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Register a new userTimely Updates By Multiple Sources: The M/M/1 Queue Revisited
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Added by
Sanjit Kaul
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Multiple sources submit updates to a monitor through an M/M/1 queue. A stochastic hybrid system (SHS) approach is used to derive the average age of information (AoI) for an individual source as a function of the offered load of that source and the competing update traffic offered by other sources. This work corrects an error in a prior analysis. By numerical evaluation, this error is observed to be small and qualitatively insignificant.
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We consider a joint sampling and scheduling problem for optimizing data freshness in multi-source systems. Data freshness is measured by a non-decreasing penalty function of emph{age of information}, where all sources have the same age-penalty function. Sources take turns to generate update packets, and forward them to their destinations one-by-one through a shared channel with random delay. There is a scheduler, that chooses the update order of the sources, and a sampler, that determines when a source should generate a new packet in its turn. We aim to find the optimal scheduler-sampler pairs that minimize the total-average age-penalty at delivery times (Ta-APD) and the total-average age-penalty (Ta-AP). We prove that the Maximum Age First (MAF) scheduler and the zero-wait sampler are jointly optimal for minimizing the Ta-APD. Meanwhile, the MAF scheduler and a relative value iteration with reduced complexity (RVI-RC) sampler are jointly optimal for minimizing the Ta-AP. The RVI-RC sampler is based on a relative value iteration algorithm whose complexity is reduced by exploiting a threshold property in the optimal sampler. Finally, a low-complexity threshold-type sampler is devised via an approximate analysis of Bellmans equation. This threshold-type sampler reduces to a simple water-filling sampler for a linear age-penalty function.
We consider a system in which an information source generates independent and identically distributed status update packets from an observed phenomenon that takes $n$ possible values based on a given pmf. These update packets are encoded at the transmitter node to be sent to the receiver node. Instead of encoding all $n$ possible realizations, the transmitter node only encodes the most probable $k$ realizations and disregards whenever a realization from the remaining $n-k$ values occurs. We find the average age and determine the age-optimal real codeword lengths such that the average age at the receiver node is minimized. Through numerical evaluations for arbitrary pmfs, we show that this selective encoding policy results in a lower average age than encoding every realization and find the age-optimal $k$. We also analyze a randomized selective encoding policy in which the remaining $n-k$ realizations are encoded and sent with a certain probability to further inform the receiver at the expense of longer codewords for the selected $k$ realizations.
This exposition presents a novel approach to solving an M/M/m queue for the waiting time and the residence time. The motivation comes from an algebraic solution for the residence time of the M/M/1 queue. The key idea is the introduction of an ansatz transformation, defined in terms of the Erlang B function, that avoids the more opaque derivation based on applied probability theory. The only prerequisite is an elementary knowledge of the Poisson distribution, which is already necessary for understanding the M/M/1 queue. The approach described here supersedes our earlier approximate morphing transformation.
BCH codes are an interesting class of cyclic codes due to their efficient encoding and decoding algorithms. In many cases, BCH codes are the best linear codes. However, the dimension and minimum distance of BCH codes have been seldom solved. Until now, there have been few results on BCH codes over $gf(q)$ with length $q^m+1$, especially when $q$ is a prime power and $m$ is even. The objective of this paper is to study BCH codes of this type over finite fields and analyse their parameters. The BCH codes presented in this paper have good parameters in general, and contain many optimal linear codes.
The notion of strong external difference family (SEDF) in a finite abelian group $(G,+)$ is raised by M. B. Paterson and D. R. Stinson [5] in 2016 and motivated by its application in communication theory to construct $R$-optimal regular algebraic manipulation detection code. A series of $(n,m,k,lambda)$-SEDFs have been constructed in [5, 4, 2, 1] with $m=2$. In this note we present an example of (243, 11, 22, 20)-SEDF in finite field $mathbb{F}_q$ $(q=3^5=243).$ This is an answer for the following problem raised in [5] and continuously asked in [4, 2, 1]: if there exists an $(n,m,k,lambda)$-SEDF for $mgeq 5$.
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