Departure Processes from GI/GI/∞ and GI/GI/c/c with Bursty Arrivals

Fumiaki MACHIHARA; Taro TOKUDA

doi:10.1587/transcom.2016EBP3297

Departure Processes from GI/GI/∞ and GI/GI/c/c with Bursty Arrivals

Fumiaki MACHIHARA, Taro TOKUDA

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When the random variable has a completely monotone density function, we call it bursty (BRST) random variable. At first, we prove that the entropy of inter-arrival time is smaller than or equal to the entropy of inter-departure time in an infinite-server system GI/GI/∞ having general renewal arrivals. On the basis of that result, we prove that a BRST/GI/∞ having bursty arrivals and the associated loss system BRST/GI/c/c have the following paradoxical behavior: In the BRST/GI/∞, the stationary number of customers as well as the inter-departure time become stochastically less variable, as the service time becomes stochastically more variable. Also for the loss system BRST/GI/c/c, the blocking probability decreases and the inter-departure time becomes stochastically less variable, as the service time becomes stochastically more variable.

Publication: IEICE TRANSACTIONS on Communications Vol.E100-B No.7 pp.1115-1123

Publication Date: 2017/07/01

Publicized: 2017/01/12

Online ISSN: 1745-1345

DOI: 10.1587/transcom.2016EBP3297

Type of Manuscript: PAPER

Category: Fundamental Theories for Communications

Authors

Fumiaki MACHIHARA
Tokyo Denki University
Taro TOKUDA
Tokyo Denki University

Keyword

GI/GI/∞, GI/GI/c/c, bursty arrival, relation between arrival and departure, entropy, paradoxical behavior

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Fumiaki MACHIHARA, Taro TOKUDA, "Departure Processes from GI/GI/∞ and GI/GI/c/c with Bursty Arrivals" in IEICE TRANSACTIONS on Communications, vol. E100-B, no. 7, pp. 1115-1123, July 2017, doi: 10.1587/transcom.2016EBP3297.
Abstract: When the random variable has a completely monotone density function, we call it bursty (BRST) random variable. At first, we prove that the entropy of inter-arrival time is smaller than or equal to the entropy of inter-departure time in an infinite-server system GI/GI/∞ having general renewal arrivals. On the basis of that result, we prove that a BRST/GI/∞ having bursty arrivals and the associated loss system BRST/GI/c/c have the following paradoxical behavior: In the BRST/GI/∞, the stationary number of customers as well as the inter-departure time become stochastically less variable, as the service time becomes stochastically more variable. Also for the loss system BRST/GI/c/c, the blocking probability decreases and the inter-departure time becomes stochastically less variable, as the service time becomes stochastically more variable.
URL: https://global.ieice.org/en_transactions/communications/10.1587/transcom.2016EBP3297/_p

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@ARTICLE{e100-b_7_1115,
author={Fumiaki MACHIHARA, Taro TOKUDA, },
journal={IEICE TRANSACTIONS on Communications},
title={Departure Processes from GI/GI/∞ and GI/GI/c/c with Bursty Arrivals},
year={2017},
volume={E100-B},
number={7},
pages={1115-1123},
abstract={When the random variable has a completely monotone density function, we call it bursty (BRST) random variable. At first, we prove that the entropy of inter-arrival time is smaller than or equal to the entropy of inter-departure time in an infinite-server system GI/GI/∞ having general renewal arrivals. On the basis of that result, we prove that a BRST/GI/∞ having bursty arrivals and the associated loss system BRST/GI/c/c have the following paradoxical behavior: In the BRST/GI/∞, the stationary number of customers as well as the inter-departure time become stochastically less variable, as the service time becomes stochastically more variable. Also for the loss system BRST/GI/c/c, the blocking probability decreases and the inter-departure time becomes stochastically less variable, as the service time becomes stochastically more variable.},
keywords={},
doi={10.1587/transcom.2016EBP3297},
ISSN={1745-1345},
month={July},}

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TY - JOUR
TI - Departure Processes from GI/GI/∞ and GI/GI/c/c with Bursty Arrivals
T2 - IEICE TRANSACTIONS on Communications
SP - 1115
EP - 1123
AU - Fumiaki MACHIHARA
AU - Taro TOKUDA
PY - 2017
DO - 10.1587/transcom.2016EBP3297
JO - IEICE TRANSACTIONS on Communications
SN - 1745-1345
VL - E100-B
IS - 7
JA - IEICE TRANSACTIONS on Communications
Y1 - July 2017
AB - When the random variable has a completely monotone density function, we call it bursty (BRST) random variable. At first, we prove that the entropy of inter-arrival time is smaller than or equal to the entropy of inter-departure time in an infinite-server system GI/GI/∞ having general renewal arrivals. On the basis of that result, we prove that a BRST/GI/∞ having bursty arrivals and the associated loss system BRST/GI/c/c have the following paradoxical behavior: In the BRST/GI/∞, the stationary number of customers as well as the inter-departure time become stochastically less variable, as the service time becomes stochastically more variable. Also for the loss system BRST/GI/c/c, the blocking probability decreases and the inter-departure time becomes stochastically less variable, as the service time becomes stochastically more variable.
ER -