Spectral Structure of <I>M</I>/<I>G</I>/1 Systems: Asymptotic Behavior and Relaxation Time

Julian KEILSON; Fumiaki MACHIHARA; Ushio SUMITA

Spectral Structure of M/G/1 Systems: Asymptotic Behavior and Relaxation Time

Julian KEILSON, Fumiaki MACHIHARA, Ushio SUMITA

Full Text Views

0

Cite this

Summary :

Let T_BP be the server busy period of an M/G/1 queueing system characterized by arrival intensity λ and service time c.d.f. A(τ). In this paper, we investigate the regularity structure of the Laplace transform σ_BP(s)=E[] on the complex s-plane. It is shown, under certain broad conditions, that finite singular points of σ_BP(s) are all branch points. Furthermore the branch point s₀ having the greatest real part is always purely negative and is of multiplicity two. The basic branch point s₀ and the associated complex structure provide a basis for an asymptotic representation of various descriptive distributions of interest. For a natural relaxation time |s₀|^-1 of the M/G/1 system, some useful bounds are obtained and the asymptotic behavior as traffic intensity approaches one is also discussed. Detailed results of engineering value are provided for two important classes of service time distributions, the completely monotone class and the Erlang class.

Publication: IEICE TRANSACTIONS on Communications Vol.E75-B No.12 pp.1245-1254

Publication Date: 1992/12/25

Publicized

Online ISSN

DOI

Type of Manuscript: Special Section INVITED PAPER (Special Issue on Teletraffic)

Category

Cite this

Copy

Julian KEILSON, Fumiaki MACHIHARA, Ushio SUMITA, "Spectral Structure of M/G/1 Systems: Asymptotic Behavior and Relaxation Time" in IEICE TRANSACTIONS on Communications, vol. E75-B, no. 12, pp. 1245-1254, December 1992, doi: .
Abstract: Let T_BP be the server busy period of an M/G/1 queueing system characterized by arrival intensity λ and service time c.d.f. A(τ). In this paper, we investigate the regularity structure of the Laplace transform σ_BP(s)=E[] on the complex s-plane. It is shown, under certain broad conditions, that finite singular points of σ_BP(s) are all branch points. Furthermore the branch point s₀ having the greatest real part is always purely negative and is of multiplicity two. The basic branch point s₀ and the associated complex structure provide a basis for an asymptotic representation of various descriptive distributions of interest. For a natural relaxation time |s₀|^-1 of the M/G/1 system, some useful bounds are obtained and the asymptotic behavior as traffic intensity approaches one is also discussed. Detailed results of engineering value are provided for two important classes of service time distributions, the completely monotone class and the Erlang class.
URL: https://global.ieice.org/en_transactions/communications/10.1587/e75-b_12_1245/_p

Copy

@ARTICLE{e75-b_12_1245,
author={Julian KEILSON, Fumiaki MACHIHARA, Ushio SUMITA, },
journal={IEICE TRANSACTIONS on Communications},
title={Spectral Structure of M/G/1 Systems: Asymptotic Behavior and Relaxation Time},
year={1992},
volume={E75-B},
number={12},
pages={1245-1254},
abstract={Let T_BP be the server busy period of an M/G/1 queueing system characterized by arrival intensity λ and service time c.d.f. A(τ). In this paper, we investigate the regularity structure of the Laplace transform σ_BP(s)=E[] on the complex s-plane. It is shown, under certain broad conditions, that finite singular points of σ_BP(s) are all branch points. Furthermore the branch point s₀ having the greatest real part is always purely negative and is of multiplicity two. The basic branch point s₀ and the associated complex structure provide a basis for an asymptotic representation of various descriptive distributions of interest. For a natural relaxation time |s₀|^-1 of the M/G/1 system, some useful bounds are obtained and the asymptotic behavior as traffic intensity approaches one is also discussed. Detailed results of engineering value are provided for two important classes of service time distributions, the completely monotone class and the Erlang class.},
keywords={},
doi={},
ISSN={},
month={December},}

Copy

TY - JOUR
TI - Spectral Structure of M/G/1 Systems: Asymptotic Behavior and Relaxation Time
T2 - IEICE TRANSACTIONS on Communications
SP - 1245
EP - 1254
AU - Julian KEILSON
AU - Fumiaki MACHIHARA
AU - Ushio SUMITA
PY - 1992
DO -
JO - IEICE TRANSACTIONS on Communications
SN -
VL - E75-B
IS - 12
JA - IEICE TRANSACTIONS on Communications
Y1 - December 1992
AB - Let T_BP be the server busy period of an M/G/1 queueing system characterized by arrival intensity λ and service time c.d.f. A(τ). In this paper, we investigate the regularity structure of the Laplace transform σ_BP(s)=E[] on the complex s-plane. It is shown, under certain broad conditions, that finite singular points of σ_BP(s) are all branch points. Furthermore the branch point s₀ having the greatest real part is always purely negative and is of multiplicity two. The basic branch point s₀ and the associated complex structure provide a basis for an asymptotic representation of various descriptive distributions of interest. For a natural relaxation time |s₀|^-1 of the M/G/1 system, some useful bounds are obtained and the asymptotic behavior as traffic intensity approaches one is also discussed. Detailed results of engineering value are provided for two important classes of service time distributions, the completely monotone class and the Erlang class.
ER -