AAge-of-Information in the Presence of Error
Kun Chen
IIIS, Tsinghua [email protected]
Longbo Huang
IIIS, Tsinghua [email protected]
Abstract —We consider the peak age-of-information (PAoI)in an
M/M/ queueing system with packet delivery error,i.e., update packets can get lost during transmissions to theirdestination. We focus on two types of policies, one is to adoptLast-Come-First-Served (LCFS) scheduling, and the other is toutilize retransmissions, i.e., keep transmitting the most recentpacket. Both policies can effectively avoid the queueing delayof a busy channel and ensure a small PAoI. Exact PAoIexpressions under both policies with different error probabilitiesare derived, including First-Come-First-Served (FCFS), LCFSwith preemptive priority, LCFS with non-preemptive priority,Retransmission with preemptive priority, and Retransmissionwith non-preemptive priority. Numerical results obtained fromanalysis and simulation are presented to validate our results. I. I
NTRODUCTION
Many information systems work in such a mode that statusupdates are first collected from a time-varying environment,and then control decisions are made based on these infor-mation. Examples include sensor networks for large-scalemonitoring [1], vehicular networks where vehicle positionand velocity information are disseminated to assist safe andintelligent transportation [2], and wireless networks wherescheduling is carried out based on channel state information[3]. A key to these systems is to ensure timely delivery of statusupdates , since out-of-date information can lead to incorrectsystem status estimation and result in severe performance loss.
Age-of-information (AoI), first proposed in [4], provides ameasure for the “freshness” of the current status information,and is an important metric for measuring quality-of-service(QoS) of a system. Different from typical performance metricssuch as delay or throughput, AoI jointly captures the latency intransmitting updates and the rate at which they are delivered.There have been various recent works on understandingAoI. [4] analyzes AoI for queueing models including
M/M/ , M/D/ and D/M/ . A more complicated case with multipleupdate sources is analyzed in [5]. [6] studies AoI in a Last-Come-First-Served (LCFS) M/M/ queueing system withor without preemption. The case when the destination mayreceive out-of-order packets is considered in [7]. In [8], theauthors introduce a notion peak age-of-information (PAoI) andconsider systems with packet management, i.e., the queue canchoose to only keep a subset of update packets. AoI in amulti-class M/G/ queueing system is studied in [9]. In [10],the authors study optimal update scheduling in a discrete-timemulti-source system. The optimal update generating policy isexplored in [11]. We notice that one common assumption made in mostaforementioned works is that update packet delivery is alwaysperfect, and AoI has been investigated mostly under the First-Come-First-Served (FCFS) principle. An exception is [6],which studies AoI under the LCFS principle, but also assumesperfect packet delivery. However, in practical systems, packettransmissions often contain errors and losses, e.g., due tointerference or buffer overflow at intermediate routers in amulti-hop network. To study the impact of such delivery errorson AoI, in this paper, we focus on an M/M/ queueing modelwhere each packet, upon service completion, arrives at thedestination with a nonzero probability. Our model captures (i)the queueing effect, which approximates the process whereupdate packets are sent over a channel or a network andcan cause congestion (This is different from [10], which alsoconsiders transmission errors), and (ii) the error component,which models the fact that update packets can get lost duringthe delivery process.We first focus on the LCFS service principle and derivethe exact PAoI for both the systems with preemptive priorityand non-preemptive priority. Intuitively, LCFS is good for tworeasons. (i) Compared to packet management schemes, e.g,[8], LCFS similarly avoids delaying new update packets withqueueing by letting them go first. This results in significantreduction of AoI compared to FCFS, especially when thechannel utilization is high. (ii) When there are errors in packettransmissions, packet management schemes can suffer severelydue to the lack of updates to deliver, while LCFS still ensuresa good delivery rate and does not affect AoI significantly.Next we analyze the PAoI under retransmission schemes.Here we do not assume feedback, since retransmissions basedon feedback may suffer from waste of time waiting forfeedback, or interference between update packets and feedbackinformation. Thus, the Retransmission policies refer to keeptransmitting the most recent packet repeatedly until a newpacket arrives. Compared to LCFS, retransmission policieshave an advantage of always transmitting the most recentupdates, at the cost of additional packet state management.We also derive the exact PAoI expressions for retransmissionwith or without preemption.In this work LCFS and Retransmission policies are bothstudied to cover various scenarios. Although utilizing retrans-missions is expected to contribute to a small AoI, it does notapply to scenarios where transmissions are not guaranteed,e.g., UDP and some wireless sensor networks. The rest ofthe paper is organized as follows. In Section II we introduce a r X i v : . [ c s . PF ] M a y he model. In Sections III, IV, V, VI and VII we presentour analysis for the FCFS, LCFS preemptive, LCFS non-preemptive, Retransmission preemptive and Retransmissionnon-preemptive cases. In Section VIII we present numericalresults. We conclude the paper in Section IX.II. S YSTEM M ODEL
We consider a system where a source transmits updates(packets) to a remote destination through a queue. The sourcegenerates packets according to a Poisson process with rate λ .The service time for each packet is exponentially distributedwith service rate µ .Different from previous works, we assume that upon servicecompletion, each packet arrives at the destination indepen-dently with probability p ∈ [0 , . Such a system is modeledby an M/M/ queueing system with packet loss, as shown inFig. 1. The packet loss model captures real-world situationswhere update packets can get lost during delivery to theirdestination, e.g., interference or buffer overflow, and has notyet been studied. source destinationpqueue 1-p Fig. 1. The packet delivery process in a queueing system with packet loss.
We study the peak age-of-information (PAoI) in this system,which is defined as follows. Suppose each update packet hasa time-stamp, marking its generation time. Denote the time-stamp of the most recently received update at time t as δ ( t ) .Then, the status age is defined as [4]: ∆( t ) (cid:44) t − δ ( t ) , and the set of peak ages is defined as: { ∆( t i ) |∃ (cid:15) > s.t. ∀ t ∈ ( t i − (cid:15) ) ∪ ( t i + (cid:15) ) , ∆( t ) < ∆( t i ) } . Then, PAoI [8] is defined to be: A P (cid:44) lim I →∞ I I (cid:88) i =1 ∆ i = E { ∆ i } , where ∆ i = ∆( t i ) is the i -th peak of ∆( t ) (See Fig. 2). Thelast equality follows from the ergodicity of ∆ i . As shown in[9], PAoI is closely related to the average AoI, but is muchmore tractable.We first introduce some useful definitions. Denote N theset of all packets, according to the order in which they arrive.For a packet n , denote a ( n ) its arrival time, d ( n ) its departuretime, and u ( n ) the time it starts to receive service. Let Φ denote the set of all successfully transmitted packets. Underthe LCFS service discipline, a successfully transmitted packetmay be outdated when arriving at the destination. Thus, wefurther define the set of informative packets Ψ as: Ψ (cid:44) { n ∈ Φ | d ( n ) − a ( n ) < ∆( d ( n )) } . That is,
Ψ = { n , n , . . . , n i , . . . } contains the packets whichoffer new information (so the system age decreases) when theyreach the destination. Regarding the evolution of the system, we define the fol-lowing random variables: X n (cid:44) a ( n + 1) − a ( n ) ,W n (cid:44) u ( n ) − a ( n ) ,S n (cid:44) d ( n ) − u ( n ) , i.e., X n is the inter-arrival time between n and n + 1 ; W n isthe waiting time of n ; S n is the “service time” of n . Note thatin the LCFS with preemptive priority case, S n may includeservice time of later packets if n is preempted by other packets.III. PA O I UNDER
FCFSFor the basic First-Come-First-Served (FCFS) case withpacket loss, PAoI can be easily obtained. Define the firstinformative packet which arrives no earlier that n as α ( n ) (cid:44) min { n i | n i ∈ Ψ , a ( n i ) ≥ a ( n ) } . Moreover, define the inter-arrival time between n and α ( n ) as ˆ X n (cid:44) a ( α ( n )) − a ( n ) , and define ˆ S n as the time duration from the moment n startsto receive service to the moment α ( n ) departs, i.e., ˆ S n (cid:44) d ( α ( n )) − u ( n ) . Note that if n ∈ Ψ , we have α ( n ) = n , ˆ X n = 0 and ˆ S n = S n .Since Φ = Ψ under FCFS, PAoI is composed of the (expected)inter-arrival time of two successfully transmitted packets, plusthe time a packet spends in the system. Thus, A F CF SP = E { X n i + ˆ X n i +1 + W n i +1 + S n i +1 | n i , n i +1 ∈ Φ } , = 1 λ + E { ˆ X n i +1 } + 1 µ − λ , where n i + 1 is n i ’s next packet and n i +1 is n i ’s next packetin Φ . Moreover, E { ˆ X n i +1 } = p E { ˆ X n i +1 | n i + 1 ∈ Φ } +(1 − p ) E { X n i +1 + ˆ X n i +2 | n i + 1 / ∈ Φ } = 0 + (1 − p )( 1 λ + E { ˆ X n i +1 } ) , where we have used E { ˆ X n i +1 } = E { ˆ X n i +2 } . Thus, E { ˆ X n i +1 } = 1 − ppλ , which implies: A F CF SP = 1 pλ + 1 µ − λ . (1)However, the FCFS policy, as discussed above, can suffer fromtraffic congestion, under which each packet will take a longtime to get through the queue and the PAoI can be poor. Thus,in this work, we focus on the Last-Come-First-Served (LCFS)as well as Retransmission policies and consider the followingtwo scheduling schemes.1) Preemptive priority: If a new packet arrives while theserver is busy, it preempts the current packet and startsservice immediately.2) Non-preemptive priority: The server always completesthe current packet and then starts serving the most recentpacket in the queue.The reasons to focus on PAoI with LCFS are as follows:(i) Intuitively, letting later packets go earlier should make ( t ) a(1) d(1) a(2) d(2)d(3)a(3) X ˆ S S ˆ Y t d(4)a(4) (a) Preemptive ( t ) a(1) a(2) a(3) d(1) d(3) d(2) t ˆ Z W d(4)a(4) ˆ Z (b) Non-preemptiveFig. 2. Evolution of status age in the LCFS M/M/ system. PAoI is dividedin different ways under the preemptive and non-preemptive cases. the status at the destination fresher. Hence, the PAoI will bemuch smaller. (ii) Compared to packet management schemeswhere packets are often dropped for queue size reduction, e.g.,[8], LCFS still transmits all packets. Thus, in the case whenerrors can occur and packets can get lost, LCFS ensures thatthe destination still gets updates more regularly, maintaininga lower level of PAoI. Note that characterizing PAoI underLCFS, even with perfect delivery, is nontrivial and has notbeen studied before, especially for the non-preemptive case.We also analyze the PAoI under Retransmission policies,which have an advantage over LCFS, as they always deliverthe most latest information. On the other hand, retransmissionrequires additional packet state management.IV. PA O I UNDER
LCFS
WITH P REEMPTIVE P RIORITY
We begin with LCFS with preemptive priority. Note thatin this case, u ( n ) = a ( n ) , ∀ n , i.e., packets get served im-mediately upon arrival. Moreover, in this case { ˆ S n } n arestatistically the same. As shown in Fig. 2(a), PAoI is theelapsed time from the moment when a packet n i ∈ Ψ arrives, until the moment when n i +1 ∈ Ψ departs (recallthat Ψ denotes the set of informative packets). Define the firstinformative packet which arrives after n ’s departure as β ( n ) (cid:44) min { n i | n i ∈ Ψ , a ( n i ) > d ( n ) } , and the inter-departure time between n and β ( n ) : ˆ Y n (cid:44) d ( β ( n )) − d ( n ) . Since the packets arriving after a ( n i ) but before d ( n i ) preempt n i and get lost upon departure (because n i ∈ Ψ ), we have (seeFig. 2(a)): A LCF S,preP = E { S n i + ˆ Y n i | n i ∈ Ψ } . (2) A. Analyzing a Service Process
Here we use S n to also denote the process of serving apacket n . For simplicity, we define the following symbols(notice that in other sections these symbols may have differentdefinitions): ˜ p (cid:44) P ( ˆ S n ≤ S n ) , ˜ t (cid:44) E { ˆ S n | ˆ S n ≤ S n } , ˜ s (cid:44) E { S n | ˆ S n > S n } , i.e., ˜ p is the probability that there exists a packet that reachesthe destination successfully during S n (including n and thepackets arriving after a ( n ) but before d ( n ) ) . We first have the following lemma, based on which we willderive ˜ t and ˜ s . Lemma 1.
For a nonnegative random variable X , an event E and a sequence of events E , E , . . . , E K which satisfies E i ∩ E j = ∅ , ∀ i (cid:54) = j and E = ∪ Kk =1 E k , we have P ( E ) E { X | E } = K (cid:88) k =1 P ( E k ) E { X | E k } . Proof. P ( E ) E { X | E } = P ( E ) (cid:90) ∞ P ( X > x | E ) dx = P ( E ) (cid:90) ∞ P ( X > x, E ) P ( E ) dx = (cid:90) ∞ K (cid:88) k =1 P ( X > x, E k ) dx = K (cid:88) k =1 P ( E k ) (cid:90) ∞ P ( X > x, E k ) P ( E k ) dx = K (cid:88) k =1 P ( E k ) E { X | E k } The probability that X n ≤ S n is λλ + µ . If that happens, thesystem will first serve packet n + 1 (during which other newpackets may come and complete service before n + 1 ), thencontinue the service of n . Based on this observation, we have: ˜ p = µλ + µ p + λλ + µ [ P ( ˆ S n +1 ≤ S n +1 ) + P ( ˆ S n +1 > S n +1 ) × P ( ˆ S n ≤ S n | X n ≤ S n , ˆ S n +1 > S n +1 )]= µλ + µ p + λλ + µ [˜ p + (1 − ˜ p )˜ p ] . (3)Furthermore, we have: ˜ p ˜ t = µλ + µ p E { S n | X n > S n , n ∈ Φ } + λλ + µ (cid:2) ˜ p E { X n +ˆ S n +1 | X n ≤ S n , ˆ S n +1 ≤ S n +1 } + (1 − ˜ p )˜ p × E { ˆ S n | X n ≤ S n , ˆ S n +1 > S n +1 , ˆ S n ≤ S n } (cid:3) = µλ + µ p λ + µ + λλ + µ (cid:2) ˜ p ( 1 λ + µ + ˜ t )+ (1 − ˜ p )˜ p ( 1 λ + µ + ˜ s + ˜ t ) (cid:3) , (4) (1 − ˜ p )˜ s = µ (1 − p ) λ + µ E { S n | X n > S n , n / ∈ Φ } + λλ + µ (1 − ˜ p ) × E { S n | X n ≤ S n , ˆ S n +1 > S n +1 , ˆ S n > S n } = µ (1 − p ) λ + µ λ + µ + λλ + µ (1 − ˜ p ) ( 1 λ + µ + 2˜ s ) . (5)In the above, we have used P ( ˆ S n ≤ S n | X n ≤ S n , ˆ S n +1 > S n +1 )= P ( ˆ S n − X n − S n +1 ≤ S n − X n − S n +1 | ˆ S n > X n + S n +1 )= ˜ p, nd that E { ˆ S n | X n ≤ S n , ˆ S n +1 > S n +1 , ˆ S n ≤ S n } = E { X n + S n +1 + ˆ S n − X n − S n +1 | X n ≤ S n , ˆ S n +1 > S n +1 , ˆ S n ≤ S n } = E { X n | X n ≤ S n } + E { S n +1 | ˆ S n +1 > S n +1 } + E { ˆ S n − X n − S n +1 | ˆ S n > X n + S n +1 , ˆ S n ≤ S n } = 1 λ + µ + ˜ s + ˜ t, since both the services and arrivals are memoryless. We getfrom (3) that: λ ˜ p + ( µ − λ )˜ p − µp = 0 , (6)which leads to: ˜ p = − ( µ − λ ) + (cid:112) ( µ − λ ) + 4 λµp λ . (7)Solving (4) and (5), and using (6) give us: (1 − ˜ p )˜ s = 1 − ˜ pµ − λ + 2 λ ˜ p , (8) ˜ p ˜ t = ˜ p + µµ − λ +2 λ ˜ p (˜ p − p ) µ − λ + λ ˜ p . (9)On the other hand, n ∈ Ψ means that only n reachesthe destination successfully during S n , which is equivalentto ˆ S n = S n . Therefore, we get: P ( n ∈ Ψ) = µλ + µ p + λλ + µ (1 − ˜ p ) × P ( ˆ S n − X n − S n +1 = S n − X n − S n +1 ) , P ( n ∈ Ψ) E { S n | n ∈ Ψ } = µλ + µ p λ + µ + λλ + µ (1 − ˜ p ) × P ( ˆ S n = S n )( 1 λ + µ + ˜ s + E { S n | n ∈ Ψ } ) , where we have used E { S n − X n − S n +1 | S n > X n + S n +1 , n ∈ Ψ } = E { S n | n ∈ Ψ } . As a result, P ( n ∈ Ψ) = µpµ + λ ˜ p E { S n | n ∈ Ψ } = 1 µ − λ + 2 λ ˜ p . B. Computing PAoI
Now consider E { ˆ Y n i | n i ∈ Ψ } = E { ˆ Y n i } . Suppose the firstpacket which arrives after d ( n i ) is ˜ n i . Since the exponentialdistribution is memoryless, the expected time from d ( n i ) to a (˜ n i ) is λ . If ˆ S ˜ n i ≤ S ˜ n i , the (expected) remaining time of ˆ Y n i from a (˜ n i ) is ˜ t . Otherwise the remaining time is ˜ s + E { ˆ Y ˜ n i } .Based on the above analysis, E { ˆ Y n i } = 1 λ + ˜ p ˜ t + (1 − ˜ p )(˜ s + E { ˆ Y ˜ n i } ) , from which we obtain: E { ˆ Y n i } = 1˜ p (cid:2) λ + ˜ p ˜ t + (1 − ˜ p )˜ s (cid:3) . Substituting (8) and (9) into the above gives us: E { ˆ Y n i } = µ ( µ − λ ) + 2 λµp + λ ( λ + µ )˜ pλµp ( µ − λ + 2 λ ˜ p ) . As a result, A LCF S,preP = E { S n i | n i ∈ Ψ } + E { ˆ Y n i } = 1 µ − λ + 2 λ ˜ p + µ ( µ − λ ) + 2 λµp + λ ( λ + µ )˜ pλµp ( µ − λ + 2 λ ˜ p )= µ ( µ − λ ) + 3 λµp + λ ( λ + µ )˜ pλµp ( µ − λ + 2 λ ˜ p ) , (10)where ˜ p is given in (7). In the case when p = 1 , the aboveresult becomes P AoI = λ + µ + λ + µ .V. PA O I UNDER
LCFS
WITH N ON -P REEMPTIVE P RIORITY
In this case, if a new packet arrives while the server is busy,it cannot interrupt the current service. From Fig. 2(b), we seethat PAoI is similarly the elapsed time from the moment whena packet n i ∈ Ψ arrives, to the moment when n i +1 departs.Define the first informative packet which arrives after n startsto receive service as γ ( n ) (cid:44) min { n i | n i ∈ Ψ , a ( n i ) > u ( n ) } . and the time duration from the moment n starts to receiveservice to the moment γ ( n ) departs as ˆ Z n (cid:44) d ( γ ( n )) − u ( n ) . Since the packets arriving after a ( n i ) but before u ( n i ) areserved before n i and get lost upon departure (because n i ∈ Ψ ),we have (see Fig. 2(b)): A LCF S,nonP = E { W n i + ˆ Z n i | n i ∈ Ψ } . (11) A. Analyzing a Service Process
We first define ¯ S n as the process since u ( n ) till the first timethe server becomes free or starts to serve a packet that arrivesno later than u ( n ) (excluding n ). Since ¯ S n is determined by theservices and arrivals after u ( n ) and independent of the systemstate at u ( n ) and the history before u ( n ) , the ¯ S n processesinduced by different packets n are identically distributed. Were-define the following symbols: ˜ p (cid:44) P ( ˆ Z n ≤ ¯ S n ) , ˜ t (cid:44) E { ˆ Z n | ˆ Z n ≤ ¯ S n } , ˜ s (cid:44) E { ¯ S n | ˆ Z n > ¯ S n } , i.e., ˜ p is the probability that there exists a packet which arrivesafter u ( n ) and reaches the destination successfully during ¯ S n .Consider ¯ S n . Suppose the number of packets arriving during S n is σ ( S n ) . We have ∀ k ≥ , p ( σ ( S n ) = k ) = ( λλ + µ ) k µλ + µ , E { S n | σ ( S n ) = k } = k + 1 λ + µ . If σ ( S n ) = k > (which is needed for ˆ Z n ≤ ¯ S n ), when n completes service, the system will serve the ( n + k ) -th packetand enter ¯ S n + k . If n + k ∈ Φ , then ˆ Z n ≤ ¯ S n and the remainingtime of ˆ Z n from d ( n ) is E { S n + k | σ ( S n ) = k, n + k ∈ Φ } = µ .If n + k / ∈ Φ , then if ˆ Z n + k ≤ ¯ S n + k , ˆ Z n ≤ ¯ S n and theremaining time of ˆ Z n from d ( n ) is E { ˆ Z n + k | σ ( S n ) = k, n + k / ∈ Φ , ˆ Z n + k ≤ ¯ S n + k } = E { ˆ Z n + k | ˆ Z n + k ≤ ¯ S n + k } = ˜ t .Similar analysis applies to the ( n + k − -th, the ( n + k − -th, · · · , and the ( n + 1) -th packet. Thus, using Lemma 1, ˜ p = ∞ (cid:88) k =1 µλ + µ ( λλ + µ ) k (cid:8) p + (1 − p )˜ p (1 − p )(1 − ˜ p )[ p + (1 − p )˜ p ] + · · · +(1 − p ) k − (1 − ˜ p ) k − [ p + (1 − p )˜ p ] (cid:9) = λλ + µ − µλ + µ λλ + µ (1 − p )(1 − ˜ p )1 − λλ + µ (1 − p )(1 − ˜ p ) , (12) ˜ p ˜ t = ∞ (cid:88) k =1 µλ + µ ( λλ + µ ) k (cid:26) p ( k + 1 λ + µ + 1 µ )+(1 − p )˜ p ( k + 1 λ + µ + ˜ t ) + (1 − p )(1 − ˜ p ) × (cid:2) p ( k + 1 λ + µ + ˜ s + 1 µ ) + (1 − p )˜ p ( k + 1 λ + µ +˜ s + ˜ t ) (cid:3) + · · · + (1 − p ) k − (1 − ˜ p ) k − × (cid:2) p ( k + 1 λ + µ + k ˜ s − ˜ s + 1 µ )+(1 − p )˜ p ( k + 1 λ + µ + k ˜ s − ˜ s + ˜ t ) (cid:3)(cid:27) = ∞ (cid:88) k =1 µλ + µ ( λλ + µ ) k (cid:26) k − (cid:88) j =0 (1 − p ) j (1 − ˜ p ) j × (cid:2) p ( k + 1 λ + µ + j ˜ s + 1 µ )+(1 − p )˜ p ( k + 1 λ + µ + j ˜ s + ˜ t ) (cid:3)(cid:27) , (13)and (1 − ˜ p )˜ s = µλ + µ λ + µ + ∞ (cid:88) k =1 µλ + µ ( λλ + µ ) k × (1 − p ) k (1 − ˜ p ) k ( k + 1 λ + µ + k ˜ s ) . (14)From (12), we get: λ (1 − p )˜ p + ( µ − λ + 2 λp )˜ p − λp = 0 , (15)which leads to ˜ p = − ( µ − λ +2 λp )+ √ ( λ + µ ) − λµ (1 − p )2 λ (1 − p ) , < p < λλ + µ , p = 1 . (16)Solving (13) and (14), and using (15) give us (1 − ˜ p )˜ s = 1 − ˜ pλ + µ − λ (1 − p )(1 − ˜ p ) , (17) ˜ p ˜ t = λp + 2 λp + ( λ − λp − µ + µp )˜ pµp [ λ + µ − λ (1 − p )(1 − ˜ p )] . (18) B. Computing PAoI
Now we compute PAoI shown in Fig. 2(b). Define π ( t ) as the number of packets in the system (including the packetbeing served) at time t . So π ( t ) = 0 means the system is freeat time t . Different from the preemptive case, here π ( a ( n i )) and π ( u ( n i )) will respectively affect W n i and ˆ Z n i , in thatthey affect the degree to which new packets need to wait forservice completion.We first compute the number of packets an arrival in Ψ seeswhen it arrives. Since Ψ is a special set of packets, they donot see exactly as what an ordinary packet will see. To thisend, we define for each kp k (cid:44) P [ π ( a ( n )) = k | n ∈ Ψ] = P [ π ( a ( n )) = k, n ∈ Ψ] P ( n ∈ Ψ) . (19)Consider the waiting time W n of packet n . If π ( a ( n )) = 0 then W n = 0 . Otherwise n needs to wait for the completionof the current service and the services of packets which arriveduring the current service, till the server starts to serve a packetarriving no later that a ( n ) . Since the exponential distribution ismemoryless, for π ( a ( n )) > , W n is the same as the process ¯ S ¯ n of a virtual packet ¯ n with u (¯ n ) = a ( n ) , and n ∈ Ψ isequivalent to ( ˆ Z ¯ n > ¯ S ¯ n ) ∩ ( n ∈ Φ) . For a steady-state M/M/ queue, we know that P [ π ( t ) = k ] = (1 − λµ )( λµ ) k . Thus, p = (1 − λµ ) p P [ n ∈ Ψ] ,p k = (1 − λµ )( λµ ) k (1 − ˜ p ) p P [ n ∈ Ψ] , k ≥ . Moreover, (cid:80) ∞ k =0 p k =1. Therefore, p = µ − λµ − λ ˜ p ,p k = µ − λµ − λ ˜ p (1 − ˜ p )( λµ ) k , k ≥ . Hence, the waiting time can be computed as: E { W n i | n i ∈ Ψ } = ∞ (cid:88) k =0 p k E { W n i | π ( a ( n i )) = k, n i ∈ Ψ } = p · − p )˜ s = λ (1 − ˜ p )( µ − λ ˜ p )[ λ + µ − λ (1 − p )(1 − ˜ p )] . For E { ˆ Z n i | n i ∈ Ψ } , define: z k (cid:44) E { ˆ Z n | π ( u ( n )) = k, n ∈ Ψ } = E { ˆ Z n | π ( u ( n )) = k } . For E { ˆ Z n | π ( u ( n )) = k } , if a packet n j arrives during S ( n ) (with probability λλ + µ ), it will wait W n j = ¯ S ¯ n j before beingserved, with ¯ n j a virtual packet defined as before. Since π ( u ( n j )) = k , if ˆ Z ¯ n j > ¯ S ¯ n j and n j / ∈ Φ , the (expected)remaining time of ˆ Z n from u ( n j ) is still z k . Otherwise nopackets arrives during S ( n ) , giving us π ( d ( n )) = k − . Basedon the above analysis, we get: z = µλ + µ (cid:2) λ + µ + 1 λ + p µ + (1 − p ) z (cid:3) (20) + λλ + µ (cid:2) λ + µ + ˜ p ˜ t + (1 − ˜ p ) p (˜ s + 1 µ )+(1 − ˜ p )(1 − p )(˜ s + z ) (cid:3) , and that for general k , z k = µλ + µ ( 1 λ + µ + z k − ) (21) + λλ + µ (cid:2) λ + µ + ˜ p ˜ t + (1 − ˜ p ) p (˜ s + 1 µ )+(1 − ˜ p )(1 − p )(˜ s + z k ) (cid:3) . Solving (20) gives us z = µ + λ + λp + λ τλ [ λ + µp − λ (1 − p )(1 − ˜ p )] , (22)where τ = ( λ + µ ) p + ( λ + µ ) p + [ λ + ( µ − λ ) p − µ ]˜ pµp [ λ + µ − λ (1 − p )(1 − ˜ p )] . (23)rom (21), we can get z k − (1 − ˜ p )(1 + λτ ) µ ˜ p = (1 − ˜ p )[ z k − − (1 − ˜ p )(1 + λτ ) µ ˜ p ] . The evolution of the LCFS queueing system shows that if apacket n sees no more than one packet when it arrives, thenthere will be only one packet in the system (packet n itself)when it starts to receive service. Thus, P [ π ( u ( n )) = 1 | n ∈ Ψ] = p + p , (24) P [ π ( u ( n )) = k | n ∈ Ψ] = p k , k ≥ . (25)Hence, E { ˆ Z n i | n i ∈ Ψ } = ∞ (cid:88) k =0 P [ π ( u ( n i )) = k | n i ∈ Ψ] E { ˆ Z n i | π ( u ( n i )) = k, n i ∈ Ψ } = ( p + p ) z + ∞ (cid:88) k =2 p k z k = p z + ∞ (cid:88) k =1 p k z k = p z + ∞ (cid:88) k =1 p k [ z k − (1 − ˜ p )(1 + λτ ) µ ˜ p ]+ ∞ (cid:88) k =1 p k (1 − ˜ p )(1 + λτ ) µ ˜ p = p z + ∞ (cid:88) k =1 p k (1 − ˜ p ) k − [ z − (1 − ˜ p )(1 + λτ ) µ ˜ p ]+ ∞ (cid:88) k =1 p k (1 − ˜ p )(1 + λτ ) µ ˜ p = µ ( µ − λ )( µ + λ + λp + λ τ ) λ ( µ − λ ˜ p )[ µ − λ (1 − ˜ p )][ λ + µp − λ (1 − p )(1 − ˜ p )]+ λ (1 − ˜ p ) (1 + λτ ) µ ( µ − λ ˜ p )[ µ − λ (1 − ˜ p )] . Therefore, PAoI can be computed as: A LCF S,nonP = λ (1 − ˜ p )( µ − λ ˜ p )[ λ + µ − λ (1 − p )(1 − ˜ p )]+ µ ( µ − λ )( µ + λ + λp + λ τ ) λ ( µ − λ ˜ p )[ µ − λ (1 − ˜ p )][ λ + µp − λ (1 − p )(1 − ˜ p )]+ λ (1 − ˜ p ) (1 + λτ ) µ ( µ − λ ˜ p )[ µ − λ (1 − ˜ p )] , (26)where τ is given in (23) and ˜ p is given in (16). From thisresult, we can see that even in the case of p = 1 , the solutionis non-trivial.VI. PA O I UNDER R ETRANSMISSION WITH P REEMPTIVE P RIORITY
In this case, we consider the case when a packet is transmit-ted repeatedly until it reaches the destination successfully or itis preempted. Thus,
Φ = Ψ . Here the “departure” of a packetmeans the moment it is transmitted successfully or preempted.Actually, since we do not assume feedback, a packet will still be served before the arrival of the next packet even ifit has been transmitted successfully, but that has no influenceto the system due to the preemptive priority. We can regardthis policy as only storing the latest packet and replace it witha new one as soon as a new packet arrives.
A. Analyzing a Service Process
We divide PAoI in the same way as in Section IV and againre-define the following symbols: ˜ p (cid:44) P ( n ∈ Ψ) , ˜ t (cid:44) E { ˆ S n | n ∈ Ψ } , ˜ s (cid:44) E { ˆ S n } , i.e., ˜ p is the probability that there is no packets arriving during ˆ S n , or ˆ S n = S n . Other than the total service time S n , wefurther use S n,k to denote the k -th service of packet n .For the ˆ S n process, in S n, , if n + 1 arrives (with prob-ability λλ + µ ), it will preempt n and the remaining time of ˆ S n from a ( n + 1) is ˆ S n +1 . Otherwise if n gets lost uponservice completion, the server will start to retransmit n and E { ˆ S n − S n, | X n > S n, , ˆ S n > S n, } = ˜ s , since the firsttransmission does not influence the following retransmissions.Based on this observation, we get: ˜ s = µλ + µ (cid:2) p λ + µ + (1 − p )( 1 λ + µ + ˜ s ) (cid:3) + λλ + µ ( 1 λ + µ + ˜ s ) , (27)which leads to: ˜ s = 1 pµ . (28)If n ∈ Ψ , we know n has been transmitted successfully beforeit is preempted. By considering the number of transmissions ittakes to successfully transmit n and using Lemma 1, we have: ˜ p = ∞ (cid:88) k =0 (1 − p ) k p ( µλ + µ ) k +1 , (29) ˜ p ˜ t = ∞ (cid:88) k =0 (1 − p ) k p ( µλ + µ ) k +1 k + 1 λ + µ . (30)By solving (29) and (30), we can get ˜ p = pµλ + pµ , (31) ˜ t = 1 λ + pµ . (32) B. Computing PAoI
Now we are going to compute the PAoI. For ˆ Y n i , theexpected time from d ( n i ) to a ( n i + 1) is λ and the processfrom a ( n i + 1) to d ( n i +1 ) is ˆ S n i +1 . Hence E { ˆ Y n i | n i ∈ Ψ } = 1 λ + ˜ s. Thus, we can compute PAoI as A RT,preP = E { S n i + ˆ Y n i | n i ∈ Ψ } = E { ˆ S n i | n i ∈ Ψ } + E { ˆ Y n i | n i ∈ Ψ } = ˜ t + 1 λ + ˜ s = 1 λ + pµ + 1 λ + 1 pµ . (33) emark: It turns out that this result corresponds to the resultunder the LCFS with preemptive priority policy with a servicerate pµ and a success probability . This is intuitive since inthe LCFS with preemptive priority case with p = 1 , eachpacket is either transmitted successfully or preempted, whilein this case each packet is still either transmitted successfullyor preempted, with a mean service time pµ .VII. PA O I UNDER R ETRANSMISSION WITH N ON -P REEMPTIVE P RIORITY
Under the Retransmission with non-preemptive priority pol-icy, the server keeps retransmitting the most recent packet nomatter it has been successfully transmitted or not. In this case,the most recent packet is kept in the queue and the server isalways busy. Since a new arrival can’t interrupt the currentservice, a packet may be replaced by a more recent packetwhen it is waiting in the queue or upon a service completion.Here the “departure” of a packet means the moment it istransmitted successfully or replaced. Let Ω denote the set ofpackets which are not replaced while waiting in the queue, i.e.,the ones that have been served before departure. Note that inthis case Φ = Ψ and Φ ⊂ Ω . A. Analyzing a Service Process
We divide PAoI in the same way as in Section V, but here W n , S n , ˆ S n and ˆ Z n are only meaningful for packet n ∈ Ω ,which means that there is no packets arriving during W n .Again we define the following notions for packet n ∈ Ω : ˜ p (cid:44) P ( n ∈ Ψ) , ˜ t (cid:44) E { ˆ S n | n ∈ Ψ } , ˜ s (cid:44) E { ˆ S n } , i.e., ˜ p is the probability that n has been transmitted success-fully before it is replaced by a more recent packet, or ˆ S n = S n .For the ˆ S n process, if n reaches the destination successfullyafter S n, , then ˆ S n ends. Otherwise the server will start totransmit another packet ˆ n ∈ Ω which arrives during S n, , orretransmit n , both resulting in the expected remaining time of ˆ S n as ˜ s . This observation gives us ˜ s = p µ + (1 − p )( 1 µ + ˜ s ) , (34)which leads to: ˜ s = 1 pµ . (35)If n ∈ Ψ , we know n has been transmitted successfully beforeit is replaced. By considering the number of transmissions ittakes to successfully transmit n and using Lemma 1, we have: ˜ p = ∞ (cid:88) k =0 (1 − p ) k ( µλ + µ ) k p, (36) ˜ p ˜ t = ∞ (cid:88) k =0 (1 − p ) k ( µλ + µ ) k p ( kλ + µ + 1 µ ) . (37)By solving (36) and (37), we can get ˜ p = p ( λ + µ ) λ + pµ , (38) ˜ t = 1 µ + (1 − p ) µ ( λ + µ )( λ + pµ ) . (39) B. Computing PAoI
Now we are going to compute the PAoI. Remember theserver is always busy and n i ∈ Ψ indicates that there is noarrivals during n i ’s waiting time. Thus, E { W n i | n i ∈ Ψ } = 1 λ + µ . Consider the period ˆ Z n i . We have E { S n i | n i ∈ Ψ } = E { ˆ S n i | n i ∈ Ψ } = ˜ t . If there is a packet ˆ n i with a ( ˆ n i ) > a ( n i ) in the queue at d ( n i ) ( ˆ n i can only arrives during the lastservice of n i before d ( n i ) , and the probability is λλ + µ ), theprocess from d ( n i ) to d ( n i +1 ) is ˆ S ˆ n i . Otherwise, the expectedtime from d ( n i ) to a ( n i +1) is λ . After that, because the serveris always busy, it still needs to complete a service of n i beforeit starts to transmit a packet arrives after n i . Based on theseobservations, we have: E { ˆ Z n i | n i ∈ Ψ } = ˜ t + λλ + µ ˜ s + µλ + µ ( 1 λ + 1 µ + ˜ s ) . So PAoI can be computed as: A RT,nonP = 1 λ + µ + ˜ t + λλ + µ ˜ s + µλ + µ ( 1 λ + 1 µ + ˜ s )= 1 µ + 1 λ + pµ + 1 λ + 1 pµ . (40)VIII. N UMERICAL RESULTS
We present numerical evaluations of PAoI under differ-ent scheduling policies, including FCFS, FCFS with packetmanagement (the
M/M/ / ∗ scheme in [8]), LCFS withpreemptive priority, LCFS with non-preemptive priority, Re-transmission with preemptive priority and Retransmission withnon-preemptive priority. Note that the M/M/ / ∗ scheme in[8] is equivalent to the LCFS with non-preemptive prioritypolicy that discards all stale packets. The service rate is set to µ = 1 while the arrival rate is varied to show performancesunder different channel utilizations ρ = λµ . The cases p = 0 . , p = 0 . and p = 1 are selected to represent different deliveryerror regimes. We present not only the results computedfrom our formulas (1), (10), (26), (33) and (40), but alsothose obtained by simulating real queueing systems with thecorresponding settings.From Fig. 3, we see that the simulation results match ourtheoretical results very well. We can see that when channelutilization is high, the PAoI under FCFS becomes very largedue to large queueing delay, while other policies effectivelyavoid this problem. On the other hand, when packet lossrate is high, FCFS with packet management suffers fromthe lack of packet deliveries but LCFS again ensures a lowPAoI, matching our intuition about the benefits of LCFS.Moreover, retransmission policies have significant reductionson PAoI compared to other policies when packet loss rateis high. But when packet loss rate is low, Retransmissionwith non-preemptive priority suffers a performance loss sinceretransmissions can also block later packets. .2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10102030405060 Channel Utilization PA o I FCFS, theoryFCFS, simulationFCFS with packet management, simulationpreemptive LCFS, theorypreemptive LCFS, simulationnon−preemptive LCFS, theorynon−preemptive LCFS, simulationpreemptive Retransmission, theorypreemptive Retransmission, simulationnon−preemptive Retransmission, theorynon−preemptive Retransmission, simulation p=0.5p=0.1p=1
Fig. 3. PAoI in different queueing systems with packet loss.
IX. C
ONCLUSION
We consider the peak age-of-information (PAoI) in an
M/M/ queueing system with packet delivery failure, a set-ting that models real-world situations with transmission errors.We derive exact PAoI expressions under different schedul-ing policies, including FCFS, LCFS with preemptive prior-ity, LCFS with non-preemptive priority, Retransmission withpreemptive priority, and Retransmission with non-preemptivepriority. Our analytical and simulation results show that theLCFS principle as well as retransmissions can successfullyavoid increments in PAoI resulting from large queueing delayand packet loss. A CKNOWLEDGMENT
The authors would like to thank Prof. Eytan Modiano atMIT for the motivating discussions and valuable comments.This work was supported in part by the National Ba-sic Research Program of China Grant 2011CBA00300,2011CBA00301, the National Natural Science Foundation ofChina Grant 61361136003, 61303195, Tsinghua Initiative Re-search Grant, Microsoft Research Asia Collaborative ResearchAward, and the China Youth 1000-talent Grant.R
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