Reducing Age-of-Information for Computation-Intensive Messages via Packet Replacement

Transcription

1 Reducing Age-of-Information for Computation-Intensive Messages via Packet Replacement arxiv: v1 [cs.it] 15 Jan 19 Jie Gong, Qiaobin Kuang, Xiang Chen and Xiao Ma School of Data and Computer Science, Sun Yat-sen University, Guangzhou 516, China School of Electronics and Information Technology, Sun Yat-sen University, Guangzhou 516, China Abstract Freshness of data is an important performance metric for real-time applications, hich can be measured by age-of-information. For computation-intensive messages, the embedded information is not available until being computed. In this paper, e study the age-of-information for computationintensive messages, hich are firstly transmitted to a mobile edge server, and then processed in the edge server to extract the embedded information. The packet generation follos zeroait policy, by hich a ne packet is generated hen the last one is just delivered to the edge server. The queue in front of the edge server adopts one-packet-buffer replacement policy, meaning that only the latest received packet is preserved. We derive the expression of average age-of-information for exponentially distributed transmission time and computing time. With packet replacement, the average age is reduced compared ith the case ithout packet replacement, especially hen the transmission rate is close to or greater than the computing rate. I. INTRODUCTION Age-of-information, defined as the time elapsed since the generation of the latest delivered update, is one of the key metrics to measure the freshness of information in realtime monitoring and control applications [1]. Existing orks mainly study the influence of queuing and transmission delay on the age performance. Hoever, in computation-intensive applications such as autonomous driving and online facial recognition, an update, e.g. an image or a section of video record, needs to be processed to reveal the status information embedded in the packet. Due to the limited computational resource in the end devices, it is urgently required to adopt mobile edge computing (MEC) [] technology to offload the computing tasks. This ork focuses on the analysis of age-ofinformation for computation-intensive messages in MEC. In the literature, age-of-information as initially studied in the elementary queuing systems such as M/M/1, D/M/1, and M/D/1 queuing models ith first-come-first-served (FCFS) discipline [1]. As FCFS may result in large queuing delay, the last-come-first-served (LCFS) queue as considered to reduce the age-of-information [3]. Then, three packet management policies ere introduced to further enhance the performance [4], here out-dated messages ere discarded as they ere less valuable for status update. Other than applying queuing analysis here update packets are generated randomly, update generation policies can be designed hen the source has access to the channel s idle/busy state. The zero-ait policy, hich generates a fresh update just as the prior update is delivered and the channel becomes idle, as proposed in [5] to completely eliminate the aiting time in the queue. The optimality of the zero-ait policy as analyzed in [6]. The impact of computation on age has been recently considered in [7] hich focuses on scheduling in computation and netorking ith centralized cloud. Nevertheless, it is still an open problem to characterize age-of-information ith computing in MEC. In MEC, each packet experiences to stages: transmission and computing, hich can be vieed as a to-hop netork. Among the multi-hop related research efforts, the optimality of the last-generated-first-served (LGFS) policy as analyzed in multi-hop netorks [8]. The age-of-information for multiflo multi-hop netorks ith interference as studied in [9]. In the multi-hop line netork ith preemptive servers and random arrivals, a simple expression of average age as obtained in [1], [11] using stochastic hybrid systems (SHS) tool. Different from existing orks, our preliminary ork [1] derived the expression for the system ith zero-ait policy in transmission stage and M/M/1 FCFS queue in computing stage. In this paper, e further consider packet replacement policy in computing stage. In particular, the edge server preserves a queue of length one. If the queue is full and a ne packet arrives at the server, the old packet in the queue is discarded and replaced by the ne one. We characterize the age-of-information by deriving the distribution of transmission time, aiting time and inter-arrival time of the successfully computed packets. Numerical results illustrate that the message going through the system ith packet replacement is fresher than that ithout packet replacement. II. SYSTEM MODEL Consider a status update system for computation-intensive messages hich are processed at mobile edge server as shon in Fig. 1. The hole procedure is divided into to stages: transmission stage and computing stage. In the transmission stage, an update packet is firstly generated from the source, and then transmitted through the channel. The generation of update packets follos zero-ait policy, by hich a ne update is generated by the source hen the transmission of the previous update is just completed. Therefore, there is no

2 Acknoledgement Δ(t) P 3 P 4 Source Destination P Channel Edge Server Packet Replacement Fig. 1. Status update system ith MEC. Q 3 ~ Q Q 1 Q 4 τ 1 τ τ 3 τ 4 τ 5 τ 6 τ 7 τ 8 t t 1 t t 1 ' t 3 t ' t 4 t 3 ' t 4 ' t 5 X X 3 t 5 ' t aiting queue in the transmission stage. In the computing stage, the packet is received and computed by the edge server so that the information embedded in the packet is revealed to the destination. Both transmission and computing times are assumed to be random and follo exponential distribution ith means 1/λ and 1/µ, respectively, leading to queuing in the computing stage. In this paper, e consider a one-packet-buffer replacement queue, i.e., at most one packet is alloed to ait in the queue, and it is replaced if a ne one arrives. In this ay, the out-dated update is discarded and the freshest update goes into the queue for computing. It is expected to reduce the age-of-information compared ith FCFS queue. Denote τ i as the generation time instant of the i-th update packet, i 1,,. As zero-ait policy is adopted, τ i is also the transmission completion time of the(i 1)-th packet. Notice that not all the packets are received by the destination due to the packet replacement in the computing stage. We mainly focus on the computed packets as they account for the average age-of-information. For ease of analysis, the computed packets are re-indexed by k 1,,. Denote t k as the completion time instant of the transmission stage for the k-th computed packet, and t k as the completion time instant of the computing stage for the k-th computed packet. At the time instant t, the index of the latest information received by the destination is denoted by K(t) : max{k t k t}. (1) As τ i(t) t K(t) for some i(t), the generation time instant of this packet can be given as U(t) : τ i(t) 1. The age-ofinformation is defined as (t) : t U(t). () The satooth shaped sample path of the random process (t) is illustrated in Fig.. It can be seen from the figure that since the fifth packet has already arrived at the edge server just before the second packet is successfully computed. Hence, the fifth packet is re-indexed as the third computed packet. The average age in the observation range (,t) is defined by t : 1 t t (τ)dτ. (3) The integration is equal to the area belo the curve (t), hich can be calculated as the summation of the areas of parallelogramsp k and trapezoidsq k,k,,k(t), here X 1 Y Z 3 T Y 3 T 3 Fig.. A sample path of age-of-information ith MEC. K(t) is defined in (1). Therefore, the average age can be ritten as t K(t) 1 t Q 1 K(t) k (P k Q k )Q, (4) K(t) 1 here Q is the area in time interval (t K(t),t). As t goes to infinity, K(t) goes to infinity as ell. Consequently, the term Q 1Q K(t) 1 goes to zero as the nominator is finite. To calculate P k and Q k, denote X k : τ i τ i 1 as the transmission time for the k-th computed packet, hereisatisfies τ i t k, Y k : t k t k 1 as the time elapsed beteen the transmission completion time instants of the k-th and (k 1)-th computed packets, T k : t k t k as the system time of the k-th computed packet in the computing stage, including aiting time and service time, and Z k : t k t k 1 as the inter-departure time of the computing stage. We have P k X k 1 Z k, (5) Q k 1 (T k Y k ) 1 T k 1, (6) K(t) t t 1 Y k t, (7) k here t t t K(t). Substituting (5)-(7) into (4), and letting t going to infinity, e have lim t E[P k]e[q k ] t E[Y k ] 1 ( E[X k 1 Z k ]E[T k Y k ] 1 ) E[Y k ] E[Y k ], (8) here E[ ] is the expectation operator, and the fact that E[Tk ] E[T k 1 ] is used. Remark 1: The inter-departure time Z k is independent of X k 1 as it only depends on the arrival process at the edge server beteen the (k 1)-th and k-th packets and the system times of these to packets in computing stage. The

3 influence of X k 1 to Z k is blocked by the one-packet-buffer replacement queue principle. Hence, e have E[X k 1 Z k ] E[X k 1 ]E[Z k ]. (9) Furthermore, e have by definition that K(t) k E[Z k ] lim Z k t K(t) 1 lim t t K(t) 1 E[Y k]. (1) Hence, to calculate the average age-of-information, e only need to consider X k, Y k and T k. Remark : The average area E[Q k ] is equal to the one in [4] for M/M/1/ queue that is derived based on hether the queue is empty or not upon departure. For completeness of description, e re-calculate the result by proposing another method, hich directly derives the distributions of X k and Y k, hich are detailed in the next section. III. CALCULATION OF AVERAGE AGE In this section, e derive the average age according to (8). By definition, e have T k W k S k, here W k and S k are the aiting time and the service time in the computing stage, respectively. As X k and Y k depend on the system time of the (k 1)-th packet, e firstly derive the distribution of W k. Then, e calculate the terms in (8) one by one. A. Distribution of W k The event that the k-th packet has zero aiting time occurs if and only if there is no arrival during the service time s of the (k 1)-th packet, hich is equivalent to the event that the inter-arrival time in computing stage is larger than s. Since the inter-arrival time of the computing stage is exponentially distributed, e have Pr(W k ) Pr(no arrival in (,s))f Sk 1 (s)ds e λs µe µs ds µ λµ. (11) For the case that W k >, We consider the probability Pr( < W k S k 1 s). Notice that in the computing stage, the aiting time of the k-th packet is no longer than the service time of the (k 1)-th packet s. If s, W k is guaranteed. Then, the event W k > happens if and only if at least one packet arrives in time duration of length s. If s > on the other hand, the event < W k happens if and only if at least one packet arrives in time duration (s,s) 1. According to the total probability formula, e have Pr( < W k ) Pr( < W k S k 1 s)f Sk 1 (s)ds (1 e λs )µe µs ds (1 e λ )µe µs ds λ λµ (1 e (λµ) ). (1) 1 In this notation, e reset the transmission completion time instant of the (k 1)-th computed packet as. This is valid throughout this section. Hence, for >, the probability density function of W k is f W () : f Wk () λe (λµ), >. (13) B. Distribution of X k Recall that X k is the transmission time for the k-th computed packet. As X k is related to the aiting and computing process of the(k 1)-th packet, e derive the distribution ofx k conditioned on W k 1 and S k 1 by analyzing the probability Pr(X k > x W k 1,S k 1 s). Given W k 1 and S k 1, e analyze the conditions on hich the event X k > x occurs. In general, if the k-th packet arrives at time instant t, X k > x occurs if there are no packet arrivals before t for a time duration longer than x and no arrivals after t until the computing completion time for the (k 1)-th packet. The detailed results are as follos: 1) < s: If < s, and x, the event X k > x occurs hen the k-th packet arrives in the small interval (t,tdt) hile at the same time no packet arrivals during time intervals (t x, t) and (tdt, s), or no packet arrivals during (,s). The probability that a single packet arrives in (t,t dt) is λdt o(dt), Hence, the probability Pr(X k > x W k 1,S k 1 s) is the integral over all possible t, i.e., Pr(X k > x W k 1,S k 1 s) x e λ(t ) e λ(s t) λdt s x e λx e λ(s t) λdte λs (14) xλe λs e λx. (15) Notice that e ignore e λdt and o(dt) as they are higher-order infinitesimal. The first integral in (14) refers to the special case that the time interval before t is shorter than x. As there are no packet arrivals during (, ) by definition, this special case also results in X k > x. If < x s, the analysis is similar to the above, but the integral range is from x to s. Hence, e have Pr(X k > x W k 1,S k 1 s) x x e λ(t ) e λ(s t) λdt s x e λx e λ(s t) λdte λs (16) λe λs e λx. (17) If s < x s, the event X k > x occurs hen the k-th packet arrives in (t,tdt) hile at the same time no packet arrivals during time intervals (,t) and (t dt, s) for x < t < s, or no packet arrivals during (, s). We have Pr(X k > x W k 1,S k 1 s) s x e λ(t ) e λ(s t) λdte λs (s x)λe λs e λs. (18)

4 Finally, for the case that x > s, the event X k > x occurs hen there are no arrival during time interval (, x). Therefore, Pr(X k > x W k 1,S k 1 s) e λ(x ). (19) ) s < : In this case, e analyze the probability in the same ay. If x s, it is easy to verify that the result is equal to (15). Similarly, the result ith < x s is the same as (18), and the result ith x > s is the same as (19). While for s < x, the event X k > x occurs hen the k-th packet arrives in (t,t dt) hile at the same time no packet arrivals during time intervals (,t) and (tdt,s) for < t < s, or no packet arrivals during (, s). We have Pr(X k > x W k 1,S k 1 s) s e λ(t ) e λ(s t) λdte λs sλe λs e λs. () 3) : This case can be vieed the extreme case of s. By setting in (15)-(19) and check the validity, e have Pr(X k > x W k 1,S k 1 s) e λx. (1) In summary, by taking the derivative of Pr(X k > x W k 1,S k 1 s) in terms of x, e can obtain the conditional probability density function f X W,S (x,s) : f Xk W k 1,S k 1 (x,s) λe λx λe λs, x min{,s}, λe λx, < x s, λe λs, max{,s} < x s, () λe λ(x ), x > s,, else. According to (11), (13) and (), by the la of total expectation, e have E[X k ] E[E[X k W k 1,S k 1 ]] Pr(W k ) µe µs E[X k W k 1,S k 1 s]ds λe (λµ) µe µs E[X k W k 1,S k 1 s]dsd Pr(W k ) µe µs xλe λx dxds λe (λµ) µe µs xf X W,S (x,s)dxdsd 1 ( 1 1 µ ρ(1ρ) ρ 3 (1ρ) 4 ρ ) (1ρ), (3) here ρ λ/µ, and the second integral is calculated by dividing the integral region based on (). C. Distribution of Y k Recall that Y k is the time elapsed beteen the transmission completion time instants of the k-th and (k 1)-th computed packets. Similar to the previous subsection, e derive the conditional probability density functionf Yk W k 1,S k 1 (y,s) by calculating the conditional probability Pr(Y k y W k 1,S k 1 s). When y < s, the event Y k y occurs if and only if there is at least one packet arrival in (,y) and no packet arrivals in (y,s), i.e., Pr(Y k y W k 1,S k 1 s) (1 e λ(y ) )e λ(s y) e λ(s y) e λs. (4) When y s, the event Y k > y happens if and only if no packet arrivals in (,y). We have Pr(Y k y W k 1,S k 1 s) 1 Pr(Y k > y W k 1,S k 1 s) 1 e λ(y ). (5) By taking derivation of Pr(Y k y W k 1,S k 1 s), e have f Y W,S (y,s) : f Yk W k 1,S k 1 (y,s) { λe λ(s y), y < s, λe λ(y ), y s. (6) To calculate E[Y k ] and E[Yk ], e derive the probability density function of Y k as f Yk (y) Pr(W k 1 ) f Y W,S (y,s)µe µs ds λe (λµ) f Y W,S (y,s)µe µs dsd ( λ µ λµ ) e λy λµ(λµ) λµ (λµ) e µy ( λ µ λµ ) λµ λ y e (λµ)y. (7) Accordingly, e can obtain E[Y k ] 1 1ρρ µ ρ(1ρ), (8) ). (9) E[Y k ] µ ( 1 1 ρ ρ(1ρ) (1ρ) 4 D. Calculation of E[T k Y k ] To calculate E[T k Y k ], e rerite T k as a function of W k 1 and S k 1. We observe the relation beteen T k 1 and Y k. If T k 1 > Y k > W k 1, i.e., the k-th computed packet arrives at the edge server hen the (k 1)-th packet is still being processed, e have W k T k 1 Y k. Otherise, W k. Hence, e have W k (T k 1 Y k ) (W k 1 S k 1 Y k ). (3)

5 / packet replacement /o packet replacement ρ Fig. 3. Performance comparison for status update via mobile edge computing ith or ithout packet replacement. µ 1. Therefore, e obtain E[T k Y k ] E[(W k S k )Y k ] E[(W k 1 S k 1 Y k ) Y k ]E[S k ]E[Y k ] (31) as S k and Y k are independent ith each other. By utilizing the distributions of W k 1 and S k 1, and the conditional probability density function f Y W,S (y,s), e can calculate that E[(W k 1 S k 1 Y k ) Y k ] s Pr(W k 1 ) µe µs (s y)yλe λ(s y) dsdy λe (λµ) s µe µs (s y)yλe λ(s y) dydsd 1 ( 1 µ 1ρ 1ρ ) (1ρ) 4. (3) Since E[S k ] 1/µ, summarizing (8), (31) and (3), e have E[T k Y k ] 1 ( µ 1 1 ρ 1ρ ) (1ρ) 4. (33) E. Average Age-of-Information Finally, according to (8)-(1), (3), (8), (9) and (33), e obtain 1 ( µ ρ 13ρ (1ρ) ρ 3 (1ρ) 4 (1ρ) ) 1ρρ. (34) IV. NUMERICAL COMPARISON In this section, the average age-of-information for computation-intensive messages ith packet replacement is compared ith that ithout packet replacement as in [1], here all the packets ait in the queue for processing ith FCFS discipline. As shon in Fig. 3, ith packet replacement, the average age-of-information decreases as ρ increases. The phenomenon is reasonable as ith the increase of the channel transmission rate, the update packet aiting for computing is fresher as it is replaced by the latest update faster. Asymptotically, the minimum average age is achieved hen ρ, hich results in min /µ. In comparison, the average age ithout replacement first decreases as ρ increases, and then increases to infinity as ρ 1. When ρ is close to 1, the queue length in edge server becomes quite long, and the age becomes large due to the long time aiting in the queue. V. CONCLUSION AND FUTURE WORK In this paper, e have derived the average age-ofinformation for to-stage mobile computing system ith zeroait and packet replacement. The stationary distributions of some random processes are obtained, including the aiting time W k before being computed, the transmission time X k for the computed packet, and the inter-arrival time Y k of to consecutive computed packets. It is shon that ith packet replacement, the average age is reduced compared ith the case ithout packet replacement, and the value tends to a minimum /µ hen the transmission rate tends to infinity. Future ork includes finding other packet generation policies instead of zero-ait to further reduce the average age, and consider the cases ith multiple users or multiple edge servers. REFERENCES [1] S. Kaul, R. Yates, and M. Gruteser, Real-time status: Ho often should one update? in Proc. of IEEE INFOCOM, Mar. 1, pp [] Y. Mao, C. You, J. Zhang, K. Huang, and K. B. Letaief, A survey on mobile edge computing: The communication perspective, IEEE Commun. Surveys Tut., vol. 19, no. 4, pp , Fourthquarter 17. [3] S. K. Kaul, R. D. Yates, and M. Gruteser, Status updates through queues, in 46th Annual Conf. Inf. Sciences and Systems (CISS), Mar. 1, pp [4] M. Costa, M. Codreanu, and A. Ephremides, On the age of information in status update systems ith packet management, IEEE Trans. Inf. Theory, vol. 6, no. 4, pp , Apr. 16. [5] R. D. Yates, Lazy is timely: Status updates by an energy harvesting source, in IEEE Int. Symp. Inf. Theory (ISIT), Jun. 15, pp [6] Y. Sun, E. Uysal-Biyikoglu, R. D. Yates, C. E. Koksal, and N. B. Shroff, Update or ait: Ho to keep your data fresh, IEEE Trans. Inf. Theory, vol. 63, no. 11, pp , Nov. 17. [7] A. Alabbasi and V. Aggaral, Joint information freshness and completion time optimization for vehicular netorks, arxiv preprint arxiv: , 18. [8] A. M. Bedey, Y. Sun, and N. B. Shroff, Age-optimal information updates in multihop netorks, in IEEE Int. Symp. Inf. Theory (ISIT), Jun. 17, pp [9] R. Talak, S. Karaman, and E. Modiano, Minimizing age-of-information in multi-hop ireless netorks, in 55th Annual Allerton Conf. Commun., Control, and Computing (Allerton), Oct. 17, pp [1] R. D. Yates, Age of information in a netork of preemptive servers, in IEEE Conf. Computer Commun. Workshops (INFOCOM WKSHPS), Apr. 18, pp [11], The age of information in netorks: Moments, distributions, and sampling, arxiv preprint arxiv: , 18. [1] Q. Kuang, J. Gong, X. Chen, and X. Ma, Age-of-information for computation-intensive messages in mobile edge computing, arxiv preprint arxiv: v3, 19.