Age-Oriented Opportunistic Relaying in Cooperative Status Update Systems with Stochastic Arrivals
^††thanks: This work is done when the first author Bohai Li is a visiting student at the Chinese University of Hong Kong.

With the rapid development of the Internet of Things (IoT) applications, timely status updates have become increasingly critical in many areas, such as telesurgery and self-driving cars [1]. However, the conventional performance metrics, e.g., throughput and delay, cannot properly characterize the timeliness of the status updates [2]. For example, throughput can be maximized by generating and transmitting the status updates as frequent as possible. However, excessive update rates may lead to network congestion, which makes the updates suffer from long transmission delay. Such long delay can be reduced by lowing the update rate. However, reducing the rate of updates overmuch can also make the monitor receive undesired outdated status updates. Motivated by this, age of information (AoI), defined as the time elapsed since the generation of the latest received status update, has been recently introduced to quantify the information freshness from the perspective of the receiver that monitors a remote process [3]. Unlike conventional metrics, AoI is related to both the transmission delay and the update generation rate [4]. As a result, AoI is a more comprehensive evaluation criterion for information freshness.

Since the AoI concept was first proposed to characterize the information freshness in a vehicular status update system [5], extensive work focusing on the analysis and optimization of AoI has appeared. Most of the existing work is concerned with the AoI of single-hop wireless networks [6, 7, 8, 9, 10, 11, 12, 13]. On the other hand, the AoI performance in multi-hop networks has also been studied in [14, 15, 16, 17]. However, all the above work overlooked the direct link between the source and the destination. Therefore, the updates from the source can only be transmitted to the destination via the relay. As far as we know, there is no existing work that designs and optimizes the average AoI of a cooperative status update system with the existence of a direct link. Such a design is indeed non-trivial. This is because delivering the status updates via the direct link will have shorter transmission time at the cost of a higher error probability, while the delivery of status updates through the two-hop relay link could be more reliable at the cost of longer transmission time.

Motivated by this gap, in this paper we consider a three-node cooperative status update system, where a source timely reports randomly generated status updates to its destination with the help of a relay. We concentrate on designing and analyzing an age-oriented opportunistic relaying (AoR) protocol to reduce the AoI of the considered system. By carefully observing the AoI evolution process, we first define some necessary time intervals to mathematically express the average AoI of the proposed protocol. By representing these time-interval definitions in terms of key system parameters, including the generation probability of the status updates and the transmission success probabilities of three links, we then attain a closed-form expression of the average AoI for the proposed AoR protocol. Given reasonable values of the transmission success probabilities, we further minimize the average AoI by optimizing the status generation probability at the source. Simulation results are then provided to validate the theoretical analysis. Furthermore, the simulation results demonstrate that the average AoI of the proposed AoR protocol is smaller than that of the non-cooperative system, especially when the source-destination link has bad channel conditions.

We consider a three-node cooperative status update system, in which the source (S) wants to send randomly generated status updates to a designated destination (D) as timely as possible with the help of a relay (R). There is a direct link between S and D. The transmission of the status updates can either go through the direct link or the S-R-D link. To quantify the timeliness of the status updates, we adopt a recently proposed metric, termed AoI, which is defined as the time elapsed since the generation of the last successfully received status update. It is assumed that the considered system is time slotted and the transmission of each status update by S/R takes exactly one time slot, whose length is normalized to one without loss of generality. All the channels suffer from block fading, i.e., the channels responses remain unchanged within one time slot but vary independently from one time slot to another.

In this section, we first express the average AoI by the defined time intervals given in Section II. By representing these time-interval definitions in terms of key system parameters, (i.e., $p$, $P_{1}$, $P_{2}$ and $P_{3}$), we then manage to attain a closed-form expression of the average AoI for the proposed AoR protocol. Given reasonable value sets of $P_{1}$, $P_{2}$ and $P_{3}$, we further minimize the average AoI by optimizing the status generation probability $p$.

In this section, we present simulation results to validate our analytical results, and demonstrate the superiority of the considered cooperative status update system over its non-cooperative counterpart in terms of achieving lower average AoI. For the non-cooperative status update system, we plot its average AoI curves according to the result given in [6, Eqs. (35)]. Each curve presented in the section is averaged over $10^{7}$ time slots.

We first study the average AoI of both cooperative and non-cooperative systems against the the transmission success probability of the S-D link (i.e., $P_{1}$), as shown in Fig. 2, for two different generation probabilities of the status updates. In both cases, we assume that the transmission success probabilities of the S-R link and the R-D link are $P_{2}\!\!=\!\!P_{3}\!\!=\!\!0.8$. It can be observed that in both cases, the performance of the considered cooperative status update system is obviously superior to that of the non-cooperative system, especially when $P_{1}$ is small. Specifically, when $p\!=\!0.5$ (or $p\!=\!0.9$), the considered cooperative system has a significant performance improvements over the non-cooperative system for the case $P_{1}\!<\!0.5$ (or $P_{1}\!<\!0.3$), respectively. This is because when the S-D link has bad channel conditions, the generated updates at S can be transmitted to D via R in our system while those in the non-cooperative system can only be retransmitted via the direct link. However, we can also observe that the AoI performance gain decreases as $P_{1}$ increases, and approaches 1 when $P_{1}$ is close to 0.8 (i.e., equal to $P_{2}$ and $P_{3}$). The rationality behind this is that when $P_{1}$ is large, most of the successfully received updates at D are transmitted via the S-D link. Therefore, R is not much helpful in reducing the average AoI. We also note that when $P_{1}$ is less than 0.27, a lower generation probability, i.e., $p\!=\!0.5$, results in a better AoI performance. On the contrary, the AoI performance of a higher generation probability, i.e., $p\!=\!0.9$, is always better when $P_{1}$ is greater than 0.27. Motivated by this interesting observation, we explore the relationship between the generation probability and the average AoI in the following.

We now investigate the average AoI against the generation probability $p$ by fixing the transmission success probabilities of the S-R link and the R-D link as $P_{2}\!=\!P_{3}\!=\!0.8$. As depicted in Fig. 3, we also consider two cases with the transmission success probability of the S-D link being 0.25 and 0.5, respectively. In the first case of $P_{1}=0.25$, it is apparent that the average AoI first decreases and then increases as the generation probability increases. This is because if the S-D link has bad channel conditions, the updates successfully delivered to D by R have a significant contribution on minimizing the average AoI. However, as S can preempt the transmission of R whenever there is a new arrival, if the updates are generated too frequently, the updates at R will be always preempted by S and cannot be transmitted to D to effectively reduce the AoI. Therefore, too high generation probabilities in turn increase the average AoI. We can also find that the average AoI when $p=0.5$ is smaller than that when $p=0.9$, which confirms our mentioned observation when $P_{1}$ is less than 0.27 in Fig. 2. In the second case of $P_{1}=0.5$, the average AoI always decreases as the generation probability increases. The monotone decreasing property demonstrates our mentioned phenomenon when $P_{1}$ is greater than 0.27 in Fig. 2. It is also worth mentioning that the optimal generation probabilities to minimize the average AoI in the case of $P_{1}\!=\!0.25$ and $P_{1}\!=\!0.5$ are $p=0.6770$ and $p=1$, respectively, which is consistent with the results calculated by using the formula in Theorem 1.

In this paper, we proposed an age-oriented opportunistic relaying (AoR) protocol to reduce the age of information (AoI) of a cooperative status update system, where the source timely reports randomly generated status updates to the destination with the help of a relay. By analyzing the evolution of AoI, we derived a closed-form expression of the average AoI of the proposed AoR protocol as a function of the generation probability of the status updates and the transmission success probability of each link. Given reasonable assumptions about the transmission success probabilities, we further figured out the optimal generation probability of the status updates that minimizes the average AoI. Simulation results validated our theoretical analysis and demonstrated the superiority of the proposed AoR protocol over the non-cooperative system in terms of minimizing the average AoI, especially when the source-destination link is not as good as the other two links.

A. Proof of Theorem 1

Recall that $\bar{\Delta}=\frac{[1-(1-p)(1-P_{3})]\cdot[1-(1-p)(1-P_{1})(1-P_{2})]}{p[pP_{1}+(1-p)P_{3}-(1-p)(1-P_{1})(1-P_{2})P_{3}]}$. To proceed, we derive the first-order derivative of $\bar{\Delta}$ with respect to (w.r.t) $p$. After some algebra manipulations, we have

where

are defined for the notation simplicity.

After a careful observation on the right hand side (RHS) of (29), we can deduce that the sign of $\frac{\partial\bar{\Delta}}{\partial p}$ is only determined by the numerator

since the denominator is always large than zero. To determine the monotonicity of the function $\bar{\Delta}$, we need to investigate the properties of the quadratic function $\kappa(p)$ on the feasible set of $p$ (i.e., [0, 1]).

Firstly, after some algebra manipulations, we note that

As $0<P_{1}<P_{2}<1$ and $0<P_{1}<P_{3}<1$, we have that $\omega<0$ and $\psi^{2}-4\chi\omega>0$. This means that the curve of $\kappa(p)$ always intersects the Y-axis at the negative half of the Y-axis and it always has two roots on the X-axis, which can be given by

where $x_{2}>x_{1}$.

To further characterize the shapes of the function $\kappa(p)$, we now investigate $\chi$ and $\psi$ for the following four possible cases:

1) When $\chi>0$ and $\psi>0$, we always have $x_{1}<0$ and $x_{2}>0$. Since $p$ is in the range of $[0,1]$, we have two sub-cases: (a) $x_{2}<1$ and (b) $x_{2}>1$. We draw the possible shapes of $\kappa(p)$ versus $p$ for the sub-case (a) and sub-case (b) in Fig. 4 (a) and Fig. 4 (b), respectively.

From Fig. 4 (a), we can see that in the sub-case (a), $\kappa(p)<0$ holds for $p\in[0,x_{2}]$, and $\kappa(p)>0$ holds for $p\in(x_{2},1]$. This means that the function $\bar{\Delta}$ is decreasing for $p\in[0,x_{2}]$ and is increasing for $p\in(x_{2},1]$. As a result, the minimum value of $\bar{\Delta}$ is achieved at $p=x_{2}$.

From Fig. 4 (b), we can see that in the sub-case (b), $\kappa(p)<0$ holds for $p\in[0,1]$. Therefore, the function $\bar{\Delta}$ is always decreasing for $p\in[0,1]$, which indicates that the value of $\bar{\Delta}$ is minimized at $p=1$.

2) When $\chi>0$ and $\psi<0$, we always have $x_{1}<0$ and $x_{2}>0$. Since $p$ is in the range of $[0,1]$, we also have two sub-cases: (a) $x_{2}<1$ and (b) $x_{2}>1$. The possible shapes of $\kappa(p)$ versus $p$ for the sub-case (a) and sub-case (b) are depicted as Fig. 5 (a) and Fig. 5 (b), respectively.

Similar to case 1), we can prove that the minimum value of $\bar{\Delta}$ is achieved by setting $p=x_{2}$ and $p=1$ for the case $x_{2}<1$ and $x_{2}>1$, respectively.

3) When $\chi<0$ and $\psi<0$, we always have $x_{1}<x_{2}<0$. Therefore, there only exists one possible shape of $\kappa(p)$ versus $p$ in this case, which is depicted as Fig. 6.

From Fig. 6, it is straightforward to see that $\kappa(0)<0$ holds for $p\in[0,1]$. This means that the function $\bar{\Delta}$ is always decreasing for $p\in[0,1]$ in this case. As a result, the minimum value of $\bar{\Delta}$ is achieved at $p=1$.

4) When $\chi<0$ and $\psi>0$, we always have $x_{2}>x_{1}>0$. Therefore, the possible shapes of $\kappa(p)$ versus $p$ can be depicted as Fig. 7. According to (30b), after some algebra manipulations, $\psi$ can be rewritten as

As $\psi>0$, $0<P_{1}<P_{2}<1$ and $0<P_{1}<P_{3}<1$, we thus have $P_{1}-P_{1}P_{3}-P_{2}P_{3}+P_{1}P_{2}P_{3}<0$, which can be rewritten as

To further validate this case, we investigate the value of $-\chi$ under the constraint of (35). We rewrite $-\chi$ as a quadratic function of $P_{1}$, which can be expressed as

where

are defined for the simplicity of notations.

As $0<P_{1}<P_{2}<1$ and $0<P_{1}<P_{3}<1$, according to (37b) and (37c), we have $\mu>0$ and $\xi<0$. In addition, if we set $P_{1}=1$, after some manipulations, we can attain that $-\chi({P_{1}})=(P_{3}-1)^{2}>0$. Similarly, if we substitute $P_{1}=\frac{P_{2}P_{3}}{1-P_{3}+P_{2}P_{3}}$ into the expression of $-\chi(P_{1})$, we can have $-\chi(P_{1})=\frac{P_{2}P_{3}(P_{2}-1)(P_{3}-1)^{2}}{(P_{2}P_{3}-P_{3}+1)^{2}}<0$. Based on the above analysis, we can draw the possible shapes of $-\chi(P_{1})$ versus $P_{1}$ for $\lambda>0$ and $\lambda<0$ in Fig. 8 (a) and Fig. 8 (b), respectively. Note that in these figures, we also show the possible positions of the point with X-coordinate being $\frac{P_{2}P_{3}}{1-P_{3}+P_{2}P_{3}}$.

From Fig. 8 (a) and Fig. 8 (b), we can see that if $P_{1}<\frac{P_{2}P_{3}}{1-P_{3}+P_{2}P_{3}}$ and $0<P_{1}<1$, $-\chi<0$ always holds. This means that we always have $\chi>0$ if $\psi>0$. This is contrary to the initial assumptions of this case (i.e., $\chi<0$ and $\psi>0$). Therefore, the case 4) is not valid.

In conclusion, we can find that in all of the possible case 1), case 2) and case 3), the minimum value of $\bar{\Delta}$ is achieved by setting $p=x_{2}$ and $p=1$ when $x_{2}<1$ and $x_{2}>1$, respectively. Therefore, based on (30a), (30b), (30c) and (33b), after some algebra manipulations, the optimal $p$ that minimizes $\bar{\Delta}$ can be given by (27).

This completes the proof of Theorem 1.