Abstract
The idea behind the recently introduced “age-of-information” performance measure of a network message processing system is that it indicates our knowledge regarding the “freshness” of the most recent piece of information that can be used as a criterion for real-time control. In this foundational paper, we examine two such measures, one that has been extensively studied in the recent literature and a new one that could be more relevant from the point of view of the processor. Considering these measures as stochastic processes in a stationary environment (defined by the arrival processes, message processing times and admission controls in bufferless systems), we characterize their distributions using the Palm inversion formula. Under renewal assumptions, we derive explicit solutions for their Laplace transforms and show some interesting decomposition properties. Previous work has mostly focused on computation of expectations in very particular cases. We argue that using bufferless or very small buffer systems is best and support this by simulation. We also pose some open problems including assessment of enqueueing policies that may be better in cases where one wishes to minimize more general functionals of the age-of-information measures.
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Notes
A point process \(\varphi \) on a product space \(S \times M\) is called M-marked (or just marked) if \(\varphi (\{s\} \times M) \in \{0,1\}\) for all \(s \in S\).
Abbreviations
- \(\delta _x\) :
-
Delta measure at the point x
- \(\overline{X}\) :
-
A random variable with density \(\mathbb {P}(X>x)/\mathbb {E}X\)
- \(T_n\) :
-
Arrival time of a message
- \(\chi _n\) :
-
Accept/reject index
- \(\psi _n\) :
-
Success/failure index
- \(\mathcal {Z}_n\) :
-
Informally, the event that the server is idle just before \(T_n\)
- \(\widetilde{\mathbb {P}}\) :
-
Informally, probability measure governing the stationary system
- \(\mathbb {P}\) :
-
Palm probability of \(\widetilde{\mathbb {P}}\) with respect to the arrival process
- \(\mathbb {P}^*\) :
-
Palm probability of \(\widetilde{\mathbb {P}}\) with respect to reading intervals beginnings \(=\mathbb {P}(\cdot |\mathcal {Z}_0)\)
- \({\mathfrak a}\) :
-
\(= \sum _n \delta _{T_n}\), arrival process as a point process
- \({\mathfrak a}(t)\) :
-
\(=\mathfrak a([0,t))\)
- U(t):
-
\(=\mathbb {E}{\mathfrak a}(t)\)
- Z(t):
-
\(= \mathbb {E}\sum _{i=0}^{{\mathfrak a}(t)-1} T_i\)
- \(W_u(t)\) :
-
\(= \mathbb {E}e^{-u T_{{\mathfrak a}(t)}}\)
- \(V_u(t)\) :
-
\(= \mathbb {E}\sum _{i=0}^{{\mathfrak a}(t)-1} e^{-u T_i}\)
- \(Q_u(t)\) :
-
\(= \mathbb {E}\big \{ (T_{{\mathfrak a}(t)}-t)\, e^{-u T_{{\mathfrak a}(t)-1}} \big \}\)
- \(\sigma _n\) :
-
Processing time of a message
- \(T_n'\) :
-
Departure time of a message either due to successful reading or not
- \(A_t\) :
-
Last arrival epoch before t
- \(S_t\) :
-
Last arrival epoch before t of a successful message
- \(D_t\) :
-
Last departure epoch before t of a successful message
- \(A^*_t\) :
-
\(=S_{D_t}\)
- \(\Delta f(t)\) :
-
\(=f(t+)-f(t-)\)
- \(\tau _n\) :
-
\(=T_{n+1}-T_n\)
- \(B_k\) :
-
Beginning of a reading interval
- \({\mathbf {R}}_k\) :
-
Duration of a reading interval
- \(B_k'\) :
-
\(=B_k+\mathbf {R}_k\)
- \(\mathbf {C}_k\) :
-
\(=B_{k+1}-B_k\), cycle length
- \(\lambda \) :
-
Arrival rate \(=1/\mathbb {E}\tau \)\(=\sum _n \widetilde{\mathbb {P}}(0<T_n<1)\)
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Acknowledgements
We thank Kostya Borovkov for reading an early draft of the paper. We also thank the anonymous reviewers who read the paper carefully, identified all typos, and critically questioned it. Their detailed comments helped us improve the paper.
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Kesidis, G., Konstantopoulos, T. & Zazanis, M.A. The distribution of age-of-information performance measures for message processing systems. Queueing Syst 95, 203–250 (2020). https://doi.org/10.1007/s11134-020-09655-x
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DOI: https://doi.org/10.1007/s11134-020-09655-x
Keywords
- Age of information
- Message processing systems
- Palm probability
- Renewal process
- Poisson process
- Performance evaluation
- Stochastic decomposition