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

RAIRO - Operations Research (2011)

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

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topHaviv, 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

top- E. Altman, T. Jimenez, R. Nunez Queija and U. Yechiali, Optimal routing among $\xb7$/M/1 queues with partial information. Stochastic Models20 (2004) 149–172 Zbl1060.60087
- E. Altman, T. Jimenez, R. Nunez Queija and U. Yechiali, A correction to Optimal routing among $\xb7$/M/1 queues with partial information. Stochastic Models21 (2005) 981 Zbl1060.60087
- F. Avram, Analytic solutions for some QBD models (2010)
- R. Hassin, On the advantage of being the first server. Management Sci.42 (1996) 618–623 Zbl0880.90049
- 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
- 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
- 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
- W. Keller-Gehring, Fast algorithm for the characteristic polynomial. Theor. Comput. Sci.36 (1985) 309–317 Zbl0565.68041
- L. Kleinrock, Queueing Systems2. John Wiley and Sons, New York (1976) Zbl0361.60082
- 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
- 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
- M. Neuts Matrix-Geometric Solutions in Stochastic Models. The John Hopkins University Press, Baltimore (1981)
- V. Ramaswami and G. Latouch, A general class of Markov processes with explicit matrix-geometric solutions. OR Spektrum8 (1986) 209–218
- S.M. Ross Stochastic Processes, 2nd edition, John Wiley and Sons, New York (1996)

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