The Power of Waiting for More than One Response in Minimizing the Age-of-Information
TThe Power of Waiting for More than One Responsein Minimizing the Age-of-Information
Yu Sang, Bin Li, and Bo Ji
Abstract —The Age-of-Information (AoI) has recently beenproposed as an important metric for investigating the timelinessperformance in information-update systems. Prior studies on AoIoptimization often consider a
Push model, which is concernedabout when and how to “push” (i.e., generate and transmit)the updated information to the user. In stark contrast, in thispaper we introduce a new
Pull model, which is more relevant forcertain applications (such as the real-time stock quotes service),where a user sends requests to the servers to proactively “pull”the information of interest. Moreover, we propose to employrequest replication to reduce the AoI. Interestingly, we findthat under this new Pull model, replication schemes capture anovel tradeoff between different levels of information freshnessand different response times across the servers, which can beexploited to minimize the expected AoI at the user’s side.Specifically, assuming Poisson updating process at the serversand exponentially distributed response time, we derive a closed-form formula for computing the expected AoI and obtain theoptimal number of responses to wait for to minimize the expectedAoI. Finally, we conduct numerical simulations to elucidate ourtheoretical results. Our findings show that waiting for more thanone response can significantly reduce the AoI in most scenarios.
I. I
NTRODUCTION
The last decades have witnessed the prevalence of smartdevices and significant advances in ubiquitous computing andthe Internet of things. This trend is forecasted to continuein the years to come [1]. The development of this trendhas spawned a plethora of real-time services that requiretimely information/status updates. One practically importantexample of such services is vehicular networks and intel-ligent transportation systems [2], [3], where accurate statusinformation (position, speed, acceleration, tire pressure, etc.)of a vehicle needs to be shared with other nearby vehiclesand road-side facilities in a timely manner in order to avoidcollisions and ensure substantially improved road safety. Moresuch examples include sensor networks for environment/healthmonitoring [4], [5], wireless channel feedback [6], news feeds,weather updates, online social networks, fare aggregating sites(e.g., Google Shopping), and stock quotes service.For systems providing such real-time services, those com-monly used performance metrics, such as throughput anddelay, exhibit significant limitations in measuring the system
This work was supported in part by the NSF under Grants CCF-1657162,CNS-1651947, and CNS-1717108.Yu Sang ([email protected]) and Bo Ji ([email protected]) are withthe Department of Computer and Information Sciences, Temple University,Philadelphia, PA, and Bin Li ([email protected]) is with the Department ofElectrical, Computer and Biomedical Engineering, University of Rhode Island,Kingston, Rhode Island. performance [7]. Instead, the timeliness of information updatesbecomes a major concern . To that end, a new metric called the
Age-of-Information (AoI) has been proposed as an importantmetric for studying the timeliness performance [2]. The AoIis defined as the time elapsed since the most recent updateoccurred (see Eq. (1) for a formal definition). Using the AoImetric introduced in [2] for vehicular networks, the work of [7]employs a simple system model to analyze and optimize thetimeliness performance of an information-update system. Thisseminal work has recently aroused dramatic interests from theresearch community and has inspired a series of interestingstudies on the AoI analysis and optimization (see [8] andreferences therein).While all prior studies consider a
Push model, which isconcerned about when and how to “push” (i.e., generate andtransmit) the updated information to the user, in this paperwe introduce a new
Pull model, under which a user sendsrequests to the servers to proactively “pull” the information ofinterest. This Pull model is more relevant for many importantapplications where the user’s interest is in the freshness ofinformation at the point when the user requests it ratherthan in continuously monitoring the freshness of information.One application of the Pull model is in the real-time stockquotes service, where a customer (i.e., user) submits a queryto multiple stock quotes providers (i.e., servers) and eachprovider responds with the most up-to-date information it has.
To the best of our knowledge, however, none of the existingwork on the timeliness optimization has considered such aPull model. In stark contrast, we focus on the Pull modeland propose to employ request replication to minimize theexpected AoI at the user’s side.
Although a similar Pullmodel is considered for data synchronization in [9], [10],the problems are quite different and request replication isnot exploited. Note that the concept of replication is notnew and has been extensively studied for various applications(e.g., cloud computing and datacenters [11], [12], storageclouds [13], parallel computing [14], [15], and databases [16],[17]).
However, for the AoI minimization problem under thePull model, replication schemes exhibit a unique property andcapture a novel tradeoff between different levels of informationfreshness and different response times across the servers.This tradeoff reveals the power of waiting for more than oneresponse and can be exploited to minimize the expected AoIat the user’s side.
Next, we explain the above key tradeoff through a com-parison with cloud computing systems. It has been observedthat in a cloud or a datacenter, the processing time of a same a r X i v : . [ c s . N I] A ug ob can be highly variable on different servers [12]. Due tothis important fact, replicating a job on multiple servers andwaiting for the first finished copy can help reduce latency[11], [12]. Apparently, in such a system it is not beneficialto wait for more copies of the job to finish, as all the copieswould give the same outcome. In contrast, in the information-update system we consider, although the servers may possessthe same type of information (weather forecast, stock prices,etc.), they could have different versions of the informationwith different levels of freshness due to the random updatingprocesses. Hence, the first response may come from a serverwith stale information; waiting for more than one response hasthe potential of receiving fresher information and thus helpsreduce the AoI. Hence, it is no longer the best to stop afterreceiving the first response (as in the other aforementioned ap-plications).
On the other hand, waiting for too many responseswill lead to a longer total waiting time and thus, also incursa larger AoI at the user’s side.
Therefore, it is challenging todetermine the optimal number of responses to wait for in orderto minimize the expected AoI at the user’s side.
In what follows, we summarize the key contributions ofthis paper.
First , for the first time we introduce the Pullmodel for studying the timeliness optimization problem andpropose to employ request replication to reduce the AoI.
Second , assuming Poisson updating process at the servers andexponentially distributed response time, we derive a closed-form formula for computing the expected AoI and obtainthe optimal number of responses to wait for to minimizethe expected AoI. Some extensions are also discussed.
Third ,we conduct extensive numerical simulations to elucidate ourtheoretical results. We also investigate the impact of the systemparameters (the updating rate, the mean response time, andthe total number of servers) on the achieved gain in the AoIreduction. Simulation results for other types of response timedistribution are also provided. Our findings show that waitingfor more than one response can significantly reduce the AoIin most scenarios.The remainder of this paper is organized as follows. We firstdescribe our new Pull model in Section II. Then, we analyzethe expected AoI under replication schemes in Section III,obtain the optimal number of responses for minimizing the ex-pected AoI in Section IV, and briefly discuss some extensionsof our work in Section V. Section VI presents the simulationresults. Finally, we conclude the paper in Section VII.II. S
YSTEM M ODEL
We consider an information-update system where a userpulls time-sensitive information from n servers. These n servers are connected to a common information source andupdate their data asynchronously . We call such a model the Pull model (see Fig. 1). Let i ∈ { , , . . . , n } be the serverindex. We assume that the information updating process atserver i is Poisson with rate λ > and is independent andidentically distributed ( i.i.d. ) across the servers. This impliesthat the inter-update time (i.e., the time duration betweentwo successive updates) at each server follows an exponential Servers Information Source Replicated Requests Responses User Updates 1 2 n Fig. 1: The Pull model of information-update systems. Notethat the arrows in the figure denote logical links rather thanphysical connections. The updates, requests, and responses areall transmitted through (wired or wireless) networks.
AoI time tu i ( t ) i ( t ) Fig. 2: An illustration of the AoI evolution at server i .distribution with mean /λ . Here, the inter-update time at aserver can be interpreted as the time required for the server toreceive information updates from the source. Let u i ( t ) denotethe time when the most recent update at server i occurs, andlet ∆ i ( t ) denote the AoI at server i , which is defined as thetime elapsed since the most recent update at this server: ∆ i ( t ) (cid:44) t − u i ( t ) . (1)Therefore, if an update occurs at a server, then the AoI at thisserver drops to zero; otherwise, the AoI increases linearly astime goes by until the next update occurs. Fig. 2 provides anillustration of the AoI evolution at server i .In this work, we consider the ( n, k ) replication scheme,under which the user sends the replicated copies of the requestto all n servers and waits for the first k responses. Let R i denote the response time for server i . Note that each servermay have a different response time, which is the time elapsedsince the request is sent out by the user until the user receivesthe response from this server. We assume that the time for therequests to reach the servers is negligible compared to the timefor the user to download the data from the servers. Hence, theresponse time can be interpreted as the downloading time. Let s denote the downloading start time, which is the same for allthe servers, and let f i denote the downloading finish time forserver i . Then, the response time for server i is R i = f i − s .We assume that the response time is exponentially distributedwith mean /µ and is i.i.d. across the servers. Note that themodel we consider above is simple, but it suffices to capturethe key aspects and novelty of the problem we study.Under the ( n, k ) replication scheme, when the user receivesthe first k responses, it uses the freshest information amonghese k responses to make certain decisions (e.g., stock tradingdecisions based on the received stock price information). Let ( j ) denote the index of the server corresponding to the j -th re-sponse received by the user. Then, set K = { (1) , (2) , . . . , ( k ) } contains the indices of the servers that return the first k responses, and the following is satisfied: f (1) ≤ f (2) ≤ · · · ≤ f ( k ) and R (1) ≤ R (2) ≤ · · · ≤ R ( k ) . Let server i ∗ be the onethat contains the freshest information (i.e., that has the smallestAoI) among these k responses when downloading starts at time s , i.e., i ∗ = argmin i ∈ K ∆ i ( s ) (or i ∗ = argmax i ∈ K u i ( s ) dueto Eq. (1)). Here, we are interested in the AoI at the user’s sidewhen it receives the k -th response, denoted by ∆( k ) , which isthe time difference between when the k -th response is receivedand when the information at server i ∗ is updated, i.e., ∆( k ) (cid:44) f ( k ) − u i ∗ ( s ) . (2)Then, there are two natural questions of interest. First, for agiven k , can one obtain a closed-form formula for computingthe expected AoI at the user’s side, E [∆( k )] ? Second, howto determine the optimal number of responses to wait for,such that E [∆( k )] is minimized? The second question can beformulated as the following optimization problem: min k ∈{ , ,...,n } E [∆( k )] . (3)We will answer these two questions in the following twosections, respectively.III. E XPECTED A O IIn this section, we focus on answering the first question andderive a closed-form formula for computing the expected AoIat the user’s side under the ( n, k ) replication scheme.We begin with the definition of ∆( k ) and rewrite Eq. (2)as follows: ∆( k ) = f ( k ) − u i ∗ ( s )= f ( k ) − s + s − u i ∗ ( s ) ( a ) = R ( k ) + s − max i ∈ K u i ( s )= R ( k ) + min i ∈ K { s − u i ( s ) } ( b ) = R ( k ) + min i ∈ K ∆ i ( s ) , (4)where (a) is from the definition of R i and i ∗ and (b) is fromthe definition of ∆ i ( t ) (i.e., Eq. (1)). As can be seen fromthe above expression, under the ( n, k ) replication scheme theAoI at the user’s side consists of two terms: (i) R ( k ) , thetotal waiting time for receiving the first k responses, and (ii) min i ∈ K ∆ i ( s ) (or ∆ i ∗ ( s ) ), the AoI of the freshest informationamong these k responses when downloading starts at time s .An illustration of these two terms and ∆( k ) is shown in Fig. 3.Taking the expectation of both sides of Eq. (4), we have E [∆( k )] = E (cid:2) R ( k ) (cid:3) + E (cid:20) min i ∈ K ∆ i ( s ) (cid:21) . (5)Intuitively, as k increases, i.e., waiting for more responses, theexpected total waiting time (i.e., the first term) increases. On u (1) ( s ) u i ⇤ ( s ) u ( k ) ( s ) s R ( k ) f ( k ) f (1) f i ⇤ R (1) R i ⇤ i ⇤ ( s ) (1) ( s ) ( k ) ( s ) Server (1) Server i * Server ( k ) time ( k ) Fig. 3: An illustration of the AoI at the user’s side and its twoterms under the ( n, k ) replication scheme.the other hand, upon receiving more responses, the expectedAoI of the freshest information among these k responses (i.e.,the second term) decreases. Hence, there is a natural tradeoffbetween these two terms, which is a unique property of ournewly introduced Pull model.Next, we formalize this tradeoff by deriving the closed-formexpressions of the above two terms as well as the expectedAoI. We state the main result of this section in Theorem 1. Theorem . Under the ( n, k ) replication scheme, the expectedAoI at the user’s side can be expressed as: E [∆( k )] = 1 µ ( H ( n ) − H ( n − k )) + 1 kλ , (6) where H ( n ) = (cid:80) nl =1 1 l is the n -th partial sum of the divergingharmonic series.Proof. We first analyze the the first term of the right-hand sideof Eq. (5) and want to show E [ R ( k ) ] = µ ( H ( n ) − H ( n − k )) . Note that the response time is exponentially distributedwith mean /µ and is i.i.d. across the servers. Hence, randomvariable R ( k ) is the k -th smallest value of n i.i.d. exponentialrandom variables with mean /µ . The order statistics results ofexponential random variables give that R (1) is an exponentialrandom variable with mean nµ and that ( R ( j ) − R ( j − ) isan exponential random variable with mean n +1 − j ) µ for any j ∈ { , , . . . , n } [18]. Hence, we have the following: E (cid:2) R ( k ) (cid:3) = E R (1) + k (cid:88) j =2 ( R ( j ) − R ( j − ) = E [ R (1) ] + k (cid:88) j =2 E (cid:2) R ( j ) − R ( j − (cid:3) = k (cid:88) j =1 n + 1 − j ) µ = 1 µ ( H ( n ) − H ( n − k )) . (7)Next, we analyze the second term of the right-hand side ofEq. (5) and want to show the following: E (cid:20) min i ∈ K ∆ i ( s ) (cid:21) = 1 kλ . (8)ote that the updating process at each server is a Poisson pro-cess with rate λ and is i.i.d. across the servers. Hence, the inter-update time for each server is exponentially distributed withmean /λ . Due to the memoryless property of the exponentialdistribution, the AoI at each server has the same distributionas the inter-update time, i.e., random variable ∆ i ( s ) is alsoexponentially distributed with mean /λ and is i.i.d. acrossthe servers [19]. Therefore, random variable min i ∈ K ∆ i ( s ) isthe minimum of k i.i.d. exponential random variables withmean /λ , which is also exponentially distributed with mean kλ . This implies Eq. (8).Combining Eqs. (7) and (8), we complete the proof. Remark.
The above analysis indeed agrees with our intu-ition: while the expected total waiting time for receiving thefirst k responses (i.e., Eq. (7)) is a monotonically increasingfunction of k , the expected AoI of the freshest informationamong these k responses (i.e., Eq. (8)) is a monotonicallydecreasing function of k .IV. O PTIMAL R EPLICATION S CHEME
In this section, we will exploit the aforementioned tradeoffand focus on answering the second question we discussed atthe end of Section II. Specifically, we aim to find the optimalnumber of responses to wait for in order to minimize theexpected AoI at the user’s side.Using the analytical result of Theorem 1, we rewrite theoptimization problem in Eq. (3) as: min k ∈{ , ,...,n } E [∆( k )] = 1 µ ( H ( n ) − H ( n − k )) + 1 kλ . (9)Let k ∗ be an optimal solution to Eq. (9). We state the mainresult of this section in Theorem 2. Theorem . An optimal solution k ∗ can be computed as: k ∗ = min (cid:40)(cid:38) µn (cid:112) ( λ + µ ) + 4 λµn + λ + µ (cid:39) , n (cid:41) . (10) Proof.
We first define D ( k ) as the difference of the expectedAoI between the ( n, k + 1) and ( n, k ) replication schemes,i.e., D ( k ) (cid:44) ∆( k + 1) − ∆( k ) for any k ∈ { , , . . . , n − } .From Eq. (6), we have that for any k ∈ { , , . . . , n − } , D ( k ) = 1( n − k ) µ − k ( k + 1) λ . (11)It is easy to see that D ( k ) is a monotonically increasingfunction of k .We now extend the domain of D ( k ) to the set of positivereal numbers and want to find k (cid:48) such that D ( k (cid:48) ) = 0 . Withsome standard calculations and dropping the negative solution,we derive the following: k (cid:48) = 2 µn (cid:112) ( λ + µ ) + 4 λµn + λ + µ . (12)Next, we discuss two cases: (i) k (cid:48) > n and (ii) < k (cid:48) ≤ n .In Case (i), we have k (cid:48) > n . This implies that D ( k ) =∆ i ( k + 1) − ∆ i ( k ) < for all k ∈ { , , . . . , n } due to the fact that D ( k ) is monotonically increasing. Hence, theexpected AoI ∆( k ) is a monotonically decreasing functionfor k ∈ { , , . . . , n } . Therefore, k ∗ = n must be an optimalsolution.In Case (ii), we have < k (cid:48) ≤ n . We consider two subcases: k (cid:48) is an integer in { , , . . . , n } and k (cid:48) is not an integer.If k (cid:48) is an integer in { , , . . . , n } , we have D ( k ) = ∆( k +1) − ∆( k ) < for k ∈ { , , . . . , k (cid:48) − } and D ( k ) = ∆( k +1) − ∆( k ) > for k ∈ { k (cid:48) + 1 , . . . , n } . Hence, the expectedAoI ∆( k ) is first decreasing (for k ∈ { , , . . . , k (cid:48) − } ) andthen increasing (for k ∈ { k (cid:48) + 1 , . . . , n } ). Therefore, thereare two optimal solutions: k ∗ = k (cid:48) and k ∗ = k (cid:48) + 1 since ∆( k (cid:48) + 1) = ∆( k (cid:48) ) (due to D ( k (cid:48) ) = 0 ).If k (cid:48) is not an integer, we have D ( k ) = ∆( k +1) − ∆( k ) < for k ∈ { , , . . . , (cid:98) k (cid:48) (cid:99)} and D ( k ) = ∆( k + 1) − ∆( k ) > for k ∈ {(cid:100) k (cid:48) (cid:101) , . . . , n } . Hence, the expected AoI ∆( k ) is firstdecreasing (for k ∈ { , , . . . , (cid:98) k (cid:48) (cid:99)} ) and then increasing (for k ∈ {(cid:100) k (cid:48) (cid:101) , . . . , n } ). Therefore, k ∗ = (cid:100) k (cid:48) (cid:101) must be an optimalsolution.Combining two subcases, we have k ∗ = (cid:100) k (cid:48) (cid:101) in Case(ii). Then, combining Cases (i) and (ii), we have k ∗ =min {(cid:100) k (cid:48) (cid:101) , n } = min (cid:26)(cid:24) µn √ ( λ + µ ) +4 λµn + λ + µ (cid:25) , n (cid:27) . Remark.
There are two special cases that are of particularinterest: waiting for the first response only (i.e., k ∗ = 1 ) andwaiting for all the responses (i.e., k ∗ = n ). In Corollary 1, weprovide a sufficient and necessary condition for each of thesetwo special cases. Corollary . (i) k ∗ = 1 is an optimal solution if and only if λ ≥ µ ( n − ; (ii) k ∗ = n is an optimal solution if and only if λ ≤ µn ( n − .Proof. The proof follows straightforwardly from Theorem 2.A little thought gives the following: k ∗ = 1 is an optimalsolution if and only if D (1) ≥ . Solving D (1) = n − µ − λ ≥ gives λ ≥ µ ( n − . Similarly, k ∗ = n is an optimalsolution if and only if D ( n − ≤ . Solving D ( n −
1) = µ − n ( n − λ ≤ gives λ ≤ µn ( n − . Remark.
The above results agree well with the intuition. Fora given number of servers, if the mean inter-update time ismuch smaller than the mean response time (i.e., λ (cid:29) µ ), thenall the servers have frequent updates and thus, the differenceof the freshness levels among the servers is small. In this case,it is not beneficial to wait for more responses. On the otherhand, if the mean inter-update time is much larger than themean response time (i.e., λ (cid:28) µ ), then one server may possessfresher information than another server. In this case, it is worthwaiting for more responses, which leads to a significant gainin the AoI reduction. V. E XTENSIONS
In this section, we discuss some extensions of our work.
Replication scheme.
So far, we have only considered the ( n, k ) replication scheme. One limitation of this scheme isthat it requires the user to send a replicated request to every Number of responses k A v g . A o I λ = 1 , μ = 200 (simulation) λ = 1 , μ = 200 (theoretical) λ = 1 , μ = 5 (simulation) λ = 1 , μ = 5 (theoretical) λ = 100 , μ = 2 (simulation) λ = 100 , μ = 2 (theoretical) (a) Exponential response time Number of responses k A v g . A o I λ = 1 , μ = 200 (simulation) λ = 1 , μ = 200 (theoretical) λ = 1 , μ = 3 (simulation) λ = 1 , μ = 3 (theoretical) λ = 20 , μ = 2 (simulation) λ = 20 , μ = 2 (theoretical) (b) Uniform response time A v g . A o I λ = 1 , μ = 200 (simulation) λ = 1 , μ = 5 (simulation) λ = 100 , μ = 3 (simulation) (c) Gamma response time Fig. 4: Simulation results of average AoI vs. the number of responses k for three different types of response time distributions.server, which may incur a large overhead when there are alarge number of servers (i.e., when n is large). Instead, a morepractical scheme would be to send the replicated requests to asubset of servers. Hence, we consider the ( n, m, k ) replicationschemes, under which the user sends a replicated request toeach of the m servers that are randomly and uniformly chosenfrom the n servers, and waits for the first k responses, where m ∈ { , , . . . , n } and k ∈ { , , . . . , m } . Making the sameassumptions as in Section II, we can derive the expected AoIat the user’s side in a similar manner. Specifically, reusing theproof of Theorem 1 and replacing n with m in the proof, wecan show the following: E [∆( k )] = 1 µ ( H ( m ) − H ( m − k )) + 1 kλ . (13) Uniformly distributed response time.
Note that our currentanalysis requires the memoryless property of the Poissonupdating process. However, the analysis can be extended tothe uniformly distributed response time. We make the sameassumptions as in Section II, except that the response time isnow uniformly distributed in the range of [ a, a + h ] with a ≥ and h ≥ . In this case, we have E [ R ( k ) ] = khn +1 + a [18].Since Eq. (8) still holds, from Eq. (5) we have E [∆( k )] = khn + 1 + a + 1 kλ . (14)Following a similar line of analysis to that in the proof ofTheorem 2, we can show that an optimal solution k ∗ can becomputed as: k ∗ = min (cid:40)(cid:38) n + 1) (cid:112) h λ + 4 hλ ( n + 1) + hλ (cid:39) , n (cid:41) . (15)VI. N UMERICAL R ESULTS
In this section, we perform extensive simulations to evaluatethe AoI performance in an information-update system with servers under the ( n, k ) replication scheme. We first describeour simulation settings. Throughout the simulations, the up-dating process at each server is assumed to be Poisson withrate λ and is i.i.d. across the servers. The user’s request for theinformation is generated at time s , which is uniformly selectedfrom the time interval [0 , T ] , where we set T = 10 /λ . Thisimplies that each server has a total of updates on average. Next, we evaluate the AoI performance through simulationsfor three types of response time distribution: exponential , uni-form , and Gamma . First, we assume that the response time isexponentially distributed with mean /µ . Fig. 4a presents howthe average AoI changes as the number of responses k variesin three representative setups, where each point represents anaverage of simulation runs. We also include plots of ourtheoretical results (i.e., Eq. (6)) for comparison. A crucialobservation from Fig. 4a is that the simulation results matchperfectly with our theoretical results. In addition, we observethree different behaviors of the average AoI performance: (i)If the inter-update time is much smaller than the responsetime (i.e., λ = 100 , µ = 2 ), then the average AoI increasesas k increases and thus, it is not beneficial to wait for morethan one response. (ii) In contrast, if the inter-update time ismuch larger than the response time (i.e., λ = 1 , µ = 200 ),then the average AoI decreases as k increases and thus, it isworth waiting for all the responses so as to achieve a smalleraverage AoI. (iii) When the inter-update time is comparableto the response time (i.e., λ = 1 , µ = 5 ), then as k increases,the AoI would first decrease and then increase. On the onehand, when k is small, the freshness of the data at the serversdominates and thus, waiting for more responses helps reducethe average AoI. On the other hand, when k becomes large,the total waiting time becomes dominant and thus, the averageAoI increases as k further increases.In Section V, we discussed the extension of our theoreticalresults to the case of uniformly distributed response time.Hence, we also perform simulations for the response timeuniformly distributed in the range of [ µ , µ ] with mean /µ .Fig. 4b presents the average AoI as the number of responses k changes. In this scenario, the simulation results also matchperfectly with the theoretical results (i.e., Eq. (14)). Also,we observe a very similar phenomenon to that in Fig. 4a onhow the average AoI varies as k increases in three differentsimulation setups.In addition, Fig. 4c presents the simulation results for theresponse time with Gamma distribution, which can be used tomodel the response time in relay networks [20]. Specifically,we consider a special class of the Gamma( r, θ ) distributionthat is the sum of r i.i.d. exponential random variables withmean θ (which is also called the Erlang distribution). Then,the mean response time /µ is equal to rθ . We fix r = 5 in λ O p t i m a l k * Optimal k * Improvement Ratio ρ I m p r o v e m en t R a t i o ρ (a) Impact of updating rate λ . μ O p t i m a l k * Optimal k * Improvement Ratio ρ I m p r o v e m en t R a t i o ρ (b) Impact of mean response time /µ . n O p t i m a l k * Optimal k * Improvement Ratio ρ I m p r o v e m en t R a t i o ρ (c) Impact of total number of servers n . Fig. 5: Impact of the system parameters on the optimal k ∗ and the corresponding improvement ratio. We consider the exponentialdistribution for the response time. In (a), we fix µ = 1 , n = 20 ; in (b), we fix λ = 1 , n = 20 ; in (c), we fix λ = 1 , µ = 10 .the simulations. Although we are unable to derive analyticalresults in this case, the observations are similar to that underthe exponential and uniform distributions.Finally, we investigate the impact of the system parameters(the updating rate, the mean response time, and the totalnumber of servers) on the optimal number of responses k ∗ andthe improvement ratio , defined as ρ (cid:44) E [∆(1)] / E [∆( k ∗ )] . Theimprovement ratio captures the gain in the AoI reduction underthe optimal scheme compared to a naive scheme of waitingfor the first response only.Fig. 5a shows the impact of the updating rate λ . We observethat the optimal number of responses k ∗ decreases as λ increases. This is because when the updating rate is large,the AoI diversity at the servers is small. In this case, waitingfor more responses is unlikely to receive a response with muchfresher information. Therefore, the optimal scheme will simplybe a naive scheme that waits only for the first response whenthe updating rate is relatively large (e.g., λ = 2 ). Fig. 5b showsthe impact of the mean response time /µ . We observe that theoptimal number of responses k ∗ increases as µ increases. Thisis because when µ is large (i.e., when the mean response timeis small), the cost of waiting for additional responses becomesmarginal and thus, waiting for more responses is likely tolead to the reception of a response with fresher information.Fig. 5c shows the impact of the total number of servers n .We observe that both the optimal number of responses k ∗ and the improvement ratio increase with n . This is becausean increased number of servers leads to more diversity gainsboth in the AoI at the servers and in the response time.VII. C ONCLUSION
In this paper, we introduced a new Pull model for studyingthe AoI minimization problem under the replication schemes.Assuming Poisson updating process and exponentially dis-tributed response time, we derived the closed-form expressionof the expected AoI at the user’s side and provided a formulafor computing the optimal solution. Not only did our workreveal a novel tradeoff between different levels of informationfreshness and different response times across the servers, butwe also demonstrated the power of waiting for more than oneresponse in minimizing the expected AoI at the user’s side.An interesting direction for future work would be to developdynamic replication schemes that do not require the knowledgeof the updating process and the response time distribution. R
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