Age and Value of Information: Non-linear Age Case

0.24 MB·

by  Antzela Kosta

Age and Value of Information: Non-linear Age Case
Antzela Kosta∗, Nikolaos Pappas∗, Anthony Ephremides∗†, and Vangelis Angelakis∗
∗ Department of Science and Technology, Linko¨ping University, Campus Norrko¨ping, 60 174, Sweden
† Electrical and Computer Engineering Department, University of Maryland, College Park, MD 20742
E-mail: {antzela.kosta, nikolaos.pappas, vangelis.angelakis}@liu.se, [email protected]
Abstract—We consider a real-time status update system con- was introduced. In [7], the authors consider the problem of
sisting of a source-destination network. A stochastic process is optimizing the PAoI by controlling the arrival rate of update
observed at the source, and samples, so called status updates,
messages and derive properties of the optimal solution for the
are extracted at random time instances, and delivered to the
M/G/1 and M/G/1/1 models.
destination. In this paper, we expand the concept of information
7 ageing by introducing the Cost of Update Delay (CoUD) metric Controlling the messages in a network can increase the
1
to characterize the cost of having stale information at the desti- performance,startingfromasimplelast-generated-first-served
0
nation. We introduce the Value of Information of Update (VoIU) (LGFS) service discipline [8], to more complicated packet
2
metricthatcapturesthereductionofCoUDuponreceptionofan
management that discards non informative packets [6], [9],
n update.TheimportanceoftheVoIUmetricliesonitstractability
[10].In[11]theauthorsintroducepacketdeadlinesasacontrol
a which enables the minimization of the average CoUD.
J mechanism and study its impact on the average age of a
4 I. INTRODUCTION M/M/1/2 system. In [12] on the other hand, the authors take
2 Our work is motivated by the need for adaptability to into consideration packet delivery errors, i.e., update packets
meet stringent timeliness constraints in communication sys- can get lost during transmissions to their destination. In [13]

]
tems, arising from sensing and actuation applications within the authors consider multiple servers where each server can
T
the IoT. Characterization of time-critical information can be be viewed as a wireless link. They prove that a preemptive
I
s. done through the so called real-time status updates that are LGFS service simultaneous optimizes the age, throughput,
c messages carrying the timestamp of their generation. Status and delay performance in infinite buffer queuing systems. In
[
updates can range from sensor observations to stock market [14] the minimization of age is done over general multihop

1 data, tracking time-varying content over a network. networks. Another control policy is to assume that the source
v A common objective of such communications systems is is monitoring the network servers idle/busy state and is able
7 to maximize the freshness of the received data. We consider to generate status updates at any time, as in [15], [16].
2
a stochastic process being observed by a source that extracts In [15] the authors define an age penalty function of a
9
6 samplesorstatusupdatesatrandomtimes.Thestatusupdates general form, to characterize the level of “dissatisfaction” for
0 are transmitted over a network in order to update the destina- data staleness. Pushing forward, we investigate the Cost of
. tion node about the evolution of the process. Update Delay (CoUD) metric for three sample case functions
1
0 To quantify freshness, the concept of Age of Information thatcanbeeasilytunedthroughaparameter.Foreachcase,we
7 (AoI)hasbeenintroducedin[1],tocharacterizethetimeliness derive tractable expressions of the average cost for a M/M/1
1 of information in a status update system. The age captures model with a first-come-first-served (FCFS) queue discipline.
:
v the elapsed time since the last received status update was Althoughin[15]penaltyfunctionsaresaidtobedetermined
i generated. More specifically, at time of observation age is by the application, we go further and associate the cost of
X
defined as the current time excluding the time at which the stalenesswiththestatisticsofthesource.Priortodefiningthis
r
a observed state was generated. Keeping average AoI small association, we first need to elaborate on the requirement of
corresponds to maintaining fresh information. smallAoI.WhyareweinterestedinsmallAoI?Considerthat
Part of AoI research has so far focused on the use of weareobservingasystemattimeinstantt.However,themost
different queuing models through which the status updates recent value of the observed process available is the one that
maybeprocessed.Theaverageagehasbeeninvestigatedin[1] hadarrivedatt−∆,forsomerandom∆.Nowassumethatthe
for the M/M/1, D/M/1 and M/D/1 queues. In [2] the authors destinationnodewantstoestimatetheinformationattimet.If
take into consideration a more dynamic feature of wireless the samples at t and t−∆ are independent, the knowledge of
networks, that is, packets traveling over the network might t−∆ is not useful for the prediction and age simply indicates
reach the destination through multiple paths. This gives rise delay. However, if the samples at t and t−∆ are correlated,
to out-of-order delivery and thus non-informative (obsolete) then the value of ∆ will affect the accuracy of the prediction.
packets [3]. The performance of the M/M/1, M/M/2, and Asmaller∆canleadtoamoreaccurateprediction.Ourwork
M/M/∞ cases is investigated in [2]. Multiple sources are is a first step towards exploring this potential usage of AoI.
studied in [4], [5], where the authors characterize how the Next, we introduce a novel metric called Value of Informa-
service facility can be shared among multiple update sources. tion of Update (VoIU) to capture the degree of importance of
In [6], the new metric of peak age of information (PAoI) the information received at the destination. A newly received
update reduces the uncertainty of the destination about the to measure the degree of importance of the information
currentvalueoftheobservedstochasticprocess,andVoIUcap- received at the destination. Intuitively, this metric depends on
tures that reduction that is directly related to the time elapsed two system parameters at time of observation: (i) the cost of
since the last update reception. Following this approach, we update delay at the destination (ii) the time that the received
take into consideration not only the probability of a reception updatewasgenerated.Thiscanbeeasilyshowntobesimilarly
event, but also the impact of the event on our knowledge of expressed as a dependence on: (i) the interarrival time of the
the evolution of the process. last two packets received (ii) the current reception time.
SmallCoUDcorrespondstotimelyinformationwhileVoIU The value of information is a bounded fraction that takes
represents the impact of the received information in reducing values in the real interval [0,1], with 0 representing the
the CoUD.Therefore,in acommunication systemit wouldbe minimum benefit of an update and 1 the maximum.
highly desirable to minimize the average CoUD, and at the Lemma 1: In a system where status updates are instanta-
same time maximize the average VoIU. To this end we obtain neously available from the source to the destination, VoIU is
the average VoIU for the M/M/1 queue and discuss how the given by:
optimal server utilization with respect to VoIU can be used V = lim V =1. (3)
i i
combined with the CoUD average analysis. t(cid:48)i→ti
The interpretation of this property is that in the extreme case
II. SYSTEMMODEL whenthesystemtimeisinsignificantandapacketreachesthe
Weconsiderasysteminwhichasourcegeneratesstatusup- destination as soon as it is generated, we assign to the VoIU
dates in the form of packets, transmitted through a network to metricthemaximumvaluereflectingthatthereceptionoccurs
aremotedestination.Thegeneratedpacketsarequeuedbefore without value loss.
transmitted over the network and are transmitted according to Finally, to explore a wide array of potential uses of the
a FCFS queue discipline. notionofcost,weexploreinthispaperthreesamplecasesfor
Upon reception of a new status update, Yates et al. [1] the f (·) function
s
definedAgeofInformation∆(t)=t−u(t)tobethedifference 
αt
of the current time instant and the time stamp of the received 
update. In this paper we expand the notion by defining Cost fs(t)= eαt−1 (4)
of Update Delay (CoUD) log(αt+1)
C(t)=f (t−u(t)) (1) for α≥0. As discussed earlier, we can not leverage CoUD if
s
we do not assume that the samples of the observed stochastic
to be a stochastic process that increases as a function of time
process are correlated. Thus, we propose the adjustment of
between updates [15]. We introduce a non-negative, mono-
CoUDaccordingtotheautocorrelationoftheprocess.Specifi-
tonically increasing category of functions f (t), to represent
s cally,iftheautocorrelationissmallwesuggesttheexponential
the evolution of cost of update delay according to the data
function, while if the autocorrelation is large the logarithmic
characteristics of the source of the new information. This
function is more suitable. For intermediate values the linear
staleness metric increases with time and ensures that the
case is a reasonable choice.
status at the destination is as timely as possible based on the The autocorrelation R(t ,t ) = E[x(t )x∗(t )] of a
1 2 1 2
autocorrelation structure of the source signal.
stochastic process is a positive definite function, that is
Update i is generated at time ti and is received by the (cid:80) β β∗R(t ,t ) > 0, for any β and β . Tuning the
destination at time t(cid:48). The cost of information absence at the i,j i j i j i j
i parameter α properly enables us to associate with accuracy
destination increases as a function f (t) of time. Note that
s the right f (·) function to a corresponding autocorrelation.
s
age as coined by Yates is a special cost case, where the cost
Next, we focus on VoIU and analyze it for each case of f (t)
s
is counted in time units, as shown in Fig. 1a. In this paper
separately.
we consider that the cost can take any form of a “payment”
A. Value of Information of Update Analysis
function that can also assign to it any relevant unit.
The ith interarrival time Y =t −t is the time elapsed We first derive useful results for the general case with-
i i i−1
between the generation of update i and the previous update out considering specific queueing models. For the first case,
generation and is a random variable. Moreover, T =t(cid:48)−t is f (t)=αt, expression (2) yields
i i i s
the system time of update i corresponding to the sum of the Y
waitingtimeatthequeueandthenetworkdelay.Notethatthe VP,i = Y +iT . (5)
i i
randomvariablesY andT arerealsystemtimemeasuresand
i i Note that for α = 1 the cost of update delay corresponds
are independent of the way we choose to calculate the cost of
to the timeliness of each status update arriving and is the so
update delay i.e., of f (t).
s called age of information. The cost reductions {D ,...,D },
At time t(cid:48), the cost C(t(cid:48)) is reset to f (t(cid:48) −t ) and we 1 n
i i s i i depicted in Fig. 1a, correspond to the interarrival times
introduce the Value of Information of Update (VoIU) i as
{Y ,...,Y }, and also the limits
1 n
f (t(cid:48) −t )−f (t(cid:48) −t )
Vi = s i f i(−t(cid:48)1−t s)i i (2) lim VP,i =1, (6)
s i i−1 Yi→+∞
C(t) C(t) C(t)
Q3
Q3 D3 Q3 D3 D2 D3
C0 Q1 DQ12 D2 Q4 Qn Dn C0 Q1 D1 Q2 D2 Q4 Qn Dn C0 Q1 DQ12 Q4 Qn Dn
t0 t1 t2 t3 t4 tn−1 tn time t0 t1 t2 t3 t4 tn−1 tn time t0 t1 t2 t3 t4 tn−1 tn time
t(cid:48) t(cid:48) t(cid:48) t(cid:48) t(cid:48) t(cid:48) t(cid:48) t(cid:48) t(cid:48) t(cid:48) t(cid:48) t(cid:48)
1 2 3 n 1 2 3 n 1 2 3 n
Y2 T2 Yn Tn Y2 T2 Yn Tn Y2 T2 Yn Tn
(a) (b) (c)
Fig. 1: Example of linear, exponential and logarithmic CoUD evolution.
lim V =0, (7) Additionally, defining the effective arrival rate as
P,i
Ti→+∞
agree with the definition. Next, for fs(t)=eαt−1, shown in λ= lim N(T) (12)
Fig. 1b, the definition of VoIU is T→∞ T
eα(Yi+Ti)−eαTi andnoticingthatN(T)→∞asT →∞,andthatthesample
V = , (8)
E,i eα(Yi+Ti)−1 average will converge to its corresponding stochastic average
due to the assumed ergodicity of V , we conclude with the
and the corresponding limits are lim V = 1, i
Yi→+∞ E,i expression
limATti→la+st∞,foVrEt,hie=ca1se−fe−(tα)Y=i. log(αt+1),depictedinFig.1c, V = lim VT =λE(cid:2)V(cid:3), (13)
s T→∞
we obtain
where E[·] is the expectation operator.
log(α(Y +T )+1)−log(αT +1)
V = i i i , (9)
L,i log(α(Y +T )+1)
i i III. COSTOFUPDATEDELAYCOMPUTATIONFORTHE
limYi→+∞VL,i = 1, and limTi→+∞VL,i = 0. The previous M/M/1SYSTEM
results can be interpreted as follows. As the interarrival times
of the received packets become large the value of information ForaM/M/1systemstatusupdatesaregeneratedaccording
of the updates takes its maximum value, underlining the to a Poisson process with mean λ and thus the interarrival
importance to have a new update as soon as possible. On the times Yi are independent and identically distributed i.i.d ex-
other hand, when the system time gets significantly large we ponential random variables with E[Y] = 1/λ. Furthermore,
expect that the received update is not as timely as we would the service times are i.i.d. exponentials with mean 1/µ and
prefer in order to maintain the freshness of the system, hence the server utilization is ρ= λ.
µ
we assign to the VoIU metric the minimum value. Additionally, the probability density function of the system
Supposethatourintervalofobservationis(0,T).Then,the time T for the M/M/1 is [17]
timeaveragevalueofinformation(normalizedbytheduration
P (t)=µ(1−ρ)e−µ(1−ρ)t, t≥0. (14)
of time interval) is given by T
N(T) Note that the variables Y and T are dependent and this
1 (cid:88)
VT = T Vi. (10) complicates the calculations of the average cost of update
i=1 delay in the general case. The time average CoUD of (1) in
Without loss of generality we assume that the first packet this scenario can be calculated as the sum of the disjoint Q ,
1
generationwasatthetimeinstantt andtheobservationbegins Q for i≥2, and the area of width T over the time interval
0 i n
att=0withanemptyqueueandthevalueC(0)=C .More- (t ,t(cid:48) ). This decomposition yields
0 n n
over,theobservationintervalendswiththeservicecompletion
of N(T) samples, with N(T) = max{n|tn ≤ T} denoting C = Q1+Tn2/2+(cid:80)Ni=(2T)Qi. (15)
the number of arrivals by time T. T T
The time average value in (10) is an important metric
Below we derive the average CoUD for the three cases of the
taken into consideration when evaluating the performance of
f (t) function that we have considered and find the optimum
s
a network of status updates and should be calculated for each
server policy for each one of them.
case of f (t) separately. The time average VoIU for the three
s For f (t) = αt, the area Q for i ≥ 2 is a trapezoid equal
considered cases can be rewritten as s i
to the difference of two triangles, hence
N(T)
N(T) 1 (cid:88)
VT = T N(T) i=1 Vi. (11) QP,i = 12α(Ti+Yi)2− 21αTi2 =α(cid:2)YiTi+ Y2i2(cid:3). (16)
Next, for f (t)=eαt−1, the area Q yields Further, using the iterated expectation and the probability
s i
density function of X, (25) implies
(cid:90) t(cid:48) (cid:90) t(cid:48)
i i
QE,i = ti−1(eα(t−ti−1)−1)dt− ti (eα(t−ti)−1)dt= E(cid:2)VP(cid:3)= µ(12−λ ρ) 2F1(1,2;3;2λλ−µ), (26)
= α1(cid:2)eα(Yi+Ti)−eαTi(cid:3)−Yi. (17) where the integral is calculated with the help of [18, 6.228]
and F is the hypergeometric function [19].
2 1
And lastly, for f (t)=log(αt+1) we obtain
s Next, for the cases f (t) = eαt−1 and f (t) = log(αt+
s s
QL,i =(cid:90) t(cid:48)i log((α(t−ti−1)+1)dt−(cid:90) t(cid:48)ilog(α(t−ti)+1)dt 1E)(cid:2),VwLe(cid:3).compute numerically the expected values E(cid:2)VE(cid:3) and
ti−1 ti
= 1(cid:2)(α(Y +T)+1)log(α(Y +T)+1)−(αT +1)log(αT +1)(cid:3)−Y. V. NUMERICALRESULTS
α i i i i i i i
(18) In this section, we evaluate the performance in terms of
The time average CoUD for the three cases can be rewritten the CoUD and VoIU metrics, as calculated in the previous
as sections. We consider a M/M/1 system model with service
Q˜ N(T)−1 1 N(cid:88)(T) rate µ = 1, therefore the server utilization ρ is equal to the
CT = T + T N(T)−1 Qi (19) arrival rate λ.
i=2 InFigure2a,weillustratethevariationoftheaverageCoUD
where, Q˜ =Q +T2/2 is a term that will vanish as T →∞. and VoIU with the server utilization ρ, for the linear case.
1 n
Recall that for f (t) = αt the VoIU is independent to the
Then, similarly to the VoIU analysis, we conclude with the s
parameter α, therefore for multiple CoUD curves corresponds
expression
C = lim C =λE(cid:2)Q(cid:3). (20) only one VoIU curve. This indicates that as far as the cost
T
T→∞ per time unit is linearly increased, higher cost leads to higher
average CoUD, but the same average VoIU. This is due to
For the linear case, using the result of [1], we obtain
the fact that we assign to each unit of time the same cost.
1 (cid:18) 1 ρ2 (cid:19) Increasing α results to proportional increase of the average
C =α 1+ + . (21)
P µ ρ 1−ρ CoUD,howevertheoptimalserverpolicyisthesameforevery
function.Moreover,theoptimalpolicywithrespecttoVoIUis
Next, for the exponential case we compute the terms
differentthantheoneforCoUDwiththeformerbeingactually
(cid:40)
E(cid:2)eαT(cid:3)= α−−µµ((11−−ρρ)) , if α−(µ−λ)<0 (22) greater. Thus, it is more appropriate to increase rather than
decrease ρ in case the optimal utilization can not be achieved.
+∞ , otherwise,
In Figure 2b the average CoUD is depicted as a function
of the server utilization for the linear, exponential, and loga-
(cid:40)
E(cid:2)eα(Y+T)(cid:3)= [α−µµ((11−−ρρ))](λα−λ) , if α−λ<0 (23) rithmic functions with parameter α=0.1. All three functions
+∞ , otherwise. have similar behaviour, with the minimum CoUD achieved
when ρ ≈ 0.5. Over all values of ρ, the exponential f
s
Afterapplyingalltherelevantexpressionsto(20),wefindthe has the greatest CoUD, followed by the linear f and then
s
average CoUD to be the logarithmic f , that is, C > C > C . However, as
s E P L
(cid:18) (cid:18) (cid:19)(cid:19) ρ deviates from the optimum, we see that the exponential
1 µ(1−ρ) λ
CE = αλ α−µ(1−ρ) α−λ +1 −1. (24) functionbecomessharperthatthelinearfunction,andthelog-
arithmic function becomes smoother. For smaller utilizations
Finally, for the logarithmic case we compute some terms of where status updates are not frequent enough and for higher
equation (18) and others are evaluated numerically. We omit utilizationwherepacketsspendmoretimeinthesystemdueto
the details due to space limit. backlogs, CoUD is increased. For this increase each function
sets its own cost per time unit that results to more intense
IV. VALUEOFINFORMATIONUPDATECOMPUTATIONFOR growth for the exponential average CoUD and less intense
THEM/M/1SYSTEM growth for the logarithmic CoUD.
Following the same procedure as in the CoUD metric we Figure2cpresentsthenumericalevaluationofthequantities
computetheaverageVoIUgivenby(13),fortheM/M/1queue λE(cid:2)VP(cid:3), λE(cid:2)VE(cid:3), and λE(cid:2)VL(cid:3) for three values of the param-
with a first-come-first-served discipline. For the fs(t) = αt eter α, 0.1, 0.5, and 1. As we shift from fs(t) = logt+1
case, the expected value E[V] conditioned on the interarrival to fs(t)=e0.1t−1 VoIU becomes greater over all ρ and all
time X =x can be obtained as functionsfollowsimilarbehaviour.Forallcases,themaximum
(cid:20) X (cid:46) (cid:21) VoIU is achieved when ρ ≈ 0.6. Note that VoI is directly
E X =x =−xµ(1−ρ)exµ(1−ρ) related to CoUD. On the average analysis, taking the linear
X+T
function as a point of reference, we see that choosing an
×Ei(−µ(1−ρ)x)), for (µ−λ)>0. (25)
exponential function would result in higher CoUD and VoIU,
20 6 0.3
CoUD1105 ffffssss((((tttt))))====1t00...551ttt 5 fffsss(((ttt)))===ealotagt-(1at+1) V0.25
VoIU00..0012..5505230.1 0fs.2(t)=at 0.3 0.S4erverU0t.i5lization0ρ.6 0.7 0.8 0.9 CCostofupdatedelay1234 ValueofInformationofUpdate0.001..512 fffsss(((ttt)))===etlo0g.1(0t-.11t+1)
0.1 ffss((tt))==lloogg((0t+.51t)+1)
0.05 0 0.05
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
ServerUtilizationρ ServerUtilizationρ ServerUtilizationρ
(a) (b) (c)
Fig. 2: Comparison of the average CoUD and VoIU vs. utilization for the M/M/1 system with µ = 1, and different function
cases.
while choosing the logarithmic function would result in lower [5] R. D. Yates and S. K. Kaul, “The age of information: Real-time
CoUD and VoIU. This tradeoff considers timeliness against status updating by multiple sources,” 2016. [Online]. Available:
http://arxiv.org/abs/1608.08622
timeliness that is combined with transmission resources (i.e.,
[6] M.Costa,M.Codreanu,andA.Ephremides,“Ageofinformationwith
bandwidth). packetmanagement,”inProc.IEEEInt.Symp.Inf.Theory(ISIT),Jun.
2014,pp.1583–1587.
VI. CONCLUSION [7] L.HuangandE.Modiano,“Optimizingage-of-informationinamulti-
class queueing system,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT),
In this study, we have considered the characterization of
Jun.2015,pp.1681–1685.
theinformationtransmittedoverasource-destinationnetwork, [8] S. K. Kaul, R. D. Yates, and M. Gruteser, “Status updates through
modelled as a M/M/1 queue. To capture freshness, we in- queues,”in46thAnnu.Conf.Inf.Sci.Syst.(CISS),Mar.2012,pp.1–6.
[9] M.Costa,M.Codreanu,andA.Ephremides,“Ontheageofinformation
troduce the CoUD metric through three cost functions that
instatusupdatesystemswithpacketmanagement,”IEEETransactions
can be chosen in relation with the autocorrelation of the onInformationTheory,vol.62,no.4,pp.1897–1910,Apr.2016.
process under observation. To characterize the importance of [10] N.Pappas,J.Gunnarsson,L.Kratz,M.Kountouris,andV.Angelakis,
“Age of information of multiple sources with queue management,” in
an update, we define VoIU that measures the reduction of
Proc.IEEEInt.Conf.Com.(ICC),Jun.2015,pp.5935–5940.
CoUD and therefore of uncertainty. CoUD and VoIU can [11] C. Kam, S. Kompella, G. D. Nguyen, J. E. Wieselthier, and
be used interchangeably depending on the application. We A. Ephremides, “Age of information with a packet deadline,” in Proc.
IEEEInt.Symp.Inf.Theory(ISIT),Jul.2016,pp.2564–2568.
analysed the relation between CoUD and VoIU and observed
[12] K. Chen and L. Huang, “Age-of-information in the presence of error,”
that convex and concave CoUD functions lead to a tradeoff inProc.IEEEInt.Symp.Inf.Theory(ISIT),Jul.2016,pp.2579–2583.
between CoUD and VoIU, while linearity reflects only on the [13] A. M. Bedewy, Y. Sun, and N. B. Shroff, “Optimizing data freshness,
throughput, and delay in multi-server information-update systems,” in
CoUD.
Proc.IEEEInt.Symp.Inf.Theory(ISIT),Jul.2016,pp.2569–2573.
Depending on the application we can choose the utilization [14] A. M. Bedewy, Y. Sun, and N. B. Shroff, “Age-optimal
that satisfies the minimum CoUD objective or the maximum information updates in multihop networks,” 2017. [Online]. Available:
https://arxiv.org/abs/1701.05711
VoIU objective. In the linear CoUD case, VoIU is indepen-
[15] Y.Sun,E.Uysal-Biyikoglu,R.Yates,C.E.Koksal,andN.B.Shroff,
dent to the cost assigned per time unit. In the exponential “Update or wait: How to keep your data fresh,” in 35th Annual IEEE
and logarithmic cases however, there is a tradeoff between Int.Conf.Comput.Commun.(INFOCOM),Apr.2016,pp.1–9.
[16] R. D. Yates, “Lazy is timely: Status updates by an energy harvesting
CoUD and VoIU. That is, the smaller the average CoUD,
source,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Jun. 2015, pp.
the smaller the average VoIU. For high correlation among the 3008–3012.
samples, choosing f (t)=logαt+1 decreases their value of [17] A.PapoulisandS.U.Pillai,Probability,randomvariables,andstochas-
s
information and equivalently choosing f (t)=eαt−1 in low ticprocesses. McGraw-Hill,Dec.2011.
s [18] I.S.Gradshteyn,I.M.Ryzhik,A.Jeffrey,andD.Zwillinger,Tableof
correlation has the opposite effect. integrals,series,andproducts. Elsevier2007.
[19] E. W. Weisstein, “From MathWorld—A Wolfram Web Resource.”
REFERENCES [Online].Available:http://mathworld.wolfram.com
[1] S.Kaul,R.Yates,andM.Gruteser,“Real-timestatus:Howoftenshould
one update?” in IEEE Int. Conf. Comp. Commun. (INFOCOM), Mar.
2012,pp.2731–2735.
[2] C. Kam, S. Kompella, G. D. Nguyen, and A. Ephremides, “Effect of
messagetransmissionpathdiversityonstatusage,”IEEETransactions
onInformationTheory,vol.62,no.3,pp.1360–1374,Mar.2016.
[3] C. Kam, S. Kompella, and A. Ephremides, “Age of information under
randomupdates,”inProc.IEEEInt.Symp.Inf.Theory(ISIT),Jul.2013,
pp.66–70.
[4] R.D.YatesandS.Kaul,“Real-timestatusupdating:Multiplesources,”
inProc.IEEEInt.Symp.Inf.Theory(ISIT),Jul.2012,pp.2666–2670.