75–111 Zbl1166.60333
  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 Zbl0091.13801
  8. W. Keller-Gehring, Fast algorithm for the characteristic polynomial. Theor. Comput. Sci.36 (1985) 309–317 Zbl0565.68041
  9. L. Kleinrock, Queueing Systems2. John Wiley and Sons, New York (1976)  Zbl0361.60082
  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  Zbl1078.60078
  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  
  12. M. Neuts Matrix-Geometric Solutions in Stochastic Models. The John Hopkins University Press, Baltimore (1981)  
  13. V. Ramaswami and G. Latouch, A general class of Markov processes with explicit matrix-geometric solutions. OR Spektrum8 (1986) 209–218 
  14. S.M. Ross Stochastic Processes, 2nd edition, John Wiley and Sons, New York (1996)  

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