Age of Information in Multi-hop Networks with Priorities
Olga Vikhrova, Federico Chiariotti, Beatriz Soret, Giuseppe Araniti, Antonella Molinaro, Petar Popovski
aa r X i v : . [ c s . N I] S e p Age of Information in Multi-hop Networkswith Priorities
Olga Vikhrova ∗ , Federico Chiariotti † , Beatriz Soret † , Giuseppe Araniti ∗ , Antonella Molinaro ∗ , Petar Popovski †∗ DIIES Department, University Mediterranea of Reggio Calabria, Italy ,emails: [email protected], [email protected], [email protected] † Department of Electronic Systems, Aalborg University, Denmark , emails: [email protected], [email protected], [email protected]
Abstract —Age of Information is a new metric used in real-timestatus update tracking applications. It measures at the destinationthe time elapsed since the generation of the last received packet.In this paper, we consider the co-existence of critical and non-critical status updates in a two-hop system, for which thenetwork assigns different scheduling priorities. Specifically, thehigh priority is reserved to the packets that traverse the twonodes, as they experience worse latency performance. We obtainthe distribution of the age and its natural upper bound termed peak age . We provide tight upper and lower bounds for priorityupdates and the exact expressions for the non-critical flow ofpackets with a general service distribution. The results givefundamental insights for the design of age-sensitive multi-hopsystems.
Index Terms —AoI, Peak AoI, IoT, multi-hop networks, priority
I. I
NTRODUCTION
The
Age of Information (AoI) [1], [2] characterizes thefreshness of the information from the receiver’s perspective,and it has been proved to be a proper metric in many real-time and context-aware Internet of Things (IoT) applications[3]. In these applications, the end receiver is interested in afresh knowledge of the remotely controlled system, rather thanthe packet delay. Besides the average age, the
Peak Age ofInformation (PAoI) [4] is a byproduct of the age process thatquantifies the worst case.There are many examples of age-sensitive IoT applications.In [5], the authors consider a Mobile Edge Computing (MEC)system and investigate the impact that pre-processing theraw data collected from sensors has in the age performance.Another example is given in [6], which addresses the problemof the optimal status update generation in a wireless systemwhere the source of updates runs applications with regular IoTtraffic and AoI-sensitive traffic. Finally, the role of satellitesin tracking applications for wide-area sensor and vehicularnetworks is growing due to their natural way to provideubiquitous coverage for the massive IoT in areas where cellularcommunications are not available or less cost-effective [7]. Asexplained in [8], Low Earth Orbit (LEO) satellites organised ina constellation may collect the status updates and forward themover the inter- or intra- satellite links to the ground station.A close examination of the above-mentioned works revealsthe common features of the tracking update systems and ex-isting research gaps. A single queuing system can capture thetimeliness of information only between two directly communi- cating instances, but it fails to give adequate results in multi-hop networks, i.e., when status updates are forwarded overone or several relay nodes. Another element is the existenceof heterogeneous requirements and paths: different servicesshould be treated according to their priority level, and statusupdates might use different entry points to the communicationsystem. This motivates us to consider a general multi-hopcommunication system with traffic arrivals at the intermediatenodes and different priorities for the status updates. For ouranalysis, we take the illustrative case of two nodes, wherestatus update packets sent via the relay (the first node) takespriority over the updates sent directly to the monitor (thesecond node) as shown in Fig. 1. Priority packets preempt allnon-priority packets in the queue of the second node but do notimpact the ongoing service. This priority policy will improvethe performance of the status updates that need the relay toreach the destination, reducing the difference in performancebetween the two paths.In this paper we obtain the distribution of the AoI andthe PAoI using the Laplace-Stiltjes Transform (LST) for thesystem of interest. We also give the distribution of the
SystemDelay of priority packets that traverse the two nodes, whilethe system delay only at one node was known before. Unlikeprevious works on AoI with packets prioritization, we considera general service time distribution and more complex systemmodel with relay. We also give closed-form expressions for theaverage AoI, PAoI and system delay of non-priority packetsand tight bounds for priority packets, while the moments ofhigher orders can be derived from the given LST expressions.The rest of the paper is organized as follows. In sectionII we introduce related works on the AoI, and describe thesystem model in section III. The metrics of interests are givenin section IV, while the numerical results are discussed insection V. The concluding remarks are given in the last section.II. R
ELATED WORKS
A system design similar to ours has been considered in [9].Authors investigate the average AoI when the status updatecan be delivered either over the less reliable direct link orover the two-hop relay link with better reliability. However,all packets at the second node have been treated equally.In [5] only average PAoI is given for the two-hop tandemexponential queues with multiple sources. Authors in [10]study the average AoI of a two-hop system with packet arrivalsnly at the first node and zero-waiting policy at the secondnode.In [11] authors derive a general formula for the stationarydistribution of the AoI in terms of the system delay andthe PAoI for a wide class of
G/G/ systems with a singlesource under the general FCFS and Last Come First Serve(LCFS) packet management policies with various preemptionand packet discarding options. However, LCFS policy can notbe applied to the systems where packets carry incrementalinformation and can not be discarded.The idea of assigning different priorities to the updatepackets has been discussed for the first time in [12]. Theaverage AoI is given for an exponential single-server systemwith a shared queue and LCFS discipline, where the arrivedpacket preempts another packet either in service or in waitingonly if it has higher priority. In [13] authors focus on a queuingsystem with k classes of priorities, different buffer sizes andqueuing disciplines. In particular, the different combinationsof infinite queues with FCFS and LCFS disciplines andqueues with a single place to wait are considered. The exactexpressions of the expected PAoI are given for the generalservice time distribution if the queues are infinite and for theexponential service time if the queue size is one, while thetight bounds have been calculated for the remaining scenarios.The above-mentioned works with the packet‘s prioritizationare limited to the single-node systems.III. S YSTEM MODEL
We consider a two-hop network with intermediate traffic.Sources generate packets with status updates according to aPoisson process with rate λ . With probability p priority packetsarrive at the first node and with probability − p all remainingnon-priority packets arrive directly to the second node, λ = pλ and λ = (1 − p ) λ . Such a network is modeled as twotandem queues connected in series with packet prioritization inthe second queue. In particular, both queues apply the generalFCFS discipline but in the queue of the second node all packetscoming originally from the first node (priority packets) pre-empt in waiting packets coming directly from the source (non-priority packets). Non-priority packets see the second node asan M/G/ queue with priorities, while priority packets find M/M/ and M/G/ queues connected in series.Service times at the first node are limited to the exponentialdistribution for the sake of mathematical tractability, i.e. toensure that the departure process from the first node is Poisson.Let b and b be the mean service times of priority and non-priority packets packets at the second node. The total systemutilization equals to the second node utilization ρ = ρ + ρ ,where ρ j = λ j b j , j = { , } . Utilization of the first node ρ = λ /µ , µ − is the mean service time at the first node.Let j, i denote packet i of priority class j . Let t j,i and t ′ j,i be the time instances of packet j, i arrival to the system(generation of a new status at source) and its departurefrom the system (updating the status at the monitor). Then Y j,i = t j,i − t j,i − denotes the random variable (RV) of packet j, i interarrival time and T j,i = t ′ j,i − t j,i corresponds to the λ , µ − λ , b λ , b priority packets non-priority packets Fig. 1. System model as two FCFS queues in tandem with priorities at thesecond node.
RV of the packet’s system delay . The AoI ∆ j,i at time t > consists of the AoI Z j,i − immediately after the departure ofthe packet j, i − and the time from t ′ j,i − to t , i.e. ∆ j,i = Z j,i − + ( t − t ′ j,i − ) . In general FCFS systems Z j,i equals tothe system delay T j,i if all packets are time-stamped on theirarrival. Therefore the PAoI A j,i = t ′ j,i − t j,i − = Y j,i + T j,i .In the ergodic system ( ρ < ), the probability densityfunction (pdf) of the AoI can be defined as f ∆ j ( x ) = λ j ( F T j ( x ) − F A j ( x )) , x ≥ , where F T j ( x ) and F A j ( x ) stand for the Probability Distribution Functions (PDFs) of thesystem delay and PAoI, respectively [11]. The Laplace-StiltjesTransform (LST) δ j ( s ) of the AoI distribution therefore yields: δ j ( s ) = λ j s ( τ j ( s ) − α j ( s )) , s > , (1)where τ j ( s ) = ∞ R e − sx dF T j ( x ) and α j ( s ) = ∞ R e − sx dF A j ( x ) .Priority and non-priority packets arrive to the system inde-pendently, their interarrival times are exponentially distributedholding the LST λ j / ( λ j + s ) . System delay T j,i depends on thepackets interarrival time Y j,i and the system delay T j,i − , italso depends on the arrival and departure processes of packetsof another class. The RV T ,i = T ,i + T ,i while T ,i and T ,i are not independent. In the next section we define thePAoI for packet j, i and then obtain the general distribution of A j for both classes of packets, the similar approach is appliedfor calculation of the total system delay T of priority packets.Let us give the known distributions of the system delaysat each node as preliminaries for further analysis. The systemdelay T at the first node (M/M/1) is exponentially distributedwith parameter θ = µ − λ , the corresponding LST τ ( s ) equals to θ/ ( θ + s ) . The LST of the system delay T ofpriority packets and system delay T of non-priority packetsat the second node are given in [14, chapter 8.6]: τ ( s ) = s (1 − ρ ) + λ (1 − β ( s )) s − λ + λ β ( s ) β ( s ) , (2) τ ( s ) = (1 − ρ )( s + λ − λ γ ( s )) s − λ + λ β ( s + λ − λ γ ( s )) β ( s ) , (3)where β ( s ) and β ( s ) are the LSTs of the service timedistributions of priority and non-priority packets at the secondnode, γ ( s ) stands for the LST of the distribution of the interval G , which elapses from the arrival of a priority packet in theempty queue of the second node until the end of continuousservice of priority packets arriving afterwards. This interval isknown as a busy period generated by a priority packet and itsLST γ ( s ) = β ( s + λ − λ γ ( s )) . The busy period G starts ABLE IL
IST OF NOTATIONS
Notation Definition k Node index ( j, i ) Packet i of priority class j t j,i Packet ( j, i ) arrival time t ′ j,i Packet ( j, i ) departure time λ j Arrival rate for class j b j Mean service time for class j ρ j Second node utilization by class j b = µ − Mean service time at the first node θ Mean system delay at the first node ρ First node utilization
RV LST Definition Y j,i Packets interarrival time S kj,i β ( s ) , β j ( s ) Service time of packet ( j, i ) at node k W jk,i ω jk ( s ) Waiting time of packet ( j, i ) at node k T jk,i τ k ( s ) , τ j ( s ) System delay of packet ( j, i ) at node k D j,i η j ( s ) Supplementary to PAoI of packet j, i in-terval as defined in Fig. 2 X ,i ξ ( s ) Supplementary to system delay of packet j, i interval as defined in Fig. 2 G j,i γ j ( s ) Busy period generated by a packet j, i ˜ Z j,i ˜ ζ j,i ( s ) Residual time of interval Z j,i A j α j ( s ) PAoI of class j ∆ j δ j ( s ) AoI of class j from the moment when a non-priority packet arrives to theempty node, therefore its LST is β ( s + λ − λ γ ( s )) . Forconvenience we give the complete list of notations in Table I.IV. A NALYSIS
A. Priority packets
When priority packet i arrives to the system it can bequeued in both nodes, queued only in one node or go throughtwo nodes without any queuing delay. The presence of non-priority packets at the second node hinders the derivation of thePAoI and system delay distributions. We assume that packet i finds the second node free of non-priority packets with theprobability − ρ .There are six cases C1–C6 that help to define system delay T ,i and PAoI A ,i of packet i in the system of interest. Let usdefine intervals D ,i (bold red line) and X ,i (bold blue line) asillustrated in Fig. 2. Let also η ( s, C m ) and ξ ( s, C m ) denote C Y ,i S ,i S ,i T ,i − T ,i − C Y ,i S ,i S ,i T ,i − T ,i − C Y ,i S ,i S ,i T ,i − T ,i − C Y ,i S ,i S ,i T ,i − T ,i − C Y ,i S ,i S ,i T ,i − T ,i − ˜ S C Y ,i S ,i S ,i T ,i − T ,i − ˜ S D ,i X ,i Fig. 2. PAoI and system delay of priority packets in cases C1 - C6. the LST of the joint distribution of intervals contributing to D ,i and X ,i for a case C m , m = { , . . . , } , respectively.We define the LST of the system delay τ ( s, C m ) and thePAoI α ( s, C m ) for each case. The resulting distributions willbe given as a sum of LSTs of the six joint distributions namely τ ( s ) = P m τ ,i ( s, C m ) and α ( s ) = P m α ,i ( s, C m ) .C1: Packet i does not experience any queuing at nodes,therefore the PAoI A ,i = D ,i + S ,i and system delay T ,i = X ,i + S ,i . This happens if T ,i − < Y ,i , T ,i − + T ,i < Y ,i + S ,i and if during the interval Y ,i + S ,i − T ,i − − T ,i − all unserved non-prioritypackets complete their service and no new non-prioritypackets arrive. Since we assume that packet i findsthe second node free of non-priority packets with theprobability − ρ and service time S ,i is independentof other intervals, the LST of both metrics can be givenas α ( s, C ) = (1 − ρ ) η ( s, C ) β ( s ) and τ ( s, C ) =(1 − ρ ) ξ ( s, C ) β ( s ) .C2: Packet i finds the second node busy with packet i − , butits queuing delay at the first node W ,i = 0 , thereforePAoI A ,i = D i, + S ,i and system delay T ,i = X ,i + S ,i like in the case C1, but D ,i = T ,i − + T ,i − , X ,i = T ,i − + T ,i − − Y ,i . This is true if T ,i −
5) = ρ τ ( s ) β ( s )( τ ( s ) − τ ( s + µ )) , (18) ξ ( s, C
5) = ρ τ ( s )( τ ( s ) − τ ( s + µ )) . (19)The resulting LST of the PAoI distribution of prioritypackets yields: α ( s ) = h λ νλ + s β ( s ) τ ( λ + s ) − ss + θ ρ β ( s ) ×× ( τ ( s ) − τ ( s + µ )(1 − νβ ( s ))) i β ( s ) , (20)where ν = 1 − ρ + ρ ˜ β ( s ) .The LST of system delay is given as follows: τ ( s ) = h τ ( s ) τ ( λ ) (cid:16) ν − λ λ − s (cid:17) + τ ( s ) ×× (cid:16) (1 − ρ ) λ λ − s + ρ τ ( s ) (cid:17)i β ( s ) . (21)Given (1) and (20)–(21) the LST of ∆ yields: δ ( s ) = β ( s ) h τ ( s ) τ ( λ ) λ s − λ ( ν + λ s (1 − ν ))++ λ λ + s β ( s ) τ ( s )(1 + λ s (1 − ν )) − ρ β ( s )1 − ρ τ ( s ) ×× τ ( µ + s ) τ ( s )(1 − νβ ( s )) + νρ ˜ β ( s ) β ( s ) i . (22)Having the LSTs (20)–(22), we can calculate the averagesystem delay, PAoI and AoI as E [ T ] = − τ ′ (0) , E [ A ] = − α ′ (0) , and E [∆ ] = − δ ′ (0) : E [ T ] = b + λ b (2) − ρ ) + b + λ b (2)1 + λ b (2)2 − ρ ) , (23)where b ( k ) j denote the k -th moments of packet j service time. E [ A ] = (cid:16) λ + b + ρ ˜ b (cid:17) τ ( λ ) − ρ ( b + bτ ( µ )) ×× (1 − ρ + ρ ˜ b ) + (1 − ρ ) (cid:16) b + E [ T ] τ ( µ ) (cid:17) ++ ρ (1 − ρ + ρ ˜ b ) (cid:16) b + E [ T ] + E [ T ] τ ( µ ) (cid:17) ++ ρ ( b + E [ T ] + E [ T ](1 − τ ( µ ))) . (24)where ˜ b = b (2)2 / b is the average residual service time ofnon-priority packets.We give lower bound E [∆ ] for the average AoI: E [∆ ] = b + 1 λ τ ( λ ) + τ ( λ ) E [ T ] + ρ E [ T ]++ ρ (cid:16) θ − ρ µ + ρ θ + 1 λ + µλ − ρ − ρ (cid:17) . (25) . Non-priority packets Non-priority packet i can start service only if the secondnode is free of priority packets, i.e. at the end of the busyperiod G ,i − or G , or if the node is empty. Let us introducethe interval Ψ ,i − = W ,i − + G ,i − , where W ,i − standsfor the waiting time of non-priority packet i − . Intervals W ,i − and G ,i − are independent, therefore the LST of Ψ ,i − can be given as ψ ( s ) = ω ( s ) β ( s + λ − λ γ ( s )) .We consider three cases to define the PAoI A ,i .B1: if Y ,i > Ψ ,i − and packet i finds the second nodeempty it immediately goes to service, therefore A ,i = Y ,i + S ,i . At the end of interval Ψ ,i − the node isempty, therefore the probability that packet i finds thenode empty upon arrival equals to − ρ .B2: if Y ,i > Ψ ,i − and packet i finds the node busy with apriority packet with probability ρ it waits until the end ofthe ongoing busy period G , thus A ,i = Y ,i + ˜ G + S ,i ,where ˜ G denotes the residual time of interval G .B3: if Y ,i < Ψ ,i − packet i finds the second node busy withnon-priority packet i − , therefore A ,i = Ψ ,i − + S ,i .The LST α ( s ) can be given as the sum of three LSTsnamely α ( s, B ) , α ( s, B ) and α ( s, B ) defined above. a) Case B1: the LST of Y ,i + S ,i if Y ,i > Ψ ,i − andthe node is free of priority packets can be given as α ( s, B ) = (1 − ρ ) λ λ + s ψ ( s + λ ) β ( s ) . (26) b) Case B2: the LST of Y ,i + ˜ G + S ,i when Y ,i > Ψ ,i − and packet i arrives during the busy period G takes α ( s, B ) = ρ λ λ + s ψ ( s + λ )˜ γ ( s ) β ( s ) , (27)where ˜ γ ( s ) = (1 − γ ( s )) / E [ G ] s stands for the distributionof the residual time of the interval G . c) Case B3: if Y ,i < Ψ ,i − the LST of the PAoI yields α ( s, B ) = ( ψ ( s ) − ψ ( s + λ )) β ( s ) . (28)The resulting LST of the PAoI distribution of non-prioritypackets gives α ( s ) = h (1 − ρ ) λ λ + s ψ ( s + λ ) + ψ ( s ) − ψ ( s + λ )++ ρ λ λ + s ψ ( s + λ )˜ γ ( s ) i β ( s ) . (29)Having (1), (3) and (29) we give the LST of the AoI distribu-tion of non-priority packets as follows: δ ( s ) = ρ − ρ τ ( s ) ˜ β ( s + λ − λ γ ( s ))++ ψ ( λ + s ) β ( s ) (cid:16) λ λ + s + ρ λ λ + s λ s (1 − ˜ γ ( s )) (cid:17) , (30)where ˜ β ( s + λ − λ γ ( s )) denotes the residual time of the busyperiod G and equals to (1 − β ( s + λ − λ γ ( s ))) /s E [ G ] . The straightforward calculation of α ′ (0) and δ ′ (0) givesthe average PAoI E [ A ] and AoI E [∆ ] : E [ A ] = b + λ b (2)1 + λ b (2)2 − ρ )(1 − ρ ) + b − ρ + ψ ( λ ) 1 λ ++ ρ ψ ( λ ) b − ρ ) . (31) E [∆ ] = ρ − ρ (cid:16) b + λ b (2)1 + λ b (2)2 − ρ )(1 − ρ ) + b − ρ (cid:17) ++ ψ ( λ ) (cid:16) λ + ρ λ (cid:16) b (3)1 /b (2)1 − ρ ) + λ b (2)1 (1 − ρ ) (cid:17)(cid:17) ++ ψ ( λ ) (cid:16) ρ ρ − ρ ) (cid:17) ( b + ψ ′ ( λ )) . (32)V. S ELECTED NUMERICAL RESULTS
The results of our analysis have been validated by MonteCarlo simulation. All data collected during the transient statehas been discarded. We model arrivals, service and departuresof packets of the reference system. We calculate theaverage PAoI, AoI and system delay for different values of p = { . , . , . , . , . } to capture the effect of the statusupdates generation rate on the AoI. The numerical resultsare given under the assumption of exponential service timewith means b = b = b = 1 for variable utilization ρ = { . , . . . , . } at the second node.The metrics of interest of priority packets are depictedin Fig. 3. The simulation results of the PAoI illustrated inFig. 3(a) show a perfect fit of our bound with the analyticalcurves, which justifies the assumption that priority packet i finds non-priority packets at the second node with the givenprobability. The results for non-priority packets in Fig. 4 areinstead exact. The given lower bound for AoI is tight whenthe system utilization is low and becomes more visible when ρ increases. In our system, the PAoI is a tight upper bound of theAoI due to the low correlation between interarrival and delayintervals of consecutive packets. The average AoI of prioritypackets decreases when the status update rate increases if thepriority system utilization ρ < . . If ρ ≥ . the AoIgradually increases demonstrating a wide U shape, the AoI ofnon-priority packets shows similar results in Fig. 4(b). Thismeans that the optimal performance can be reached.Besides the average AoI the average PAoI and system delayof non-priority packets are shown in Fig. 4(a) and Fig. 4(c)respectively. Again the average PAoI is a very tight upperbound for the AoI. Due to the non-priority packets preemptionin waiting the average system delay rapidly increases when theutilization at the second node increases. Both PAoI and AoIof non-priority packets depend on the system delay more thanthat of priority packets. If non-priority packets may tolerate acertain packet error rate also due to the discarding of outdatedpackets the AoI could be improved if a newly arrived non-priority packet replaces the previously queued packet. .1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9020406080100120 A v e r age PA o I Priority, p = 0.1(sim), p = 0.1Priority, p = 0.3(sim), p = 0.3Priority, p = 0.5(sim), p = 0.5Priority, p = 0.7(sim), p = 0.7Priority, p = 0.9(sim), p = 0.9 (a) Average PAoI A v e r age A o I Priority, p = 0.1(sim), p = 0.1Priority, p = 0.3(sim), p = 0.3Priority, p = 0.5(sim), p = 0.5Priority, p = 0.7(sim), p = 0.7Priority, p = 0.9(sim), p = 0.9 (b) Average AoI A v e r age sys t e m de l a y Priority, p = 0.1(sim), p = 0.1Priority, p = 0.3(sim), p = 0.3Priority, p = 0.5(sim), p = 0.5Priority, p = 0.7(sim), p = 0.7Priority, p = 0.9(sim), p = 0.9 (c) Average system delayFig. 3. Average PAoI (a), AoI (b) and system delay (c) of priority packets A v e r age PA o I Non-Priority, p = 0.1(sim), p = 0.1Non-Priority, p = 0.3(sim), p = 0.3Non-Priority, p = 0.5(sim), p = 0.5Non-Priority, p = 0.7(sim), p = 0.7Non-Priority, p = 0.9(sim), p = 0.9 (a) Average PAoI A v e r age A o I Non-Priority, p = 0.1(sim), p = 0.1Non-Priority, p = 0.3(sim), p = 0.3Non-Priority, p = 0.5(sim), p = 0.5Non-Priority, p = 0.7(sim), p = 0.7Non-Priority, p = 0.9(sim), p = 0.9 (b) Average AoI A v e r age sys t e m de l a y Non-Priority, p = 0.1(sim), p = 0.1Non-Priority, p = 0.3(sim), p = 0.3Non-Priority, p = 0.5(sim), p = 0.5Non-Priority, p = 0.7(sim), p = 0.7Non-Priority, p = 0.9(sim), p = 0.9 (c) Average system delayFig. 4. Average PAoI (a), AoI (b) and system delay (c) of non-priority packets
VI. C
ONCLUSIONS
In this paper we have investigated the timeliness of thestatus updates in a multi-hop IoT tracking system with twonodes and different entry points for priority and non-prioritytraffic. We have derived the distribution of AoI, PAoI andsystem delay in terms of LST and have given closed-formexpressions for their first moments. We have obtained theexact expressions for non-priority packets and tight boundsfor priority flow of packets. In our system, PAoI is a tightupper bound for both classes of traffic.The extension to N hops requires an exponential servicetime at first N − hops while the last hop that aggregates trafficfrom all previous hops holds general service time distribution.Such an assumption is in line with many multi-hop systemsfrom the reference literature. Other possible research directionsare the extension to more priority levels, LCFS discipline withpackets discarding, and age-aware packet management.R EFERENCES[1] S. Kaul, R. Yates, and M. Gruteser, “Real-time status: How often shouldone update?” in , 2012, pp. 2731–2735.[2] S. Kaul, M. Gruteser, V. Rai, and J. Kenney, “Minimizing age of infor-mation in vehicular networks,” in , 2011, pp. 350–358.[3] M. A. Abd-Elmagid, N. Pappas, and H. S. Dhillon, “On the role of age ofinformation in the internet of things,”
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