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Inequalities between time and customer averages for HNB(W)UE arrival processes

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  21 March 2024

Shigeo Shioda Open the ORCID record for Shigeo Shioda [Opens in a new window]  and
Kana Nakano
Shigeo Shioda*
Affiliation:
Chiba University
Kana Nakano*
Affiliation:
Chiba University
*
*Postal address: Graduate School of Engineering, Chiba University, 1-33 Yayoi, Inage, Chiba 263-8522, Japan.
*Postal address: Graduate School of Engineering, Chiba University, 1-33 Yayoi, Inage, Chiba 263-8522, Japan.
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Abstract

We show that for arrival processes, the ‘harmonic new better than used in expectation’ (HNBUE) (or ‘harmonic new worse than used in expectation’, HNWUE) property is a sufficient condition for inequalities between the time and customer averages of the system if the state of the system between arrival epochs is stochastically decreasing and convex and the lack of anticipation assumption is satisfied. HNB(W)UE is a wider class than NB(W)UE, being the largest of all available classes of distributions with positive (negative) aging properties. Thus, this result represents an important step beyond existing result on inequalities between time and customer averages, which states that for arrival processes, the NB(W)UE property is a sufficient condition for inequalities.

Keywords

Time averagecustomer averagestochastic orderHNBUEHWBUEpiecewise exponential distribution

MSC classification

Primary: 60K25: Queueing theory 68U35: Information systems (hypertext navigation, interfaces, decision support, etc.)
Secondary: 60K05: Renewal theory 60G10: Stationary processes
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 21
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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