Age of Information: The Gamma Awakening

Transcription

1 Age of Information: The Gamma Awakening Eie Najm and Rajai Nasser LTHI, EPFL, Lausanne, Switzerand Emai: {eie.najm, arxiv: v [cs.it] 5 Apr 06 Abstract Status update systems is an emerging fied of study that is gaining interest in the information theory community. We consider a scenario where a monitor is interested in being up to date with respect to the status of some system which is not directy accessibe to this monitor. However, we assume a source node has access to the status and can send status updates as packets to the monitor through a communication system. We aso assume that the status updates are generated randomy as a Poisson process. The source node can manage the packet transmission to minimize the age of information at the destination node, which is defined as the time eapsed since the ast successfuy transmitted update was generated at the source. We use queuing theory to mode the source-destination ink and we assume that the time to successfuy transmit a packet is a gamma distributed service time. We consider two packet management schemes: LCFS with preemption and LCFS without preemption. We compute and anayze the average age and the average peak age of information under these assumptions. Moreover, we extend these resuts to the case where the service time is deterministic. I. INTRODUCTION In status update systems, one or severa sources send information updates to one or severa monitors at a certain effective rate. Naturay, the goa of this process is to ensure that the status updates are as timey as possibe at the receiver side. For this purpose, [] uses the term age, to refer to the time eapsed since the generation at instant ut of the newest packet avaiabe at the receiver. Formay, the age of such packet is t t ut and the timeiness requirement at the monitor corresponds to a sma average age. Indeed, rea-time status updating can be modeed as a source feeding packets at rate to a queue which deivers them to the monitor with some deay. Hence, the requirement at the destination transates into finding the optima transmission scheme and/or the optima effective update rate at the source that minimizes im τ τ τ 0 tdt. However, numerous factors affect the evauation of such as the mode of the source update process, the number of sources, the mode of the queue, the number of queues avaiabe, etc. Kau et a. in [] sove one aspect of the probem where they consider a singe source generating packets as a rate Poisson process feeding them to a singe First Come First Served FCFS queue with exponentia service time. Using Kenda notation, this is an FCFS M/M/ system. Moreover, the authors aso consider the cases of deterministic source and exponentia service time, i.e., FCFS D/M/ system, as we as random source and deterministic service time, i.e., FCFS M/D/ system. Yates and Kau in [] generaize the probem soved in [] by considering the presence of mutipe sources sending updates through the same FCFS queue to the same monitor. Aong the same generaization direction as in [], one may ask: what woud happen if we increase the number of queues avaiabe, i.e., if the source is abe to serve mutipe updates at the same time? This question is tacked in [3], where a singe Poisson process is sending updates over an infinite number of queues with exponentia service time. However, in these aforementioned works, the authors mosty consider FCFS queues. One woud focus on Last Come First Served LCFS type of queues since they are intuitivey more suitabe for the probem in hand: we are interested in deivering the newest update to the monitor, which means we gain more by sending the youngest packet in the queue first. This idea is deveoped in [4] where the authors derive an expression for by treating the foowing two modes whie assuming exponentia interarriva and service time: i LCFS queue without preemption; if the queue is busy, any new update wi have to wait in a buffer of size. This means that the new update wi repace any oder packet aready waiting to be served. ii LCFS with preemption, where contrary to the first case, any new update wi prompt the source to drop the packet being served and start transmitting the newcomer. In [4], it is shown that an LCFS queue with preemption achieves a ower average age compared to the mode without preemption. However, both modes outperform the FCFS mode presented in []. In this paper, we aso consider these ast two schemes in order to derive cosed form expressions for. However, the main novety is the assumption of a gamma distribution for the service time in age of information probems. The motivation for such a distribution is twofod: Based on the cassica appications of gamma distributions in queuing theory, these distributions can be seen as a reasonabe approximation if we want to mode reay networks. Indeed, in such network, a transmitter and a receiver are separated byk reays with each reay taking an exponentia amount of time to compete transmission to the next hop. This means that the tota transmission time is the sum of k independent exponentia random variabes which induces a gamma distribution. As we wi see ater, a deterministic random variabe can be seen as the imit of a sequence of gamma distributed random variabes. Therefore, one can study the performance of the LCFS-based schemes under deterministic service time by taking the imit of the resut obtained for a gamma distributed service time. Athough this is an indirect method of cacuating, it is simper than the direct approach. This paper is organized as foows: in Section II we present

2 the preiminary resuts that wi be used throughout our work and define the average peak age as an aternative metric. In Section III we derive the cosed form expressions for both the average age and the average peak age when assuming an LCFS scheme with preemption. On the other hand, Section IV computes the formuas for these quantities when considering an LCFS queue without preemption. In these ast two sections the service time is assumed to be gamma distributed. However, in Section V we cacuate the two ages for a deterministic service time for each of the two schemes. Finay, Section VI presents numerica simuations that vaidate our theoretica resuts. II. A. Genera definitions PRELIMINARIES As we have seen, our two schemes of interest are LCFS with preemption and LCFS without preemption. The variation of the instantaneous age for these two scenarios is given in Figure. The saw-tooth pattern depicted in those figures is due to the foowing behavior of the age. Let t i be the time the i th packet is generated and et t i be the time the ith packet is received if it is successfuy received. Moreover, without oss of generaity, we assume the beginning of observation is at time t 0 and that the queue is empty at this instant with an initia age of 0. The age t increases ineary with time and is set to a smaer vaue when a packet is received. Hence, the instantaneous age is equa to the current time minus the generation time of the newest of the received packets. It is important to note that in both schemes of interest, some packets might be dropped. Hence we ca the packets that are not dropped, and thus deivered to the receiver, as successfuy received packets or successfu packets. In addition to that, we aso define:ii i to be the true index of thei th successfuy received packet, ii Y i t I i+ t I i to be the interdeparture time between two consecutive successfuy received packets, iii X i t Ii+ t Ii to be the interarriva time between the successfuy transmitted packet and the next generated one which may or may not be successfuy transmitted, so f X x e x, iv T i to be the system time, or the time spent by the i th successfu packet in the queue and v N τ max{n;t In τ}, the number of successfuy received packets in the interva [0,τ]. B. Computing the Average Age Using these quantities and Figures a and b, the authors in [4] show that im τ τ τ 0 tdt e EQ i, where e is the effective update rate and EQ i is the expected vaue of the area Q i at steady state. Hence, we need to determine these two quantities. Computing the effective rate: As stated in [5], e P{packet is received successfuy} 3 where P{packet is received successfuy} is the probabiity that a packet in the queue wi be deivered to the receiver. Computing EQ i : Based on Figures a and b, it was shown in [4] that Y EQ i ET i Y i +E C. Computing the Average Peak Age i. 4 Another metric of interest is the average peak age. We define the peak age as P i im t, t t I i t<t I i which is the vaue of the instantaneous age just before it is reduced by the reception of the i th successfu packet. From Figures a and b, we can deduce that the peak age can be written as P i T i +Y i. Therefore, the average peak age is given by: D. Defining the service time EP i ET i +EY i. 5 A the above resuts were obtained without any assumption on the service time. However, as we have discussed before, this paper studies two modes for the service time: a gamma distributed service time with parameters k, and a deterministic service time. Here is a brief description of the gamma distribution. Definition. A random variabe S with gamma distribution Γk, has the foowing probabiity density function: f S s sk e s k Γk. The Erang distribution Ek, is a specia case of the gamma distribution where k N. Such random variabe has a mean of ES k and a variance VarS k. These quantities wi come in handy ater on. Another important property of gamma random variabes is given by the foowing emma: Lemma. Suppose S n Γk n, n is a sequence of random variabes such that ES n µ, for some µ > 0. Then the sequence S n converges in distribution to a deterministic variabe Z as k becomes very arge, i.e, where Z µ S n d Z, as k, with probabiity. The above emma obviousy sti hods if S n Ek n, n. This emma provides an additiona motivation for studying the average age and the average peak age under the assumption of a gamma distributed service time since we can easiy extend the resuts to the deterministic service time mode by etting k.

3 t Δt Age Age 0 Q Q Q 3 Q N Δ 0 Q Q4 QNτ Q Q 3 t t t 3 t 4 t 5 t 6 t 7 t' t' 5 t' t 8 7 t' t N 8 t' N T Y T Y a Age of information for LCFS with preemption scheme t N - t t t Nτ- t t 3 t 4 t 5 t t' t' 6 t' 4 t' t 6 Nτ t' Nτ T Y T Y W 3 S 3 b Age of information for LCFS without preemption scheme t Fig.. Variation of the instantaneous age for both schemes III. AGE OF INFORMATION FOR LCFS WITH PREEMPTION In this section we wi compute the average age and the average peak age EP k for the Last Come First Served LCFS scheme with preemption and a gamma distributed service time. As we have seen before, in this scenario any packet being served is preempted if a new packet arrives and the new packet is served instead. Hence, the number of packets in the queue can be modeed as a continuous-time two-state semi-markov chain depicted in Figure. The 0-state corresponds to the state where the queue is empty and no packet is being served whie the -state corresponds to the state where the queue is fu and is serving one packet. However, given that the interarriva time between packets is exponentiay distributed with rate then one spends an exponentia amount of time X in the 0-state before jumping with probabiity to the other state. Once in the -state, two independent cocks are started: the gamma distributed service time cock of the packet being served and the rate memoryess cock of the interarriva time between the current packet and the next one to be generated. We jump back to the 0-state if the service time cock happens to tick before that of the interarriva time. Given that the interarriva times between packets are i.i.d as we as the service time of each packet, then the probabiity to jump from the -state to the 0-state does not depend on the index of the current packet. Hence, the jump from the -state to the 0-state occurs with probabiity p PS < X, where S is a generic gamma distributed service time and X is a generic rate memoryess interarriva time which is independent of S. On the other hand, if the interarriva time cock happens to tick before the service time cock then the current packet being served is preempted and the new generated packet takes its pace in the queue. Therefore, we stay in the -state and the two cocks are started anew independenty from before. This expains the p probabiity seen in Figure for staying in the -state. Given that the probabiity p wi be usefu in the computation of the average age as we as the average peak age, we start by deriving its expression here: p PS < X k. 6 + Now we are ready to derive the two age metrics. A. Average age We start by deriving the expression for the average age. We need to compute two quantities for this purpose: EQ i and the effective rate e. Computing EQ i : Using 4, we obtain: Y EQ i ET i Y i +E i Y ET i EY i +E i. 7 The second equaity comes from the fact that T i and Y i are independent since the interarriva time is exponentia and hence memoryess. In fact, thei th successfu packet eaves the queue empty and hence Y i ˆX i +Z i where ˆX i X i T i is the remaining of the interarriva time between the departure of the i th successfu packet and the arriva of the next generated one and Z i is the time for a new packet to be successfuy deivered. Z i does not overap with T i and thus is independent from it. As for ˆX i, we aso get that it is independent of T i. To prove this we notice that for a successfuy received packet i the joint distribution f Xi,T i x,t can be written as { 0 if x < t f Xi,T i x,t f X,Sx,t PS<X if x > t, 8 where X and S are the generic independent interarriva time and service time respectivey. Now, using a change of variabe we get f ˆXi,T i ˆx,t f Xi T i,t i ˆx,t f Xi,T i ˆx+t,t { 0 if x < 0 f X,Sˆx+t,t PS<X if x > 0 { 0 if x < 0 hˆxgt if x > shows that ˆXi and T i are indeed independent. Moreover, one can show that ˆXi is exponentia with rate. Given that ˆX i and Z i are both independent from T i, then Y i and T i are aso independent. From now on we wi drop the subscript index since at steady state T i and T i have same the distribution, which is aso the case for Y i and Y i. The foowing emma wi be used to evauate 7:

4 Fig.. 0, p -p Semi-Markov chain representing the queue for LCFS with preemption Lemma. Let G be gamma distributed with parameters k, and F be a rate exponentia random variabe independent of G. Then, conditioned on the event {G < F}, the distribution of G becomes gamma with parameters. k, + f G/G<F t tk e t+ kγk. 0 + Proof: In order to prove this Lemma we wi compute the probabiity density function f G G<F : Pt G < t+ǫ G < F f G G<F t im ǫ 0 ǫ a Pt G < t+ǫpg < F t < G < t+ǫ im ǫ 0 ǫpg < F PF > t f G t PG < F b f G t e t p tk e e t t k Γk + tk e t+ kγk + k where a is obtained by appying Bayes rue and in b, p is given by 6. In order to appy Lemma, we first notice that for a given packet i, the event {S i < X i } is equivaent to the event {packet i was successfuy received}. Hence the probabiity P PS i < α S i < X i is the probabiity that the service time of the i th packet is ess than α given that this packet was successfuy transmitted. However, since the service times and interarriva times are i.i.d then P does not depend on the index i. Now since T is the service time of a successfu packet then this eads us to PT < α PS i < α S i < X i PS < α S < X, where S and X are the generic service and interarriva time respectivey. By repacing G by S and F by X in Lemma, we deduce that the system time T is gamma distributed with parameters. Therefore, k, + ET k +. Now we turn our attention to the distribution ofy for which we compute its moment generating function. Before going further in our anaysis, we state the foowing emma. Lemma 3. Let G be gamma distributed with parameters k, and F be a rate exponentia random variabe independent of G. If F is a random variabe such that φ F s p where p PF < α PF < α F < G, then the moment generating function of F is given by s s+ s k + k., 3 Proof: We first start by computing the probabiity density function of X. Pt F < t+ǫ F < G f X t im ǫ 0 ǫ Pt F < t+ǫpf < G t F < t+ǫ im ǫ 0 ǫpf < G e t PG > t, p where p + k. So now we can cacuate the moment generating function of F. φ F s 0 0 p f F te st dt p e t PG > te st dt s 0 PG < te st dt d Using integration by parts and the fact that dtpg < t f G t tk e t k Γk, we get φ F s p s s+ s k Lemma 4. The moment generating function of Y is given by: φ Y s k. 4 s+ s Proof: By observing Figure we notice that Y is the smaest time needed to go from the 0-state back to the 0- state. Hence Y can be written as Y X+W where X is the generic interarriva time and W is the time spent in the -state before the first jump back to the 0-state. So W can be written as: S with probabiity p X +S with probabiity pp W X +X +S with probabiity p p. M X j +S, 5 j0

5 where X j is such that PX j < α PX < α X < S, S is such that PS < α PS < α S < X and M is a geometricp random variabe which is independent of X j and S, and which gives the number of discarded packets before the first successfu reception. Appying Lemmas and 3 on S and X respectivey and using the fact that M,S and X j are a mutuay independent, it foows that φ W s E e s M j0 X j φ S s k E φ X s M + + s + φ X s j p p j + s j0 k s k. 6 s+ s Moreover, since X and W are independent and φ X s s, we get using 6 φ Y s φ X sφ W s s+ s k. Now that we have found φ Y we can compute the first two moments of Y as EY +k and EY + k + k+ k. Combining these resuts with, we obtain, EQ i +k. 7 Computing the effective rate: Using 3 we get k e p. 8 + Now we are ready to compute the average age: We concude Proposition. The average age in the LCFS with preemption scheme assuming Γk, service time is given by: e EQ i +k. 9 Proof: Using 7 and 8. B. Average peak age Proposition. The average peak age in the LCFS with preemption scheme assuming Γk, service time is given by: EP i ET+EY k + + +k. 0 Proof: Using 5, and the vaue of EY. Fig. 3. A k,/ A,/ A 0,/ A 3 3,/... A k-,/ k IV. A ',/ A,/ A 3,/ ' 3'... A k-,/ k' A k,/ eve 0 eve Markov chain representing the queue for LCFS without preemption AGE OF INFORMATION FOR LCFS WITHOUT PREEMPTION Another interesting scheme worth to study is the LCFS without preemption. In this scenario, we assume that the queue has a buffer of size and we wait for the packet being served to finish before serving a new one. If whie serving a packet a new update arrives, it repaces any packet waiting in the buffer. In this section we wi derive a cosed form expression for the average age and the average peak age EP k for LCFS without preemption whie assuming an Erang distribution for the service time with parameter k,. An Erang distribution is nothing but a specia case of the gamma distribution where k N. Moreover, an Erang distribution k, can be seen as the sum of k independent memoryess random variabes A j, each with rate. Using this observation, we mode the state of the queue as a two-eve Markov chain as shown in Figure 3. As in the previous section, we wi denote the generic rate interarriva time by X and the generic Erang distributed service time by S k j A j. Using this notation, we notice that the service time can be represented as the succession of k exponentia-time steps that need to be accompished for a successfu reception. Hence, a packet in state j {,...,k} or j {,...,k } is a packet competing his j th step out of a tota of k. Moreover, the 0-state represents an empty queue, a the states of eve 0 represent an empty buffer and those of eve represent a fu buffer. After spending an exponentia amount of time in the 0-state, we can ony jump to the -state once a new update arrives. Using the memoryess property of the exponentia distribution, we can describe the evoution of this packet in the queue as foows: at state j {,..., k}, two exponentia cocks start simutaneousy. One cock denoted A j of rate and another one denoted Λ j of rate. If cock A j ticks first then the packet jump to state j+ and the buffer stays empty. Otherwise it jumps to state j since now the buffer is fu. On the other hand, if the packet is at state j and the A j cock ticks first then the packet jump to state j + without updating the buffer. However, if the Λ j ticks first then the packet stays in state j but we update the buffer with the new arriva.

6 A. Average age Proposition 3. The average age in the LCFS without preemption scheme assuming Erang Ek, service time is: k+ +3k k q k + +k +k+ k k+k +k + + +k+ k+ + +k ++ k +k+ k Proof: As in the previous section we need to compute the effective rate given by 3 and EQ i given by 4. Computing EQ i : Foowing the same ine of thoughts as in Section III, we wi cacuate ET i Y i by expressing it as the average of two conditionay independent variabes given some set { of events. For this end we define the famiy of events Ψ i j A i j > Λi j ; k j+ Ai }, < X where j k. Hence Ψ i j is the event that during the service time of the ith successfu packet a new update arrived at the j th step of the service time i.e, state j or j and then no new update arrived for the remainder of the service time. The superscript i is used to indicate that we are deaing with the i th successfu packet. Forj 0,Ψ i 0 is the event that theith successfu packet eaves the queue empty. Note that for everyi, {Ψ i j, j k} is a partition of the probabiity space. It is sufficient to condition on the event Ψ i 0 in order to ensure conditiona independence between T i and Y i. This is due to the foowing fact: given Ψ i 0, we know that the i th successfu packet eft the queue empty and hence we have a situation identica to that of the with preemption case see Section III and T i and Y i are independent. On the other hand, given Ψ i 0, the buffer is not empty and thus a new packet wi be served directy after the departure of the i th successfu packet. In this case, the interdeparture time Y i is simpy the service time of the i th successfu packet whose vaue is independent of T i W i + S i, where W i the waiting time and S i is the service time of the i th successfu packet see Figure b. Athough conditioning on Ψ i 0 is enough to obtain independence between T i and Y i, we wi need to condition on the two independent events Ψ i j and Ψ i in order to be abe to cacuate the conditiona expectation of T i. However, it is cear that conditioning on these two events aso eads to the independence between T i and Y i. Hence we get ET i Y i E Ti Ψ i j Ψ i E Yi Ψ i j Ψ i j,0 PΨ i j PΨ i. We start by computing E T i Ψ i E W i Ψ i j Ψ i +E Si Ψ i j Ψ i. j Ψ i The waiting time of the i th successfu packet doesn t depend on Ψ i j since they are disjoint in time, but it does depend on Ψ i. In fact, given Ψ i 0, the i th successfu packet wi not wait and start service upon arriva since the i th successfu packet eft the queue empty. However, given Ψ i with 0, the i th successfu packet arrived when the i th successfu packet was at state or of its service time. In order to find the distribution of W i conditioned { on Ψ i we introduce the foowing event: Ψ i,n n g Λi,g < Ai, n+ g Λi,g > } k m Ai m, where {Λ i,g } g is the sequence of interarriva times after the i th successfu packet enters state. Notice that Ψ i,n is the event that exacty n updates arrived when the i th successfu packet was in state or and then no more updates were generated for the remainder of the service time. Hence Ψ i nψ i,n. So conditioned on Ψ i,n we have W i m A i A i m n g n g Λ i,g Λ i,g + m+ A i m 3 It can be shown that, conditioned on { n g Λi,g < Ai }, A i n g Λi,g has an exponentia distribution with rate. This means that under this condition aone, W i has the same distribution as the sum of k + independent exponentia random variabes with rate {. If we further condition on n+ g Λi,g > } k m Ai m and use Lemma, we deduce that conditioned on Ψ i,n, W i has a gamma distribution with parameters k +, + we concude that if we condition on Ψ i as Γ k +, +. Therefore,. Now since Ψ i nψ i,n,, W i is distributed E { W i Ψ i j Ψ i 0 if 0 k + + if 0. 4 Now we turn our attention to E S i Ψ i j Ψ i. One first notices that the service time S i of the i th successfu packet is independent of its arriva time given by the event Ψ i since we assumed independence between service time and interarriva time. Hence, E S i Ψ i j Ψ i E S i Ψ i j. For the case j 0, we get E S i Ψ i 0 E ma i m k + m A i m < X 5 where the ast equaity is obtained by appying Lemma with

7 G k m Ai m and F X. As for the case j 0, we get E S i Ψ i j E A i m Ai j > Λ i j, A i m < X j m m mj+ EA i m +EAi j A i j > Λ i +E mj+ A i m mj+ A i j m < X a j + + k j k +j 6 + where the third term in a is obtained by appying Lemma with G k mj+ Ai m and F X. Therefore, combining 5 and 6 we get, k + if 0,j 0 E T i Ψ i j Ψ i k++j + if 0,j > 0 k +. + if > 0,j 0 k ++j + if > 0,j > 0 7 Now we need to compute E Y i Ψ i j Ψ i. For this end, observe that Y i is independent of Ψ i given that they don t overap in time. Moreover, for j 0, the i th successfu packet eaves the queue empty and thus we wi need to wait an exponentia amount of timex of ratebefore the i th successfu packet arrives and is served directy. Hence, conditioned on Ψ i 0, Y i has same distribution as X +S with X and S independent. On the other hand, for j 0, the i th successfu packet eaves the queue with another packet waiting in the buffer ready to be served. Thus in this case, Y i is simpy the service time of the i th successfu packet. To sum up, E { Y i Ψ i j Ψ i +k if j 0 8 k if j > 0 To compute ET i Y i we sti need the probabiity PΨ i j. For j > 0, we use the fact that Ψ i j is the intersection of two independent events and find that PΨ i j. As for j 0, we have aready seen in Section III + k j+ k. that PΨ i 0 p + These probabiities are independent of the index i and thus we can find PΨ i by repacing j by in the previous expressions. Combining this resuts with 7, 8 we obtain after some tedious cacuations ET i Y i k kk + +k+qk k+k +k +q k+ qk k qk+ 9 with q +. The ast term to compute in order to obtain EQ i is EYi EY i Ψi 0 PΨ i 0 +EYi Ψi 0 PΨ i 0. Based on our previous observations we know that EYi Ψi 0 EX +S and EYi Ψi 0 ES. Using these facts we get +k EYi k +k +q k. 30 Combining 9 and 30, we finay get EQ i k+ +3k +q k+ k+k +k k +q k qk k qk+. 3 Computing the effective rate: To cacuate the effective rate we first observe that the event {packet is successfuy received} is equivaent to the event {packet passes by the - state}. Hence if we uniformize the Markov chain so that the time spent at each state is exponentia with rate +, we get e + π where π is the steady-state probabiity of the -state in the uniformized Markov chain. The anaysis of q q such chain [6], chapter 5 givesπ q k+ +k q. Therefore, e + k +k+ k. 3 Finay, repacing EQ i and e in e EQ i by their expressions in 3 and 3, we obtain our resut. B. Average peak age Proposition 4. The average peak age in the LCFS without preemption scheme assuming Erang Ek, service time is: EP i +k k + k+. 33 Proof: We know that EP i ET i + EY i. We cacuate these two terms as foows ET i j,0 E T i Ψ i j Ψ i PΨ i j PΨ i +k qk+ + +k, 34 where we used 7 for the ast equaity. For EY i we wi ony condition on Ψ i 0. Hence using 8, we get EY i EY i Ψ i 0 PΨ i 0 +EY i Ψ i 0 PΨ i 0 k + qk. 35 Thus, combining the above two resuts we obtain our resut.

8 V. AGE OF INFORMATION FOR DETERMINISTIC SERVICE TIME In order to compute the four ages of interest under a deterministic service time assumption, we use Lemma. For that, we fix the mean of the service times S n to ES n µ, for some µ > 0, and et k. It is beyond the scope of this paper to show that if S n d Z, as k then we aso have convergence in the average ages, i.e, Sn Z. Here Sn refers to the average age corresponding to service time S n. However, we wi use this resut to derive the different ages. A. LCFS with preemption Letting k in 9 and 0, we get B. LCFS without preemption e/µ EP i µ + e/µ Letting k in and 33, we get +ρ ρ e ρ +ρ+ρe ρ +3ρ +ρe ρ EP i + e ρ µ where ρ µ. VI. NUMERICAL RESULTS In this section we show that the theoretica resuts obtained in the previous sections match the simuations. We aso compare the performance of the two transmission schemes of interest as we as the effect of the parameter k on each of them. First it is worthy to specify that a simuations were done using gamma distributed service times with a having the same mean k, except for the deterministic case where the service time is fixed to. Figure 4 presents the average age under LCFS with preemption scheme and gamma distributed service time. Two observations can be made based on this pot: i the theoretica curves given by 9 and 36 coincide with the empirica curves and ii as the vaue of k increases, the average age increases for a vaues of. This means that, under LCFS with preemption, the average age assuming deterministic service time k is higher than the average age assuming a reguar gamma distributed service. In particuar, it is higher than the average age assuming memoryess time. This observation can be expained by the fact that the probabiity of packet being preempted is given k by p + refer to Section III which is an increasing function of k. Therefore, as k increases the receiver wi have to wait on average a onger time ti a new update is deivered since the preempting rate becomes higher. This anaysis is true for any vaue of, hence the phenomenon seen in Figure 4. In a parae setting, Figure 5 presents the average age under LCFS without preemption. In this case aso two observations Average age Average age, k7 Average age, k Average age, k3.5 Average age, k Fig. 4. Average age for gamma service time S with ES, different k and LCFS with preemption can be made: i the theoretica curves given by and 38 match the empirica resuts and ii as the vaue of k increases, the average age decreases for amost a except for vaues cose to 0 where a distributions behave simiary. This difference in performance is especiay seen at high. We give here a quick intuition that expains this behavior. When is high, the time where the queue is empty goes to 0 and thus the queue is aways transmitting. This aso means that on average the waiting time W i goes to 0. Given these two observations, one can say that the system time T i and the interdeparture time Y i wi have amost the same distribution as the service time, whie being amost independent. Thus EQ i ES + ES. As for the effective rate e, since the queue is amost aways busy, the average rate at which the receiver gets new update is nothing but the inverse of the average service time, i.e e ES. Therefore, ES + ES ES + 3k. This resut which is aso obtained by taking the imit over in is decreasing with k. Hence the behavior seen in Figure 5. Next, we compare the performance of the two transmission schemes in two modes: for gamma distributed and deterministic service time. Figure 6 shows the average age under LCFS with and without preemption when the service time is taken to be gamma distributed with k. In this case we notice that for sma the two schemes perform simiary. However, for s around, the LCFS with preemption scheme performs sighty better before being outperformed by the LCFS without preemption scheme at high s. Practicay, this means that if one is using a medium whose service time is modeed as a gamma random variabe, the best strategy among the considered ones is not to preempt whie increasing the update generation rate as much as possibe. This strategy aso appies when the service time is deterministic as seen in Figure 7. In fact, we observe that for deterministic service time and for a vaues of, the average age and the average peak age for the LCFS without preemption scheme are smaer than the average age and average peak age for the LCFS with preemption respectivey. VII. CONCLUSION We considered the gamma distribution as a mode for the service time in status update systems. We computed and anayzed the average and average peak age of information

9 Average age Average age, k Average age, k6 Average age, k7 Average age, k Fig. 5. Average age for gamma service time S with ES, different k and LCFS without preemption [] R. Yates and S. Kau, Rea-time status updating: Mutipe sources, in Information Theory Proceedings ISIT, 0 IEEE Internationa Symposium on, Juy 0, pp [3] C. Kam, S. Kompea, and A. Ephremides, Age of information under random updates, in Information Theory Proceedings ISIT, 03 IEEE Internationa Symposium on, Juy 03, pp [4] S. Kau, R. Yates, and M. Gruteser, Status updates through queues, in Information Sciences and Systems CISS, 0 46th Annua Conference on, March 0, pp. 6. [5] M. Costa, M. Codreanu, and A. Ephremides, On the age of information in status update systems with packet management, CoRR, vo. abs/ , 05. [Onine]. Avaiabe: [6] S. M. Ross, Stochastic Processes Wiey Series in Probabiity and Statistics, nd ed. Wiey, Feb Average age for LCFS without preemption Average age for LCFS with preemption Average age Fig. 6. Average age for gamma service time S with k and ES Average peak age for LCFS with preemption Average age for LCFS with preemption Average peak age for LCFS without preemption Average age for LCFS without preemption 7 Age Fig. 7. Average age and average peak age for deterministic service time under two schemes: LCFS with preemption and LCFS without preemption. This aowed us to evauate these metrics for deterministic service time. This suggests that considering gamma distributions for simiar probems can be a good idea since the Gamma distributions or at east Erang distributions are practicay reevant as they can be used to mode the tota service time for reay networks. ACKNOWLEDGMENT The authors woud ike to thank Emre Teatar for hepfu discussions. REFERENCES [] S. Kau, R. Yates, and M. Gruteser, Rea-time status: How often shoud one update? in INFOCOM, 0 Proceedings IEEE, March 0, pp.