Steady-state probabilities of the PH/PH/1 queue

Leboudec, Jean-Yves

doi:10.1007/bf01159088

Cited by 11 publications

(3 citation statements)

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“…A.,., = 7oa op (Iii e ･･･ op Ii.,) X (I2i X ･･･ op I2.,); [9]. Then (2,2) shall be brought Like Lemma 3,1, we shall prove the following lemma as we solve (2,2).…”

Section: 1 Balanceequationsmentioning

confidence: 91%

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Matrix product-form solutions of stationary probabilities in tandem queues

Luh

1999

Journal of the Operations Research Society of Japan

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}Jlt:jirnoto et al., have proved that the tail of the jeint queue length distribution in a twe-stage tandem queueing system has the geometric decay property, We continue to investigate the properties of stationary distributiens in this tandem queueing system. Under the same conditions proposed by them, it is further shown that the stationary probability of the saturated states in the P"IPfflci-IP"lc2 queue has a Iinear combination of product-forms. The method of linear combinatien of product-forms is presented in a QBD process with a countable number of phases in eac:h level, We show that each component of these products cam be expressed in terms of roots ef the associated characteristic polynomials which involve only the Laplace-Stielt.ies transforrms of' the interarrival and service t,ime distributions, '1'akahashi [61 t/ested various types of models wit･h various trafiic intensities at stage 1 and 2. 436

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“…A.,., = 7oa op (Iii e ･･･ op Ii.,) X (I2i X ･･･ op I2.,); [9]. Then (2,2) shall be brought Like Lemma 3,1, we shall prove the following lemma as we solve (2,2).…”

Section: 1 Balanceequationsmentioning

confidence: 91%

“…Observe that (3,8), (3,9), (3.11) and (3,I2) has Jb.ll.I) + .ib.11 + .JbJ> + . This equaa/ion inust siun up all stat,ionary probabilities of possible st,at,es in the system to an unity.…”

Section: Xvd Have Obtainedmentioning

confidence: 99%