Correlation Coefficient Analysis of the Age of Information in Multi-Source Systems
aa r X i v : . [ c s . PF ] S e p Correlation Coefficient Analysis of theAge of Information in Multi-Source Systems
Yukang Jiang ∗ , Kiichi Tokuyama ∗ , Yuichiro Wada † , Moeko Yajima ∗ ∗ Tokyo Institute of Technology, [email protected], { tokuyama.k.aa, yajima.m.ad } @m.titech.ac.jp † FUJITSU LABORATORIES LTD. / RIKEN AIP, [email protected]
Abstract —This paper studies the age of information (AoI)on an information updating system such that multiple sourcesshare one server to process packets of updated information. Insuch systems, packets from different sources compete for theserver, and thus they may suffer from being interrupted, beingbacklogged, and becoming stale. Therefore, in order to graspstructures of such systems, it is crucially important to study ametric indicating a correlation of different sources. In this paper,we aim to analyze the correlation of AoIs on a single-serverqueueing system with multiple sources. As our contribution, weprovide the closed-form expression of the correlation coefficient of the AoIs. To this end, we first derive the Laplace-Stieltjestransform of the stationary distribution of each AoI for themultiple sources. Some nontrivial properties on the systems arerevealed from our analysis results.
Index Terms —Age of information, queueing theory, single-server queues, multiple sources, correlation coefficient, stationarydistribution.
I. I
NTRODUCTION
In recent years, we can see the real-time information updat-ing systems in many places because of the ever-increasingdemand of controlling time-critical information throughoutnetwork systems. The typical examples are monitoring systemsof weather reports, vehicular status update systems that assistself-driving of cars, remote controlling of construction machin-ery, etc. In such systems, various kinds of status are displayedon equipped monitors (e.g., temperature, humidity, and airpressure in weather reports; position, velocity, and accelerationin vehicular status). When updated status of one kind iscaptured by a sensor, a packet is generated by the associatedsource of an updated information. Thereafter, it is processedby one of servers, and then displayed on the correspondingmonitor. In addition, while all the servers are busy withprocessing, a newly arriving packet is backlogged and becomesoutdated. Note that these situations occur in practice if thearrival frequency of packets is beyond the processing powerof servers. Owing to the above properties of the systems, theinformation displayed on the monitor is not always up-to-date.Therefore, the freshness of the displayed information shouldbe quantified and managed for those information updatingsystems.From such backgrounds, a performance metric called theage of information (AoI) was proposed [1]. To define the AoI,let η ( t ) denote the timestamp of the generation time of the information displayed on the monitor at time t . Then, the AoIat time t is defined by ∆ t := t − η ( t ) . The AoI defined by ∆ t can indicate the freshness, because theabove expression means the elapsed time from the generationof the information displayed on the monitor. We note that, ina system with multiple sources, the AoI is defined for eachsource.The AoI in queueing systems have been studied in recentyears. We here focus on previous studies which investigatedthe AoIs in queueing systems with multiple informationsources. Yates and Kaul provided the pioneering study in[2]. They considered single-server queueing systems wheretwo sources share one server to process updated information,and derived the userful expression of the average AoI ofeach information source. They also studied the first-come-first-served (FCFS) M/M/1 system as a special case, and derivedthe closed-form expression of the average AoI of each source.Kaul and Yates [3] showed that two-source M/M/1/1 systemswith preemption outperform two-source FCFS M/M/1 systemsin terms of the average AoI, which triggered further studies forsuch without-queue systems. Najm and Telatar [4] derived theaverage AoI and the peak AoI of each source in two-sourceM/G/1 systems with preemption. The systems with generalmultiple information sources were investigated by Yates andKaul [5]. They considered the average AoI on M/M/1/1 andM/M/1/2 systems, in both of which multiple sources shareone server. In case of M/M/1/2, they utilized stochastic hybridsystems to discard waiting packets, and then they successfullyreduced complexity of their analysis.As described above, most of the works handling multipleinformation sources have been devoted to analyzing the av-erage AoI of each source. To the best of our knowledge,no previous works try analytical studies about the correlationof the AoIs from different information sources. In order tograsp structures of the information updating systems with themultiple sources, it is crucially important to study a metricindicating a correlation of different sources in addition to theAoIs of the individual sources. Considering the correlationleads to a better management of an information updatingsystem with multiple sources.In this paper, we study the correlation of AoIs of a statusupdating system with multiple sources, which is modeled byn M/M/1/1 queueing system with preemption. Our model isassumed to consist of multiple information sources and thecorresponding monitors and to share one server. Our modeldescribed above is defined in detail in Section II.The contribution of this paper is as follows. We firstderive the Laplace-Stieltjes transform (LST) of the stationarydistribution of each AoI in our model. The LST enables toobtain the mean and the variance of the AoI. Next, assumingthat the number of sources is two, we provide the closed-formexpression of the correlation coefficient of the AoIs, which isthe main contribution of this paper. Furthermore, using thisresult, we reveal some nontrivial properties on our model.The rest of this paper is organized as follows. In Section II,we describe the model of our investigating information updat-ing systems. Sections III and IV present analysis results. InSection III, we derive the LST for each AoI. In Section IV,assuming that the number of sources is two, we obtain thecorrelation coefficient of the AoIs. Numerical experiments areconducted in Section V. Finally, this paper is concluded inSection VI. II. S
YSTEM M ODEL
We consider an information updating system such that mul-tiple sources generate information packets of updated status.Generated packets are immediately transmitted to an M/M/1/1queueing system, which is illustrated in Fig. 1. Note thatan M/M/1/1 queueing system has only one server and nobuffer space. After being processed in the server, the packetsare directly sent to monitors, and then the monitors displaythe updated information. Each source has the correspondingmonitor, and updated information of a source is displayed onits corresponding monitor. The number of sources is denotedas K ∈ N , and K := { , , . . . , K } denotes the set of typeindexes of sources.Packets are generated from source k ( k ∈ K ) according tothe time-homogeneous Poisson process with rate λ k . Servicetimes of packets are assumed to be independent and identicallydistributed (i.i.d.) with the exponential distribution havingmean /µ ; that is, packets from all sources have the sameservice time distribution. Besides, preemption is assumed inour system; that is, the packet which currently occupies theserver will be pushed out if a new packet arrives before itsservice completion. We refer to a packet which completesits service without being pushed out as the valid packet .Henceforth, we refer to generation times of packets as arrivaltimes. In addition, we define λ := P k ∈ K λ k as the total arrivalrate of the K sources.III. A O I FOR EACH SOURCE
In this section, we derive the LST of stationary AoI of eachsource. For k ∈ K and n ∈ Z := { , ± , ± , . . . } , let α k,n denote the n th arrival time of the packet of source k , and S k,n denote the service time of the packet of source k whicharrives at α k,n . We define A k ( t ) as the AoI of source k ∈ K (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6) (cid:1)(cid:2)(cid:3)(cid:4)(cid:2)(cid:3) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:2)(cid:6) (cid:1) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:2)(cid:6) (cid:2) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:2)(cid:6) (cid:3) (cid:1)(cid:1)(cid:1) (cid:1) ✁ (cid:1)(cid:2)(cid:3)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:3)(cid:8)(cid:9)(cid:6)(cid:7)(cid:10)(cid:3)(cid:11)(cid:12)(cid:7)(cid:13)(cid:11)(cid:14)(cid:3)(cid:15)(cid:6)(cid:7) (cid:4) (cid:16)(cid:6)(cid:3)(cid:5)(cid:4)(cid:15)(cid:6)(cid:7)(cid:3)(cid:8)(cid:9)(cid:6)(cid:7)(cid:17)(cid:7) (cid:5)(cid:4) (cid:6) 1,2,⋯, (cid:11)(cid:12) Fig. 1: Our investigating system: a packet is generated fromone of the K sources, processed by the server, and thendisplayed on the corresponding monitor. The flow is modeledaccording to M/M/1/1 queueing model with preemption.at time t ∈ R . Using these notations, we have the followingexpression, for k ∈ K and t ∈ R . A k ( t ) = t − max n ∈ Z { α k,n ; α k,n +1 − α k,n > S k,n , t > β k,n } , where β k,n := α k,n + S k,n . Samples paths of A k ( t ) areillustrated in Figs. 2 and 3. { ( α k,n , S k,n ) } n ∈ Z is the stationaryand ergodic marked point process. Thus, we define A k , k ∈ K ,as the random variable following the stationary distribution of { A k ( t ) } t ∈ R .In addition, we define some notations related to validpackets. For k ∈ K and n ∈ Z , we define α ∗ k,n and S ∗ k,n as the arrival time and service time of the n th valid packet ofsource k , respectively. We also define β ∗ k,n as the n th departuretimes of the valid packet; that is, β ∗ k,n = α ∗ k,n + S ∗ k,n . Withoutloss of generality, we assume that · · · < β ∗ k, ≤ < β ∗ k, < β ∗ k, < · · · . The service time distribution of valid packets is obtained asfollows
Lemma 1
The service time of a valid packet of source k ∈ K follows the exponential distribution having mean / ( λ + µ ) .Proof. Let X a denote a random variable following theexponential distribution with mean /a for a > . Note thatpackets arrive according to the Poisson process with rate λ ifwe ignore types of sources. In addition, a packet is valid ifno other packets arrive until its service is completed. Thus, itfollows that, for k ∈ K and n ∈ Z , P ( S ∗ k,n > x ) = P ( X µ > x | X λ > X µ ) = e − ( λ + µ ) x , x ≥ . ✷ Using this lemma, we obtain the LST of each source.
Theorem 1
The Laplace-Stieltjes transform of the stationaryAoI of source k ∈ K , denoted by e A k ( s ) , is given by e A k ( s ) = λ k µ ( s + λ )( s + µ ) − ( λ − λ k ) µ , s ≥ . ✁✂✄☎ (cid:2) ✁✂✄☎ (cid:1) (cid:0)✆✝✞✟✠ (cid:1) (cid:0)✆✝✞✟✡✆✝✞☛✠ (cid:1) ✁✂✄☞✌☎ (cid:2) ✁✂✄☞✌☎ (cid:3) ✁✂✍✎✏✑ (cid:4) ✁✂✄ (cid:5) ✁✂✍✎✏✑ (cid:6) ✁✂✄ (cid:1) (cid:5) ✁✂✍✎✏✑✒✓✔✔ ✔ (cid:1) ✕✖✗✘ (cid:1) ✕✖✗✙ (cid:1) ✕✖✗✙✚✙ (cid:1) ✕✖✗✙✚✘ (cid:2) ✛ (cid:3)(cid:4)(cid:5) (cid:7) (cid:1) (cid:0)✆✝✞✜✢ (cid:3)(cid:6) ✛✣✤✥ (cid:5) (cid:3)(cid:6) ✛✣✤✟✠✥ (cid:5)(cid:1) (cid:0)✆✝✞ Fig. 2: A sample path of the AoI process A k ( t ) ( k ∈ K ). Proof.
We define A + k,n (resp. A − k,n ) as the AoI of theimmediately after (resp. before) the n th update of source k .Let e A + k ( s ) (resp. e A − k ( s ) ) denote the LSTs of the stationarydistributions of { A + k,n } (resp. { A − k,n } ). It follows from [7]that, for k ∈ K and s ≥ , e A k ( s ) = λ ∗ k e A + k ( s ) − e A − k ( s ) s , (1)where λ ∗ k denotes the arrival rate of valid packets of source k .Note here that A + k,n is equivalent to the service time of thevalid packet arriving at α ∗ k,n . Thus, from Lemma 1, we obtain e A + k ( s ) = λ + µs + λ + µ . (2)We define R k,n as the interval time of n th and ( n +1) st updatesof source k ; that is, R k,n = β ∗ k,n +1 − β ∗ k,n . The followingrelation holds for k ∈ K and n ∈ Z . A − k,n = A + k,n − + R k,n − . (3)We also have E [e − sR k,n ] = λ k µ ( s + λ )( s + µ ) − ( λ − λ k ) µ , (4)which is shown in Appendix A. Applying (2) and (4) to (3)yields e A − k ( s ) = λ + µs + λ + µ λ k µ ( s + λ )( s + µ ) − ( λ − λ k ) µ . (5)Furthermore, a packet of source k ∈ K is valid with probability µ/ ( λ + µ ) , independently other packets. We then have λ ∗ k = λ k µλ + µ . (6) Consequently, applying (2), (5), and (6) to (1), we obtainTheorem 1. ✷ Furthermore, using Theorem 1, we can easily obtain theexpectation and variance of each AoI.
Corollary 1
For k ∈ K , we have E [ A k ] = 1 λ k (cid:26) λµ (cid:27) , V [ A k ] = 1 λ k (cid:26) λ − λ k µ + λ µ (cid:27) . IV. C
ORRELATION C OEFFICIENT
In this section, assuming that K = 2 , we derive the corre-lation coefficient of stationary AoIs. For n ∈ Z , let α n denotethe n th arrival time of packets of either sources 1 or 2, andlet S n denote the service time of the packet arriving at α n .In addition we define some notations related to valid pack-ets. We define α ∗ n and S ∗ n as the arrival time and service time ofthe n th valid packet of either sources 1 or 2, respectively. Wealso define β ∗ n as the n th departure times of the valid packetof either sources 1 or 2; that is, β ∗ n = α ∗ n + S ∗ n . Without lossof generality, we assume that · · · < β ∗ ≤ < β ∗ < β ∗ < · · · . We define R n as the interval time of n th and ( n +1) st updates;that is, R n = β ∗ n +1 − β ∗ n . We obtain the following lemma. Lemma 2 R n follows the convolution of two independentrandom variables following exponential distributions havingmean /λ and /µ . The proof of Lemma 2 is shown in Appendix B.We first consider the AoI of source k ∈ { , } immediatelyafter source 1 or 2 is updated. We define A † k,n := A k ( β ∗ n ) for k = 1 , and n ∈ Z . We obtain the following lemma. Lemma 3 { A † ,n } , { A † ,n } , and { A † ,n A † ,n } are stationaryand ergodic. In addition, we have, for n ∈ Z , E [ A † k,n ] = 1 λ + µ + λ − λ k λ k (cid:18) λ + 1 µ (cid:19) , k = 1 , , E [ A † ,n A † ,n ] = 2 (cid:18) λ + µ (cid:19) + (cid:18) λ λ λ − (cid:19) (cid:18) λ + 1 µ (cid:19) λ + µ . Proof.
For n ∈ Z , let C n denote the type of the valid packetarriving at α ∗ n . For k = 1 , and n ∈ Z , we define D k,n := min { m ≤ n ; C m = k } , which means that the D k,n th valid packet is the last packetwhich arrives from source k before the n th update. Note that D k,n = n if the valid packet arriving at α ∗ n is generated bysource k . Using this notation, we have A † k,n = S ∗ D k,n + P n − j = D k,n R j , (7) ✁✂✄☎✆✝(cid:0)✞✟✠✡☛ ☞✞✟✠✡☛ ☞✌☛ (cid:2) ✍✎✏✑ ✒✒✓✔✕✖✗ (cid:3) ☞✞✟✠✡✘✙☛(cid:0)✞✟✠✡✘✙☛ ☞✌✚✙☛ (cid:4) ✍✎✏✑✛✆✜ (cid:4) ✄✜ (cid:2) ✄☎✆(cid:0)✌✚✙☛ (cid:0)✌☛ (cid:1) ✁✂✍✎✏✑✝ (cid:1) ✆✂✍✎✏✑✝ (cid:1) ✆✂✄☎✆✝ (cid:1) ✁✂✄✝ (cid:1) ✆✂✄✝ (cid:4) ✄☎✆✜ (cid:4) ✍✎✏✑✜ ✒✒✒✒✒✒✓✙✕✖✗✓✢✕✖✗ ✒✒✒ Fig. 3: Two sample paths of the AoI processes A k ( t ) ( k = 1 , ).which implies that { A † ,n } , { A † ,n } and { A † ,n A † ,n } are sta-tionary and ergodicity.Since all packets have the same service time distribution, { C n } are the i.i.d. random variables such that P ( C n = 1) = λ /λ and P ( C n = 2) = λ /λ . We then have P ( D k,n = m ) = (cid:26) ( λ k /λ )(1 − λ k /λ ) n − m , m ≤ n, , m > n. (8)Note that { D k,n } is independent of { S ∗ n } and { R n } . It thenfollows from (7) and (8) that, for n ∈ Z and k = 1 , , E [ A † k,n ] = E h E h S ∗ D k,n + P n − j = D k,n R j (cid:12)(cid:12) D k,n ii = E [ S ] + λλ k (cid:18) − λλ k (cid:19) E [ R ] . Applying Lemmas 1 and 2 to the above, we obtain E [ A † k,n ] = 1 λ + µ + λλ k (cid:18) − λ k λ (cid:19) (cid:18) λ + 1 µ (cid:19) . In addition, we have, for m, ℓ ∈ Z , P ( D ,n = m, D ,n = ℓ )= ( λ /λ )( λ /λ ) n − ℓ , m = n, ℓ < n, ( λ /λ )( λ /λ ) n − m , m < n, ℓ = n, , otherwise. (9)Note that S ∗ n does not depend on S ∗ n − m and R ℓ for ℓ = n − ,but depends on R n − . Thus, from (7) and (9), we obtain E [ A † ,n A † ,n ]= E h E [ A † ,n A † ,n | D ,n , D ,n ] i = E [ S ∗ ] + (cid:26) λ λ λ − (cid:27) E [ S ∗ ] E [ R ] + E [ S ∗ R ] , By definitions of R n , we have S ∗ R = S ∗ { ( α ∗ + S ∗ ) − ( α ∗ + S ∗ ) } . Thus, it follows from Lemma 1 and (6) that E [ S ∗ n R n − ] = E [ S ∗ ] E [ α ∗ − α ∗ ] + E [( S ∗ ) ] − E [ S ∗ ] E [ S ∗ ]= (cid:18) λ + 1 µ (cid:19) (cid:18) λ + µ (cid:19) + (cid:18) λ + µ (cid:19) . (10)Applying Lemmas 1, 2 and (10) to the above, we obtain E [ A † ,n A † ,n ] = 2 (cid:18) λ + µ (cid:19) + (cid:18) λ λ λ − (cid:19) (cid:18) λ + 1 µ (cid:19) λ + µ . ✷ Using Lemma 3, we obtain the main theorem of this paper.
Theorem 2
The correlation coefficient of AoIs in 2-sourceM/M/1/1 push-out queue, denoted by ρ , is given by ρ = − λ λ µλ p ( λ + 2 λ µ + µ )( λ + 2 λ µ + µ ) (11) Proof.
Using the pointwise ergodic theorem (see, e.g., [6,Theorem 1.6.4]) yields E [ A A ] = lim T →∞ T Z T A ( t ) A ( t )d t. (12)Dividing the integral in (12) by update times { β n } , we have E [ A A ] = lim T →∞ ∆( T ) + lim T →∞ N ( T ) T · N ( T ) N ( T ) X n =1 F n , (13)where N ( t ) denotes the total number of updates in [0 , t ) and δ ( T ) = − T Z β ∗ A ( t ) A ( t )d t + 1 T Z Tβ ∗ N ( T ) A ( t ) A ( t )d t,F n = Z β ∗ n β ∗ n − A ( t ) A ( t )d t. (14)Note that, for t ∈ [ β ∗ n , β ∗ n +1 ) , A ( t ) A ( t ) = (cid:16) t − β ∗ n + A † ,n (cid:17) (cid:16) t − β ∗ n + A † ,n (cid:17) . (15)We estimate the right-hand side of (13). From (15), we have | ∆( T ) · T | ≤ R ( A † , + R )( A † , + R ) (16) + R N ( T ) ( A † ,N ( T ) + R N ( T ) )( A † ,N ( T ) + R N ( T ) ) . It follows from Lemmas 2 and 3 that R n and A k ( β ∗ n ) are finitew.p.1. Thus, it follows from (16) that lim T →∞ ∆( T ) = 0 . (17)Furthermore, { R n } is the i.i.d. random variables, because thesystem becomes empty at time β ∗ n , n ∈ Z . Thus, it followsfrom the elementary renewal theorem and Lemma 2 that lim T →∞ N ( T ) T = 1 E [ R ] = (cid:18) λ + 1 µ (cid:19) − . (18)urthermore, using the pointwise ergodic theorem, we have lim T →∞ N ( T ) N ( T ) X n =1 F n = lim N →∞ N N X n =1 F n = E [ F ] . (19)where the first equation holds because it follows from (18) that lim T →∞ N ( T ) = ∞ . Substituting (17)–(19) into (13) yields E [ A A ] = (cid:18) λ + 1 µ (cid:19) − · E [ F ] . (20)Next, we calculate E [ F ] . Applying (15) to (14), we have F n = R n R n A † ,n + A † ,n ) + R n A † ,n A † ,n . Using Lemmas 2 and 3, it follows from the above that E [ F ] = E [ R ]3 + E [ R ]2 E [ A † ,n + A † ,n ] + E [ A † ,n A † ,n ] E [ R ∗ ]= λ λ λ (cid:18) λ + 1 µ (cid:19) − (cid:18) λ + 1 µ (cid:19) λ + µ . Combining the above and (20), we obtain E [ A A ] = λ λ λ (cid:18) λ + 1 µ (cid:19) − (cid:18) λ + 1 µ (cid:19) λ + µ . Consequently, from the above and Corollary 1, we obtain ρ = E [ A A ] − E [ A ] E [ A ] p V [ A ] V [ A ]= − λ λ µλ p ( λ + 2 λ µ + µ )( λ + 2 λ µ + µ ) . ✷ From Theorem 2, we can see some nontrivial properties onthe systems as follows. The proof of Corollary 2 is omitteddue to the page restriction.
Corollary 2
The following statements holds. (i)
AoIs of sources 1 and 2 have a negative correlation; thatis, ρ < . (ii) When any of λ , λ , or µ approaches infinity, the corre-lation coefficient ρ converges to zero. (iii) The minimum value of the correlation coefficient ρ is − / , which is achieved when λ / λ / µ . V. N
UMERICAL R ESULTS
In this section, we provide numerical results of the corre-lation coefficient presented in Theorem 2, and confirm thestatements presented in Corollary 2 through the numericalresults. Fig. 4 shows the correlation coefficients which isnumerically computed with several cases using Theorem 2,where the x -axis represents the value of λ , the arrival rate ofsource 1. Note that the parameter µ is fixed for each curve inFig. 4, and λ is fixed as λ = 2 .From Fig. 4, we observe that ρ is always negative, that is,the two AoIs in our model always have a negative correlation,which implies (i) in Corollary 2. We also find that the curves in (cid:1)(cid:2) (cid:3)(cid:3) (cid:4) (cid:5) (cid:6) (cid:7) (cid:8) (cid:2)(cid:9) (cid:10) (cid:1)(cid:2) (cid:4) (cid:11)(cid:11) (cid:8) (cid:12) (cid:8) (cid:4) (cid:9) (cid:7) (cid:10) (cid:1) (cid:1) Fig. 4: Numerically computed results of ρ with several patternsof the parameter µ , and with fixed parameter λ as λ = 2 .the figures are all convergent to zero, so that (ii) in Corollary 2is likely to hold with respect to λ . Moreover, we see that thecorrelation coefficient has a minimum value with respect to λ . This means that a certain arrival rate of packets from onesource gives the strongest negative correlation with the othersource. Furthermore, we see from the figure that the smallestminimum value of ρ is seen when µ = 4 . Actually, the smallestvalue in the figure is − / , and is achieved when ( λ , λ , µ ) =(2 , , , which corresponds to (iii) in Corollary 2.VI. C ONCLUSION
In this paper, we considered a correlation of the AoIs of twosources sharing one server to process information, and derivedthe closed form expression of the correlation coefficient, onthe model of M/M/1/1 queueing systems. In addition, wealso derived the expression of the LST of the stationarydistribution of each AoI on the assumption that multiple K sources share the one server. From our analysis, we foundthat the correlation coefficient is always negative, and thatthe correlation coefficient has a certain minimum value. Thisindicates that there is always a negative correlation betweenthe two AoIs, and the strongest negative correlation is achievedby adjusting the parameters introduced in our model.For further study, it would be expected that the correlationcoefficient is investigated with more generalized assumptionsfor modeling the systems because our model adopts an ele-mental M/M/1/1 queueing systems with the common servicerate for all the sources. To consider the correlation amongseveral sources is also an interesting research point.A PPENDIX
AThis appendix is devoted to the proof of (4). For k ∈ K and n ∈ Z , let N k,n denote the number of arriving packets ofsource k in (0 , β ∗ n ] . We define M k,n as the number of arrivingpackets of source k in ( β ∗ n , β ∗ n +1 ] ; that is, M k,n = N k,n +1 − N k,n . In addition, we define U k,n := α k,n +1 − α k,n .et also J k,n denote the length from the time that the n thvalid packet departs to the time that a new packet arrivesfrom source k ; that is J k,n = U k,N k,n − S k,N k,n . We havethe following relation (see Fig. 2). R k,n = J k,n + M k,n − X ℓ =1 U k,N k,n + ℓ + S k,N k,n +1 . which leads to E (cid:2) e − sR k,n (cid:12)(cid:12) M k,n (cid:3) = E (cid:2) e − sJ k,n (cid:3) · E h e − sS k,Nk,n +1 i × Q M k,n − ℓ =1 E h e − sU k,Nk,n + ℓ (cid:12)(cid:12) M k,n i . (21)For ℓ = 1 , . . . , M k,n − , the ( N k,n + ℓ ) -th arrival packet isnot valid. We then have, for ℓ = 1 , . . . , M k,n − , E h e − sU Nk,n + ℓ (cid:12)(cid:12) M k,n i = E (cid:2) e − sX λk (cid:12)(cid:12) min ℓ ∈ K X λ ℓ ≤ X µ (cid:3) = λ + µs + λ + µ s + λλ λ k s + λ k , (22)where X a denotes a random variable following the exponentialdistribution with mean /a for a > . Furthermore, it followsfrom the memoryless property that E (cid:2) e − sJ k,n (cid:3) = E h e − s ( X λk − X µ ) (cid:12)(cid:12) X λ k > X µ i = λ k s + λ k . (23)Note that S k,N k,n +1 is the service time of a valid packet. Thus,it follows from Lemma 1 that E [e − sS k,Nk,n +1 ] = λ + µs + λ + µ . (24)Applying (22)–(24) to (21), we obtain E (cid:2) e − sR k,n | M k,n (cid:3) = λ + µs + λ + µ λ k s + λ k × (cid:18) λ + µs + λ + µ λ k s + λ k s + λλ (cid:19) M k,n − . (25)Finally, we show the distribution function of M k,n . A packetis valid if no packets (of all sources) arrive before its servicecompletion. It then follows from the independence of { α k,n } and { S k,n } that a packet is valid with probability µ/ ( λ + µ ) ,independently other packets. Therefore, for any n ∈ N and k ∈ K , M k,n follows the geometric distribution on N withparameter µ/ ( λ + µ ) . Thus, from (25), we obtain E [e − sR k,n ] = λ k µ ( s + λ )( s + µ ) − ( λ − λ k ) µ . (26)A PPENDIX
BThis appendix is devoted to the proof of Lemma 2. Wederive the moment generation function of R n . Let N n denotethe total number of packets arriving from either source 1 and2 in (0 , β ∗ n ] . In addition, we define U n := α n +1 − α n . We define M n := N n +1 − N n and J n := U N n − S N n . Assimilar way to (21), we have E (cid:2) e − sR n (cid:12)(cid:12) M n (cid:3) = E (cid:2) e − sJ n (cid:3) · E h e − sS Nn +1 i × Q M n − ℓ =1 E (cid:2) e − sU Nn + ℓ (cid:12)(cid:12) M n (cid:3) . (27)For ℓ = 1 , . . . , M n − , the ( N n + ℓ ) -th arrival packet is notvalid. We then have, for ℓ = 1 , . . . , M n − , E (cid:2) e − sU Nn + ℓ (cid:12)(cid:12) M n (cid:3) = E (cid:2) e − sX λ | X λ ≤ X µ (cid:3) = λ + µs + λ + µ . (28)Furthermore, it follows from the memoryless property that E (cid:2) e − sJ n (cid:3) = E h e − s ( X λ − X µ ) (cid:12)(cid:12) X λ > X µ i = λs + λ . (29)Note that S N n +1 is the service time of a valid packet. Thus,it follows from Lemma 1 that E h e − sS Nn +1 i = λ + µs + λ + µ . (30)Applying (28)–(30) to (27), we obtain E (cid:2) e − sR n (cid:12)(cid:12) M n (cid:3) = (cid:18) λ + µs + λ + µ (cid:19) M n λs + λ . (31)As similar to (21), for any n ∈ N , M n follows the geometricdistribution on N with parameter µ/ ( λ + µ ) . Thus, from (31),we obtain E [e − sR n ] = λs + λ µs + µ , which means that R n follows the convolution of X λ and X µ .R EFERENCES[1] S. Kaul, R. Yates, and M. Gruteser, Real-time status: How often shouldone update?. in
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