zbMATH — the first resource for mathematics

Tandem Brownian queues. (English) Zbl 1139.60045
The authors analyze a two-node tandem queue with Brownian input. Section 2 presents a detailed description of the two-node tandem queue, as well as a closely related two-node parallel queue. It also gives formal implicit expressions for the over-flow probabilities, and contains a brief discussion of Schilder’s sample-path large-deviations theorem. In Section 3 the two-node parallel system is analyzed. The authors derive an exact expression of the joint distribution function, large-buffer asymptotics, and the most probable path. Then they argue that the two-node parallel queue is closely related to the two-node tandem queue. Exploiting this property they obtain in Section 4 the desired results for the tandem systems. In Section 5 they discuss their results, and identify some open research questions.

MSC:
 60K25 Queueing theory (aspects of probability theory) 90B22 Queues and service in operations research 60J65 Brownian motion
Keywords:
tandem queue; Brownian inpun
Full Text:
References:
 [1] Addie R, Mannersalo P, Norros I (2002) Most probable paths and performance formulae for buffers with Gaussian input traffic. Euro Trans Telecomm 13:183–196 · doi:10.1002/ett.4460130303 [2] Adler R (1990) An introduction to continuity, extrema, and related topics for general Gaussian processes. IMS Lect Notes-Monogr Ser 12 · Zbl 0747.60039 [3] Bahadur R, Zabell S (1979) Large deviations of the sample mean in general vector spaces. Ann Probab 7:587–621 · Zbl 0424.60028 · doi:10.1214/aop/1176994985 [4] Debicki K, Mandjes M, van Uitert M (2007a) A tandem queue with Lévy input: a new representation of the downstream queue length. To appear in: Probab Eng Infor Sci · Zbl 1114.60074 [5] Debicki K, Dieker A, Rolski T (2007b) Quasi-product form for Lévy-driven fluid networks. To appear in: Math Oper Res · Zbl 1341.60111 [6] Dembo A, Zeitouni O (1998) Large deviations techniques and applications, 2nd edn. Springer, New York · Zbl 0896.60013 [7] Deuschel J-D, Stroock D (1989) Large deviations. Academic, London [8] Ganesh A, O’Connell N, Wischik D (2004) Big queues. Springer Lecture Notes in Mathematics 1838 · Zbl 1044.60001 [9] Harrison JM, Williams RJ (1992) Brownian models of feedforward queueing networks: quasireversibility and product form solutions. Ann Appl Probab 2:263–293 · Zbl 0753.60071 · doi:10.1214/aoap/1177005704 [10] Kella O, Whitt W (1992) A tandem fluid network with Lévy input. In: Bhat UN, Basawa IV (eds) Queueing and related models. Oxford University Press, New York pp 112–128 · Zbl 0783.60089 [11] Majewski K (1998) Heavy traffic approximations of large deviations of feedforward queueing networks. Queueing Sys 28:125–155 · Zbl 0909.90146 · doi:10.1023/A:1019147006084 [12] Mandjes M (2004) Packet models revisited: tandem and priority systems. Queueing Sys 47:363–377 · Zbl 1073.60090 · doi:10.1023/B:QUES.0000036397.20364.70 [13] Mandjes M, van Uitert M (2005) Sample-path large deviations for tandem and priority queues with Gaussian inputs. Ann Appl Probab 15:1193–1226 · Zbl 1069.60079 · doi:10.1214/105051605000000133 [14] Norros I (1999) Busy periods of fractional Brownian storage: a large deviations approach. Adv Perform Anal 2:1–20 [15] Reich E (1958) On the integrodifferential equation of Takács I. Ann Math Stat 29:563–570 · Zbl 0086.33703 · doi:10.1214/aoms/1177706632
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.