Age-based Scheduling: Improving Data Freshness for Wireless Real-Time Traffic

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1 Age-based Scheduing: Improving Data Freshness for Wireess Rea-Time Traffic Ning Lu Department of CS Thompson Rivers University amoops, BC, Canada Bo Ji Department of CIS Tempe University Phiadephia, PA, USA Bin Li Department of CB University of Rhode Isand ingston, Rhode Isand, USA ABSTRACT We consider the probem of scheduing rea-time traffic with hard deadines in a wireess ad hoc network. In contrast to existing rea-time scheduing poicies that merey ensure a minima timey throughput, our design goa is to provide guarantees on both the timey throughput and data freshness in terms of age-of-information AoI), which is a newy proposed metric that captures the age of the most recenty received information at the destination of a ink. The main idea is to introduce the AoI as one of the driving factors in making scheduing decisions. We first prove that the proposed scheduing poicy is feasibiity-optima, i.e., satisfying the per-traffic timey throughput requirement. Then, we derive an upper bound on a considered data freshness metric in terms of AoI, demonstrating that the network-wide data freshness is guaranteed and can be tuned under the proposed scheduing poicy. Interestingy, we revea that the improvement of network data freshness is at the cost of sowing down the convergence of the timey throughput. xtensive simuations are performed to vaidate our anaytica resuts. Both anaytica and simuation resuts confirm the capabiity of the proposed scheduing poicy to improve the data freshness without sacrificing the feasibiity optimaity. CCS CONCPTS Networks Ad hoc networks; Network performance anaysis; YWORDS Data freshness, wireess scheduing, age of information, rea-time traffic, ad hoc networks ACM Reference Format: Ning Lu, Bo Ji, and Bin Li Age-based Scheduing: Improving Data Freshness for Wireess Rea-Time Traffic. In Mobihoc 18: The ighteenth ACM Internationa Symposium on Mobie Ad Hoc Networking and Computing, June 6 9, 018, Los Angees, CA, USA. ACM, New York, NY, USA, 10 pages. Permission to make digita or hard copies of a or part of this work for persona or cassroom use is granted without fee provided that copies are not made or distributed for profit or commercia advantage and that copies bear this notice and the fu citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or repubish, to post on servers or to redistribute to ists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. Mobihoc 18, June 6 9, 018, Los Angees, CA, USA 018 Association for Computing Machinery. ACM ISBN /18/06... $ INTRODUCTION Scheduing wireess network traffic with hard packet deadines pays an important roe in providing quaity-of-service QoS) guarantees for mission-critica systems, such as wireess sensor monitoring, vehice patooning, industria automation, and other cyberphysica system. Consider the scenario where mutipe sensors measure variabes of a pant, and a measurements are required to send to a centra controer through a wireess network for monitoring and contro purposes. To keep the usefuness of the measurements, each data packet in the network has to be sent in a rea-time manner 1 ; otherwise, it woud become stae and thus useess. Too many useess measurements might not be toerabe and coud affect the stabiity and contro performance of the system. Rea-time scheduing agorithms aim at scheduing deadine-constrained packets in wireess networks to guarantee a minimum portion of packets to be deivered on time i.e., timey throughput) for each traffic fow. It is worth noting that imposing a hard deadine on each packet s reception guarantees that the received packet at the time of reception is usefu; whie maintaining a certain timey throughput assures that a sufficient number of usefu packets are received during a fixed period of time on average. The design of rea-time scheduing agorithms has attracted many research attentions recenty e.g., 4, 5, 7, 11]). Athough providing QoS guarantees such that the destination receives timey packets at a desired rate is crucia, it might not be sufficient to maintain information freshness for reatime systems see 3]). For exampe, in the above scenario, the most recent measurement of a variabe coud become out-of-date to the controer if it waits too ong before the next update, which can ead to disturbance in monitoring and contro. Hence, measurements from sensors are expected to update timey at the controer s side, indicating that the data freshness at the information sink aso has an important roe to pay. Recenty, the age-of-information AoI) is proposed to capture the freshness of information updates in a system, which has received great attention because of rapid depoyment of rea-time appications 14]. By definition e.g., given in 8]), AoI is the amount of time eapsed since the most recent update at the destination) was generated at the source). It naturay characterizes the data freshness from the receiver s perspective. Motivated by the AoI research and the importance of information freshness at the receiver for rea-time appications e.g., maintaining fresh measurements at the controer in wireess monitoring), we are particuary interested in designing rea-time scheduing agorithms that guarantee 1 There exist a number of factors that prevent the rea-time deivery of packets, such as queueing deay due to channe contention among transmitters, and packet oss due to channe fuctuation.

2 Mobihoc 18, June 6 9, 018, Los Angees, CA, USA N. Lu, B. Ji, and B. Li a minimum portion of packets deivered on time as we as improve the AoI of rea-time traffic. To the best of our knowedge, none of existing works has taken into consideration of providing both timey throughput and data freshness in terms of AoI) guarantees for deadine-constrained traffic. The current focus of AoI research is on AoI anaysis and optimization under different network settings e.g.,, 6, 8, 14]). In 6], adota et a. proposed a greedy poicy AoI-Greedy) for scheduing deadine-constrained traffic in a symmetric wireess network. However, this work ony focuses on the AoI minimization without providing any guarantees on timey throughput, and is specificay suitabe for broadcast i.e., fuy connected) networks. It is worth mentioning that our work is aso reated to improving the short-term performance of rea-time traffic i.e., how often the packets are deivered to the receiver). Quantifying and optimizing short-term performance of rea-time traffic fows remains open due to the fact that the anaysis of transient behavior short-term) of a system is more difficut than the anaysis of asymptotic behavior ong-term). As an effort to this end, in 3], Hou proposed to introduce a penaty on each fow if its short-term performance is beow some specified requirement, and empoy Brownian motion approximation to optimize the overa penaty of the system. We consider our work to be an effort towards that end too, where the AoI can be considered as a metric that captures the short-term performance. In this paper, we consider the probem of scheduing deadineconstrained rea-time traffic in an ad hoc wireess network with guarantees on both timey throughput and data freshness in terms of AoI at the receivers. Particuary, we adopt a frame-based mode : the network nodes communicate in frames, each of which consists of a fixed number of consecutive time sots of equa duration; for each ink, packets arrive at the beginning of each frame and need to be transmitted by the end of the frame; otherwise the packets wi be dropped. To provide the guarantee on timey throughput, we use a virtua-queue technique in designing the scheduing agorithm; whie to improve the performance of data freshness, we introduce AoI as one of the driving factors in making scheduing decisions. We provide an exampe in Section to iustrate how the AoI performance can be improved through making appropriate scheduing decisions on packet deivery. The contributions of this work can be summarized as foows: We propose a rea-time scheduing poicy that provides guarantees on both the timey throughput and the data freshness for deadine-constrained traffic in ad hoc networks, by introducing the AoI as one of the driving factors in making scheduing decisions. We prove that the proposed scheduing poicy is feasibiityoptima. We aso derive an upper bound on the network data freshness defined in Section, which demonstrates that the data freshness at the receivers is guaranteed in the network. Particuary, we revea that the improvement of data freshness is at the cost of sowing down the convergence of the timey throughput. We show that trading the system performance i.e., convergence speed of the timey throughput) for The probem of rea-time scheduing under genera interference, channe, packet arriva, and deadine modes coud be extremey difficut. A more modest goa in research community is to attack this probem under the frame-based mode, which is first considered in 4]. data freshness can be done by tuning a contro parameter. The introduction of this contro parameter makes the traditiona virtua-queue-ength-based poicies e.g., 4, 5, 7, 11]) and the AoI-Greedy poicy specia cases of our proposed poicy. The remainder of the paper is organized as foows. Section introduces the system mode. Section 3 presents the scheduing design and shows that the proposed poicy is feasibiity-optima. We aso anayze the AoI performance and investigate the trade-off between the data freshness and the convergence speed of the timey throughput in Section 3. Simuation resuts are given in Section 4. We discuss the distributed impementation of the proposed scheduing poicy in Section 5. Section 6 provides concuding remarks. SYSTM MODL We consider scheduing rea-time traffic fows in an ad hoc wireess network with L inks. Assume that time is sotted and each consecutive T time sots are grouped into one frame. ach ink is associated with one rea-time fow, and we assume that packets arrive at each ink ony at the beginning of each frame, and each packet has to be deivered by the end of the frame; otherwise, it wi be dropped. Let A kt ] denote the number of packet arrivas at ink {1,,, L} in frame k, where k = 0, 1,,, which are independenty distributed over inks and identicay distributed over time with mean λ, and A kt ] A max for some A max <. Note that we aso use AkT ] A kt ]) L to denote the arriva vector in frame k. The timey throughput requirement on each fow is that each ink guarantees a minimum timey throughput of λ 1 γ ), where γ 0, 1) is the maximum aowabe packet dropping rate due to deadine expiry. For exampe, on average at most 0% of packets can be dropped or a timey throughput is at east 0.8 λ at ink when γ = 0.. We consider that L inks share one common communication channe, and each ink experiences an independent bock fading, where the channe state is constant during a frame and can vary over frames. We use a random variabe C kt ] to capture the channe state information CSI) of ink in frame k, which is the maximum number of packets that can be deivered in a time sot if ink is schedued. We assume that CkT ] C kt ]) L are independenty and identicay distributed over time with C kt ] C max,, k, for some C max <. We consider the case where the CSI is known to the scheduer via channe probing at the beginning of each frame. Since nearby inks cannot be schedued simutaneousy due to interference, scheduing is required to determine a subset of inks that coud transmit simutaneousy without interfering with each other in each time sot, caed feasibe schedue denoted by St] S t]) L, where S t] = 1 if ink is schedued in sot t and S t] = 0, otherwise. We use S to denote the set of a feasibe schedues. A key technique in the design is the use of virtua queues 1]. Assume that each ink maintains a virtua queue to keep track of the number of dropped packets due to deadine expiry. Let V kt ] denote the virtua queue ength of ink at the beginning of frame k. The evoution of virtua queue is given by { } V k + 1)T ] = max V kt ] + I kt ] D kt ], 0, 1)

3 Age-based Scheduing: Improving Data Freshness for Wireess Rea-Time Traffic Mobihoc 18, June 6 9, 018, Los Angees, CA, USA where I kt ] 0is the increase of the virtua queue and is equa to the number of dropped packets in frame k, i.e., k+1)t 1 I kt ] A kt ] min C kt ]S t], A kt ] ; ) and the decrease of the virtua queue is D kt ] with mean γ λ, and D kt ] < D max for some D max <. According to 9], if there is a scheduing poicy that can stabiize a virtua queues, there aways exist non-negative numbers αa, c; s 0, s 1,...,s T 1 ) such that αa, c; s 0, s 1,...,s T 1 ) = 1, a, c, 3) s 0,s 1,...,s T 1 S λ 1 γ ) P A a) P C c) a c s 0,s 1,...,s T 1 S { T 1 αa, c; s 0, s 1,...,s T 1 ) min c s τ,, a },. 4) τ =1 where s τ s τ, ) L, τ {0, 1,...,T 1}, P Aa) = Pr{AkT ] = a} and P C c) = Pr{CkT ] = c}. The inequaity 4) states that the average service provided to ink during a frame the right hand side of 4)) shoud be no ess than the average amount of traffic that needs to be deivered the eft hand side of 4)), in order to meet the timey throughput requirement of rea-time traffic. A scheduing agorithm is said to be feasibiity-optima if, for any given maximum aowabe dropping rate vector γ = γ ) L and channe state C, for any arriva process that ies stricty within the maxima satisfiabe region Λγ, C), it stabiizes a virtua queues in the sense that a virtua queue processes are positive recurrent. The maxima satisfiabe region is defined as foows: Λγ, C) = { λ : αa, c; s 0, s 1,...,s T 1 ) 0, such that 3) and 4) hod } 5). In addition to providing guarantees on timey throughput, we are aso interested in improving data freshness of the network. To this end, we adopt the AoI in designing the rea-time scheduing poicy. We denote R kt ] the AoI of ink up to frame k, i.e, the number of frames eapsed up to frame k since the most recent successfu packet deivery on ink. The smaer R kt ], the better data freshness. Note that the use of frame as the unit of AoI can be seen in reated work 6]. This is aso because we assume that the arrivas ony happen at the beginning of each frame. To capture the dynamics of R kt ], we introduce H kt ] to denote the event that at east one packet is deivered at ink in frame k, i.e., k+1)t 1 H kt ] min C kt ]S t], A kt ] > 0. In addition, et I H kt] denote the indicator variabe such that I H kt] = 1ifeventH kt ] happens, and I H kt] = 0 otherwise. Thus, the evoution of AoI of ink can be described as foows. { 1 ifih kt] = 1; R k + 1)T ] = 6) R kt ] + 1 if I H kt] = 0. As we can see, R kt ] grows ineary unti ink has at east one packet deivery in a frame and then it drops to one at the beginning of the next frame. Figure 1: xampe of AoI dynamics of a ink We provide an exampe in Fig. 1 to show how to improve AoI performance of a ink by appropriatey scheduing packet deivery. As shown in Fig. 1a), the AoI of a ink updates at the beginning of every frame. The deivery of packets a, b, and e resut in the decrease of AoI. If we ony focus on the time window shown in the figure 11 frames), Fig. 1a) shows an average deivery ratio of 60% 3 packet deiveries over 5 packet arrivas) with an average AoI of.7 frames. Whie in Fig. 1b), with the same packet arriva pattern and deivery ratio, the schedue of deivery of packet d instead of e eads to a drop of average AoI to frames, indicating a better performance on data freshness. In this work, we are interested in designing feasibiity-optima scheduing agorithms which can aso improve the network data freshness, which is measured as the tota weighted sum of the average steady-state AoI over a inks in the network, i.e., L ω R ], where ω 0 is a weighting parameter associated with ink and gives the preference of ink towards the information freshness, and R denotes the steady-state AoI of ink under some stabiizing scheduing poicy. Next, we wi deveop the scheduing poicy that not ony uses virtua-queue engths, but aso considers the AoI in making scheduing decisions. 3 AG-BASD RAL-TIM SCHDULING In this section, we first present the proposed scheduing poicy and then show that it is feasibiity-optima. After that, we investigate the performance of data freshness under our poicy. At ast, we examine the trade-off between the data freshness and the convergence speed of the timey throughput. 3.1 Agorithm Description The proposed Age-based rea-time AoI-RT) scheduing agorithm is given in Agorithm 1. The scheduing decision is made on a per-frame basis. The idea is that, at the beginning of frame k, a maximum weighted schedue

4 Mobihoc 18, June 6 9, 018, Los Angees, CA, USA Agorithm 1: AoI-RT Agorithm In each frame k, the AoI-RT agorithm seects a schedue {S k+1)t 1 t]} such that {S k+1)t 1 t]} arg max 1 ζ )V kt ] + ζβ R kt ]) St] S,,...,k+1)T 1 k+1)t 1 min C kt ]S t], A kt ]. where 0 ζ 1 and β 0,, are some contro parameters. {S k+1)t 1 t]} is seected among a possibe schedues, by treating k+1)t 1 1 ζ )V kt ] + ζβ R kt ]) min C kt ]S t], A kt ] k+1)t 1 in Agorithm 1 as the weight of a schedue {St]} over the frame k. The weight can be considered as a tota sum of ink weight" over a inks. For each ink, the ink weight under a certain schedue is the production of 1 ζ )V kt ] + ζβ R kt ]) and the service schedued to ink. Here, the virtua-queue ength V kt ] and the weighted AoI β R kt ] are the two driving factors in making scheduing decisions. Note that the AoI-RT agorithm is simiar to the traditiona MaxWeight agorithm. The difference ies in that the weight now is the inear combination of the virtua-queue ength and the weighted AoI parameterized by ζ. Note that ζ is a parameter to contro the data freshness performance of the agorithm. In particuar, with ζ = 0, the AoI-RT agorithm coincides with the pure virtua-queue-ength-based VQLbased) scheduing agorithms e.g., 4, 5, 7, 11]), which is feasibiityoptima but provides no guarantees on data freshness in terms of AoI; whie with ζ = 1, the AoI-RT agorithm coincides with the greedy agorithm proposed in 6] AoI-Greedy) that optimizes AoI but provides no guarantees on timey throughput. We wi show that our poicy with 0 < ζ < 1 can provide guarantees on both timey-throughput and AoI. It is worth noting that β is a parameter reated to the ink preference towards the data freshness. Whie our age-based poicy is simiar to the reguar scheduer proposed in 10], where the weight of a ink is a combination of its congestion eve queue-ength) and its time-since-ast-service TSLS), there are three major differences: 1) the reguar scheduer strives to improve the service reguarity from the sender s perspective i.e., how often a ink is schedued) whie our proposed AoI-RT agorithm improves the data freshness from the receiver s point of view i.e., how often the packets are deivered to the receiver). Hence, guaranteeing TSLS is different from guaranteeing AoI; ) the reguar scheduer may serve a ink even if it is empty whie the AoI-RT agorithm aways serves non-empty inks, and since the reguar scheduer may serve a ink even if it is empty, it may ead to the waste of service; and 3) the reguar scheduer is not designed for rea-time traffic and therefore does not provide guarantees on timey throughput. N. Lu, B. Ji, and B. Li 3. Feasibiity Optimaity In this subsection, we wi show that the proposed AoI-RT agorithm is feasibiity-optima, i.e., providing guarantees on timey throughput of rea-time traffic. Proposition 3.1. The AoI-RT agorithm with any parameter 0 ζ < 1, is feasibiity-optima, i.e., for any arriva process that ies stricty within the maxima satisfiabe region Λγ, C), the AoI-RT agorithm stabiizes the system, i.e., im sup 1 1 V kt ]] B ϵ1 ζ ), 7) where B ζg L β + 1 ζ LA max+d max),g min{a max,tc max }, ϵ is some positive constant satisfying that λ + ϵ1 sti ies within the maxima satisfiabe region, and 1 is an L dimensiona vector of a ones. Proof. We appy the Lyapunov-drift anaysis to obtain the feasibiity optimaity. Consider the Lyapunov function W VkT ], RkT ]) 1 ζ V kt ] + ζg β R kt ], 8) where VkT ] V kt ]) L and RkT ] R kt ]) L. It is shown in Appendix A that there exists a positive constant ϵ > 0 satisfying λ + ϵ1 Λγ, C) such that ΔW kt ] W Vk + 1)T ], Rk + 1)T ]) W VkT ], RkT ]) VkT ], RkT ] ] ϵ 1 ζ )V kt ] + B. 9) Taking the expectation on the both sides of 9) and summing over k = 0, 1,..., 1, we have the desired resut. Proposition 3.1 estabishes the feasibiity optimaity of the AoI- RT agorithm. As we can see from 7), a the virtua queues are stabiized as ong as 0 ζ < 1. However, when ζ = 1, the sum of average virtua queue engths becomes unbounded, which impies the AoI-Greedy poicy is not abe to meet timey throughput requirements. In the foowing subsection as we as simuation experiments, we wi show that the proposed agorithm is aso capabe of improving the information freshness by reducing the AoI. 3.3 Data Freshness Guarantee We anayze the AoI performance of our proposed agorithm in this subsection. Specificay, we derive an upper bound on the network data freshness defined in Section under the AoI-RT agorithm, which demonstrates that the data freshness under our poicy can be guaranteed. We aso provide a ower bound on the network data freshness under genera poicies Upper Bound Anaysis. heo By Proposition 3.1, we have that V kt ] and R kt ] wi converge to V and R in distribution in the steady state, where V and R

5 Age-based Scheduing: Improving Data Freshness for Wireess Rea-Time Traffic Mobihoc 18, June 6 9, 018, Los Angees, CA, USA denote the virtua-queue ength and AoI of ink under the AoI- RT agorithm in the steady state, respectivey. Then, we have the foowing proposition regarding the AoI performance. Proposition 3.. The network data freshness under the AoI-RT agorithm by choosing β = ω λ 1 γ ), is upper bounded by the foowing: ω R ] G β 1 + ϵ ϵ) ζ 1 ) A + D ], 10) where ϵ > 0 satisfies λ + ϵ1 Λγ, C), and A and D have the same distribution as A kt ] and D kt ], respectivey for a. Reca that G min{a max,tc max }. Proof. See Appendix B for the detais. Note that when ω, λ, and γ are given for a inks, we can aways find β for the AoI-RT agorithm such that ω = β λ 1 γ ) hods. From Proposition 3., we can see that the considered network data freshness is upper bounded. The second term in 10) contains the second moment of the packet arriva processes and the channe fuctuations, showing that a better performance on data freshness in terms of AoI is dependent on ess variations over the wireess channe and packet arrivas. Athough a the aforementioned factors are non-controabe, the AoI-RT agorithm provides a way of tuning data freshness performance via a controabe parameter ζ. It can be seen that the performance of data freshness improves with the increase of ζ. It is worth noting that in the extreme case when ζ = 0, the AoI-RT agorithm reduces to the pure VQL-based scheduing agorithms, and the upper bound given by Proposition 3. goes to infinity. This impies that the data freshness is not guaranteed by the conventiona VQL-based scheduing poicy even though it is feasibiity-optima; whie in the extreme case when ζ = 1, the second term disappears such that the AoI-Greedy poicy yieds the best AoI performance Lower Bound Anaysis. heo Next, we derive a fundamenta ower bound on the network data freshness for a set of scheduing poicies Π that can stabiize the system. Proposition 3.3. Under any poicy π Π, the network data freshness is ower bounded by the foowing: ω R π ) ] 1 L ) ω ω )1 +, 11) Ω where R π ) denotes the AoI of ink under poicy π in the steady state, and Ω max {Sτ ]} T 1 : T 1 S τ ])>0 ω. Proof. See Appendix C for the detais. Speciay, for a fuy-connected non-fading network with ω = ω and A kt ] = A for a and k, and each frame of one time sot, Proposition 3.3 impies L ω R π ) ] 1 LL + 1)ω. Further, et per-ink timey throughput requirement λ 1 γ ) = 1 L1+ϵ) for a. From Proposition 3., when ζ = 1, L ω R ] ωgl. Therefore, the network data freshness has an order of ΘL ) under the AoI-Greedy poicy in the considered fuy connected network, which aso demonstrates the tightness of our upper and ower bounds on the network data freshness in this particuar case. 3.4 Convergence Speed of Timey Throughput In this subsection, we first estabish the foowing proposition on the convergence speed of timey throughput, i.e., how fast the running average of the instantaneous timey throughput converges to the target average timey throughput with respect to the observation window size the number of frames observed). Proposition 3.4. The convergence speed of timey throughput under the AoI-RT agorithm is upper bounded as foows. 1 1 FkT ] λ1 γ) 1 V 1 0] + 1 V T ]] + 1 ) ) Var A + Var D, where A, D have the same distribution as A kt ], D kt ], respectivey for a, 1 stands for the 1 norm, FkT ] F kt ]) L denotes the instantaneous throughput vector, and { k+1)t 1 } F kt ] min C kt ]S t], A kt ]. Proof. Reca the dynamics of virtua queues 1). We have 1 1 FkT ] λ1 γ) 1 1 = 1 FkT ] 1 1 AkT ] DkT ]) AkT ] DkT ]) λ1 γ) 1 = 1 ) V0] VT ] AkT ] DkT ]) λ1 γ) 1 1 V0] VT ] AkT ] DkT ]) λ1 γ) 1 = 1 V 0] + 1 V T ]] A kt ] D kt ]) λ 1 γ ). 1)

6 Mobihoc 18, June 6 9, 018, Los Angees, CA, USA N. Lu, B. Ji, and B. Li a) Improvement of network data freshness b) Timey throughput performance c) Data freshness vs. virtua-queue ength ρ = 0.1) Figure : 100-ink fuy connected network Note that each term in the ast summation term in 1) can be upper bounded as foows. 1 1 ) A kt ] D kt ]) λ 1 γ ) 1 1 A kt ] D kt ]) λ 1 γ )) 13) ) 1 1 = Var A kt ] D kt ]) = 1 1 Var A kt ] D kt ]) 14) = 1 ) )) Var A + Var D. The inequaity 13) is due to Jensen s inequaity and the convexity of quadratic form. The equaity 14) is due to the independence of A kt ] and D kt ] for a and k. Thus, 1 1 A kt ] D kt ]) λ 1 γ ) 1 ) ) Var A + Var D. 15) We have the desired resut by substituting 15) to 1). It is shown from Proposition 3.4 that the convergence speed of timey throughput depends on the virtua-queue ength, and the random variations of the packet arriva process and the departure process of the virtua queue. Given the observation window size, the increase of the tota expected virtua-queue ength sows down the convergence of timey throughput. Note that Proposition 3.1 aso provides an upper bound on the expected tota virtua-queue ength in the steady state, which increases ineary with contro parameter ζ. Whie from Proposition 3., it is known that we can improve the information freshness by increasing ζ. The trade-off between data freshness and the convergence speed of timey throughput is now cear: when increasing ζ under the AoI-RT scheduing agorithm, the network data freshness is improved at the cost of sowing down the convergence of timey throughput. The existence of this trade-off is due to the introduction of AoI in making scheduing decisions. It is interesting that we coud significanty improve the data freshness in terms of AoI if a ong convergence period is toerabe to the rea-time appications such as wireess sensor monitoring with soft timey throughput requirements 13]). We wi aso examine the trade-off in simuations to show how much AoI performance can be improved by sighty sowing down the convergence of timey throughput. 4 SIMULATION RSULTS In this section, we present two sets of simuations to demonstrate the performance of the proposed AoI-RT agorithm. In the first set of simuations, we consider a 100-ink fuy connected network e.g., ceuar networks). Particuary, among 100 inks, there are N 1 inks of type 1 each with a packet arriva Bernoui) rate of 0.3 N 1, and N inks of type each with an arriva Bernoui) rate of 0.3 N. Note that N 1 + N = 100. The ratio of N 1 to N 1 + N, denoted by ρ, can vary and we consider three different vaues of ρ:0.1, 0.3, and 0.5. It can be seen that ρ = 0.1 and ρ = 0.3 ead to the scenarios where two types of inks are of heterogeneous traffic; whie when ρ = 0.5, a inks have the same arriva rate. In addition, the frame has a ength of one time sot, and for a inks, γ = 0.5, β = 1, and C kt ] = 1 if schedued. The simuation parameters are seected carefuy such that the arriva process ies in the maxima satisfiabe region Λγ, C). The improvement of network data freshness in terms of AoI under our poicy 0 < ζ < 1), the VQL-based poicy ζ = 0), and the AoI-Greedy poicy ζ = 1) is shown in Fig. a). The y-axis gives the improvement ratio of the network data freshness under different vaues of ζ to the network data freshness when ζ = 0. As it can be seen, the performance of data freshness improves with the increase of ζ. From Fig. a), we can aso see that when the network presents certain traffic heterogeneity the arriva rate difference between two types), a better improvement ratio can be obtained. For exampe, when ρ = 0.1 and ζ = 1, the improvement ratio is about 0% compared to 15% when ρ = 0.5 no heterogeneity) and ζ = 1. This is because with traffic heterogeneity, it is easier for the scheduing agorithm to boost the AoI performance of inks with ower traffic by reducing the service to inks with higher traffic. The investigation of capabiity of meeting the timey throughput requirement is given in Fig. b), where the reative change is defined by the

7 Age-based Scheduing: Improving Data Freshness for Wireess Rea-Time Traffic Mobihoc 18, June 6 9, 018, Los Angees, CA, USA a) AoI histogram comparison ink 5) b) CCDF of AoI ink 5) c) Convergence speed of timey throughput Figure 3: 10-ink ad hoc network varying ζ is shown in Fig. 3c). As expected, when ζ = 0, the timey throughput has the fastest convergence to the requirement. However, if the data freshness is more important to appications as is often the case), we coud trade for data freshness by increasing ζ. Overa, the simuations demonstrate the abiity of the proposed AoI-RT poicy to improve the data freshness whie providing timey throughput guarantees. Again, the VQL-based poicy can ony meet the timey throughput requirement; whie the AoI-Greedy poicy can ony guarantee the data freshness at the receivers. Figure 4: Confict graph of the 10-ink ad hoc network difference between the average timey throughput of a ink and the timey throughput requirement) divided by the timey throughput requirement. It can be seen ceary that the reative change remains cose to zero when varying ζ, showing that our poicy can aso provide the guarantee on timey throughput whie improving AoI as shown in Fig. a)). However, when ζ = 1, the timey throughput requirement cannot be satisfied under the AoI-Greedy poicy. For exampe, when ρ = 0.1, the average timey throughput of type 1 inks is over 10% beow the requirement. Fig. c) shows as expected that with the increase of ζ, the network data freshness improves; whie the tota average virtua-queue ength increases i.e., the convergence of timey throughput sows down 3 ). In the second set of simuations, we consider a 10-ink ad hoc network in Fig. 4 with Bernoui arrivas: λ 5 = 0.1 and λ = 0.6 for a other. The frame has a ength of two time sots. For a inks, γ = 0., β = 1, and C kt ] is Bernoui with mean of 0.95 for a k. We first investigate the AoI performance of ink 5 with ower traffic). Fig. 3a) gives histograms of AoI over a the simuated time frames with normaized frequencies for ink 5 when ζ = 0 and ζ = , respectivey. It iustrates that the occurrence of arge vaues of AoI e.g., > 50 frames) becomes ess often when ζ = , indicating a better performance of data freshness. The compementary cumuative distribution function CCDF) of AoI is further shown in Fig. 3b), showing that when ζ = , the probabiity of seeing a arge vaue of AoI is much smaer comparing the case where ζ = 0 e.g., can be amost 10 times ess for a AoI vaue of 75). The convergence speed of timey throughput when 3 Due to the ack of space, we ony show the convergence of timey throughput in the second set of simuations, and both sets of simuations yied the same insight. 5 DISTRIBUTD IMPLMNTATION The proposed AoI-RT scheduing poicy is centraized such that a centraized entity shoud compute the scheduing decision for every frame gobay with necessary message passing. It is interesting to design ow-compexity distributed impementations of the AoI-RT poicy such that the scheduing decision can be made ocay at each ink with itte or without message passing. We wi consider the distributed impementation of our poicy as future works. However, we woud ike to provide some discussions here. In fuy connected networks, ony one ink can be schedued for transmission at a time, which significanty reduces the scheduing decision space. A greedy soution that seects the maximum weighted schedue for every sot coud be optima and an agorithm simiar to the FCSMA proposed in 9] coud be used to impement such a greedy soution. Whereas for genera ad hoc networks, the design of distributed agorithms coud be very chaenging. In 11], an asymptoticay optima agorithm is proposed for ad hoc networks with periodic packet arrivas and perfect channe conditions. Simiary, by utiizing the Markov approximation technique originay proposed in 1], we coud design distributed agorithms of the AoI-RT poicy for some specia cases of ad hoc networks, which are amenabe to ow-compexity impementations. 6 CONCLUSION In this work, we proposed the AoI-RT agorithm for scheduing rea-time traffic with hard deadines in ad hoc networks, which provides guarantees on both timey throughput and data freshness in terms of AoI. The agorithm takes the virtua-queue ength and AoI of each ink as driving factors in making scheduing decisions, which has been proved to be feasibiity-optima. We aso derived

8 Mobihoc 18, June 6 9, 018, Los Angees, CA, USA N. Lu, B. Ji, and B. Li an upper bound of the data freshness metric to show that the performance of data freshness can be guaranteed under our poicy. Particuary, we showed that the data freshness can be traded from the convergence speed of the timey throughput, which is done by tuning scheduing parameters. Simuation resuts further demonstrate the abiity of our poicy to improve the data freshness whie providing the timey throughput guarantee. Future works incude the distributed impementations of the proposed scheduing poicy. ACNOWLDGMNTS This work is supported by the Natura Sciences and ngineering Research Counci of Canada Discovery Grant: RGPIN ; and the foowing US Nationa Science Foundation grants: CNS , CNS , and CCF RFRNCS 1] Chen, M., Liew, S. C., Shao, Z., and ai, C. Markov approximation for combinatoria network optimization. I transactions on information theory 59,10 013), ] Costa, M., Codreanu, M., and phremides, A. On the age of information in status update systems with packet management. I Transactions on Information Theory 6, 4 016), ] Hou, I.-H. On the modeing and optimization of short-term performance for rea-time wireess networks. In Proc. of I INFOCOM San Francisco, CA, USA, Apri 016). 4] Hou, I.-H., Borkar, V., and umar, P. A theory of qos for wireess. In Proc. of I INFOCOM Rio de Janeiro, Brazi, Apri 009). 5] Jaramio, J. J., Srikant, R., and Ying, L. Scheduing for optima rate aocation in ad hoc networks with heterogeneous deay constraints. I Journa on Seected Areas in Communications 9, 5 011), ] adota, I., Uysa-Biyikogu,., Singh, R., and Modiano,. Minimizing the age of information in broadcast wireess networks. In Proc. of I Aerton Conference Monticeo, IL, USA, Octorber 016). 7] ang, X., Hou, I., Ying, L., et a. On the capacity requirement of argest-deficitfirst for scheduing rea-time traffic in wireess networks. In Proc. of ACM MobiHoc Hangzhou, China, June 015). 8] au, S., Yates, R., and Gruteser, M. Rea-time status: How often shoud one update? In Proc. of I INFOCOM Orando, FL, USA, Apri 01). 9] Li, B., and ryimaz, A. Optima distributed scheduing under time-varying conditions: A fast-csma agorithm with appications. I Transactions on Wireess Communications 1, 7 013), ] Li, B., Li, R., and ryimaz, A. Throughput-optima scheduing design with reguar service guarantees in wireess networks. I/ACM Transactions on Networking 3, 5 015), ] Lu, N., Li, B., Srikant, R., and Ying, L. Optima distributed scheduing of reatime traffic with hard deadines. In Proc. of I CDC Las Vegas, NV, USA, December 016). 1] Neey, M. J. Stochastic network optimization with appication to communication and queueing systems. Synthesis Lectures on Communication Networks 3, 1 010), ] Seno, L., Cena, G., Vaenzano, A., and Zunino, C. Bandwidth management for soft rea-time contro appications in industria wireess networks. I Transactions on Industria Informatics 13, 5 017), ] Sun, Y., Uysa-Biyikogu,., Yates, R., oksa, C.., and Shroff, N. B. Update or wait: How to keep your data fresh. I Transactions on Information Theory 63, ), A PROOF OF INQUALITY 9) The evoution of R kt ] 6) can be rewritten in a more compact form, i.e., R k + 1)T ] = R kt ]1 I H kt]) + 1 = R kt ] R kt ]I H kt] ) To make our proof more concise, we omit the time index kt and use x + to denote xk + 1)T ] for any x. Hence, 16) becomes R + = R R I H ) Mutipying β on both sides of 16), and summing over a inks, we have β R + = β R β R I H + β. 18) We are now ready to prove the inequaity 9). Indeed, we have ΔW W V +, R + ) WV, R) V, R] 1 ζ = V + ) + ζg β R + 1 ζ 1 ζ V ζg ] β R V, R V + I D ) V V, R] ] + ζg β R + β R V, R, 19) where the ast step foows from the dynamics of virtua queues 1) and the fact that max{, 0}) ). By pugging 18) into 19), we have ΔW 1 ζ V + I D ) V V, R] ] + ζg β β R I H V, R = 1 ζ ) V I D ) + 1 ] I D ) V, R ] + ζg β ζg β R I H V, R ] 1 ζ ) λ 1 γ )V + B ζg β R I H V, R k+1)t 1 1 ζ )V min C S t], A V, R, 0) where the ast step is true for B that is defined in Proposition 3.1 and uses the fact that I denotes the number of dropped packets see )). Note that H is the event that at east one packet is deivered at ink in frame k. WeuseG min{a max,tc max } to represent the maximum number of packets that can be deivered within one frame, and therefore we have ] β R I H V, R 1 G β R min { k+1)t 1 } ] C S t], A V, R. 1)

9 Age-based Scheduing: Improving Data Freshness for Wireess Rea-Time Traffic Mobihoc 18, June 6 9, 018, Los Angees, CA, USA By pugging 1) into 0), we have ΔW 1 ζ ) λ 1 γ )V + B k+1)t 1 1 ζ )V + ζβ R ) min C S t], A V, R. ) Since the arriva process is stricty within the maxima satisfiabe region, there exists an ϵ > 0 such that λ 1 γ ) ϵ + P A a) P C c) a c s 0,s 1,...,s T 1 S { T 1 αa, c; s 0, s 1,...,s T 1 ) min c s τ,, a },. 3) τ =1 Therefore, we have 1 ζ ) λ 1 γ )V ϵ 1 ζ )V + 1 ζ )V P A a) P C c) a c { T αa, c; s 0, s 1,...,s T 1 ) min c s τ,, a s 0,s 1,...,s T 1 S τ =1 4) Next, we focus on the second term of the right-hand-side of the above inequaity. 1 ζ )V P A a) P C c) a c { T αa, c; s 0, s 1,...,s T 1 ) min c s τ,, a s 0,s 1,...,s T 1 S τ =1 P A a) P C c) αa, c; s 0, s 1,...,s T 1 ) a c s 0,s 1,...,s T 1 S ) { T 1 ζ )V + ζβ R min c s τ,, a τ =1 P A a) P C c) a c ) { k+1)t max 1 ζ )V + ζβ R min C S t], A St] S,,...,k+1)T 1 k+1)t 1 = 1 ζ )V + ζβ R ) min C S t], A V, R, 5) where the second ast step foows from the definition of the AoI-RT agorithm. By combining 4) and 5) and substituting them into ),we have the desired resut. B PROOF OF PROPOSITION 3. We consider the foowing quadratic Lyapunov function:w V, R) 1 ζ L V. Then, we have ΔW V, R) = W V +, R + ) WV, R) V, R] ] 1 ζ = V + ) 1 ζ V V, R 1 ζ V + I D ) V ] V, R 1 ζ ) V A D ) V, R] + 1 ζ I + D ] V, R { k+1)t ] 1 ζ )V min C S t], A V, R. Taking expectation on both sides with respect to the steady-state distribution of V, R), i.e., V, R), and using the fact that ΔW V, R)] = 0, we have 0 1 ζ ) λ 1 γ ) V ] + 1 ζ A + D ] 1 ζ )V T min {A ],C S τ,, which impies 1 ζ )V T min {A ],C S τ, 1 ζ ) λ 1 γ ) V ] + 1 ζ A + D ]. By adding ζ { }] L β R min A,C T 1 S τ, on both sides, we have { 1 ζ )V + ζβ R T ) 1 } ] min A,C S τ, 1 ζ ) λ 1 γ ) V ] + 1 ζ A + D ] + ζ β R T ] min {A,C S τ,. 6) Since the arriva process is stricty within the maxima satisfiabe region Λγ, C) and the optima soution aways occurs at extreme points in inear programming, under our proposed AoI-RT agorithm, we have 1 + ϵ)λ 1 γ )1 ζ )V + ζβ R ) k+1)t 1 ) ] ζ )V + ζβ R min {C S t], A V, R.

10 Mobihoc 18, June 6 9, 018, Los Angees, CA, USA N. Lu, B. Ji, and B. Li Then, taking expectation on both sides of above inequaity with respect to the steady-state distribution of V, R), we have 1 + ϵ)λ 1 γ )1 ζ )V + ζβ R ) { 1 ζ )V + ζβ R T ) 1 } ] min A,C S τ,. 7) By seecting ω = β λ 1 γ ), and pugging 7) into 6), we have ω R ] = β λ 1 γ ) R ] ϵ a) G 1 + ϵ b) = G 1 + ϵ β R T min {A ],C S τ, + 1 ζ A + D ] ζ 1 + ϵ) β R I H ] + β + 1 ζ ζ 1 + ϵ) ϵ) ζ 1 ) A + D ] A + D ], where step a) uses the fact that G denotes the maximum number of packets that can be deivered within one frame; and b) uses the foowing equaity: β R I H ] = β, 8) which is from taking expectation on both sides of 18) with respect to the steady-state distribution of R. C PROOF OF PROPOSITION 3.3 For any poicy π, given H π ) { { : min A,C T 1 Sπ ) } τ, we have π ) ω R = ω ω R π ) H π ) H π ) } > 0, ω ω R π ) ), H π ) H π ) which is due to Cauchy-Schwarz inequaity. This impies ω R π ) ) H π ) ω R π ) ) H π ) H π ) ω. 9) Mutipying ω on both sides of 16), summing over a inks, and taking expectation on both sides, we have π ) ω R = ω. 30) H π ) Taking expectation on both sides of 9) and pugging in 30),we have ω R π ) ) H π ) ω R π ) ) H π ) H π ) ω H π ) ω R π ) ]) L ) ω ] = ], 31) H π ) ω H π ) ω where the second inequaity is due to Jensen s inequaity and the fact that дa,b) = a b is convex. Next, we obtain the foowing regarding the quadratic term of R + from 17), ) ) ω R + = ω R R I H + 1 = ω R + ω R I H ω R I H + ω R ω R I H + ω = ω R ω R + ω R ω R + ω, H H 3) where H { : I H = 1}. Taking expectation on both sides of 3) with respect to the steady-state distribution of V, R), we have ω R π ) ] π ) = ω R + ω R π ) ) ω. H π ) H π ) After pugging in 30), we have ω R π ) ] = 1 ω R π ) ) + 1 ω. 33) H π ) By pugging 31) into 33), we have ω R π ) ] ) 1 L ω ω + ] 1. 34) H π ) ω ] Since H π ) ω is upper bounded by the maximum of the summation of ω over a schedued inks over a frame regardess of scheduing poicies, i.e., ω max ω, 35) H π ) {Sτ ]} T 1 : T 1 S τ ])>0 the proposition hods.