The Age of information (AoI) was proposed in the literature to quantify the freshness of information. The majority of the work done in this area has theoretically evaluated AoI and its Peak (PAoI). In this paper, a method for obtaining the value of AoI and PAoI from experiments is proposed. We conducted an experiment emulating an M/M/1 queue and used the proposed method to evaluate AoI and PAoI. The values were compared to the expressions presented previously in the literature. Our results show that the proposed method is accurate for the M/M/1 queue. A statistical test was conducted to confirm the reliability of this conclusion

Arshad, Kamran

Barakat, Basel

Keates, Simeon

Wassell, Ian J

Yassine, Hachem

English

Wassell, Ian

Greenwich Academic Literature Archive

How to Measure the Average and Peak Age ofInformation in Real Networks?Basel BarakatSchool of EngineeringUniversity of GreenwichKent, United Kingdombb141@gre.ac.ukHachem YassineDepartment of Engineering ScienceUniversity of OxfordOxford, United Kingdomhachem.yassine@eng.ox.ac.ukSimeon KeatesSchool of Engineering and the Built EnvironmentEdinburgh Napier UniversityEdinburgh, UKS.Keates@napier.ac.ukIan WassellComputer LaboratoryUniversity of CambridgeCambridge, United Kingdomijw24@cam.ac.ukKamran ArshadElectrical Engineering DepartmentAjman UniversityAjman, UAEAbstract—The Age of information (AoI) was proposed in theliterature to quantify the freshness of information. The majorityof the work done in this area has theoretically evaluated AoI andits Peak (PAoI). In this paper, a method for obtaining the valueof AoI and PAoI from experiments is proposed. We conducted anexperiment emulating an M/M/1 queue and used the proposedmethod to evaluate AoI and PAoI. The values were compared tothe expressions presented previously in the literature. Our resultsshow that the proposed method is accurate for the M/M/1 queue.A statistical test was conducted to confirm the reliability of thisconclusion.I. INTRODUCTIONOver the decades, continuous breakthroughs in commu-nication technologies gave birth to a range of applicationswith different requirements. Many Internet of Things (IoT)applications are based on receiving updates about the status ofa remote agent to help in decision making. Examples includewireless channel estimation [1], telehealth, environmental andindustrial monitoring, and military battlefield coordination. Forsome of these applications, it is crucial that, at any point intime, the status that the decision maker has, is up to dateand represents as much as possible the current status at thesource. There is a subtle difference between this requirementand the traditional low latency (delay) requirement becausethe latter is seen purely from the perspective of networkperformance, while the former is seen from the destination’sperspective. In other words, low-latency is not equivalent tothe freshest updates at the receiver; it very much depends onhow frequently the updates are being generated at the source.To address this requirement, a new metric called Age ofInformation (AoI) was introduced in [7] to measure andquantify the freshness of information from the receiver’sperspective. It is defined as the time since the last updatereceived was generated. The main difference between AoI andconventional network delay metrics is that AoI is observedfrom the receiver’s perspective, while the delay is observedfrom the network’s perspective. The Peak Age of Information(PAoI), introduced in [3], is another metric related to AoI andrepresents the worst case AoI. It is defined as the maximumtime elapsed since the preceding piece of information wasgenerated. The PAoI metric has a simpler formulation andis a more mathematically tractable metric [3]. Consequently,modeling PAoI has gained attention in the literature such asin [3], [4]. Also, minimizing the PAoI by optimizing networkfunctions was extensively investigated [5].However, the majority of work on AoI and PAoI has beentheoretical and assumed simple queuing models to derivetheoretical results about these metrics. In [6], an emulation-based validation of the theoretical models was presented. Morerecently, [11] presented experimental results that validated thenon-monotonous nature of AoI as a function of link utilization.However, in both papers, no clear explanation was providedas to how exactly the metrics were evaluated from the exper-iment. This paper aims to bridge this gap and provide a clearand concise tutorial for experimental researchers that wish toevaluate these metrics on real networks. The contributions ofthis paper can be summarized as follows:• we provide an intuitive formulation of how the AoI andPAoI metrics can be estimated from the recorded time-stamps in an experiment,• we validate these expressions by performing an experi-ment comprising an M/M/1 queue, and• we present a simple methodology for conducting experi-ments.The rest of the paper is organized as follows. In sectionII, we define the new metrics and some related quantities. Insection III, we present the proposed method to estimate thedelay, AoI, and PAoI. In section IV, we present a case studyto validate our method and we compare the proposed estimatesagainst their theoretical counterparts. We conclude the paperin section V.II. DEFINITIONS AND PREVIOUS WORKA. Age of InformationConsider a destination and an information source that isgenerating updates at discrete times (possibly by samplinga process) and then instantaneously transmitting them to thedestination through a communication network. We denote byti the time at which the ith update was generated/transmittedat the source and by ri the time at which it was received at thedestination. We define Xi = ti− ti−1 as the time between thegeneration of updates i and i−1, i.e., the updates inter-arrivaltime. We also define the delay time (system) Ti = ri − ti asthe time it took, from the generation of ith update, until itsreception at the destination. The Age of Information (AoI) attime t, denoted ∆(t), is defined as the time elapsed since thelast received update at the destination was generated at thesource. Mathematically, it is given by ∆(t) = t− u(t), whereu(t) is the generation time of the last received update at timet. Fig. 1 illustrates an example of how the information ageevolves with time as a sawtooth function [7].B. Time Average Peak AoIFor time interval (0, T ), where T is assumed, for simplicity,to coincide with the receipt of the nth update, i.e., T = rn,where rn is the time that the last piece of information, n, wasreceived in the time interval. The average time delay in thisperiod can be written asTT =1nn∑i=1Ti. (1)The peak age of an update is its age at the time of receiptof the next update [3], i.e., the peak age of the ith update is∆(ri+1) = Xi + Ti. Therefore, it is possible to define thetime-average peak AoI in the period T as followsPT =1n− 1n−1∑i=1∆(ri+1). (2)C. Time Average AoIThe time average AoI in the interval (0, T ), denoted ∆¯T ,is the area under the sawtooth function normalized by theobservation period (T ), and it is given by∆T =1T∫ T0∆(t) dt. (3)By a geometric argument, the area under the curve can bere-written as the sum of the of the areas in Fig. 1. Startingfrom t = 0, these are the areas of the polygon Q1, trapezoidsQi for 2 ≤ i ≤ n, and the triangle of length Tn between tnand rn. the first update (i = 1) shown in Fig. 1. Hence,∆T =Q1 +∑ni=2Qi + T2n/2T (4)Age	of	Information(∆)Time1234 5234 5212 52641264Pi-27234 82Pi-1Pi9234 92Fig. 1. Age of Information as a function of time. The updates inter-arrivaltime are referred to as Xi and the delay (system time) is Ti, i.e., the servicetime plus the queuing time. The PAoI of information i is represented by Pi.The time of generating update i is ti and the time of receiving it is ri.The trapezoid area Qi can be written as the area differencebetween the right isosceles triangles of legs Ti + Xi and Ti,respectively. This isQi =12(Ti +Xi)2 − 12T 2i (5)= TiXi +X2i2. (6)In the literature, there is an interest in the stationary casewhen T → +∞. It can be seen that the delay time convergesto E[T ] and the peak average AoI converges toP = limT→+∞PT = E[X + T ] (7)while it can be shown [7] that the average AoI converges to∆ = limT→+∞∆T = λ(E[XT ] +E[T 2]2), (8)where X is the random variable representing the inter-arrivaltime of updates at the source (generation) with its rate λ, andT is the random variable representing the system time (delay)of an update. The expectations in the expressions depend onthe network. To abstract the details of the underlying network,it is common to assume idealized queuing models such as theM/M/1, M/D/1, D/M/1 [3], [7]. Each model makes a differentassumption about the update generation (X) and system time(T ) which is composed of waiting time in the queue andservice time in the network.III. ESTIMATION OF THE METRICS FROM EXPERIMENTSConsider a setup in which the source transmits an updateat generation time (ti) and that the receiver records its timeof receipt (ri). It is recommended that all calculations beperformed after the end of the experiment as the time tocalculate the metrics might affect the accuracy of the loggedtimings.The delay that the ith update exhibits is calculated by,Ti = ri − ti. (9)The expected delay of (9) can be estimated using the samplemedian of the vector that contains all the delays of all theupdates, i.e.,T(1→n) =[T1, T2, ..., Tn], (10)where n is the total number of updates communicated. Thesample median is employed instead of the sample averagebecause in some cases, the network protocol might re-transmitsome packets, as in TCP/IP and HARQ protocols, if they suf-fered from significant errors. The re-transmission will increasethe delay time hence would significantly affect the mean value.Similarly, the average PAoI can be estimated using thesample median of the vector containing the PAoI of all thestatus updates communicated in the experiment, i.e.,P(1→n) =[P1, P2, ..., Pn]. (11)To use the initial definition Pi = Xi + Ti, requires firstfinding the inter-arrival time and the delays. In the following,we provide an easier formulation using only the times ofgeneration and receipt as shown in Fig. 2. In the figure, it isclear that the peak age of an update stretches from its time ofgeneration until the time of receipt of the next update. Hence,Pi can be evaluated asPi = ri+1 − ti. (12)!" #" !"$% #"$%&"'" '"$%("Fig. 2. Peak Age of Information measuring method illustration. Where ti isthe time that information i was generated, ri is the time in which informationi was received by the server, where Xi represents the updates inter-arrivaltime. The Peak Age can be considered as the difference between the time ofreceiving the next update and the time of generating the update.Finally, to estimate the average AoI the areas of Qi shouldbe calculated. By using the area of a trapezoid it can be shownthatQi =12(((ti+1 − ti) + (ri − ti))2 − (ri − ti)2). (13)The average AoI can then be estimated as follows∆ =∑ni=2Qi(rn − t2) . (14)Therefore, in this section, we can express the metrics ofinterest in terms of the observable time-stamps coming froman experiment.IV. TESTED CASE STUDYIn this section, we validate our estimators by emulating anM/M/1 network and comparing the measured to the theoret-ical results. We start by reviewing the theoretical results forM/M/1; then we give our validation setup before presentingand discussing the results.A. M/M/1 QueueThe method of measurement was tested on an M/M/1 queue,where the inter-arrival time and the service time follow anexponential distribution with rates λ and µ, respectively. Theexpected delay of such model, denoted E[T ], is given by [10],E[T ] = E[W + S]=λµ(µ− λ) +1µ, (15)where E[W ] = λ/µ(µ − λ) is the expected waiting time inthe queue and E[S] = 1/µ is the expected service time. From(15) and the fact that E[X] = 1/λ, the theoretical M/M/1average peak AoI is [3],PM/M/1 = E[X + T ]=1µ(1 +1ρ+ρ1− ρ), (16)where ρ = λ/µ is the link utilization. Finally, the average AoIof an M/M/1 is given by [10],∆M/M/1 =1µ(1 +1ρ+ρ21− ρ). (17)B. Experimental SetupTo validate the proposed method a simple network em-ploying an M/M/1 queue was emulated locally. A Client-Server model was used as shown in Fig. 3. The updates weresent using TCP/IP from the client to the server. The clienttransmitted the instantaneous time-stamps of when the updateswere generated, i.e., ti. Upon sending an update, the clientsleeps for a random duration that is exponentially distributedwith inter-arrival rate (λ). The rate λ was varied between 1and 8 updates per second. Fig. 4 presents the flow-chart of theclient’s behaviour.Client Server!" #"QueueFig. 3. Network System model showing the Client, where the time-stampof generating updates i, i.e., (ti). The Server saves the time of receiving theupdate (ri).StartInitializeImport libraries, set IP address and socket portGenerate Time-stamptSend UpdateSleepFor the inter-arrivaltimeFig. 4. Client’s flow chart.At the server side, the time of receipt ri is recorded foreach packet i along with the transmit time embedded in theupdate, ti. To emulate the service time, which is exponentiallydistributed with rate µ = 10, the server sleeping time was used.In particular, upon receipt of the update, the server will sleepfor a random period (exponentially distributed), and then wakeup and terminate the session with the client and record the timeas the time of receipt ri. After receiving a predefined numberof updates, (we used one thousand updates per client in theexperiments), the experiment terminated and the estimationspresented in the previous section were performed.The experiments were done using Apple MacOS with a2.2 GHz Intel Core i7 processor and a 16 GB 1600 MHzDDR3 Memory. To make sure the clients and server remainedtightly synchronized during the experiment, a single clock forall measurements was used. The updates were communicatedusing a Python 3.6 socket module [8]. To obtain the time-stamps, a Python time module [9] was used and the readingswere obtained by using the object time.time().C. Results and DiscussionsFig. 5 presents the results for the delay time and comparesit with the theoretical, given by (5). It can be seen that theexperimental and theoretical results are in good agreement.In particular, the percentage error does not exceed 5%. Thefirst row of Table I shows the results of a statistical test thatvalidates the assumption that the delay (i.e., waiting time plusservice time) can be accurately estimated using our method.00.10.20.30.40.50.62 3 4 5 6 7 8Delay Time ( T) (s)Arrival Rate λ ( s-1 )Empirical TheoreticalFig. 5. Delay versus Arrival rate for M/M/1 queue calculated theoreticallyfrom (15) and measured in the experiment using the median of (10).Next, we move to consider the PAoI estimator which wascompared with its theoretical counterpart given by (16). Theresults in Fig. 6 show that the estimated PAoI is in very goodagreement with the theoretical values. Thus, we can arguethat the experimental model proposed can obtain the PAoI anddelay time for the proposed system model accurately.0.30.40.50.60.70.80.911.11.21 2 3 4 5 6 7 8Peak Age ( P) M/M/1(s)Arrival Rate λ ( s-1 )Empirical TheoreticalFig. 6. Peak Age versus Arrival rate for M/M/1 queue calculated theoreticallyfrom 16 [3] and obtained experimentally using (11).Finally, the estimated and theoretical average AoI are shownin Fig. 7 as a function of the arrival rate. The results show thatthe estimated and theoretical results are in agreement.0.30.350.40.450.50.550.60.652 3 4 5 6 7 8Average Age  M/M/1(s)Arrival Rate λ ( s-1 )Empirical TheoreticalFig. 7. Average Age versus Arrival rate for M/M/1 queue calculatedtheoretically from (7) [7] and obtained experimentally using (14).To assess the significance of the results, a student t-test wasused to calculate the p-values and the percentage errors foreach of the estimators. Table I presents the t-test p-values andthe mean percent error values for the delay time, PAoI, andAoI measurements. As shown in Table I, the difference in themeans can be described as not significant. Thus, it can beconcluded that the proposed method is accurate. Further, itcan be concluded that the number of readings is sufficient toprecisely calculate the delay time, PAoI and AoI. Hence, thismethod was used in our paper [12] to evaluate the PAoI ofZero-wait policy.TABLE IP VALUES AND PERCENT ERRORS FOR M/M/1, VALIDATING THEPROPOSED METHOD FOR EVALUATING THE PEAK AND AVERAGE AGEParameter p-Value Percentage errorDelay (T) 0.73 6.46Peak (PAoI) 0.39 4.16Age (AoI) 0.39 4.46V. CONCLUSIONS AND FUTURE WORKThe AoI is a novel metric suggested to measure the fresh-ness of information. A considerable amount of work has beendone to evaluate and optimize AoI and PAoI theoretically. Inthis paper, we tried to motivate more experimental work onthe AoI by making it straightforward to estimate the metricsfrom experiments. The proposed method was validated on anemulated M/M/1 queue. We showed that the proposed methodcould achieve estimates that are very close to the theoreticalcounterparts. The obvious next step is to test the accuracy ofestimators in experiments involving different queuing models.Furthermore, it is worth investigating the implications ofconsidering the AoI and PAoI on real-life applications suchas remotely controlled robots.REFERENCES[1] B. Barakat and K. Arshad, “An adaptive hybrid scheduling algorithm forLTE-Advanced,” in 2015 22nd International Conference on Telecommu-nications (ICT), 2015, pp. 91-95.[2] M. Costa, M. Codreanu, and A. Ephremides, “On the Age of Informationin Status Update Systems With Packet Management,” IEEE Trans. Inf.Theory, vol. 62, no. 4, pp. 1897-1910, Apr. 2016.[3] L. Huang and E. Modiano, “Optimizing age-of-information in a multi-class queueing system,” in 2015 IEEE International Symposium onInformation Theory (ISIT), 2015, pp. 1681-1685.[4] E. Najm and R. Nasser, ”Age of information: The gamma awakening,”2016 IEEE International Symposium on Information Theory (ISIT),Barcelona, 2016, pp. 2574-2578.[5] A. Kosta, N. Pappas, and V. Angelakis, “Age of Information: A NewConcept, Metric, and Tool,” Found. Trends Netw., vol. 12, no. 3, pp.162-259, 2017.[6] C. Kam, S. Kompella, and A. Ephremides, “Experimental evaluation ofthe age of information via emulation,” in Proceedings - IEEE MilitaryCommunications Conference MILCOM, 2015, vol. 2015-Decem, pp.1070-1075.[7] S. Kaul, R. Yates, and M. Gruteser, “Real-time status: How often shouldone update?,” in Proceedings - IEEE INFOCOM, 2012, pp. 2731-2735.[8] “18.1. socket - Low-level networking interface Python 3.3.7 documenta-tion.” [Online]. Available: https://docs.python.org/3.3/library/socket.html.[Accessed: 06-Mar-2018].[9] “16.3. time - Time access and conversions - Python 3.6.4 documen-tation.” [Online]. Available: https://docs.python.org/3/library/time.html.[Accessed: 07-Mar-2018].[10] D. Bertsekas and R. Gallager, Data Networks. Prentice Hall, 1992.[11] C. Snmez, S. Baghaee, A. Ergii and E. Uysal-Biyikoglu, “Age-of-Information in Practice: Status Age Measured Over TCP/IP ConnectionsThrough WiFi, Ethernet and LTE,” 2018 IEEE International Black SeaConference on Communications and Networking (BlackSeaCom), Ba-tumi, 2018, pp. 1-5.[12] B. Barakat, S. Keates, I. J. Wassell, and K. Arshad, “Isthe Zero-Wait Policy Always Optimum for Information Freshness(Peak Age) or Throughput?,” in IEEE Communications Letters,doi:10.1109/LCOMM.2019.2907935, (in press).

How to measure the average and peak age of information in real networks?

Apollo (Cambridge)

How to Measure the Average and Peak Age ofInformation in Real Networks?Basel BarakatSchool of EngineeringUniversity of GreenwichKent, United Kingdombb141@gre.ac.ukHachem YassineDepartment of Engineering ScienceUniversity of OxfordOxford, United Kingdomhachem.yassine@eng.ox.ac.ukSimeon KeatesSchool of Engineering and the Built EnvironmentEdinburgh Napier UniversityEdinburgh, UKS.Keates@napier.ac.ukIan WassellComputer LaboratoryUniversity of CambridgeCambridge, United Kingdomijw24@cam.ac.ukKamran ArshadElectrical Engineering DepartmentAjman UniversityAjman, UAEAbstract—The Age of information (AoI) was proposed in theliterature to quantify the freshness of information. The majorityof the work done in this area has theoretically evaluated AoI andits Peak (PAoI). In this paper, a method for obtaining the valueof AoI and PAoI from experiments is proposed. We conducted anexperiment emulating an M/M/1 queue and used the proposedmethod to evaluate AoI and PAoI. The values were compared tothe expressions presented previously in the literature. Our resultsshow that the proposed method is accurate for the M/M/1 queue.A statistical test was conducted to confirm the reliability of thisconclusion.I. INTRODUCTIONOver the decades, continuous breakthroughs in commu-nication technologies gave birth to a range of applicationswith different requirements. Many Internet of Things (IoT)applications are based on receiving updates about the status ofa remote agent to help in decision making. Examples includewireless channel estimation [1], telehealth, environmental andindustrial monitoring, and military battlefield coordination. Forsome of these applications, it is crucial that, at any point intime, the status that the decision maker has, is up to dateand represents as much as possible the current status at thesource. There is a subtle difference between this requirementand the traditional low latency (delay) requirement becausethe latter is seen purely from the perspective of networkperformance, while the former is seen from the destination’sperspective. In other words, low-latency is not equivalent tothe freshest updates at the receiver; it very much depends onhow frequently the updates are being generated at the source.To address this requirement, a new metric called Age ofInformation (AoI) was introduced in [7] to measure andquantify the freshness of information from the receiver’sperspective. It is defined as the time since the last updatereceived was generated. The main difference between AoI andconventional network delay metrics is that AoI is observedfrom the receiver’s perspective, while the delay is observedfrom the network’s perspective. The Peak Age of Information(PAoI), introduced in [3], is another metric related to AoI andrepresents the worst case AoI. It is defined as the maximumtime elapsed since the preceding piece of information wasgenerated. The PAoI metric has a simpler formulation andis a more mathematically tractable metric [3]. Consequently,modeling PAoI has gained attention in the literature such asin [3], [4]. Also, minimizing the PAoI by optimizing networkfunctions was extensively investigated [5].However, the majority of work on AoI and PAoI has beentheoretical and assumed simple queuing models to derivetheoretical results about these metrics. In [6], an emulation-based validation of the theoretical models was presented. Morerecently, [11] presented experimental results that validated thenon-monotonous nature of AoI as a function of link utilization.However, in both papers, no clear explanation was providedas to how exactly the metrics were evaluated from the exper-iment. This paper aims to bridge this gap and provide a clearand concise tutorial for experimental researchers that wish toevaluate these metrics on real networks. The contributions ofthis paper can be summarized as follows:• we provide an intuitive formulation of how the AoI andPAoI metrics can be estimated from the recorded time-stamps in an experiment,• we validate these expressions by performing an experi-ment comprising an M/M/1 queue, and• we present a simple methodology for conducting experi-ments.The rest of the paper is organized as follows. In sectionII, we define the new metrics and some related quantities. Insection III, we present the proposed method to estimate thedelay, AoI, and PAoI. In section IV, we present a case studyto validate our method and we compare the proposed estimatesagainst their theoretical counterparts. We conclude the paperin section V.II. DEFINITIONS AND PREVIOUS WORKA. Age of InformationConsider a destination and an information source that isgenerating updates at discrete times (possibly by samplinga process) and then instantaneously transmitting them to thedestination through a communication network. We denote byti the time at which the ith update was generated/transmittedat the source and by ri the time at which it was received at thedestination. We define Xi = ti− ti−1 as the time between thegeneration of updates i and i−1, i.e., the updates inter-arrivaltime. We also define the delay time (system) Ti = ri − ti asthe time it took, from the generation of ith update, until itsreception at the destination. The Age of Information (AoI) attime t, denoted ∆(t), is defined as the time elapsed since thelast received update at the destination was generated at thesource. Mathematically, it is given by ∆(t) = t− u(t), whereu(t) is the generation time of the last received update at timet. Fig. 1 illustrates an example of how the information ageevolves with time as a sawtooth function [7].B. Time Average Peak AoIFor time interval (0, T ), where T is assumed, for simplicity,to coincide with the receipt of the nth update, i.e., T = rn,where rn is the time that the last piece of information, n, wasreceived in the time interval. The average time delay in thisperiod can be written asTT =1nn∑i=1Ti. (1)The peak age of an update is its age at the time of receiptof the next update [3], i.e., the peak age of the ith update is∆(ri+1) = Xi + Ti. Therefore, it is possible to define thetime-average peak AoI in the period T as followsPT =1n− 1n−1∑i=1∆(ri+1). (2)C. Time Average AoIThe time average AoI in the interval (0, T ), denoted ∆̄T ,is the area under the sawtooth function normalized by theobservation period (T ), and it is given by∆T =1T∫ T0∆(t) dt. (3)By a geometric argument, the area under the curve can bere-written as the sum of the of the areas in Fig. 1. Startingfrom t = 0, these are the areas of the polygon Q1, trapezoidsQi for 2 ≤ i ≤ n, and the triangle of length Tn between tnand rn. the first update (i = 1) shown in Fig. 1. Hence,∆T =Q1 +∑ni=2Qi + T2n/2T(4)Age	of	Information(∆)Time1234 5234 5212 52641264Pi-27234 82Pi-1Pi923492Fig. 1. Age of Information as a function of time. The updates inter-arrivaltime are referred to as Xi and the delay (system time) is Ti, i.e., the servicetime plus the queuing time. The PAoI of information i is represented by Pi.The time of generating update i is ti and the time of receiving it is ri.The trapezoid area Qi can be written as the area differencebetween the right isosceles triangles of legs Ti + Xi and Ti,respectively. This isQi =12(Ti +Xi)2 − 12T 2i (5)= TiXi +X2i2. (6)In the literature, there is an interest in the stationary casewhen T → +∞. It can be seen that the delay time convergesto E[T ] and the peak average AoI converges toP = limT→+∞PT = E[X + T ] (7)while it can be shown [7] that the average AoI converges to∆ = limT→+∞∆T = λ(E[XT ] +E[T 2]2), (8)where X is the random variable representing the inter-arrivaltime of updates at the source (generation) with its rate λ, andT is the random variable representing the system time (delay)of an update. The expectations in the expressions depend onthe network. To abstract the details of the underlying network,it is common to assume idealized queuing models such as theM/M/1, M/D/1, D/M/1 [3], [7]. Each model makes a differentassumption about the update generation (X) and system time(T ) which is composed of waiting time in the queue andservice time in the network.III. ESTIMATION OF THE METRICS FROM EXPERIMENTSConsider a setup in which the source transmits an updateat generation time (ti) and that the receiver records its timeof receipt (ri). It is recommended that all calculations beperformed after the end of the experiment as the time tocalculate the metrics might affect the accuracy of the loggedtimings.The delay that the ith update exhibits is calculated by,Ti = ri − ti. (9)The expected delay of (9) can be estimated using the samplemedian of the vector that contains all the delays of all theupdates, i.e.,T(1→n) =[T1, T2, ..., Tn], (10)where n is the total number of updates communicated. Thesample median is employed instead of the sample averagebecause in some cases, the network protocol might re-transmitsome packets, as in TCP/IP and HARQ protocols, if they suf-fered from significant errors. The re-transmission will increasethe delay time hence would significantly affect the mean value.Similarly, the average PAoI can be estimated using thesample median of the vector containing the PAoI of all thestatus updates communicated in the experiment, i.e.,P(1→n) =[P1, P2, ..., Pn]. (11)To use the initial definition Pi = Xi + Ti, requires firstfinding the inter-arrival time and the delays. In the following,we provide an easier formulation using only the times ofgeneration and receipt as shown in Fig. 2. In the figure, it isclear that the peak age of an update stretches from its time ofgeneration until the time of receipt of the next update. Hence,Pi can be evaluated asPi = ri+1 − ti. (12)!" #" !"$% #"$%&"'" '"$%("Fig. 2. Peak Age of Information measuring method illustration. Where ti isthe time that information i was generated, ri is the time in which informationi was received by the server, where Xi represents the updates inter-arrivaltime. The Peak Age can be considered as the difference between the time ofreceiving the next update and the time of generating the update.Finally, to estimate the average AoI the areas of Qi shouldbe calculated. By using the area of a trapezoid it can be shownthatQi =12(((ti+1 − ti) + (ri − ti))2 − (ri − ti)2). (13)The average AoI can then be estimated as follows∆ =∑ni=2Qi(rn − t2). (14)Therefore, in this section, we can express the metrics ofinterest in terms of the observable time-stamps coming froman experiment.IV. TESTED CASE STUDYIn this section, we validate our estimators by emulating anM/M/1 network and comparing the measured to the theoret-ical results. We start by reviewing the theoretical results forM/M/1; then we give our validation setup before presentingand discussing the results.A. M/M/1 QueueThe method of measurement was tested on an M/M/1 queue,where the inter-arrival time and the service time follow anexponential distribution with rates λ and µ, respectively. Theexpected delay of such model, denoted E[T ], is given by [10],E[T ] = E[W + S]=λµ(µ− λ)+1µ, (15)where E[W ] = λ/µ(µ − λ) is the expected waiting time inthe queue and E[S] = 1/µ is the expected service time. From(15) and the fact that E[X] = 1/λ, the theoretical M/M/1average peak AoI is [3],PM/M/1 = E[X + T ]=1µ(1 +1ρ+ρ1− ρ), (16)where ρ = λ/µ is the link utilization. Finally, the average AoIof an M/M/1 is given by [10],∆M/M/1 =1µ(1 +1ρ+ρ21− ρ). (17)B. Experimental SetupTo validate the proposed method a simple network em-ploying an M/M/1 queue was emulated locally. A Client-Server model was used as shown in Fig. 3. The updates weresent using TCP/IP from the client to the server. The clienttransmitted the instantaneous time-stamps of when the updateswere generated, i.e., ti. Upon sending an update, the clientsleeps for a random duration that is exponentially distributedwith inter-arrival rate (λ). The rate λ was varied between 1and 8 updates per second. Fig. 4 presents the flow-chart of theclient’s behaviour.Client Server!" #"QueueFig. 3. Network System model showing the Client, where the time-stampof generating updates i, i.e., (ti). The Server saves the time of receiving theupdate (ri).StartInitializeImport libraries, set IP address and socket portGenerate Time-stamptSend UpdateSleepFor the inter-arrivaltimeFig. 4. Client’s flow chart.At the server side, the time of receipt ri is recorded foreach packet i along with the transmit time embedded in theupdate, ti. To emulate the service time, which is exponentiallydistributed with rate µ = 10, the server sleeping time was used.In particular, upon receipt of the update, the server will sleepfor a random period (exponentially distributed), and then wakeup and terminate the session with the client and record the timeas the time of receipt ri. After receiving a predefined numberof updates, (we used one thousand updates per client in theexperiments), the experiment terminated and the estimationspresented in the previous section were performed.The experiments were done using Apple MacOS with a2.2 GHz Intel Core i7 processor and a 16 GB 1600 MHzDDR3 Memory. To make sure the clients and server remainedtightly synchronized during the experiment, a single clock forall measurements was used. The updates were communicatedusing a Python 3.6 socket module [8]. To obtain the time-stamps, a Python time module [9] was used and the readingswere obtained by using the object time.time().C. Results and DiscussionsFig. 5 presents the results for the delay time and comparesit with the theoretical, given by (5). It can be seen that theexperimental and theoretical results are in good agreement.In particular, the percentage error does not exceed 5%. Thefirst row of Table I shows the results of a statistical test thatvalidates the assumption that the delay (i.e., waiting time plusservice time) can be accurately estimated using our method.00.10.20.30.40.50.62 3 4 5 6 7 8Delay Time ( T) (s)Arrival Rate λ ( s-1 )Empirical TheoreticalFig. 5. Delay versus Arrival rate for M/M/1 queue calculated theoreticallyfrom (15) and measured in the experiment using the median of (10).Next, we move to consider the PAoI estimator which wascompared with its theoretical counterpart given by (16). Theresults in Fig. 6 show that the estimated PAoI is in very goodagreement with the theoretical values. Thus, we can arguethat the experimental model proposed can obtain the PAoI anddelay time for the proposed system model accurately.0.30.40.50.60.70.80.911.11.21 2 3 4 5 6 7 8Peak Age ( P) M/M/1(s)Arrival Rate λ ( s-1 )Empirical TheoreticalFig. 6. Peak Age versus Arrival rate for M/M/1 queue calculated theoreticallyfrom 16 [3] and obtained experimentally using (11).Finally, the estimated and theoretical average AoI are shownin Fig. 7 as a function of the arrival rate. The results show thatthe estimated and theoretical results are in agreement.0.30.350.40.450.50.550.60.652 3 4 5 6 7 8Average Age  M/M/1(s)Arrival Rate λ ( s-1 )Empirical TheoreticalFig. 7. Average Age versus Arrival rate for M/M/1 queue calculatedtheoretically from (7) [7] and obtained experimentally using (14).To assess the significance of the results, a student t-test wasused to calculate the p-values and the percentage errors foreach of the estimators. Table I presents the t-test p-values andthe mean percent error values for the delay time, PAoI, andAoI measurements. As shown in Table I, the difference in themeans can be described as not significant. Thus, it can beconcluded that the proposed method is accurate. Further, itcan be concluded that the number of readings is sufficient toprecisely calculate the delay time, PAoI and AoI. Hence, thismethod was used in our paper [12] to evaluate the PAoI ofZero-wait policy.TABLE IP VALUES AND PERCENT ERRORS FOR M/M/1, VALIDATING THEPROPOSED METHOD FOR EVALUATING THE PEAK AND AVERAGE AGEParameter p-Value Percentage errorDelay (T) 0.73 6.46Peak (PAoI) 0.39 4.16Age (AoI) 0.39 4.46V. CONCLUSIONS AND FUTURE WORKThe AoI is a novel metric suggested to measure the fresh-ness of information. A considerable amount of work has beendone to evaluate and optimize AoI and PAoI theoretically. Inthis paper, we tried to motivate more experimental work onthe AoI by making it straightforward to estimate the metricsfrom experiments. The proposed method was validated on anemulated M/M/1 queue. We showed that the proposed methodcould achieve estimates that are very close to the theoreticalcounterparts. The obvious next step is to test the accuracy ofestimators in experiments involving different queuing models.Furthermore, it is worth investigating the implications ofconsidering the AoI and PAoI on real-life applications suchas remotely controlled robots.REFERENCES[1] B. Barakat and K. Arshad, “An adaptive hybrid scheduling algorithm forLTE-Advanced,” in 2015 22nd International Conference on Telecommu-nications (ICT), 2015, pp. 91-95.[2] M. Costa, M. Codreanu, and A. Ephremides, “On the Age of Informationin Status Update Systems With Packet Management,” IEEE Trans. Inf.Theory, vol. 62, no. 4, pp. 1897-1910, Apr. 2016.[3] L. Huang and E. Modiano, “Optimizing age-of-information in a multi-class queueing system,” in 2015 IEEE International Symposium onInformation Theory (ISIT), 2015, pp. 1681-1685.[4] E. Najm and R. Nasser, ”Age of information: The gamma awakening,”2016 IEEE International Symposium on Information Theory (ISIT),Barcelona, 2016, pp. 2574-2578.[5] A. Kosta, N. Pappas, and V. Angelakis, “Age of Information: A NewConcept, Metric, and Tool,” Found. Trends Netw., vol. 12, no. 3, pp.162-259, 2017.[6] C. Kam, S. Kompella, and A. Ephremides, “Experimental evaluation ofthe age of information via emulation,” in Proceedings - IEEE MilitaryCommunications Conference MILCOM, 2015, vol. 2015-Decem, pp.1070-1075.[7] S. Kaul, R. Yates, and M. Gruteser, “Real-time status: How often shouldone update?,” in Proceedings - IEEE INFOCOM, 2012, pp. 2731-2735.[8] “18.1. socket - Low-level networking interface Python 3.3.7 documenta-tion.” [Online]. Available: https://docs.python.org/3.3/library/socket.html.[Accessed: 06-Mar-2018].[9] “16.3. time - Time access and conversions - Python 3.6.4 documen-tation.” [Online]. Available: https://docs.python.org/3/library/time.html.[Accessed: 07-Mar-2018].[10] D. Bertsekas and R. Gallager, Data Networks. Prentice Hall, 1992.[11] C. Snmez, S. Baghaee, A. Ergii and E. Uysal-Biyikoglu, “Age-of-Information in Practice: Status Age Measured Over TCP/IP ConnectionsThrough WiFi, Ethernet and LTE,” 2018 IEEE International Black SeaConference on Communications and Networking (BlackSeaCom), Ba-tumi, 2018, pp. 1-5.[12] B. Barakat, S. Keates, I. J. Wassell, and K. Arshad, “Isthe Zero-Wait Policy Always Optimum for Information Freshness(Peak Age) or Throughput?,” in IEEE Communications Letters,doi:10.1109/LCOMM.2019.2907935, (in press).

How to Measure the Average and Peak Age of Information in Real Networks?

Sunderland University Institutional Repository

Bar a k a t ,  Bas el,  Yassine,  H a c h e m,  Kea t e s,  Sim eo n,  Wass ell,  Ian  a n d  Arsh a d,  Ka m r a n  (2019)  H ow  to  M e a s u r e  t h e  Aver a g e  a n d  Pe ak  Age  of  Info r m a tion  in  Re al  N e t wo rks?  In:  E u rop e a n  Wirele s s  2 0 1 9;  2 5 t h  E u rop e a n  Wirel es s  Confe r e nc e .  IEEE, p p.  1-5.  ISBN 9 7 8 3 8 0 0 7 4 9 4 8 5  Downloa d e d  fro m: h t t p://su r e . s u n d e rl a n d. ac.uk/id/e p rin t /14 1 8 3/U s a g e  g u i d e l i n e sPle a s e  r ef e r  to  t h e  u s a g e  g uid elines  a t  h t t p://su r e . s u n d e rl a n d. ac.uk/policies.h t ml  o r  al t e r n a tively  con t ac t  s u r e@s u n d e rl a n d. ac.uk.How to Measure the Average and Peak Age ofInformation in Real Networks?Basel BarakatSchool of EngineeringUniversity of GreenwichKent, United Kingdombb141@gre.ac.ukHachem YassineDepartment of Engineering ScienceUniversity of OxfordOxford, United Kingdomhachem.yassine@eng.ox.ac.ukSimeon KeatesSchool of Engineering and the Built EnvironmentEdinburgh Napier UniversityEdinburgh, UKS.Keates@napier.ac.ukIan WassellComputer LaboratoryUniversity of CambridgeCambridge, United Kingdomijw24@cam.ac.ukKamran ArshadElectrical Engineering DepartmentAjman UniversityAjman, UAEAbstract—The Age of information (AoI) was proposed in theliterature to quantify the freshness of information. The majorityof the work done in this area has theoretically evaluated AoI andits Peak (PAoI). In this paper, a method for obtaining the valueof AoI and PAoI from experiments is proposed. We conducted anexperiment emulating an M/M/1 queue and used the proposedmethod to evaluate AoI and PAoI. The values were compared tothe expressions presented previously in the literature. Our resultsshow that the proposed method is accurate for the M/M/1 queue.A statistical test was conducted to confirm the reliability of thisconclusion.I. INTRODUCTIONOver the decades, continuous breakthroughs in commu-nication technologies gave birth to a range of applicationswith different requirements. Many Internet of Things (IoT)applications are based on receiving updates about the status ofa remote agent to help in decision making. Examples includewireless channel estimation [1], telehealth, environmental andindustrial monitoring, and military battlefield coordination. Forsome of these applications, it is crucial that, at any point intime, the status that the decision maker has, is up to dateand represents as much as possible the current status at thesource. There is a subtle difference between this requirementand the traditional low latency (delay) requirement becausethe latter is seen purely from the perspective of networkperformance, while the former is seen from the destination’sperspective. In other words, low-latency is not equivalent tothe freshest updates at the receiver; it very much depends onhow frequently the updates are being generated at the source.To address this requirement, a new metric called Age ofInformation (AoI) was introduced in [7] to measure andquantify the freshness of information from the receiver’sperspective. It is defined as the time since the last updatereceived was generated. The main difference between AoI andconventional network delay metrics is that AoI is observedfrom the receiver’s perspective, while the delay is observedfrom the network’s perspective. The Peak Age of Information(PAoI), introduced in [3], is another metric related to AoI andrepresents the worst case AoI. It is defined as the maximumtime elapsed since the preceding piece of information wasgenerated. The PAoI metric has a simpler formulation andis a more mathematically tractable metric [3]. Consequently,modeling PAoI has gained attention in the literature such asin [3], [4]. Also, minimizing the PAoI by optimizing networkfunctions was extensively investigated [5].However, the majority of work on AoI and PAoI has beentheoretical and assumed simple queuing models to derivetheoretical results about these metrics. In [6], an emulation-based validation of the theoretical models was presented. Morerecently, [11] presented experimental results that validated thenon-monotonous nature of AoI as a function of link utilization.However, in both papers, no clear explanation was providedas to how exactly the metrics were evaluated from the exper-iment. This paper aims to bridge this gap and provide a clearand concise tutorial for experimental researchers that wish toevaluate these metrics on real networks. The contributions ofthis paper can be summarized as follows:• we provide an intuitive formulation of how the AoI andPAoI metrics can be estimated from the recorded time-stamps in an experiment,• we validate these expressions by performing an experi-ment comprising an M/M/1 queue, and• we present a simple methodology for conducting experi-ments.The rest of the paper is organized as follows. In sectionII, we define the new metrics and some related quantities. Insection III, we present the proposed method to estimate thedelay, AoI, and PAoI. In section IV, we present a case studyto validate our method and we compare the proposed estimatesagainst their theoretical counterparts. We conclude the paperin section V.II. DEFINITIONS AND PREVIOUS WORKA. Age of InformationConsider a destination and an information source that isgenerating updates at discrete times (possibly by samplinga process) and then instantaneously transmitting them to thedestination through a communication network. We denote byti the time at which the ith update was generated/transmittedat the source and by ri the time at which it was received at thedestination. We define Xi = ti− ti−1 as the time between thegeneration of updates i and i−1, i.e., the updates inter-arrivaltime. We also define the delay time (system) Ti = ri − ti asthe time it took, from the generation of ith update, until itsreception at the destination. The Age of Information (AoI) attime t, denoted ∆(t), is defined as the time elapsed since thelast received update at the destination was generated at thesource. Mathematically, it is given by ∆(t) = t− u(t), whereu(t) is the generation time of the last received update at timet. Fig. 1 illustrates an example of how the information ageevolves with time as a sawtooth function [7].B. Time Average Peak AoIFor time interval (0, T ), where T is assumed, for simplicity,to coincide with the receipt of the nth update, i.e., T = rn,where rn is the time that the last piece of information, n, wasreceived in the time interval. The average time delay in thisperiod can be written asTT =1nn∑i=1Ti. (1)The peak age of an update is its age at the time of receiptof the next update [3], i.e., the peak age of the ith update is∆(ri+1) = Xi + Ti. Therefore, it is possible to define thetime-average peak AoI in the period T as followsPT =1n− 1n−1∑i=1∆(ri+1). (2)C. Time Average AoIThe time average AoI in the interval (0, T ), denoted ∆̄T ,is the area under the sawtooth function normalized by theobservation period (T ), and it is given by∆T =1T∫ T0∆(t) dt. (3)By a geometric argument, the area under the curve can bere-written as the sum of the of the areas in Fig. 1. Startingfrom t = 0, these are the areas of the polygon Q1, trapezoidsQi for 2 ≤ i ≤ n, and the triangle of length Tn between tnand rn. the first update (i = 1) shown in Fig. 1. Hence,∆T =Q1 +∑ni=2Qi + T2n/2T(4)Age	of	Information(∆)Time1234 5234 5212 52641264Pi-27234 82Pi-1Pi923492Fig. 1. Age of Information as a function of time. The updates inter-arrivaltime are referred to as Xi and the delay (system time) is Ti, i.e., the servicetime plus the queuing time. The PAoI of information i is represented by Pi.The time of generating update i is ti and the time of receiving it is ri.The trapezoid area Qi can be written as the area differencebetween the right isosceles triangles of legs Ti + Xi and Ti,respectively. This isQi =12(Ti +Xi)2 − 12T 2i (5)= TiXi +X2i2. (6)In the literature, there is an interest in the stationary casewhen T → +∞. It can be seen that the delay time convergesto E[T ] and the peak average AoI converges toP = limT→+∞PT = E[X + T ] (7)while it can be shown [7] that the average AoI converges to∆ = limT→+∞∆T = λ(E[XT ] +E[T 2]2), (8)where X is the random variable representing the inter-arrivaltime of updates at the source (generation) with its rate λ, andT is the random variable representing the system time (delay)of an update. The expectations in the expressions depend onthe network. To abstract the details of the underlying network,it is common to assume idealized queuing models such as theM/M/1, M/D/1, D/M/1 [3], [7]. Each model makes a differentassumption about the update generation (X) and system time(T ) which is composed of waiting time in the queue andservice time in the network.III. ESTIMATION OF THE METRICS FROM EXPERIMENTSConsider a setup in which the source transmits an updateat generation time (ti) and that the receiver records its timeof receipt (ri). It is recommended that all calculations beperformed after the end of the experiment as the time tocalculate the metrics might affect the accuracy of the loggedtimings.The delay that the ith update exhibits is calculated by,Ti = ri − ti. (9)The expected delay of (9) can be estimated using the samplemedian of the vector that contains all the delays of all theupdates, i.e.,T(1→n) =[T1, T2, ..., Tn], (10)where n is the total number of updates communicated. Thesample median is employed instead of the sample averagebecause in some cases, the network protocol might re-transmitsome packets, as in TCP/IP and HARQ protocols, if they suf-fered from significant errors. The re-transmission will increasethe delay time hence would significantly affect the mean value.Similarly, the average PAoI can be estimated using thesample median of the vector containing the PAoI of all thestatus updates communicated in the experiment, i.e.,P(1→n) =[P1, P2, ..., Pn]. (11)To use the initial definition Pi = Xi + Ti, requires firstfinding the inter-arrival time and the delays. In the following,we provide an easier formulation using only the times ofgeneration and receipt as shown in Fig. 2. In the figure, it isclear that the peak age of an update stretches from its time ofgeneration until the time of receipt of the next update. Hence,Pi can be evaluated asPi = ri+1 − ti. (12)!" #" !"$% #"$%&"'" '"$%("Fig. 2. Peak Age of Information measuring method illustration. Where ti isthe time that information i was generated, ri is the time in which informationi was received by the server, where Xi represents the updates inter-arrivaltime. The Peak Age can be considered as the difference between the time ofreceiving the next update and the time of generating the update.Finally, to estimate the average AoI the areas of Qi shouldbe calculated. By using the area of a trapezoid it can be shownthatQi =12(((ti+1 − ti) + (ri − ti))2 − (ri − ti)2). (13)The average AoI can then be estimated as follows∆ =∑ni=2Qi(rn − t2). (14)Therefore, in this section, we can express the metrics ofinterest in terms of the observable time-stamps coming froman experiment.IV. TESTED CASE STUDYIn this section, we validate our estimators by emulating anM/M/1 network and comparing the measured to the theoret-ical results. We start by reviewing the theoretical results forM/M/1; then we give our validation setup before presentingand discussing the results.A. M/M/1 QueueThe method of measurement was tested on an M/M/1 queue,where the inter-arrival time and the service time follow anexponential distribution with rates λ and µ, respectively. Theexpected delay of such model, denoted E[T ], is given by [10],E[T ] = E[W + S]=λµ(µ− λ)+1µ, (15)where E[W ] = λ/µ(µ − λ) is the expected waiting time inthe queue and E[S] = 1/µ is the expected service time. From(15) and the fact that E[X] = 1/λ, the theoretical M/M/1average peak AoI is [3],PM/M/1 = E[X + T ]=1µ(1 +1ρ+ρ1− ρ), (16)where ρ = λ/µ is the link utilization. Finally, the average AoIof an M/M/1 is given by [10],∆M/M/1 =1µ(1 +1ρ+ρ21− ρ). (17)B. Experimental SetupTo validate the proposed method a simple network em-ploying an M/M/1 queue was emulated locally. A Client-Server model was used as shown in Fig. 3. The updates weresent using TCP/IP from the client to the server. The clienttransmitted the instantaneous time-stamps of when the updateswere generated, i.e., ti. Upon sending an update, the clientsleeps for a random duration that is exponentially distributedwith inter-arrival rate (λ). The rate λ was varied between 1and 8 updates per second. Fig. 4 presents the flow-chart of theclient’s behaviour.Client Server!" #"QueueFig. 3. Network System model showing the Client, where the time-stampof generating updates i, i.e., (ti). The Server saves the time of receiving theupdate (ri).StartInitializeImport libraries, set IP address and socket portGenerate Time-stamptSend UpdateSleepFor the inter-arrivaltimeFig. 4. Client’s flow chart.At the server side, the time of receipt ri is recorded foreach packet i along with the transmit time embedded in theupdate, ti. To emulate the service time, which is exponentiallydistributed with rate µ = 10, the server sleeping time was used.In particular, upon receipt of the update, the server will sleepfor a random period (exponentially distributed), and then wakeup and terminate the session with the client and record the timeas the time of receipt ri. After receiving a predefined numberof updates, (we used one thousand updates per client in theexperiments), the experiment terminated and the estimationspresented in the previous section were performed.The experiments were done using Apple MacOS with a2.2 GHz Intel Core i7 processor and a 16 GB 1600 MHzDDR3 Memory. To make sure the clients and server remainedtightly synchronized during the experiment, a single clock forall measurements was used. The updates were communicatedusing a Python 3.6 socket module [8]. To obtain the time-stamps, a Python time module [9] was used and the readingswere obtained by using the object time.time().C. Results and DiscussionsFig. 5 presents the results for the delay time and comparesit with the theoretical, given by (5). It can be seen that theexperimental and theoretical results are in good agreement.In particular, the percentage error does not exceed 5%. Thefirst row of Table I shows the results of a statistical test thatvalidates the assumption that the delay (i.e., waiting time plusservice time) can be accurately estimated using our method.00.10.20.30.40.50.62 3 4 5 6 7 8Delay Time ( T) (s)Arrival Rate λ ( s-1 )Empirical TheoreticalFig. 5. Delay versus Arrival rate for M/M/1 queue calculated theoreticallyfrom (15) and measured in the experiment using the median of (10).Next, we move to consider the PAoI estimator which wascompared with its theoretical counterpart given by (16). Theresults in Fig. 6 show that the estimated PAoI is in very goodagreement with the theoretical values. Thus, we can arguethat the experimental model proposed can obtain the PAoI anddelay time for the proposed system model accurately.0.30.40.50.60.70.80.911.11.21 2 3 4 5 6 7 8Peak Age ( P) M/M/1(s)Arrival Rate λ ( s-1 )Empirical TheoreticalFig. 6. Peak Age versus Arrival rate for M/M/1 queue calculated theoreticallyfrom 16 [3] and obtained experimentally using (11).Finally, the estimated and theoretical average AoI are shownin Fig. 7 as a function of the arrival rate. The results show thatthe estimated and theoretical results are in agreement.0.30.350.40.450.50.550.60.652 3 4 5 6 7 8Average Age  M/M/1(s)Arrival Rate λ ( s-1 )Empirical TheoreticalFig. 7. Average Age versus Arrival rate for M/M/1 queue calculatedtheoretically from (7) [7] and obtained experimentally using (14).To assess the significance of the results, a student t-test wasused to calculate the p-values and the percentage errors foreach of the estimators. Table I presents the t-test p-values andthe mean percent error values for the delay time, PAoI, andAoI measurements. As shown in Table I, the difference in themeans can be described as not significant. Thus, it can beconcluded that the proposed method is accurate. Further, itcan be concluded that the number of readings is sufficient toprecisely calculate the delay time, PAoI and AoI. Hence, thismethod was used in our paper [12] to evaluate the PAoI ofZero-wait policy.TABLE IP VALUES AND PERCENT ERRORS FOR M/M/1, VALIDATING THEPROPOSED METHOD FOR EVALUATING THE PEAK AND AVERAGE AGEParameter p-Value Percentage errorDelay (T) 0.73 6.46Peak (PAoI) 0.39 4.16Age (AoI) 0.39 4.46V. CONCLUSIONS AND FUTURE WORKThe AoI is a novel metric suggested to measure the fresh-ness of information. A considerable amount of work has beendone to evaluate and optimize AoI and PAoI theoretically. Inthis paper, we tried to motivate more experimental work onthe AoI by making it straightforward to estimate the metricsfrom experiments. The proposed method was validated on anemulated M/M/1 queue. We showed that the proposed methodcould achieve estimates that are very close to the theoreticalcounterparts. The obvious next step is to test the accuracy ofestimators in experiments involving different queuing models.Furthermore, it is worth investigating the implications ofconsidering the AoI and PAoI on real-life applications suchas remotely controlled robots.REFERENCES[1] B. Barakat and K. Arshad, “An adaptive hybrid scheduling algorithm forLTE-Advanced,” in 2015 22nd International Conference on Telecommu-nications (ICT), 2015, pp. 91-95.[2] M. Costa, M. Codreanu, and A. Ephremides, “On the Age of Informationin Status Update Systems With Packet Management,” IEEE Trans. Inf.Theory, vol. 62, no. 4, pp. 1897-1910, Apr. 2016.[3] L. Huang and E. Modiano, “Optimizing age-of-information in a multi-class queueing system,” in 2015 IEEE International Symposium onInformation Theory (ISIT), 2015, pp. 1681-1685.[4] E. Najm and R. Nasser, ”Age of information: The gamma awakening,”2016 IEEE International Symposium on Information Theory (ISIT),Barcelona, 2016, pp. 2574-2578.[5] A. Kosta, N. Pappas, and V. Angelakis, “Age of Information: A NewConcept, Metric, and Tool,” Found. Trends Netw., vol. 12, no. 3, pp.162-259, 2017.[6] C. Kam, S. Kompella, and A. Ephremides, “Experimental evaluation ofthe age of information via emulation,” in Proceedings - IEEE MilitaryCommunications Conference MILCOM, 2015, vol. 2015-Decem, pp.1070-1075.[7] S. Kaul, R. Yates, and M. Gruteser, “Real-time status: How often shouldone update?,” in Proceedings - IEEE INFOCOM, 2012, pp. 2731-2735.[8] “18.1. socket - Low-level networking interface Python 3.3.7 documenta-tion.” [Online]. Available: https://docs.python.org/3.3/library/socket.html.[Accessed: 06-Mar-2018].[9] “16.3. time - Time access and conversions - Python 3.6.4 documen-tation.” [Online]. Available: https://docs.python.org/3/library/time.html.[Accessed: 07-Mar-2018].[10] D. Bertsekas and R. Gallager, Data Networks. Prentice Hall, 1992.[11] C. Snmez, S. Baghaee, A. Ergii and E. Uysal-Biyikoglu, “Age-of-Information in Practice: Status Age Measured Over TCP/IP ConnectionsThrough WiFi, Ethernet and LTE,” 2018 IEEE International Black SeaConference on Communications and Networking (BlackSeaCom), Ba-tumi, 2018, pp. 1-5.[12] B. Barakat, S. Keates, I. J. Wassell, and K. Arshad, “Isthe Zero-Wait Policy Always Optimum for Information Freshness(Peak Age) or Throughput?,” in IEEE Communications Letters,doi:10.1109/LCOMM.2019.2907935, (in press).

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How to Measure the Average and Peak Age of Information in Real Networks?

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