Age Analysis of Status Updating System with Probabilistic Packet Preemption
The age of information (AoI) metric was proposed to measure the freshness of messages obtained at the terminal node of a status updating system. In this paper, the AoI of a discrete time status updating system with probabilistic packet preemption is investigated by analyzing the steady state of a th...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-06-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/24/6/785 |
| _version_ | 1797487711208079360 |
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| author | Jixiang Zhang Yinfei Xu |
| author_facet | Jixiang Zhang Yinfei Xu |
| author_sort | Jixiang Zhang |
| collection | DOAJ |
| description | The age of information (AoI) metric was proposed to measure the freshness of messages obtained at the terminal node of a status updating system. In this paper, the AoI of a discrete time status updating system with probabilistic packet preemption is investigated by analyzing the steady state of a three-dimensional discrete stochastic process. We assume that the queue used in the system is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>e</mi><mi>r</mi><mo>/</mo><mi>G</mi><mi>e</mi><mi>o</mi><mo>/</mo><mn>1</mn><mo>/</mo><msup><mn>2</mn><mo>*</mo></msup><mo>/</mo><mi>η</mi></mrow></semantics></math></inline-formula>, which represents that the system size is 2 and the packet in the buffer can be preempted by a fresher packet with probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>. Instead of considering the system’s AoI separately, we use a three-dimensional state vector <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>l</mi><mo>)</mo></mrow></semantics></math></inline-formula> to simultaneously track the real-time changes of the AoI, the age of a packet in the server, and the age of a packet waiting in the buffer. We give the explicit expression of the system’s average AoI and show that the average AoI of the system without packet preemption is obtained by letting <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>η</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. When <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula> is set to 1, the mean of the AoI of the system with a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>e</mi><mi>r</mi><mo>/</mo><mi>G</mi><mi>e</mi><mi>o</mi><mo>/</mo><mn>1</mn><mo>/</mo><msup><mn>2</mn><mo>*</mo></msup></mrow></semantics></math></inline-formula> queue is obtained as well. Combining the results we have obtained and comparing them with corresponding average continuous AoIs, we propose a possible relationship between the average discrete AoI with the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>e</mi><mi>r</mi><mo>/</mo><mi>G</mi><mi>e</mi><mi>o</mi><mo>/</mo><mn>1</mn><mo>/</mo><mi>c</mi></mrow></semantics></math></inline-formula> queue and the average continuous AoI with the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>/</mo><mi>M</mi><mo>/</mo><mn>1</mn><mo>/</mo><mi>c</mi></mrow></semantics></math></inline-formula> queue. For each of two extreme cases where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>η</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>η</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, we also determine the stationary distribution of AoI using the probability generation function (PGF) method. The relations between the average AoI and the packet preemption probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>, as well as the AoI’s distribution curves in two extreme cases, are illustrated by numerical simulations. Notice that the probabilistic packet preemption may occur, for example, in an energy harvest (EH) node of a wireless sensor network, where the packet in the buffer can be replaced only when the node collects enough energy. In particular, to exhibit the usefulness of our idea and methods and highlight the merits of considering discrete time systems, in this paper, we provide detailed discussions showing how the results about continuous AoI are derived by analyzing the corresponding discrete time system and how the discrete age analysis is generalized to the system with multiple sources. In terms of packet service process, we also propose an idea to analyze the AoI of a system when the service time distribution is arbitrary. |
| first_indexed | 2024-03-09T23:51:37Z |
| format | Article |
| id | doaj.art-77e21d4d19e347e9bfdd2c8f9ffb0edd |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-09T23:51:37Z |
| publishDate | 2022-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-77e21d4d19e347e9bfdd2c8f9ffb0edd2023-11-23T16:33:05ZengMDPI AGEntropy1099-43002022-06-0124678510.3390/e24060785Age Analysis of Status Updating System with Probabilistic Packet PreemptionJixiang Zhang0Yinfei Xu1School of Information Science and Engineering, Southeast University, Nanjing 210096, ChinaSchool of Information Science and Engineering, Southeast University, Nanjing 210096, ChinaThe age of information (AoI) metric was proposed to measure the freshness of messages obtained at the terminal node of a status updating system. In this paper, the AoI of a discrete time status updating system with probabilistic packet preemption is investigated by analyzing the steady state of a three-dimensional discrete stochastic process. We assume that the queue used in the system is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>e</mi><mi>r</mi><mo>/</mo><mi>G</mi><mi>e</mi><mi>o</mi><mo>/</mo><mn>1</mn><mo>/</mo><msup><mn>2</mn><mo>*</mo></msup><mo>/</mo><mi>η</mi></mrow></semantics></math></inline-formula>, which represents that the system size is 2 and the packet in the buffer can be preempted by a fresher packet with probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>. Instead of considering the system’s AoI separately, we use a three-dimensional state vector <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>l</mi><mo>)</mo></mrow></semantics></math></inline-formula> to simultaneously track the real-time changes of the AoI, the age of a packet in the server, and the age of a packet waiting in the buffer. We give the explicit expression of the system’s average AoI and show that the average AoI of the system without packet preemption is obtained by letting <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>η</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. When <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula> is set to 1, the mean of the AoI of the system with a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>e</mi><mi>r</mi><mo>/</mo><mi>G</mi><mi>e</mi><mi>o</mi><mo>/</mo><mn>1</mn><mo>/</mo><msup><mn>2</mn><mo>*</mo></msup></mrow></semantics></math></inline-formula> queue is obtained as well. Combining the results we have obtained and comparing them with corresponding average continuous AoIs, we propose a possible relationship between the average discrete AoI with the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mi>e</mi><mi>r</mi><mo>/</mo><mi>G</mi><mi>e</mi><mi>o</mi><mo>/</mo><mn>1</mn><mo>/</mo><mi>c</mi></mrow></semantics></math></inline-formula> queue and the average continuous AoI with the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>/</mo><mi>M</mi><mo>/</mo><mn>1</mn><mo>/</mo><mi>c</mi></mrow></semantics></math></inline-formula> queue. For each of two extreme cases where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>η</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>η</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, we also determine the stationary distribution of AoI using the probability generation function (PGF) method. The relations between the average AoI and the packet preemption probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>, as well as the AoI’s distribution curves in two extreme cases, are illustrated by numerical simulations. Notice that the probabilistic packet preemption may occur, for example, in an energy harvest (EH) node of a wireless sensor network, where the packet in the buffer can be replaced only when the node collects enough energy. In particular, to exhibit the usefulness of our idea and methods and highlight the merits of considering discrete time systems, in this paper, we provide detailed discussions showing how the results about continuous AoI are derived by analyzing the corresponding discrete time system and how the discrete age analysis is generalized to the system with multiple sources. In terms of packet service process, we also propose an idea to analyze the AoI of a system when the service time distribution is arbitrary.https://www.mdpi.com/1099-4300/24/6/785age of informationdiscrete time status updating systemprobabilistic preemptionprobability generation functionstationary distribution |
| spellingShingle | Jixiang Zhang Yinfei Xu Age Analysis of Status Updating System with Probabilistic Packet Preemption Entropy age of information discrete time status updating system probabilistic preemption probability generation function stationary distribution |
| title | Age Analysis of Status Updating System with Probabilistic Packet Preemption |
| title_full | Age Analysis of Status Updating System with Probabilistic Packet Preemption |
| title_fullStr | Age Analysis of Status Updating System with Probabilistic Packet Preemption |
| title_full_unstemmed | Age Analysis of Status Updating System with Probabilistic Packet Preemption |
| title_short | Age Analysis of Status Updating System with Probabilistic Packet Preemption |
| title_sort | age analysis of status updating system with probabilistic packet preemption |
| topic | age of information discrete time status updating system probabilistic preemption probability generation function stationary distribution |
| url | https://www.mdpi.com/1099-4300/24/6/785 |
| work_keys_str_mv | AT jixiangzhang ageanalysisofstatusupdatingsystemwithprobabilisticpacketpreemption AT yinfeixu ageanalysisofstatusupdatingsystemwithprobabilisticpacketpreemption |