Joint performance analysis of ages of information in a multi-source pushout server
aa r X i v : . [ c s . PF ] J un Joint Performance Analysis of Ages ofInformation in a Multi-source Pushout Server
Yukang Jiang and Naoto Miyoshi
Abstract
Age of information (AoI) has been widely accepted as a measure quantifying freshness of statusinformation in real-time status update systems. In many of such systems, multiple sources share a limitednetwork resource and therefore the AoIs defined for the individual sources should be correlated witheach other. However, there are not found any results studying the correlation of two or more AoIs in astatus update system with multiple sources. In this work, we consider a multi-source system sharing acommon service facility and provide a framework to investigate joint performance of the multiple AoIs.We then apply our framework to a simple pushout server with multiple sources and derive a closed-formformula of the joint Laplace transform of the AoIs in the case with independent M/G inputs. We furthershow some properties of the correlation coefficient of AoIs in the two-source system.
Index Terms
Age of information, multi-source status update systems, joint Laplace transform, multi-sourcepushout server, correlation coefficient, Palm calculus, stationary framework.
I. I
NTRODUCTION
Freshness of status information is crucial in real-time status update systems seen, for example,in weather reports, autonomous driving, stock market trading and so on.
Age of information (AoI) has been widely accepted in this decade as a measure quantifying the freshness of informationin status update systems where information sources transmit packets containing status updates todestination monitors through a communication network. Specifically, the AoI is defined as the
Y. Jiang and N. Miyoshi are with Department of Mathematical and Computing Science, Tokyo Institute of Technology, Tokyo,Japan. E-mail: [email protected] support of the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research (C) 19K11838 isgratefully acknowledged. elapsed time since the information currently displayed on a monitor is generated and timestampedat the source. In many of such systems, multiple sources share a limited network resourceand therefore the AoIs defined for the individual sources should be correlated with each other.However, to the best of the knowledge of the authors, there are no results studying the correlationof two or more AoIs in a status update system with multiple sources (except for the first author’spreliminary work [1]). To investigate the correlation is essential when, for example, we considera nonlinear penalty function of multiple AoIs. In this work, we consider a multi-source systemsharing a common service facility and provide a framework to investigate joint performance ofthe AoIs defined for the individual sources.
A. Related work
Since the advent in [2], [3], a large amount of literature has emerged on development of the AoIconcept due to its importance and availability in a wide range of information communicationsystems. A complete review falls out of the reach of this paper and we only highlight somerelevant results to ours. Interested readers are referred to recent monographs [4], [5] andreferences therein.The AoI was first introduced in [2] in the context of vehicular networks and queueing-theoretictechnique was applied in [3] to analyze the mean AoI under the ergodicity assumption. Theseresults were then extended to various multi-source systems in [6]–[8]. A more tractable metric, peak AoI (PAoI) —the AoI immediately before an update, was introduced in [9], [10], whichcharacterized not only the mean but also the probability distribution of the PAoI for variousqueueing systems under the ergodicity. Not only the mean, expected nonlinear functions of theAoI were examined in [11], [12].In the early stage of the development, they assumed that time intervals of packet generationsand their service times are either exponentially distributed or deterministic; that is, theyconsidered M/M, M/D or D/M inputs in the queueing notation. Recently, some researchers havechallenged to incorporate more general probability distributions. In [13], gamma distributedservice times were assumed for Poisson arrival systems and [14], [15] considered moregeneral service time distributions in multi-source systems with independent Poisson arrivals;that is, they treated independent M/G input processes. Furthermore, [16]–[18] studied moregeneral frameworks of packet arrival and service time processes and derived general formulassatisfied by the stationary distribution and its Laplace transform of the AoI, where [16], [17] adopted the technique of sample path analysis while [18] worked based on the Palm calculuswithin the stationary framework (see, e.g., [19] for the Palm calculus). However, there are noresults confronting joint performance of multiple AoIs in multi-source systems within generalframeworks.
B. Contribution
In this paper, we consider a multi-source system sharing a service facility as in [6]–[8],[14], [15] and provide a framework to investigate joint performance of the AoIs defined for theindividual sources. We first consider a general multi-source system, where the time sequencerepresenting service completions (and status updates) follows a stationary point process on thereal line, and derive a formula satisfied by the joint Laplace transform of the stationary AoIs. Atool for our analysis is the Palm calculus within the stationary framework as in [18], where wedo not require the ergodic assumption but, once the ergodicity is assumed, we can obtain thesame results as those from the sample path analysis by the ergodic theorem. Our formula is sogeneral and is applicable to many multi-source systems. We then apply this formula to a simplepushout server, where the system has a single server and each generated packet is immediatelystarted for service without waiting; that is, the ongoing service of another packet (if any) isinterrupted and replaced by the new one. In the case with independent M/G input processes, wederive a closed-form formula of the joint Laplace transform of the AoIs. Furthermore, we revealsome properties of the correlation coefficient of the AoIs in the two-source system.
C. Organization
The rest of the paper is organized as follows. In the next section, we describe a general multi-source system and derive a formula satisfied by the joint Laplace transform of the multiple AoIs.We also provide a formula for a single-source system, which corresponds to the review of theresults in [16], [17] within our stationary framework. In Section III, we apply our formula toa multi-source pushout server. We first confirm in III-A that our multi-source pushout serveris definitely within our framework and then derive a closed-form formula of the joint Laplacetransform of the AoIs in the case with independent M/G inputs in III-B. Some properties of thecorrelation coefficient of the AoIs in the two-source system are revealed in III-C. These propertiesof the correlation coefficient are confirmed through numerical experiments in Section IV. Finally,concluding remarks and future work are discussed in Section V.
II. G
ENERAL MULTI - SOURCE SYSTEM
In this section, we consider a general multi-source single-server system. There are K ( ∈ N = { , , . . . } ) sources generating packets with different kinds of status information and each sourcehas its dedicated monitor. The set of the sources is denoted by K = { , , . . . , K } . The systemhas a single server and, after a packet from source k ∈ K is processed by the server, the statusinformation in the packet is immediately displayed on monitor k . We note that not all packetsare completed for service and some may be discarded and lost (due to buffer overflows, packetdeadlines or pushouts). We, for the moment, ignore such lost packets and focus on those beingcompleted for service.Let Ψ denote a point process on R = (−∞ , ∞) counting the times at each of which the serviceof a packet is completed and the status information on one of the monitors is updated, and let { U n } n ∈ Z denote the corresponding time sequence, where Z = {· · · , − , , , , . . . } . For n ∈ Z , let C n and D n denote respectively the source and the delay of the packet whose service is completedat U n ; that is, the status information updated at U n is generated and timestamped at U n − D n by source C n . For each k ∈ K , let Ψ k denote a sub-process of Ψ counting the times of servicecompletions of source k packets; that is, Ψ k ( B ) = Õ n ∈ Z { k } ( C n ) B ( U n ) , B ∈ B( R ) , (1)where E denotes the indicator function of a set E . Clearly, Ψ k , k ∈ K , are mutually disjoint andsatisfy Ψ = Í Kk = Ψ k . We impose the following assumption on the marked point process Ψ C , D corresponding to {( U n , C n , D n )} n ∈ Z . Assumption 1:
1) The point process Ψ is simple almost surely in a probability P ( P -a.s.), where the rule ofsubscripts is such that · · · < U − < U ≤ < U < · · · conventionally (see, e.g., [19]).2) The marked point process Ψ C , D is stationary in P with mark space K × [ , ∞) .3) The point process Ψ has positive and finite intensity λ Ψ = E [ Ψ ( , ]] , where E denotes theexpectation with respect to P .4) P Ψ ( C = k ) > for all k ∈ K , where P Ψ denotes the Palm probability for Ψ .5) P Ψ k ( D < ∞) = for all k ∈ K , where P Ψ k denotes the Palm probability for Ψ k .Note that the Palm probability P Ψ is well defined under Assumptions 1-2) and 3) (see [19])and that P Ψ ( U = ) = . Furthermore, under Assumption 1-2), {( C n , D n )} n ∈ Z is stationary in P Ψ . Assumption 1-4) does not restrict us since we can redefine K by K \ { k } if P Ψ ( C = k ) = for some k ∈ K . The intensity λ Ψ k of the sub-process Ψ k is given by λ Ψ k = λ Ψ P Ψ ( C = k ) foreach k ∈ K , so that the Palm probability P Ψ k is also well defined under 1-2)–4) and satisfies P Ψ k (·) = P Ψ (· | C = k ) . Let { U k , n } n ∈ Z , k ∈ K , denote the time sequence corresponding to Ψ k satisfying · · · < U k , − < U k , ≤ < U k , < · · · , and let also D k , n denote the delay of thesource k packet whose service is completed at U k , n (note that D k , = D P Ψ k -a.s.). Then, the AoIprocess { A k ( t )} t ∈ R for source k ∈ K is defined as (see [7], [8] for AoI in multi-source systems) A k ( t ) = D k , n + t − U k , n , t ∈ [ U k , n , U k , n + ) , n ∈ Z . (2)This definition indicates that the AoI of source k represents the elapsed time since the informationcurrently displayed on monitor k is generated and timestamped; that is, it is set to the delay of asource k packet at its service completion time and increases linearly until the status informationfrom source k is next updated. Lemma 1:
For each k ∈ K , the marked point process Ψ k , D corresponding to {( U k , n , D k , n )} n ∈ Z and also the AoI processes { A k ( t )} t ∈ R are jointly stationary with the marked point process Ψ C , D under Assumption 1. Furthermore, A k ( ) is P -a.s. finite under the same assumption. Proof:
The proof relies on a technical discussion within the stationary framework and isgiven in Appendix A.While the AoI just after an update is equal to the delay D k , n = A ( U k , n ) , that just before anupdate is called the PAoI (see [9], [10]); that is, for each k ∈ K , the sequence { P k , n } n ∈ Z of PAoIsis defined as P k , n = A ( U k , n −) , n ∈ Z , and is also stationary in P Ψ k . Prior to the joint performanceanalysis of A ( t ) , . . . , A K ( t ) , t ∈ R , we review the results of [16], [17] for a single-source single-server system within our stationary framework, which is also useful in the marginal performanceanalysis of multi-source systems. Proposition 1 (Cf. [16], [17]):
Consider a single-source system satisfying Assumption 1 with K = , where A ( t ) = A ( t ) , D n = D , n and P n = P , n in (2). Then, the stationary distribution ofthe AoI satisfies P ( A ( ) ≤ x ) = λ Ψ ∫ x (cid:0) P Ψ ( P > u ) − P Ψ ( D > u ) (cid:1) d u , x ≥ . (3)Let L A ( s ) = E [ e − sA ( ) ] , L D ( s ) = E Ψ [ e − sD ] and L P ( s ) = E Ψ [ e − sP ] , s ∈ R , denote the Laplacetransforms of A ( ) , D and P , respectively, where E Ψ denotes the expectation with respect tothe Palm probability P Ψ . Then, the following relation holds; s L A ( s ) = λ Ψ (cid:0) L D ( s ) − L P ( s ) (cid:1) , (4) for s ∈ R such that the Laplace transforms on both the sides exist. Proof:
Applying the Palm inversion formula (see [19, p. 20]), P ( A ( ) ≤ x ) = λ Ψ E Ψ (cid:20)∫ [ , U ) [ , x ] ( A ( t )) d t (cid:21) = λ Ψ E Ψ (cid:20)∫ [ D , P ) [ , x ] ( u ) d u (cid:21) = λ Ψ E Ψ [ P ∧ x − D ∧ x ] , x ≥ , where a ∧ b = min ( a , b ) for a , b ∈ R and the second equality follows from (2); that is, A ( t ) increases linearly from D to P for t ∈ [ U , U ) . The last expression above immediately derives(3) since P is distributionally equal to P in P Ψ . The formula (4) is obtained from (3) as L A ( s ) = ∫ ∞ e − sx P ( A ( ) ∈ d x ) = λ Ψ ∫ ∞ e − sx (cid:0) P Ψ ( P > x ) − P Ψ ( D > x ) (cid:1) d x , using E [ e − sX ] = − s ∫ ∞ e − sx P ( X > x ) d x for a random variable X and s ∈ R such that E [ e − sX ] is finite. Remark 1:
Formulas (3) and (4) respectively correspond to the results of Theorem 14 (i)and (ii) in [17]. As opposed to [16], [17], however, Proposition 1 does not require the ergodicityof {( D n , P n )} n ∈ Z . These formulas suggest that we can obtain the stationary distribution of theAoI and the corresponding Laplace transform once the stationary distributions of the delay andPAoI are available. The formula (3) also implies that the stationary distribution of the AoI hasthe density function λ Ψ (cid:0) P Ψ ( P > x ) − P Ψ ( D > x ) (cid:1) , x ≥ . Taking x → ∞ in (3), we have E Ψ [ P ] = E Ψ [ D ] + / λ Ψ , which intuitively makes sense since a PAoI is the sum of a delay andits subsequent interdeparture time (see (2)).Proposition 1 can be applied to the marginal distribution of each AoI in a system with K ≥ . Corollary 1:
For a system with K sources satisfying Assumption 1, the marginal stationarydistribution of the AoI for source k ∈ K satisfies P ( A k ( ) ≤ x ) = λ Ψ k ∫ x (cid:0) P Ψ k ( P k , > u ) − P Ψ k ( D k , > u ) (cid:1) d u , x ≥ . (5)Let L A k ( s ) = E [ e − sA k ( ) ] , L D k ( s ) = E Ψ k [ e − sD k , ] and L P k ( s ) = E Ψ k [ e − sP k , ] , s ∈ R , denote theLaplace transforms of A k ( ) , D k , and P k , , respectively, where E Ψ k denotes the expectation withrespect to P Ψ k . Then, the following relation holds; s L A k ( s ) = λ Ψ k (cid:0) L D k ( s ) − L P k ( s ) (cid:1) , (6) for s ∈ R such that the Laplace transforms on both the sides exist.Next, we consider the joint performance of multiple AoIs. Let L A on R K denote the jointLaplace transform of A ( ) , . . . , A K ( ) ; that is, L A ( s ) = E (cid:20) exp (cid:18) − K Õ k = s k A k ( ) (cid:19) (cid:21) , s = ( s , . . . , s K ) ∈ R K . (7)Note that L A does not always exist on the whole space of R K . We derive a general formulasatisfied by L A as far as it exists, which is applicable to the analysis of many multi-sourcesingle-server systems satisfying Assumption 1. Let ( η , . . . , η K ) denote a random permutation of K satisfying U = U η , > U η , > · · · > U η K , . (8)That is, η j represents the source such that the status information on its monitor at time is the j th newest among K (the information on monitor η is most recently updated while monitor η K displays the oldest information). Note that U η , , . . . , U η K , satisfy the relation; − U η k , = − U η , + k Õ j = ( U η j − , − U η j , ) , k = , . . . , K . (9)In the following, we write s H = Í i ∈ H s i for a nonempty subset H ⊂ K with s = s K = Í Ki = s i and η [ k ] = { η k , η k + , . . . , η K } given the random permutation ( η , . . . , η K ) satisfying (8). Theorem 1:
For a K -source single-server system satisfying Assumption 1, the joint Laplacetransform L A of the stationary AoIs A ( ) , . . . , A K ( ) satisfies s L A ( s ) = λ Ψ E Ψ (cid:20) (cid:0) − e − sU (cid:1) K − Ö k = exp n − s η k D η k , − s η [ k + ] ( U η k , − U η k + , ) o exp (cid:0) − s η K D η K , (cid:1) (cid:21) , (10)for s = ( s , . . . , s K ) ∈ R K such that the expectation on the right-hand side exists. Proof:
Applying the Palm inversion formula to (7) and then using (2), we have L A ( s ) = λ Ψ E Ψ (cid:20)∫ [ , U ) exp (cid:18) − K Õ k = s k A k ( t ) (cid:19) d t (cid:21) = λ Ψ E Ψ (cid:20) K Ö k = e − s k ( D k , − U k , ) ∫ [ , U ) e − st d t (cid:21) . (11)It is immediate that s ∫ [ , U ) e − st d t = − e − sU . Furthermore, the relation (9) on the event { U = U η , = } implies that K Ö k = e − s k ( D k , − U k , ) = K Ö k = exp (cid:0) − s η k ( D η k , − U η k , ) (cid:1) = K Ö k = exp (cid:0) − s η k D η k , (cid:1) K Ö k = k Ö j = exp (cid:0) − s η k ( U η j − , − U η j , ) (cid:1) = K Ö k = exp (cid:0) − s η k D η k , (cid:1) K Ö j = exp (cid:0) − s η [ j ] ( U η j − , − U η j , ) (cid:1) . Plugging this into (11) derives (10).
Remark 2:
We can easily confirm that (10) agrees with (4) in Proposition 1 when K = ( Î k = · = in this case) since U = and P = D + U P Ψ -a.s. On the other hand,it is not so straightforward to show that (10) with s j = for all j ∈ K \ { k } agreeswith (6) in Corollary 1 because (10) is based on the Palm inversion formula with respectto P Ψ (with the integral on [ U , U ) ) while (6) is on that with respect to P Ψ k (with theintegral on [ U k , , U k , ) ). Therefore, Corollary 1 still makes sense in marginal analysis of multi-source systems. Note that the terms in the product in (10) are evaluated in mutually disjointintervals ( U η K , − D η K , , U η K , ] , ( U η K , , U η K − , ] , . . . , ( U η , , ] and ( , U ] on the event { U = } .This property can make formula (10) useful for analysis of a class of multi-source systems suchthat the sequence of service completion times forms a regenerative process or an embeddedMarkov chain. III. A PPLICATION TO A MULTI - SOURCE PUSHOUT SERVER
In this section, we apply the results in the preceding section to a pushout server with K sources. We first confirm that the system satisfies Assumption 1 in the preceding section andthen derive a closed-form formula for the joint Laplace transform of the AoIs in the case withindependent M/G input processes. We further reveal some properties of the correlation coefficientof the AoIs in the two-source system. A. Multi-source pushout server
We here describe the system consisting of a pushout server and K sources with the dedicatedmonitors. Let Φ denote a point process on R counting the times at each of which a packet isgenerated and timestamped by any one of the sources and let { T n } n ∈ Z denote the correspondingtime sequence. For each n ∈ Z , c n and S n denote respectively the source and the required servicetime of the packet generated at T n . For each k ∈ K , let Φ k denote the sub-process of Φ countingthe generation times of source k packets; that is, Φ k ( B ) = Õ n ∈ Z { k } ( c n ) B ( T n ) , B ∈ B( R ) . We impose the following assumption on the marked point process Φ c , S corresponding to {( T n , c n , S n )} n ∈ Z . Assumption 2:
1) The point process Φ is P -a.s. simple, where { T n } n ∈ Z is numbered as · · · < T − < T ≤ < T < · · · conventionally.2) The marked point process Φ c , S is stationary in P with mark space K × [ , ∞) .3) The point process Φ has positive and finite intensity λ = E [ Φ ( , ]] .4) P Φ ( c = k ) > for all k ∈ K , where P Φ denotes the Palm probability for Φ .5) P Φ k ( S ≤ τ ) > for all k ∈ K , where τ n = T n + − T n , n ∈ Z , and P Φ k denotes the Palmprobability for Φ k .The Palm probability P Φ in Assumption 2-4) is well defined under 2-2) and 3) and so are P Φ k , k ∈ K , under 2-2)–4) since the intensity λ k of Φ k is given by λ k = λ P Φ ( c = k ) ∈ ( , ∞) .Note here that P Φ k (·) = P Φ (· | c = k ) holds for k ∈ K .The system has a single server and each generated packet is immediately started for servicewithout waiting. If another one is in service at the generation time of a packet, the service isinterrupted and replaced by the new one (the interrupted packet is pushed out and lost). Thereis no priority among the sources and the service of any packet can be interrupted by the nextgenerated one from the same or other sources. The probability that a packet generated at source k is completed for service without interruption is then given by P Φ k ( S ≤ τ ) , which is positivefor all k ∈ K under Assumption 2-5). When the service for a packet is completed withoutinterruption, the status information carried by the packet is displayed on the monitor dedicatedto the source of that packet. Then, the marked point process Ψ C , D , representing the servicecompletions considered in the preceding section, is expressed in terms of Φ c , S as Ψ C , D ( B × { k } × E ) = Õ n ∈ Z B ( T n + S n ) E ∩[ ,τ n ] ( S n ) { k } ( c n ) , B ∈ B( R ) , k ∈ K , E ∈ B([ , ∞)) . (12) Lemma 2:
When the input marked point process Φ c , S satisfies Assumption 2, then the outputprocess Ψ C , D satisfies Assumption 1 with λ Ψ = λ P Φ ( S ≤ τ ) = K Õ k = λ k P Φ k ( S ≤ τ ) , (13) P Ψ ( C = k ) = P Φ ( c = k | S ≤ τ ) = λ k P Φ k ( S ≤ τ ) Í Kj = λ j P Φ j ( S ≤ τ ) . (14) Proof:
The proof is also based on the stationary framework and is given in Appendix B.
B. Multi-source M/G/1/1 pushout server
In this subsection, by specifying the input point process as independent homogeneousPoisson processes and assuming independence in the service times, we derive a closed-form formula for the joint Laplace transform of the AoIs A ( ) , . . . , A K ( ) . We assume that Φ , . . . , Φ K are mutually independent homogeneous Poisson processes with positive and finiteintensities λ , . . . , λ K . The superposition theorem for Poisson processes (see, e.g., [20, p. 20],[21, p. 36]) then implies that Φ = Í Kk = Φ k is also a homogeneous Poisson process withintensity λ = Í Kk = λ k . We further assume that service times S n , n ∈ Z , depend only on theirsources and, when the sources of packets are given, the service times are mutually independentand independent of Φ k , k ∈ K . Namely, for any m ∈ N , n , . . . , n m ∈ Z , k , . . . , k m ∈ K , and E , . . . , E m ∈ B([ , ∞)) , we have P Φ (cid:0) c n = k , S n ∈ E , . . . , c n m = k m , S n m ∈ E m (cid:1) = λ m m Ö j = λ k j P Φ kj ( S ∈ E j ) . Let L S , k denote the Laplace transform of service times of source k packets; that is, L S , k ( s ) = E Φ k [ e − sS ] , s ∈ R (it may be infinite for some s < ), where we note that E Φ k [·] = E Φ [· | c = k ] and E Φ [·] = E [· | T = ] . In this setup, Assumption 2 is satisfied and the probability that asource k packet is completed for service without interruption is given by P Φ k ( S ≤ τ ) = E Φ k (cid:2) P Φ k ( S ≤ τ | S ) (cid:3) = E Φ k [ e − λ S ] = L S , k ( λ ) , where the second equality follows from the independent increments property of a Poisson process.Then, (13) and (14) in Lemma 2 are respectively rewritten as λ Ψ = λ L S ( λ ) = Í Kk = λ k L S , k ( λ ) and P Ψ ( C = k ) = λ k L S , k ( λ )/ (cid:0) λ L S ( λ ) (cid:1) with L S ( s ) = λ − Í Kk = λ k L S , k ( s ) . Furthermore, we usethe following notation such that, for a nonempty subset H ⊂ K , L S , H ( s ) = E Φ [ e − sS | c ∈ H ] = λ H Õ k ∈ H λ k L S , k ( s ) , with λ H = Í k ∈ H λ k . Note that L S , K = L S and L S , { k } = L S , k for k ∈ K .First, we consider the marginal Laplace transform of the AoI for each source. Proposition 2:
For the K -source M/G/1/1 pushout server described above, the marginal Laplacetransform L A k of the stationary AoI A k ( ) of source k is given by L A k ( s ) = λ k L S , k ( s + λ ) s + λ k L S , k ( s + λ ) , s ≥ , k ∈ K . (15) Proof:
We use (6) in Corollary 1. Note that P k , = D k , + ( U k , − U k , ) by (2) and thedefinition of the PAoI. In our M/G/1/1 pushout server, since D k , and U k , − U k , are mutuallyindependent and U k , − U k , is also independent of { C = k } on the event { U = U k , = } , (6)is reduced to s L A k ( s ) = λ Ψ k L D k ( s ) (cid:0) − E Ψ [ e − sU k , ] (cid:1) . (16)First, Neveu’s exchange formula (see [19, p. 21]) implies that λ Ψ k L D k ( s ) = λ k E Φ k (cid:20)Õ n ∈ Z e − sD k , n [ , T k , ) ( U k , n ) (cid:21) = λ k E Φ k (cid:2) e − sS { S ≤ τ } (cid:3) = λ k E Φ k (cid:2) e − sS P Φ k ( S ≤ τ | S ) (cid:3) = λ k L S , k ( s + λ ) , (17)where { T k , n } n ∈ Z denotes the sub-sequence of { T n } n ∈ Z corresponding to Φ k satisfying · · · < T k , ≤ < T k , < · · · and P Φ k ( T k , = ) = . The second equality in (17) follows from the observationthat there exists at most one service completion of a source k packet during [ T k , , T k , ) and itoccurs only when the packet generated at T k , is completed for service without interruption.Next, we consider E Ψ [ e − sU k , ] in (16). Note that there may be one or more service completionsduring ( U , U k , ) , but if any, they must be of the sources in K \{ k } . Since U m = U + Í m − n = ( U n + − U n ) for m ≥ (where Í n = · = ) and the server is always reset at U n , n ∈ Z , we have E Ψ [ e − sU k , ] = ∞ Õ m = E Ψ (cid:2) e − sU k , { U k , = U m } (cid:3) = E Ψ (cid:2) e − sU { C = k } (cid:3) ∞ Õ m = (cid:0) E Ψ (cid:2) e − sU { C ∈K\{ k }} (cid:3) (cid:1) m − . (18)We solve E Ψ (cid:2) e − sU { C = k } (cid:3) above. Let B n , n ∈ Z , denote the time length of the busy periodstarting at T n and ending at the next service completion. Since T is independent of the event { U = } due to the independent increments property of a Poisson process and B n is initializedat T n , we have E Ψ (cid:2) e − sU { C = k } (cid:3) = E Ψ (cid:2) e − s ( T + B ) { C = k } (cid:3) = E [ e − sT ] E Φ (cid:2) e − sB { C = k } (cid:3) . (19)Here, E [ e − sT ] = λ /( s + λ ) is immediate since Φ is a homogeneous Poisson process withintensity λ . In considering E Φ [ e − sB { C = k } ] , we note that there may be one or more pushed-out services in a busy period. Let M = min { n = , , , . . . | S n ≤ τ n } , which representsthe index of the first packet completed for service after T . Note that M < ∞ P Φ -a.s. since P Φ ( S ≤ τ ) = L S ( λ ) > . Then, { M = n } = { S i > τ i , i = , , . . . , n − S n ≤ τ n } and B = Í M − i = τ i + S M . Since ( τ n , S n ) , n ∈ Z , are mutually independent and identically distributed,we have E Φ (cid:2) e − sB { C = k } (cid:3) = ∞ Õ n = E Φ (cid:2) e − sB { C = k } { M = n } (cid:3) = E Φ (cid:2) e − sS { S ≤ τ } { c = k } (cid:3) ∞ Õ n = (cid:0) E Φ (cid:2) e − s τ { S >τ } (cid:3) (cid:1) n . (20)Here, a similar way to obtaining (17) leads to E Φ (cid:2) e − sS { S ≤ τ } { c = k } (cid:3) = λ k λ L S , k ( s + λ ) , and on the other hand, E Φ (cid:2) e − s τ { S >τ } (cid:3) = E Φ (cid:2) E Φ (cid:2) e − s τ { S >τ } | S (cid:3) (cid:3) = λ E Φ (cid:20)∫ S e −( s + λ ) x d x (cid:21) = λ (cid:0) − L S ( s + λ ) (cid:1) s + λ . Plugging these into (20), and then to (19), we obtain E Ψ (cid:2) e − sU { C = k } (cid:3) = λ k L S , k ( s + λ ) s + λ L S ( s + λ ) . (21)The same discussion as above except for replacing { k } by K \ { k } derives E Ψ (cid:2) e − sU { C ∈K\{ k }} (cid:3) = λ K\{ k } L S , K\{ k } ( s + λ ) s + λ L S ( s + λ ) , (22)and further plugging (21) and (22) into (18), we have E Ψ [ e − sU k , ] = λ k L S , k ( s + λ ) s + λ k L S , k ( s + λ ) . (23) Finally, substitution of (17) and (23) into (16) yields (15).We can obtain the marginal moments of A k ( ) , k ∈ K , from (15) in any order as far as theyexist. For example, the first two moments are given by E [ A k ( )] = − d L A k ( s ) d s (cid:12)(cid:12)(cid:12)(cid:12) s = = λ k L S , k ( λ ) , (24) E [ A k ( ) ] = d L A k ( s ) d s (cid:12)(cid:12)(cid:12)(cid:12) s = = (cid:0) + λ k L ( ) S , k ( λ ) (cid:1) λ k L S , k ( λ ) , where L ( m ) S , k denotes the m th derivative of L S , k . The variance and the coefficient of variation arethen given by Var [ A k ( )] = + λ k L ( ) S , k ( λ ) λ k L S , k ( λ ) , (25) CV ( A k ( )) = q + λ k L ( ) S , k ( λ ) , where the numerator of (25) is definitely nonnegative since − ax e − bx ≥ whenever a > , b > and a / b ≤ e (in our case, λ k ≤ e λ always holds). Remark 3:
When K = , our system corresponds to the one considered in [18, Sec. 4] andindeed (15) and (24) are the same as those presented in Corollary 4 in [18]. Furthermore, when theservice time distributions are common to all sources, our system is reduced to the one consideredin [15] and (24) becomes equal to [15, eq. (8)] when L S , k = L S for all k ∈ K . In addition, italso agrees with [6, eq. (47)] and [8, Theorem 2 (a)] when the service time distributions are acommon exponential one with mean µ − ; that is, substituting L S , k ( λ ) = L S ( λ ) = ( + ρ ) − with ρ = λ / µ = Í Kk = λ k / µ , we have E [ A k ( )] = ( + ρ )/ λ k , k ∈ K .We now tackle one of our main purposes in this paper; that is, we derive a closed-formexpression for the joint Laplace transform of the AoIs. Theorem 2:
For the K -source M/G/1/1 pushout server described above, the joint Laplacetransform L A of the stationary AoIs A ( ) , . . . , A K ( ) is given by L A ( s ) = λ · · · λ K Õ ( j ,..., j K )∈ σ [K] K Ö k = L S , j k ( s j [ k ] + λ ) s j [ k ] + λ j [ k ] L S , j [ k ] ( s j [ k ] + λ ) , s = ( s , . . . , s K ) ∈ [ , ∞) K , (26)where σ [K] denotes the set of all permutations of K and j [ k ] = { j k , j k + , . . . , j K } for apermutation ( j , . . . , j K ) ∈ σ [K] . Proof:
We use (10) in Theorem 1. Due to the independent increments property of a Poissonprocess and the independence of the service times, the behaviors of the server before and after packet generations and also those before and after service completions are independent.Therefore, (10) becomes s L A ( s ) = λ Ψ (cid:0) − E Ψ [ e − sU ] (cid:1) × Õ ( j , ..., j K )∈ σ [K] K − Ö k = E Ψ (cid:2) exp (cid:8) − s j k D j k , − s j [ k + ] ( U j k , − U j k + , ) (cid:9) { η k = j k } (cid:3) × E Ψ (cid:2) exp (cid:0) − s j K D j K , (cid:1) { η K = j K } (cid:3) . (27)For E Ψ [ e − sU ] above, the same discussion as obtaining (21) shows E Ψ [ e − sU ] = λ L S ( s + λ ) s + λ L S ( s + λ ) . (28)We next consider the last term E Ψ (cid:2) exp (cid:0) − s j K D j K , (cid:1) { η K = j K } (cid:3) in (27). The stationarity of { U n + − U n } n ∈ Z in P Ψ enables us to use a similar discussion to obtaining (17) and E Ψ (cid:2) exp (cid:0) − s j K D j K , (cid:1) { η K = j K } (cid:3) = λ Ψ jK λ Ψ L D jK ( s j K ) = λ j K L S , j K ( s j K + λ ) λ L S ( λ ) , (29)where we use λ Ψ = λ L S ( λ ) . Thus, it remains to solve E Ψ (cid:2) exp (cid:8) − s j k D j k , − s j [ k + ] ( U j k , − U j k + , ) (cid:9) { η k = j k } (cid:3) . Similar to considering (18), there may be one or more service completions during ( U j k + , , U j k , ) ,but if any, they must be of the sources j , j , . . . , j k by the definition of η k , k ∈ K . Therefore,since the server is always reset at U n , n ∈ Z , the above term is equal to ∞ Õ m = E Ψ h exp (cid:8) − s j k D j k , − s j [ k + ] ( U j k , − U j k + , ) (cid:9) { η k = j k } { Ψ ( U jk + , , U jk , ) = m } i = E Ψ h exp (cid:8) − s j k D − s j [ k + ] U (cid:9) { C = j k } i ∞ Õ m = (cid:16) E Ψ (cid:2) exp (cid:8) − s j [ k + ] U (cid:9) { C ∈ j [ k ]} (cid:3) (cid:17) m , (30)where j [ k ] = { j , . . . , j k } = K \ j [ k + ] . Similar to obtaining (21) and (22), we have E Ψ h exp n − s j k D − s j [ k + ] U o { C = j k } i = E (cid:2) e − s j [ k + ] T (cid:3) ∞ Õ n = (cid:16) E Φ (cid:2) e − s j [ k + ] τ { S >τ } (cid:3) (cid:17) n E Φ (cid:2) e − s j [ k ] S { S ≤ τ } { c = j k } (cid:3) = λ j k L S , j k ( s j [ k ] + λ ) s j [ k + ] + λ L S ( s j [ k + ] + λ ) , where we note that s j [ k ] = s j k + s j [ k + ] , and E Ψ (cid:2) exp (cid:8) − s j [ k + ] U (cid:9) { C ∈ j [ k ]} (cid:3) = λ j [ k ] L S , j [ k ] ( s j [ k + ] + λ ) s j [ k + ] + λ L S ( s j [ k + ] + λ ) . Therefore, (30) amounts to(30) = λ j k L S , j k ( s j [ k ] + λ ) s j [ k + ] + λ j [ k + ] L S , j [ k + ] ( s j [ k + ] + λ ) . (31)Finally, plugging (28), (29) and (31) into (27) and using λ Ψ = λ L S ( λ ) , we obtain (26). Remark 4:
Unfortunately, it is complicated to show that (26) with s j = for j ∈ K \ { k } agrees with (15) except for the case of small K . We thus focus on the two-source system in thenext subsection. C. Correlation coefficient in the two-source system
In this subsection, we investigate the correlation coefficient of AoIs in the two-source system.When K = , (26) in Theorem 2 is reduced to L A ( s , s ) = λ λ s + λ L S ( s + λ ) Õ k = L S , k ( s k + λ ) L S , − k ( s + λ ) s k + λ k L S , k ( s k + λ ) , s ≥ , s ≥ . (32)In this case, we can easily confirm that both L A ( s , ) and L A ( , s ) agree with (15). Theexpectation of the product A ( ) A ( ) is obtained from (32) as E (cid:2) A ( ) A ( ) (cid:3) = ∂ ∂ s ∂ s L A ( s , s ) (cid:12)(cid:12)(cid:12)(cid:12) s = s = = λ L S ( λ ) Õ k = L ( ) S , k ( λ )L S , k ( λ ) + Ö k = λ k L S , k ( λ ) . Therefore, combining this with (24), we have the covariance of A ( ) and A ( ) as Cov (cid:0) A ( ) , A ( ) (cid:1) = λ L S ( λ ) Õ k = L ( ) S , k ( λ )L S , k ( λ ) , from which we can see that A ( ) and A ( ) are negatively correlated since the first derivativeof the Laplace transform for a nonnegative random variable is always nonpositive. Furthercombination with (25) gives the correlation coefficient; CC ( A ( ) , A ( )) = λ λ (cid:0) L ( ) S , ( λ ) L S , ( λ ) + L S , ( λ ) L ( ) S , ( λ ) (cid:1) λ L S ( λ ) q(cid:0) + λ L ( ) S , ( λ ) (cid:1) (cid:0) + λ L ( ) S , ( λ ) (cid:1) . (33) We note that when the service time distributions are a common exponential one with mean µ − ; that is, L S , k ( s ) = L S ( s ) = ( + s / µ ) − for k = , , (33) is indeed reduced to that obtainedin [1]. Some properties of the correlation coefficient are collected in the following proposition. Proposition 3:
1) For k = , , lim λ k ↓ CC ( A ( ) , A ( )) = and, if L S , − k ( s ) = O (L S , k ( s )) as s → ∞ , then lim λ k →∞ CC ( A ( ) , A ( )) = .2) Suppose that we can choose the service time distribution under the constraint that L S , = L S , and λ = λ + λ is fixed. Then, CC ( A ( ) , A ( )) takes the minimum value whenthe service times are deterministic and equal to λ − . Furthermore, this minimum value isbounded below by − (cid:0) ( e − ) (cid:1) − ≈ − . , which is realized if and only if λ = λ = λ / .3) When the service times follow a common gamma distribution with shape parameter α > and rate parameter µ > (that is, the mean service time is α / µ ) in both the sources, then lim µ ↓ CC ( A ( ) , A ( )) = lim µ →∞ CC ( A ( ) , A ( )) = .4) Suppose that λ = λ + λ is fixed and that the service times follow a common gammadistribution with shape parameter α > and rate parameter µ > . Then, CC ( A ( ) , A ( )) takes the minimum value when µ = αλ . Furthermore, this minimum value is boundedbelow by − (cid:2) (cid:0) ( + / α ) α + − (cid:1) (cid:3) − , which is realized if and only if λ = λ = λ / . Proof:
1) The convergence as λ k ↓ is immediate from (33). For the convergence as λ k → ∞ , it is sufficient by symmetry to show that lim λ →∞ CC ( A ( ) , A ( )) = when L S , ( s ) = O (L S , ( s )) as s → ∞ . Dividing the numerator and the denominator on the right-hand side of(33) by λ L S , ( λ ) , we have CC ( A ( ) , A ( )) = λ (cid:18) L ( ) S , ( λ ) L S , ( λ )L S , ( λ ) + L ( ) S , ( λ ) (cid:19)(cid:18) + λ L S , ( λ ) λ L S , ( λ ) (cid:19)q(cid:0) + λ L ( ) S , ( λ ) (cid:1) (cid:0) + λ L ( ) S , ( λ ) (cid:1) . (34)Since S is nonnegative, e − sS ≤ and s S e − sS ≤ e − for s ≥ . Therefore, the dominatedconvergence theorem leads to L S , k ( s ) → and s L ( ) S , k ( s ) → as s → ∞ (and of course, L ( ) S , k ( s ) → as s → ∞ ). Hence, the numerator on the right-hand side of (34) goes to zero as λ → ∞ when L S , ( s ) = O (L S , ( s )) while the denominator goes to one.2) When L S , = L S , = L S , we have from (33) that − CC ( A ( ) , A ( )) = λ λ λ vt(cid:18) − L ( ) S ( λ ) − λ (cid:19) (cid:18) − L ( ) S ( λ ) − λ (cid:19) ≥ λ λ λ p ( e λ − λ )( e λ − λ ) , where the inequality follows from −L ( ) S ( λ ) ≤ ( e λ ) − and the equality holds only when the servicetimes are deterministic and equal to λ − . Furthermore, applying the inequality of arithmetic andgeometric means λ / = ( λ + λ )/ ≥ √ λ λ twice, the last expression above is bounded belowby λ λ λ p ( e λ − λ )( e λ − λ ) ≥ s e ( e − ) λ λ λ + ≥ ( e − ) , where both the equalities hold if and only if λ = λ .3) Since L S , ( s ) = L S , ( s ) = L S ( s ) = ( + s / µ ) − α , (33) is reduced to CC ( A ( ) , A ( )) = − αλ λ λ q (cid:0) ( µ + λ ) α + / µ α − αλ (cid:1) (cid:0) ( µ + λ ) α + / µ α − αλ (cid:1) , (35)which clearly goes to zero as µ ↓ . Furthermore, the above expression is also rewritten as CC ( A ( ) , A ( )) = − αλ λ / µλ q (cid:0) ( + λ / µ ) α + − αλ / µ (cid:1) (cid:0) ( + λ / µ ) α + − αλ / µ (cid:1) , which is confirmed to go to zero as µ → ∞ .4) From (35), we have − CC ( A ( ) , A ( )) = λ λ λ s(cid:18) ( µ + λ ) α + α µ α − λ (cid:19) (cid:18) ( µ + λ ) α + α µ α − λ (cid:19) ≥ λ λ λ s (cid:18) λ (cid:16) + α (cid:17) α + − λ (cid:19) (cid:18) λ (cid:16) + α (cid:17) α + − λ (cid:19) , where the inequality follows from ( µ + λ ) α + / µ α ≥ αλ ( + / α ) α + and the equality holds when µ = αλ . Furthermore, applying the inequality of arithmetic and geometric means λ / = ( λ + λ )/ ≥ √ λ λ , we can see that the last expression above is bounded below by (cid:0) ( + / α ) α + − (cid:1) and this lower bound is realized if and only if λ = λ .IV. N UMERICAL EXPERIMENTS
We here confirm the properties of the correlation coefficient of AoIs proved in Proposition 3through numerical experiments. Throughout the experiments, we use the two-source M/G/1/1pushout server with a common service time distribution. Figure 1 plots the values of the -0.3-0.25-0.2-0.15-0.1-0.05 0 0 5 10 15 20 C o rr e l a t i on C oe ff i c i en t λ Gamma ( α = .5)ExponentialGamma ( α = 2)Deterministic (a) λ = . , / E Φ [ S ] = . -0.3-0.25-0.2-0.15-0.1-0.05 0 0 5 10 15 20 C o rr e l a t i on C oe ff i c i en t λ Gamma ( α = .5)ExponentialGamma ( α = 2)Deterministic (b) λ = . , / E Φ [ S ] = . Fig. 1: Values of correlation coefficient of AoIs in the two-source M/G/1/1 pushout server fordifferent values of λ .correlation coefficient for different values of λ when the values of λ and E Φ [ S ] are fixed, whereFigure 1a) and 1b) show the cases of ( λ , / E Φ [ S ]) = ( . , . ) and ( λ , / E Φ [ S ]) = ( . , . ) ,respectively. Note that / E Φ [ S ] = µ / α when the service times are gamma distributed withshape parameter α > and rate parameter µ > . From these figures, we can see property 1)in Proposition 3; that is, the correlation coefficient goes to zero as λ ↓ and as λ → ∞ .As another property, we remark that the correlation coefficient takes the minimum value at λ = / E Φ [ S ] − λ in the case of Figure 1b), but it does not in the case of 1a). It does notcontradict properties 2) and 4) in Proposition 3; that is, when λ and λ are given, the correlationcoefficient takes the minimum value when / E Φ [ S ] = λ = λ + λ , but when λ and E Φ [ S ] aregiven, it does not always take the minimum value at either λ = / E Φ [ S ] − λ or λ = λ .Figure 2 plots the values of the correlation coefficient of AoIs for different values of / E Φ [ S ] when λ and λ are fixed. Figure 2a) and 2b) show respectively the cases of λ = λ and λ , λ while the value of λ = λ + λ remains the same in both the figures. From these figures, we canobserve the properties 2), 3) and 4) in Proposition 3; that is, the correlation coefficient goes tozero as / E Φ [ S ] ↓ and as / E Φ [ S ] → ∞ ; when λ = λ + λ is fixed, the correlation coefficienttakes the minimum value when the service times are deterministic and equal to λ − , in addition,when the service times are gamma distributed, it takes the minimum value when / E Φ [ S ] = λ ,where these minimum values are further minimized when λ = λ = λ / . -0.3-0.25-0.2-0.15-0.1-0.05 0 0 5 10 15 20 25 30 C o rr e l a t i on C oe ff i c i en t Φ [S ]Gamma ( α = .5)ExponentialGamma ( α = 2)Deterministic (a) λ = λ = . -0.3-0.25-0.2-0.15-0.1-0.05 0 0 5 10 15 20 25 30 C o rr e l a t i on C oe ff i c i en t Φ [S ]Gamma ( α = .5)ExponentialGamma ( α = 2)Deterministic (b) λ = . , λ = . Fig. 2: Values of correlation coefficient of AoIs in the two-source M/G/1/1 pushout server fordifferent values of / E Φ [ S ] . V. C ONCLUSION
In this paper, we have considered multi-source status update systems and have provided aframework to investigate the joint performance of multiple AoIs, which are defined for theindividual sources. Specifically, we have derived a general formula satisfied by the joint Laplacetransform of the stationary AoIs. Then, we have applied this formula to a multi-source pushoutserver and have shown a closed-form formula of the joint Laplace transform of the AoIs inthe case with independent M/G inputs. Furthermore, we have revealed some properties of thecorrelation coefficient of AoIs in the two-source system. In the future, we expect that our generalformula will be utilized to evaluate the joint performance of AoIs in many multi-source statusupdate systems and will become useful for development of various systems in the real world.A
PPENDIX AP ROOF OF L EMMA ( Ω , F , P ) .A family of shift operators { θ t } t ∈ R is defined on ( Ω , F ) such that θ t : Ω → Ω is measurableand bijective satisfying θ s ◦ θ t = θ s + t for s , t ∈ R , where θ is the identity; so that θ − t = θ − t for t ∈ R . The probability measure P is assumed to be invariant to { θ t } t ∈ R (in other words, { θ t } t ∈ R preserves P ) in the sense that P ◦ θ − t = P for t ∈ R . Then, we can assume that the marked point process Ψ C , D satisfying Assumption 1 is compatible with { θ t } t ∈ R in the sensethat ( U n , C n , D n ) ◦ θ t = ( U Ψ ( , t ] + n − t , C Ψ ( , t ] + n , D Ψ ( , t ] + n ) for each n ∈ Z and t ∈ R , where Ψ ( , t ] = − Ψ ( t , ] for t < conventionally (see [19]).In the setup above, the proof of the first assertion in Lemma 1 is achieved by showing that, foreach k ∈ K , the marked point process Ψ k , D and the AoI process { A k ( t )} t ∈ R are both compatiblewith { θ t } t ∈ R . We first confirm Ψ k , D . By (1), Ψ k , D satisfies Ψ k , D ( B × E ) = Õ n ∈ Z B ( U k , n ) E ( D k , n ) . = Õ n ∈ Z B ( U n ) E ( D n ) { k } ( C n ) , B ∈ B( R ) , E ∈ B([ , ∞)) . The compatibility of Ψ C , D with { θ t } t ∈ R implies that, for each n ∈ Z and t ∈ R , there exists aunique n ′ ∈ Z such that ( U n − t , C n , D n ) = ( U n ′ , C n ′ , D n ′ ) ◦ θ t and then Ψ k , D (( B + t ) × E ) = Õ n ∈ Z B + t ( U n ) E ( D n ) { k } ( C n ) = Õ n ′ ∈ Z (cid:0) B ( U n ′ ) E ( D n ′ ) { k } ( C n ′ ) (cid:1) ◦ θ t = Ψ k , D ( B × E ) ◦ θ t , where B + t = { s + t | s ∈ B } for t ∈ R and B ∈ B( R ) ; that is, Ψ k , D is compatible with { θ t } t ∈ R .Therefore, we can show from (2) that A k ( s + t ) = A k ( s ) ◦ θ t for any s , t ∈ R in a similar way;that is, { A k ( t )} t ∈ R is also compatible with { θ t } t ∈ R .We next show the second assertion in Lemma 1. Since the events { D k , < ∞} and { U k , − U k , < ∞} are P Ψ k -a.s. and { θ U k , n } n ∈ Z -invariant under Assumption 1, (2) ensures that the event { A k ( ) < ∞} is P Ψ k -a.s. and { θ U k , n } n ∈ Z -invariant as well. Hence, [19, p. 51, Property 1.6.2] says that A k ( ) < ∞ P -a.s. A PPENDIX BP ROOF OF L EMMA Ψ is inherited from Φ . We next check 1-2). In the same setting asin Section A, we can assume that the marked point process Φ c , S satisfying Assumption 2 iscompatible with { θ t } t ∈ R in the sense that ( T n , c n , S n ) ◦ θ t = ( T Φ ( , t ] + n − t , c Φ ( , t ] + n , S Φ ( , t ] + n ) foreach n ∈ Z and t ∈ R , where Φ ( , t ] = − Φ ( t , ] for t < . Then, we can see from (12) that Ψ C , D (( B + t ) × { k } × E ) = Ψ C , D ( B × { k } × E ) ◦ θ t holds for any t ∈ R , k ∈ K , B ∈ B( R ) and E ∈ B([ , ∞)) ; that is, Ψ C , D is also compatible with { θ t } t ∈ R . We now confirm 1-3) with (13). Let χ ( t ) denote the indicator that χ ( t ) = when the serveris occupied by a packet at time t and χ ( t ) = otherwise. Then, χ ( t ) satisfies χ ( t ) = Õ n ∈ Z [ T n , T n + τ n ∧ S n ) ( t ) , t ∈ R , which shows that { χ ( t )} t ∈ is also jointly stationary with Φ c , S . Furthermore, { χ ( t )} t ∈ R satisfies χ ( ) = χ ( ) + Φ ( , ] − Ψ ( , ] − Õ n ∈ Z ( , ] ( T n ) { S n − >τ n − } , where the last term on the right-hand side represents the number of pushouts during ( , ] . Takingthe expectations on both the sides above, the stationarity of { χ ( t )} t ∈ R implies λ Ψ = λ − E (cid:20)Õ n ∈ Z ( , ] ( T n ) { S n − >τ n − } (cid:21) = λ − λ P Φ ( S > τ ) = λ P Φ ( S ≤ τ ) , (36)and (13) holds.For 1-4) with (14), the Neveu’s exchange formula (see [19, p. 21]) shows that λ Ψ P Ψ ( C = k ) = λ E Φ (cid:20)Õ n ∈ Z { C n = k } [ , T ) ( U n ) (cid:21) = λ P Φ ( c = k , S ≤ τ ) , (37)where the second equality follows from the observation that there exists at most one servicecompletion during [ , T ) and it occurs only when the packet generated at is completed forservice without interruption. Equation (14) then follows from (36) and (37).Finally, 1-5) is immediate from Neveu’s exchange formula as λ Ψ k P Ψ k ( D < ∞) = λ k E Φ k (cid:20)Õ n ∈ Z { D n < ∞} [ , T k , ) ( U k , n ) (cid:21) = λ k P Φ k ( S ≤ τ ) , where { T k , n } n ∈ Z denotes the sub-sequence of { T n } n ∈ Z corresponding to Φ k satisfying · · · < T k , ≤ < T k , < · · · and the second equality follows from a similar observation to (37). Hence, 1-5)holds since the equality λ Ψ k = λ Ψ P Ψ ( C = k ) = λ k P Φ k ( S ≤ τ ) is obtained by (13) and (14).R EFERENCES [1] Y. Jiang, K. Tokuyama, Y. Wada, and M. Yajima, “Correlation coefficient analysis of the age of information in multi-sourcesystems,” 2020, submitted for presentation at a conference.[2] S. Kaul, M. Gruteser, V. Rai, and J. Kenney, “Minimizing age of information in vehicular networks,” in ,2011, pp. 350–358. [3] S. Kaul, R. Yates, and M. Gruteser, “Real-time status: How often should one update?” in , 2012, pp. 2731–2735.[4] A. Kosta, N. Pappas, and V. Angelakis, “Age of information: A new concept, metric, and tool,” Foundations and Trendsin Networking , vol. 12, pp. 162–259, 2017.[5] Y. Sun, I. Kadota, R. Talak, E. Modiano, and R. Srikant,
Age of Information: A New Metric for Information Freshness .Morgan & Claypool, 2019.[6] S. K. Kaul, R. D. Yates, and M. Gruteser, “Status updates through queues,” in , 2012.[7] R. D. Yates and S. K. Kaul, “Real-time status updating: multiple sources,” in , 2012, pp. 2666–2670.[8] ——, “The age of information: Real-time status updating by multiple sources,”
IEEE Transactions on Information Theory ,vol. 65, pp. 1807–1827, 2019.[9] M. Costa, M. Codreanu, and A. Ephremides, “Age of information with packet management,” in , 2014, pp. 1583–1587.[10] ——, “On the age of information in status update systems with packet management,”
IEEE Transactions on InformationTheory , vol. 62, pp. 1897–1910, 2016.[11] A. Kosta, N. Pappas, A. Ephremides, and V. Angelakis, “Age and value of information: Non-linear age case,” in , 2017, pp. 326–330.[12] ——, “The cost of delay in status updates and their value: Non-linear ageing,”
IEEE Transactions on Communications ,2020, Early Access.[13] E. Najm and R. Nasser, “Age of information: The gamma awakening,” in , 2016, pp. 2574–2578.[14] L. Huang and E. Modiano, “Optimizing age-of-information in a multi-class queueing system,” in , 2015, pp. 1681–1685.[15] E. Najm and E. Telatar, “Status updates in a multi-stream M/G/1/1 preemptive queue,” in
IEEE Conference on ComputerCommunications (INFOCOM 2018) Workshops , 2018, pp. 124–129.[16] Y. Inoue, H. Masuyama, T. Takine, and T. Tanaka, “The stationary distribution of the age of information in FCFS single-server queues,” in , 2017, pp. 571–575.[17] ——, “A general formula for the stationary distribution of the age of information and its application to single-serverqueues,”
IEEE Transactions on Information Theory , vol. 65, pp. 8305–8324, 2019.[18] G. Kesidis, T. Konstantopoulos, and M. Zazanis, “The distribution of age-of-information performance measures for messageprocessing systems,” 2019, arXiv:1904.05924 [cs.PF].[19] F. Baccelli and P. Brémaud,
Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences , 2nd ed.Springer, 2003.[20] G. Last and M. Penrose,