—Age of Information (AoI) measures the freshness of data in mission critical Internet-of-Things (IoT) applications i.e., industrial internet, intelligent transportation systems etc. In this paper, a new system model is proposed to estimate the average AoI (AAoI) in an ultra-reliable low latency communication (URLLC) enabled wireless communication system with decode- and-forward relay scheme over the quasi-static Rayleigh block fading channels. Short packet communication scheme is used to meet both reliability and latency requirements of the proposed wireless network. By resorting ﬁnite block length information theory, queuing theory and stochastic processes, a closed-form expression for AAoI is obtained. Finally, the impact of the system parameters, such as update generation rate, block length and block length allocation factor on the AAoI are investigated. All results are validated by the numerical results. Index Terms —Age-of-Information, ﬁnite block length regime, latency, reliability, ultra-reliable low latency communications (URLLC) and 5GB. I. I NTRODUCTION The concept of age of information (AoI) as a data freshness metric was introduced in the early 90s by data scientists to evaluate data in terms of temporal consistency in real-time databases. The term named “AoI” was initially used by Kaul and Marco in their work on vehicular communication networks [1],With the rapid advancements of information technology, the idea of AoI was spread to other research domains such as wireless communication, since the dissemination of timely information became increasingly critical for modern wireless communication applications [2], [3]. On the other hand, wireless cooperative communication technologies have been vigorously investigated in the telecommunication domain due to its added advantages, i.e, improved reliability of transmissions and latency requirement of the network [4]. Existing works in the literature related to AoI is concerned with single-hop wireless networks assuming each communication nodes is capable of transmitting data with an inﬁnite block length regime [5]–[7]. However, transmitting information involving an inﬁnite-block length regime is impractical in the presence of URLLC-enabled wireless communication networks. Therefore, short-packet data transmission is used to meet both reliability andlatency requirements of the wireless communication networks. Further, it is also noteworthy that in short-packet communication scenario, the Shannon-Hartley Capacity theorem is no longer applicable [8]. In addition, integrating ﬁnite-block length regime into wireless communication networks provide a precise framework to determine the relationship between the latency and the reliability between communication nodes. To the best of the authors knowledge, there is no existing work found that studies the AAoI performance metric of a wireless decode-and-forward (DF) relay scheme under ﬁnite block length regime. To ﬁll this research gap, in this paper studies the AAoI in a DF relay-assisted wireless communication system considering a ﬁnite block length regime over a quasi-static Raleigh fading channel conditions. Consequently, AAoI in wireless networks is analyzed using queuing theory and stochastic processes, and then a closed- form expression for AAoI is derived. Finally, in order to estimate AAoI in the proposed system, a closed-form expression of the block error probability is incorporated into derived closed-form expression of AAoI. Our numerical simulation results proved that by selecting a small packet size when the update size is equal or less than 10bits always help to maintain the URLLC with low AAoI in the proposed system. However, if the update size is greater than 10bits, optimal block length that achieved lowest AAoI is approximately equals to 20channel uses. It is also noteworthy that the lowest AAoI is achieved when the number of update generation rate at the source is equals to 22updates per second when the each update size is equals to 100bits. The remaining part of this paper is organized as follows. In Section II, the system model is presented with the derivations for closed-form expression of the block error probability for the proposed system. In the section III, AAoI of the proposed system is derived. In Section IV , numerical simulation results are provided to verify the mathematical derivations derived in Section II and III. Finally, paper concludes in Section V . SR D𝐻𝑆𝑅𝐻𝑅𝐷Fig. 1. System model for URLLC-enabled relay system II. S YSTEM MODEL In this paper, a URLLC-enable DF cooperative relaying system is proposed as illustrated in Fig.1, where the source (S)sends its newly generated updates to the destination (D)via aid of the relay (R). In this proposed cooperative communication system, each transmission block is split into two separate time slots. The source sends data to the relay in the ﬁrst time slot. Then, relay decodes and re-transmits the received data to the destination during the second time slot. It is assumed that there is no direct transmission between the source and the destination due to unfavourable line-of-sight (LoS) connection between them. The transmitted signal by the source, the received signal at the relay, the transmitted signal by the relay, and the received signal at the destination are denoted by X1,Y1,X2andY2, respectively. The received signal at each communication node can be written as Y1=p PSHSRX1+WSR; (1) Y2=p PRHRDX2+WRD; (2) whereHijis the channel coefﬁcient of the channel between nodeito nodejwherei2 fS;Rgandj2 fR;Dg. The symbol Wijdenotes the independent and identically distributed Addictive white Gaussian noise (AWGN) of the channel with zero mean and 2variance. In this work, transmit power of the communication node iis modelled as Pi='iP; (3) wherePis the total transmission power constraint of the system and 0<'i1is the power allocation factor of each communication node. The distance between the source and the destination is considered as dand the distance between the source and the relay is represented by d , where 0< < 1. The small scale channel gain is denoted as gij=jh2 ijj, wherehijCN (0;1)denotes the Rayleigh fading channel coefﬁcient. The probability density function (PDF) of the small scale channel gain gijcan be expressed as fgij(z) =ez: (4)The large scale channel gain ijbetween the communication nodes can be written as 10log( ij) = 20log(dij) + 20log4fc C ; (5) wherefcanddijare the carrier frequency and distance between the communication nodes iandjrespectively. The variableCdenotes the speed of light in the space. Considering small and large scale channel gains, the channel coefﬁcient between the communication nodes can be written as Hij=p ijgij: (6) The normalized received signal-to-noise ratio (SNR) jat the receiving node jcan be written as j= ijgijPi 2 ij: (7) The total block length nallocated for the proposed system is is allocated for each transmission block according to the predeﬁned allocation factor ijbetween two communication nodes. The block length of each transmission block can be calculated as nij=ijn: (8) A. Block Error Probability in the Finite-Block Length Regime Considering DF relay protocol, the overall decoding error probability can be written as "="R+"D(1"R); (9) where"Ris the block error probability at the relay, "Dis the block error probability at the destination. Due to the static nature of the communication channels, it is assumed that the fading coefﬁcients stay constant over the duration of each transmission block. Following Polyanskiy’s results on short packet communication [8] and assuming that the receiver has perfect channel state information, the expectation of the block error probability of a given block length can be written as "j=E" Q nijC( j)kp nijV( j)!# ; (10) where E[:]is the expectation operator, Q(x) = 1p 2R1 xet2 2dtandV( j)is the channel dispersion, which can be written V( j) =log22e 2(11 (1+ j)2). The variableC( j)denotes the channel capacity of a complex AWGN channel and it is given by C( j) = log2(1 + j). The number of bits per block represents by k. Moreover, under the Rayleigh fading channel conditions, "jcan be formulated as "j=Z1 0f j(z)Q nijC( j)kp njV( j)! dz; (11) wheref j(z)denotes the PDF of the received SNR jat thejthcommunication node. Due to the complexity of the Q-function, it is difﬁcult to get a closed-form expression for the overall decoding error probability. Thus, using the approximation technique given in [9] and [10], (11) can be approximated as "jZ1 0f j(z)j(z); (12) where j(z)denotes the linear approximation of Q nij:C( j)kp nij:V( j) , this can be expressed as [10] j(z) =8 < :1; j j; 1 2 jpnij( j j); j< j<j; 0; jj;(13) where j=1 2r 22k nij1; j= 2k nij1; j= j1 2 jpnij andj= j+1 2: jpnij. Using (12) and (13), a closed-form expression for the block error probability can be derived as "ij1 jpnijPi ij 2 e i2 ijPiej2 ijPi! :(14) Finally, substituting (14) in (9), a closed-form expression for the overall error probability can be obtained as "1 R DpnSRRD'S'R(P)2 SR RD 4  e R2 SR'SPeR2 SR'SP e D2 RD'RPeD2 RD'RP : (15) III. A VERAGE AGE OF INFORMATION OF THE PROPOSED URLLC- ENABLED RELAYING SCHEME If the generation time of the freshest update received at time stamptisg(t), then AoI can be deﬁned as a random process as (t) =tg(t): (16) ሻ𝚫(𝒕 𝒈𝟐 𝒈𝟐 𝑫𝟏 𝑫𝟐 𝑫𝟑𝒕 𝒀𝟏𝒀𝟐 𝑿𝟏𝑸𝟏 𝑸𝟎 ∆𝟎 Fig. 2. Evolution of Age of Information ((t))with the timeAs illustrated in the Fig. 2, it is assumed that at t= 0 the measurements of the AoI starts and the AoI at the destination is set to (0) =  0. The information source generate updates at time stamps g1;g2;:::; and the destination receive these updates at time stamps D1;D2;:::;. As illustrated in Fig. 2, data update iis transmitted from the source at time stamp t=giand it is successfully delivered to the destination at time stampDi=gi+Yi. Therefore, the AoI at the destination can be calculated as (Di) =Digi: (17) AoI increases linearly until the next update is delivered to the destination. Similarly, AoI just before i+ 1 update is successfully delivered, can be written as (D i+1) =Yi+Xi: (18) Then, at the time stamp Di+1, the AoI drops as follows  (Di+1) =Yi+1: (19) Considering the observation time interval as [0;Tn]for the proposed system, upper bound of the time interval can be given as a function of Xi, which can be written as Tn=n1X i=0Xi: (20) For the considered time period, time average AoI can be computed using the area under (t)and it can be treated as a sequence of trapezoids Qiwhen 0in1as shown in the Fig. 2. Similarly, the time average age of the proposed system can be given as Tn=1 TnZTn 0(t)d(t) =Pn1 i=0Qi Tn; (21) where Qi=ZDi+Xi Di(t)d(t) =YiXi+1 2X2 i: Similar to the work presented in [11], the time average age (Tn) tends to ensemble average age when n! 1 , i.e., which can be expressed as a= lim n!1Tn: (22) Since E[Yi] =E[Yi+1],Xiand update generation time difference have the same distribution. Thus, substituting (20) and (21) to (22), the AAoI can be written as AAoI = a=E[X2] 2+E[XY] E[X]: (23) In (23), E[XY]is difﬁcult to estimate without proper approximation since YandXare dependent random variables. Thus, it is assumed that the communication node’s data buffers are equipped with M=G= 1queue, where the source generate updates according to a Poisson process at the rate while the service time of the update is geometrically distributed. The system delay (Y)can be calculated as a summation of the service time (s)and waiting time (w)of the queue, which can be written as Y=s+w: (24) SinceXandsare i.i.d random variables, AAoI is given by the (23) can be re-formulate as a=1 2E X2 E[X]+E[s] +E[wX] E[X]: (25) SincewandXare dependent random variables, Pollaczek–Khinchine [12], [13] formula is used to further simplify (25). Considering M/G/1 queue policy, the relationship between expectation of the variables wands have been formulated as E[w] =E[s2] 2(E[X]E[s]); (26) The correlation term E[wX]can be evaluated as [14], [15] E[wX] =E[X](E[X]E[s]) E[es]+E[s2]E[X] 2(E[X]E[s]) E[X2] 2;(27) whereis the update generation rate, where E[X] =1 and E[X2] =2 2. Then, using (27) and (25), AAoI can be written as a=E[s] +E[s2] 2(E[X]E[s])+E[X]E[s] E[es]: (28) The variable sin (28) depends on number of re- transmissions between the communication nodes. The total number of transmission Rneeded for the reliable transmission of an update is geometrically distributed with transmission success rate and its probability mass function can be given by PR(m)= (1")"m1;m= 1;2:::: (29) where"ijdenotes the block error probability. The service time for an update, which is transmitted using one transmission block is given by s= (nT+ )R; (30) wherenandRdenote block length and number of transmission needed for reliable communication, respectively, is the channel induced delay and Tis the symbol duration. Since all of these variables are constants and the service time is a geometric random variable, thus, we have E(s) =nT+ 1"; (31) E(s2) =(nT+ )2(1 +") (1")2; (32) E(es) =(1")e(nT+ ) 1"e(nT+ ): (33)Finally, by using (15), (28), (31), (32) and (33) the AAoI of the proposed system can be derived as a=(nT+ ) 1"+(nT+ )2(1+") (1")2 2(1 (nT+ ) 1")+(1 (nT+ ) 1") (1")e(nT+ ) 1"e(nT+ ):(34) IV. N UMERICAL SIMULATION RESULTS In this section, numerical simulation results are presented to verify the theoretical analysis presented in this paper. First, the effect of the update generation rate on the AAoI is investigated. Unless otherwise stated, the simulation parameters are set as: d= 1 km,P= 23 dBm,n= 300 ,SR=RD= 0:5, 'S='R= 0:5, = 0:5,2=167 dBm,fc= 6 GHz, C= 3108m/s,k= 100 andT= 1104s. It is assumed that the channel induced delay ( ) is negligible compared to nT. 10010110-1100101 Simulated Theoretical Number of updates generated per second Fig. 3. AAoI as a function of update generate rate when K= 100 bits. In Fig. 3, AAoI is plotted as a function of update generation rate whenk= 100 bits. It can be observed from Fig. 3 that the network has a higher AAoI when the updates generation rate is either lower or greater than the optimal value of 22updates per second. This phenomena can be easily explained as follows. When the update generation rate is lower than the optimal value, it increases time difference between the reception of two consecutive updates causing an increase in AAoI. On the other hand, if the update generation rate signiﬁcantly larger than the optimal value, it is impossible to stabilize the queue. Thus, packets are waiting for a longer period of time in the queue leading to a higher AAoI. Furthermore, it can be clearly seen from the ﬁgure that the simulation results are approximately similar to their analytical counterparts. Fig. 4 illustrates the effect of the total block length non the AAoI performance of the proposed system. It is observed from the ﬁgure that when the update size (k)is signiﬁcantly small, the AAoI increase with the increases in block length. This is mainly due to the fact that when the block length 15 20 25 30 35 40 45 500.0160.0180.020.0220.0240.0260.0280.03 Update Size k=120 bits Update Size k=110 bits Update Size k =10 bitsFig. 4. AAoI as a function total block length n is increased, it causes increment in the service time for an update. On the other hand, an update consists of large number of bits has a higher AAoI when it is transmitted using block length less than 20channel uses due to the high block error probability. It is also noteworthy that when the update size is larger than 10bits, increase in block length towards its optimal value 20channel uses decreases the AAoI due to the decease in error probability. However, further increase in block length after the optimal value has led to increase in AAoI due to the impact of the service time on AAoI is dominant compared to the decrease in block error probability. This result proved that transmitting small size update less than 10bits always help to maintain minimum AAoI for mission critical application. However, if the update size is greater than 10bits, lowest AAoI is achieved only when the block length is equals to 20 channel uses. In Fig. 5, AAoI is plotted as a function of the block length allocation factor of the S-Rlink(SR)for different power allocation factors. It is shown that when the block length allocation factor increases, the AAoI decreases until it reaches the optimal value. This is mainly due to the fact that increase in block allocation factor decreases the block error probability at the Rcompared to the block error probability atD. Thus, overall block error probability is reduced, which can be easily understood by referring to (9). When the block length allocation factor between 0:3-0:7, the relay scheme experience lower AAoI, due to the moderate block error probabilities and stability of the queue. However, after the value 0:7the AAoI increases rapidly since the small block length allocation for a any channel increase the overall error probability. In addition, it is also shown that equal power allocation in communication nodes results in better AAoI regardless of the block length. 0.2 0.3 0.4 0.5 0.6 0.7 0.810-2100102104106 Power allocation factor - 0.5 Power allocation factor - 0.3 Power allocation factor- 0.7Fig. 5. AAoI as a function of block length allocation factor SRfor different power allocation factors at the S V. CONCLUSIONS This paper proposed a new model to measure the AAoI in a DF relay operating under ultra-reliable and low-latency constraint. For this purpose, the relationship between the block error probability and the AAoI was formulated under the ﬁnite block length regime. In addition, the impact of update size on the AAoI was investigated and it has been shown that maintaining a small update size (short packet communication) minimize the AAoI in the relay network. In addition, the impact of the update generate rate, the block length allocation factor and the power allocation factor on the AAoI was investigated. VI. A CKNOWLEDGEMENT This work is funded by the Seed Funding program of Sri Lanka Technological Campus, Sri Lanka grant no. RRSG/20/A7 and by the Russian Foundation Basic Research grant№19-37-90037.