Multicast With Prioritized Delivery: How Fresh is Your Data?

Transcription

1 Multicast With Prioritized Delivery: How Fresh is Your Data? Jing Zhong, Roy D Yates and Emina Solanin Department of ECE, Rutgers University, {ingzhong, ryates, eminasolanin}@rutgersedu arxiv:885738v [csit 7 Aug 8 Abstract We consider a multicast networ in which real-time status updates generated by a source are replicated and sent to multiple interested receiving nodes through independent lins The receiving nodes are divided into two groups: one priority group consists of nodes that require the reception of every update pacet, the other non-priority group consists of all other nodes without the delivery requirement Using age of information as a freshness metric, we analyze the time-averaged age at both priority and non-priority nodes For shifted-exponential lin delay distributions, the average age at a priority node is lower than that at a non-priority node due to the delivery guarantee However, this advantage for priority nodes disappears if the lin delay is exponential distributed Both groups of nodes have the same time-averaged age, which implies that the guaranteed delivery of updates has no effect the time-averaged freshness I INTRODUCTION The analysis of information freshness arises from a variety of real-time status updating systems, in which update messages generated by the sources are sent to interested receivers through a communication system For instance, the real-time information updates of autonomous cars are broadcast to nearby vehicles and infrastructures Similarly, live video captured for remote surgery is required to be available at the doctor with ultra low delay In these systems, the nowledge of the source state at the receiver is desired to be as fresh as possible This leads to the introduction and analysis of an Age of Information (AoI) freshness metric [ [8 Age of information, or simply age, measures the time difference between now and when the most recent update was generated At any time t, if the most recent update at the receiver is generated at time u(t), then the instantaneous age at the receiver is t u(t) In early wor on age analysis [, it was shown that the source should limit its update rate in order to avoid queueing delay caused by overloading the system with first-come firstserved (FCFS) policy Given the observation of unnecessary waiting in FCFS systems, subsequent research looed at lastcome first-served (LCFS) queueing systems that discard older updates as soon as a new update comes [3, [5, and lastgenerated first-server (LGFS) policy with preemption in service for multihop networs [6 When the source has no nowledge of the service system state, allowing pacet preemption at the queue provides lower average age in general However, these results are limited to the case where the update arrival process is given In [4, the authors consider a different scenario where the system state is available at the source such that a new update is generated only after the service of the previous update is completed A lazy updating scheme is proved to be age-optimal, indicating that the source should wait for a short period before sending a new update if the service time of the previous update is too small The analysis in [4 applies only to systems without preemption In this wor, we consider an update multicast system in which real-time update messages generated by the source are broadcast to a set of nodes through iid lins with random networ delays The receiving nodes are categorized into two groups The priority group consists of nodes that require the delivery of every update, while all other nodes without the delivery requirement are regarded as the non-priority group Once a node receives an entire update message, it acnowledges the source by sending instantaneous feedbac This model arises in a variety of delay-sensitive applications, eg vehicle networs where the update messages are popular and simultaneously request by large numbers of users Some receiving nodes require the history of all updates for the purpose of data aggregation and processing; thus the delivery of every update message is crucial Our wor is closely related to the LCFS and LGFS systems with preemption and state-dependent updating Since any new update generated by the source leads to the termination of the previous update, each lin is equivalent to a LCFS queue with preemption in service Thus, the instantaneous feedbac enables the source to submit new updates based on the state of the queue, either replacing staled updates with a fresh update or waiting for the service of the current update to be completed We assume each update pacet is divided into small chuns and encoded with a rateless code to overcome channel erasure in the multicast networ In this case, the number of chuns corresponding to an update is required to reach a certain minimum level for the update to be successfully decoded Moreover, the source is also able to instantaneously terminate an update in the middle of transmission Here we consider a simple updating scheme that exploits instantaneous feedbac Once the current update is delivered to all the nodes in the priority group, the source terminates the transmission of the current update and broadcasts a fresh update Our goal here is to evaluate the average age for both types of nodes II PROBLEM FORMULATION We consider a status updating system with a single source broadcasting time-stamped updates to multiple nodes through independent lins with random delays, as shown in Fig Each update message is time-stamped when it is generated at the source, and it taes time X i to be successfully delivered to

2 Source + Node Node Node Node + Fig : System diagram: source broadcasts status updates to n nodes through iid channels The nodes in the priority group are shaded The transmission of update + is initiated only after update is delivered to all nodes in the group node i The priority group consists of nodes,,, and the source guarantees the delivery of every update to all of these priority nodes In this wor, we assume there is an instantaneous feedbac channel from every node i bac to the source, and node i acnowledges the source instantly as soon as the update is delivered to the node i When all nodes in the priority group report receiving the update, this update is considered completed and the transmissions of this update to all other nodes are terminated The source immediately generates the next update + and repeats the multicast process When most recently received update at time t at node i is time-stamped at time u i (t), the status update age or simply the age, is the random process i (t) t u i (t) When an update reaches node i, u i (t) is advanced to the timestamp of the new update message The time average of age process at a node, which is also called the age of information, is defined as i lim τ τ τ i (t) () In this wor, we derive the average age at both the priority node and the non-priority nodes, and we will show that the average age depends on the order statistics of the random lin delay X i Definition The -th order statistic of random variables X,,X n, denoted X :n, is the -th smallest variable For shifted exponential X with CDF F X (x) e λ(x c), X :n has expectation and variance µ :n E[X :n c+ λ (H n H n ), (a) σ :n Var[X :n λ ( Hn H (n ) ), (b) where H n and H n are the generalized harmonic numbers defined as H n n and H n n III PRIORITY NODES We start by evaluating the average age at a single node in the priority group Fig depicts a sample path of the age over time at some node i in priority group with nodes Update begins transmission at time t and is timestamped T Here we remar that X i is the service time to deliver the P (t) Y A i T X i X i X i Y A i T T Fig : Sample path of the age P (t) for node i within the priority group with nodes Update delivery instances are mared by update to node i Since the X i are iid for all i and, the i (t) processes are statistically identical and each node i has the same average age i If one node gets an update earlier than any of the other nodes, it has to wait for an idle period until that update is delivered to all priority nodes The transmission time of an update to all nodes, which we call a service interval, is given by Y max(x,,x ) X : (3) We denote that update goes into service at time T and gets delivered to all nodes at time T T + Y Using similar techniques as in [9, we represent the area under the age sawtooth as the concatenation of the polygonsa i,,a i, thus the average age is P lim J J J A i J lim J J Y i It follows from Fig that E[A E[Y (4) A i Y X i +X i/+x i (Y X i )+(Y X i ) / Y X i +Y / (5) Since X i is independent of the transmission time Y of the previous update, E[A E[YE[X+E [ Y / (6) Denoting µ E[X, (), (4) and (6) yield the next theorem Theorem The average age at an individual node in the priority group is P µ+ µ : + σ : µ : Note that Theorem is valid for any distribution of X In terms of the Euler-Mascheroni constant γ 577, we also have the following result Corollary For shifted exponential (λ, c) service time X, the average age at an individual node in the priority group is lower bounded by P 3c + λ log +γ +, (7) λ t

3 E (t) Y Y + Y + T T T T + T + T +3 Fig 3: Sample path of the age E(t) in the non-priority group: successful update deliveries (at times mared by ) occur in intervals,,, and +3 Updates to the node are terminated in intervals + and + Proof appears in the Appendix Corollary indicates that the average age in the priority group P is independent of the number of nodes n in the system, and it behaves almost lie a logarithmic function as the number of nodes increases W l A l IV NON-PRIORITY NODES For a node in the non-priority group, the transmission of the current update is terminated right after the delivery of the update to all the nodes in the priority group That is, a nonpriority node i fails to receive the update if and only if the service time X i is larger than the service times of all the nodes in the priority group Let s further denote that the priority nodes have service times X,X,,X, and the non-priority node has service time X + For iid service time X, the probability that X + is the largest among all + nodes is simply q /( + ), since the ran of X + among + random variables is uniform from to + Here we also refer to q as the failure probability for a non-priority node If update is not delivered to a node i, the node waits for a service interval Y until the source generates the next update Suppose an update is delivered to node i during service interval and the next successful update delivery to node i is in service interval +M l In this case, M l is a geometric rv with probability mass function (PMF) P M (m) q m ( q), and first and second moments E[M q +, (8) E [ M +q ( +)( +) ( q) (9) We remar that M l and Y are independent An example of the age process is shown in Figure 3 The update is delivered in service interval with end time T, and the node + waits for M l 3 service intervals until the next successful delivery in interval + 3 We represent the shaded trapezoid area as A l and the length in time between two service t intervals with successful updates as W l +M l Y The time-averaged age for non-priority node is then E lim L L L l A l L lim L L l W E[A l E[W () Denote the random variable as the service time of a successful update sent to a non-priority node with CDF F (x) F X X <Y (x) Evaluating Fig 3, we have A l ( W l + ) W l + W l () Since and W are independent, the average age in () is E ( E[W E[ W [ ) +E E[W E[ W E[W +E [ () We first define Y S as the length of a service interval given that the update is successfully delivered to the non-priority node +, and Y F as the length of a service interval given that the update is failed to be delivered Thus, Y S and Y F have PDFs f YS (y) f Y Y X+ (y) and f YF (y) f Y Y<X+ (y), respectively Furthermore, E[Y qe[y F +( q)e[y S Lemma W has first and second moments E[W E[ME[Y, (3) E [ W (E[M )Var[Y F +Var[Y S + ( E [ M E[M+ ) (E[Y F ) +(E[Y S ) +(E[M )E[Y F E[Y S (4) Proof of the lemma appears in the Appendix Lemma leads to the following result for non-priority nodes Theorem The average age at an individual node in the nonpriority group is where we denote E µ i:+ +δ ()+δ (), δ () σ :+ +σ +:+ ( +)µ : + δ () µ :+ +µ +:+ +µ +:+µ :+ ( +)µ : Proof In (), indicates the service time of a non-priority node + given that X + < max(x,,x ) This condition implies X + cannot be the largest among all + nodes Thus, E[ E[X + X + < X +:+ µ i:+ (5) The claim follows by substituting (3), (4) and (5) bac into (), and replacing E[M and E [ M by (8) and (9)

4 4 3 4 non-priority priority Fig 5: Average age versus the exponential shift c with λ The priority group size (a) exponential X (b) shifted exponential X with c Fig 4: Average age versus the priority group size circle mars the priority group and cross mars the non-priority group The lower bound for priority group is shown as dashed line For exponential service times, Theorems and yield the next claim Theorem 3 For exponential service time X, the average age is the same for both priority and non-priority nodes and is given by E P λ + H λ + H λh, (6) where is the priority group size Theorem 3 implies that the average age is identical for both groups regardless of whether an update is delivered to a node or not V EVALUATION Figures 4a and 4b compare the simulation results of the average age for the priority group P and the non-priority group E as a function of the priority group size In Fig 4a, the lin delay to every node i is exponentially distributed with different λ The average age curves for both groups overlap with each other and increases monotonically, which matches Theorem 3 The lower bound on the average age for the priority group in Corollary captures the trend for varying, and becomes tighter for sufficiently large Fig 4b shows the similar result for shifted exponential delay with c For small group size, there is a significant difference between the average age for two groups As increases, the age for non-priority group E decreases slightly in the beginning and climbs up after a certain We also observe that the age difference between two groups vanishes for large enough Fig 5 depicts the average age as a function of the shift parameter c for shifted exponential delay X In Fig 5 with exponential rate λ, both groups have almost linear increasing average age for different the constant shift c The two curves start at the same point for c, and the difference in slopes leads to a larger gap between two curves as c increases VI CONCLUSION In this wor, we examine a status updating multicast networ where the receivers are prioritized in terms of pacet deliveries The average age at each receiver depends on the order statistics of the random lin service time If the service time is identically distributed as exponential for every lin, we show analytically and numerically that the average age at a priority node with pacet delivery guarantee is the same as that without the guarantee The difference between two types of nodes arises if the exponential service time is mixed with a non-zero constant time shift The analysis in this wor is limited to exponential class service time, but we believe the difference between two types of nodes is related to the hazard rate of the service distribution, which potentially determines when the source should preempt the service of the current update with a fresh update APPENDIX Proof Corollary Substituting (a) and (b) into Theorem gives P 3c + λ + H λ + H λ c+λh (7) Note that H is monotonically increasing for n Z + and lim H π /6 Thus, given λ and c, lim H λ c+λh (8) The harmonic number is given asymptotically by H log+ γ +O( ), which can be lower bounded by H log +γ, for Z > (9) Thus, (7) is given by substituting (8) and (9) into (7)

5 Proof Lemma We note the sequence Y,,Y +Ml and the number of summation terms M l are dependent Since M l is geometric, the event M l m indicates a sequence of m consecutive failures followed by a success Thus,Y is identical to Y F for {,, +M l } and the last variable in the sequence Y +Ml is identical to Y S This implies [ m M E[W P M (m)e Y i m ( ) P M (m) (m )E[Y F +E[Y S E[Y F (E[M )+E[Y S () Substituting (8) into () yields E[W + ( + E[Y F+ ) + E[Y S E[ME[Y () For the second moment, we write E [ W in total expectation as E [ [ W ( m ) M P M (m)e Y i m [ m ( [ m ) P M (m) Var Y i + E Y i Since the random variables Y i are independent, we let [ m ω P M (m)var Y i ( ) P M (m) (m )Var[Y F +Var[Y S () ( ) Var[Y F E[M +Var[Y S (3) Similarly, we have ( [ m ) ω P M (m) E Y i ( P M (m) (m )E[Y F +E[Y S ( E [ M E[M+ ) (E[Y F ) +(E[M )E[Y F E[Y S +(E[Y S ) (4) The claim follows by substituting (3) and (4) in () Proof Theorem 3 For priority nodes, we obtain the average age by substituting (a) and (b) to Theorem with c, which directly yields (6) For non-priority nodes, the first term in Theorem is δ () µ i:+ (5a) ) H + λ H + H + i λ λ H i (5b) (5c) H + + λ λ (H + ) (5d) λ + λ H + λ (5e) In (5d), we use the series identity of Harmonic numbers H i ( + )(H + ) Similarly, substituting (a) and (b) into δ () and δ () gives δ () (H (+) )+H (+) ( +)λh H λh λ( +), (6) H δ () + (H + ) H+ + ( +) λh ( +) λh + + (H + )H + λh (7) We note that δ () in (5e) and δ () in (7) only contain first order harmonic numbers, thus we combine two terms and rewrite H + H +/(+), which gives δ ()+δ () λ + H λ + λ( +) (8) H The claim is given by the sum of (6) and (8) ACKNOWLEDGMENT Part of this research is based upon wor supported by the National Science Foundation under grants CNS-4988 and and CCF-774 REFERENCES [ S Kaul, R D Yates, and M Gruteser, Real-time status: How often should one update? in Proc INFOCOM, Apr, pp [ C Kam, S Kompella, and A Ephremides, Age of information under random updates, in Proc IEEE Int Symp Inform Theory, Jul 3, pp 66 7 [3 M Costa, M Codreanu, and A Ephremides, Age of information with pacet management, in Proc IEEE Int Symp Inform Theory, 4, pp [4 Y Sun, E Uysal-Biyioglu, R Yates, C E Kosal, and N B Shroff, Update or wait: How to eep your data fresh, in Proc INFOCOM, 6 [5 E Nam and R Nasser, Age of information: The gamma awaening, in Proc IEEE Int Symp Inform Theory, 6, pp [Online Available: [6 A M Bedewy, Y Sun, and N B Shroff, Optimizing data freshness, throughput, and delay in multi-server information-update systems Proc IEEE Int Symp Inform Theory, 6 [7 I Kadota, E Uysal-Biyioglu, R Singh, and E Modiano, Minimizing the Age of Information in broadcast wireless networs Proc Allerton Conf on Commun, Control and Computing, pp , 6 [8 R D Yates, E Nam, E Solanin, and J Zhong, Timely updates over an erasure channel, in Proc IEEE Int Symp Inform Theory, 7 [9 J Zhong, E Solanin, and R D Yates, Status updates through multicast networs, in Proc Allerton Conf on Commun, Control and Computing, 7, pp