A General Formula for the Stationary Distribution of the Age of Information and Its Application to Single-Server Queues

This paper considers the stationary distribution of the age of information (AoI) in information update systems. We first derive a general formula for the stationary distribution of the AoI, which holds for a wide class of information update systems. The formula indicates that the stationary distribu...

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Published in:	IEEE transactions on information theory Vol. 65; no. 12; pp. 8305 - 8324
Main Authors:	Inoue, Yoshiaki, Masuyama, Hiroyuki, Takine, Tetsuya, Tanaka, Toshiyuki
Format:	Journal Article
Language:	English
Published:	New York IEEE 01.12.2019 The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:	Age of information Delays FCFS Information age Markov analysis Monitoring non-preemptive LCFS peak AoI Preempting preemptive LCFS Queueing analysis Queues Queuing theory Servers single-server queues stationary distribution Upgrading
ISSN:	0018-9448, 1557-9654
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Abstract	This paper considers the stationary distribution of the age of information (AoI) in information update systems. We first derive a general formula for the stationary distribution of the AoI, which holds for a wide class of information update systems. The formula indicates that the stationary distribution of the AoI is given in terms of the stationary distributions of the system delay and the peak AoI. To demonstrate its applicability and usefulness, we analyze the AoI in single-server queues with four different service disciplines: first-come first-served (FCFS), preemptive last-come first-served (LCFS), and two variants of non-preemptive LCFS service disciplines. For the FCFS and the preemptive LCFS service disciplines, the GI/GI/1, M/GI/1, and GI/M/1 queues are considered, and for the non-preemptive LCFS service disciplines, the M/GI/1 and GI/M/1 queues are considered. With these results, we further show comparison results for the mean AoI's in the M/GI/1 and GI/M/1 queues under those service disciplines.
AbstractList	This paper considers the stationary distribution of the age of information (AoI) in information update systems. We first derive a general formula for the stationary distribution of the AoI, which holds for a wide class of information update systems. The formula indicates that the stationary distribution of the AoI is given in terms of the stationary distributions of the system delay and the peak AoI. To demonstrate its applicability and usefulness, we analyze the AoI in single-server queues with four different service disciplines: first-come first-served (FCFS), preemptive last-come first-served (LCFS), and two variants of non-preemptive LCFS service disciplines. For the FCFS and the preemptive LCFS service disciplines, the GI/GI/1, M/GI/1, and GI/M/1 queues are considered, and for the non-preemptive LCFS service disciplines, the M/GI/1 and GI/M/1 queues are considered. With these results, we further show comparison results for the mean AoI’s in the M/GI/1 and GI/M/1 queues under those service disciplines.
Author	Masuyama, Hiroyuki Takine, Tetsuya Inoue, Yoshiaki Tanaka, Toshiyuki
Author_xml	– sequence: 1 givenname: Yoshiaki orcidid: 0000-0002-2483-7652 surname: Inoue fullname: Inoue, Yoshiaki email: yoshiaki@comm.eng.osaka-u.ac.jp organization: Department of Information and Communications Technology, Graduate School of Engineering, Osaka University, Suita, Japan – sequence: 2 givenname: Hiroyuki surname: Masuyama fullname: Masuyama, Hiroyuki email: masuyama@i.kyoto-u.ac.jp organization: Department of Systems Science, Graduate School of Informatics, Kyoto University, Kyoto, Japan – sequence: 3 givenname: Tetsuya orcidid: 0000-0001-7499-6929 surname: Takine fullname: Takine, Tetsuya email: takine@comm.eng.osaka-u.ac.jp organization: Department of Information and Communications Technology, Graduate School of Engineering, Osaka University, Suita, Japan – sequence: 4 givenname: Toshiyuki orcidid: 0000-0001-5962-9119 surname: Tanaka fullname: Tanaka, Toshiyuki email: tt@i.kyoto-u.ac.jp organization: Department of Systems Science, Graduate School of Informatics, Kyoto University, Kyoto, Japan
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References	ref13 ref12 ref15 ref14 ref11 ref10 ref2 zemanian (ref27) 1965 ref1 ref16 kleinrock (ref17) 1975; 1 ref23 ref26 ref25 ref20 tijms (ref19) 1994 asmussen (ref22) 2003 ref8 feller (ref21) 1971; 2 gurland (ref24) 1967; 21 ref7 ref9 ref4 ref3 ref6 ref5 shaked (ref18) 2006
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SubjectTerms	Age of information Delays FCFS Information age Markov analysis Monitoring non-preemptive LCFS peak AoI Preempting preemptive LCFS Queueing analysis Queues Queuing theory Servers single-server queues stationary distribution Upgrading
Title	A General Formula for the Stationary Distribution of the Age of Information and Its Application to Single-Server Queues
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