Age- and Deviation-of-Information of Time-Triggered and Event-Triggered Systems

We estimate the MGF of the sojourn time using the approach from [17, 28]. It follows from (5) for $n\geq 1$ and $\theta>0$ that

	$\displaystyle\mathsf{M}_{T}(\theta,n)\leq$	$\displaystyle\mathsf{E}\bigl{[}e^{\theta\max_{\nu\in[1,n]}\{S(\nu,n)-A(\nu,n)\}}\bigr{]}$
	$\displaystyle=$	$\displaystyle\mathsf{E}\biggl{[}\max_{\nu\in[1,n]}\bigl{\{}e^{\theta(S(\nu,n)-A(\nu,n))}\bigr{\}}\biggr{]}$
	$\displaystyle\leq$	$\displaystyle\mathsf{E}\Biggl{[}\sum_{\nu=1}^{n}e^{\theta(S(\nu,n)-A(\nu,n))}\Biggr{]}$
	$\displaystyle=$	$\displaystyle\sum_{\nu=1}^{n}\mathsf{E}\bigl{[}e^{\theta S(\nu,n)}\bigr{]}\mathsf{E}\bigl{[}e^{-\theta A(\nu,n)}\bigr{]},$

where we used independence of $S(\nu,n)$ and $A(\nu,n)$. By insertion of the envelope parameters we have

	$\displaystyle\mathsf{M}_{T}(\theta,n)\leq$	$\displaystyle e^{\theta(\overline{\sigma}_{S}(\theta)+\overline{\rho}_{S}(\theta))}\sum_{\nu=1}^{n}\Bigl{(}e^{-\theta(\underline{\rho}_{A}(-\theta)-\overline{\rho}_{S}(\theta))}\Bigr{)}^{n-\nu}$
	$\displaystyle\leq$	$\displaystyle e^{\theta(\overline{\sigma}_{S}(\theta)+\overline{\rho}_{S}(\theta))}\sum_{\nu=0}^{\infty}\Bigl{(}e^{-\theta(\underline{\rho}_{A}(-\theta)-\overline{\rho}_{S}(\theta))}\Bigr{)}^{\nu},$

where $\sum_{\nu=0}^{\infty}x^{\nu}=1/(1-x)$ if $x<1$ concludes the proof, implying the stability condition $\underline{\rho}_{A}(-\theta)>\overline{\rho}_{S}(\theta)$.

We use the same essential steps to estimate the MGF of the AoI. From (6) we have for $n\geq 1$ and $\theta>0$ that

	$\displaystyle\mathsf{M}_{\Delta}(\theta,n)\leq$	$\displaystyle\sum_{\nu=1}^{n}\mathsf{E}\bigl{[}e^{\theta S(\nu,n+1)}\bigr{]}\mathsf{E}\bigl{[}e^{-\theta A(\nu,n)}\bigr{]}$
	$\displaystyle+$	$\displaystyle\mathsf{E}\bigl{[}e^{\theta S(n+1,n+1)}\bigr{]}\mathsf{E}\bigl{[}e^{\theta A(n,n+1)}\bigr{]}$
	$\displaystyle\leq$	$\displaystyle e^{\theta(\overline{\sigma}_{S}(\theta)+2\overline{\rho}_{S}(\theta))}\sum_{\nu=1}^{n}\Bigl{(}e^{-\theta(\underline{\rho}_{A}(-\theta)-\overline{\rho}_{S}(\theta))}\Bigr{)}^{n-\nu}$
	$\displaystyle+$	$\displaystyle e^{\theta(\overline{\sigma}_{S}(\theta)+\overline{\rho}_{S}(\theta)+\overline{\rho}_{A}(\theta))}.$

Again, $\underline{\rho}_{A}(-\theta)>\overline{\rho}_{S}(\theta)$ achieves convergence if $n\rightarrow\infty$. ∎

By definition of the event-triggered system we have $C(A(n))=n\alpha$. Using (1), we also have $C(D(n+1))=\max\{\nu\geq 0:E(\nu)\leq D(n+1)\}$. Further, for the last expression we know that $\nu\geq(n+1)\alpha$, since $D(n+1)\geq A(n+1)$ and hence $C(D(n+1))\geq C(A(n+1))=(n+1)\alpha$. By insertion into (4) it holds for $n\geq 0$ that

$$\Phi(n)=\max\{\nu\geq(n+1)\alpha:E(\nu)\leq D(n+1)\}-n\alpha.$$

With a variable substitution it follows that

$$\Phi(n)=\alpha+\max\{\nu\geq 0:E((n+1)\alpha+\nu)\leq D(n+1)\}.$$

We use $D(n+1)=A(n+1)+T(n+1)$ and $A(n+1)=E((n+1)\alpha)=\sum_{m=1}^{(n+1)\alpha}I(m)$ to obtain

$$\Phi(n)=\alpha+\max\Biggl{\{}\nu\geq 0:\sum_{m=(n+1)\alpha+1}^{(n+1)\alpha+\nu}I(m)\leq T(n+1)\Biggr{\}}.$$

Now, choose some $\Phi_{\varepsilon}\in\mathbb{N}_{0}\geq\alpha-1$. The case $\Phi(n)>\Phi_{\varepsilon}$ occurs iff $\nu=\Phi_{\varepsilon}-\alpha+1$ satisfies the condition above, i.e., $\sum_{m=(n+1)\alpha+1}^{(n+1)\alpha+\nu}I(m)\leq T(n+1)$. It follows that

$$\mathsf{P}[\Phi(n)>\Phi_{\varepsilon}]=\mathsf{P}\Biggl{[}T(n+1)-\sum_{m=(n+1)\alpha+1}^{(n+1)\alpha+\Phi_{\varepsilon}-\alpha+1}I(m)\geq 0\Biggr{]}.$$

With Chernoff’s theorem (7) we have $\mathsf{P}[X\geq 0]\leq\mathsf{M}_{X}(\theta)$ for $\theta>0$ so that

$$\mathsf{P}[\Phi(n)>\Phi_{\varepsilon}]\leq\mathsf{M}\Biggl{[}T(n+1)-\sum_{m=(n+1)\alpha+1}^{(n+1)\alpha+\Phi_{\varepsilon}-\alpha+1}I(m)\Biggr{]}(\theta).$$

The result of Th. 1 follows for iid inter-event times $I(m)$. Note that for iid inter-event times $T(n+1)$ is independent of events that occur after $A(n+1)=E((n+1)\alpha)$.

For time-triggered systems, we have the additional difficulty that the generation of messages is not synchronized with the occurrence of events. Instead, at time $t\geq 0$, e.g., $t=A(n)$, we only know that the last event occurred at time $E(C(t))$ and the next event occurs at time $E(C(t)+1)=E(C(t))+I(C(t)+1)$. We denote $J(t)$ the residual inter-event time at time $t\geq 0$ until the next event occurs, i.e., $J(t)=E(C(t)+1)-t$. It follows that

$$J(t)=E(C(t))+I(C(t)+1)-t.$$

(14)

First, we formalize an intermediate result. Consider some times $t,\tau\geq 0$. From (1) we have

$$C(t+\tau)=\max\{\nu\geq C(t):E(\nu)\leq t+\tau\}.$$

(15)

For $\nu\geq C(t)+1$ we can write

	$\displaystyle E(\nu)=$	$\displaystyle E(C(t))+\sum_{m=C(t)+1}^{\nu}I(m)$
	$\displaystyle=$	$\displaystyle t+J(t)+\sum_{m=C(t)+2}^{\nu}I(m),$		(16)

where we use (14) in the second step. By insertion of (16) for $\nu\geq C(t)+1$ into (15) and noting that the case $\nu=C(t)$ is trivial, we obtain that

		$\displaystyle C(t+\tau)$
	$\displaystyle=$	$\displaystyle\max\Biggl{\{}\nu\geq C(t):J(t)1_{\nu\geq C(t)+1}+\sum_{m=C(t)+2}^{\nu}I(m)\leq\tau\Biggr{\}}$
	$\displaystyle=$	$\displaystyle C(t)+\max\Biggl{\{}\nu\geq 0:J(t)1_{\nu\geq 1}+\sum_{m=C(t)+2}^{C(t)+\nu}I(m)\leq\tau\Biggr{\}},$

where $1_{(.)}$ is the indicator function that is one if the argument is true and zero otherwise.

Next, we insert $D(n+1)=A(n)+\Delta(n)$ from (3) into (4) and with the previous result we obtain by substitution of $t=A(n)$ and $\tau=\Delta(n)$ for $n\geq 1$ that

$$\Phi(n)=C(A(n)+\Delta(n))-C(A(n))=\\ \max\Biggl{\{}\nu\geq 0:J(A(n))1_{\nu\geq 1}+\sum_{m=C(A(n))+2}^{C(A(n))+\nu}I(m)\leq\Delta(n)\Biggr{\}}.$$

Now, choose some $\Phi_{\varepsilon}\in\mathbb{N}_{0}$. The case $\Phi(n)>\Phi_{\varepsilon}$ occurs iff $\nu=\Phi_{\varepsilon}+1$ satisfies the condition above. It follows that

$$\mathsf{P}[\Phi(n)>\Phi_{\varepsilon}]\\ =\mathsf{P}\Biggl{[}\Delta(n)-J(A(n))-\sum_{m=C(A(n))+2}^{C(A(n))+\Phi_{\varepsilon}+1}I(m)\geq 0\Biggr{]}.$$

With Chernoff’s theorem (7) we have for $\theta>0$ that

$$\mathsf{P}[\Phi(n)>\Phi_{\varepsilon}]\\ \leq\mathsf{M}\Biggl{[}\Delta(n)-J(A(n))-\sum_{m=C(A(n))+2}^{C(A(n))+\Phi_{\varepsilon}+1}I(m)\Biggr{]}(\theta).$$

The result of Th. 1 follows for iid inter-event times $J(A(n))$ and $I(m)$. We note that in a time-triggered system $\Delta(n)$ is independent of the occurrence of events. ∎

In Fig. 6(a) we consider exponential network service times with parameter $\mu$ and a deterministic sensor signal, i.e., periodic events with deterministic inter-event times $1/\lambda=2$. This degenerate case serves as a reference. In this case both, the time-triggered and the event-triggered system, sent updates periodically. We choose $\alpha=\lambda w$ to ensure the same network utilization resulting in identical delay and AoI bounds.

The DoI bounds differ slightly since update messages are synchronized with the occurrence of sensor events in the event-triggered system but not in the time-triggered system. This is reflected by the residual inter-event time $J(t)$ in Th. 1. Since deterministic inter-event times are not memoryless, we estimate $\mathsf{M}_{J(t)}(-\theta)<1$ for $\theta>0$ by $1$ and obtain with Th. 1 for the time-triggered system that

$$\Phi_{\varepsilon}=\frac{\ln\varepsilon-\ln\mathsf{M}_{\Delta}(\theta)}{\ln\mathsf{M}_{I}(-\theta)}=\frac{\lambda(\ln\mathsf{M}_{\Delta}(\theta)-\ln\varepsilon)}{\theta}=\lambda\Delta_{\varepsilon},$$

where we inserted the MGF $\mathsf{M}_{I}(-\theta)=e^{-\theta/\lambda}$ of the deterministic inter-event time $1/\lambda$ and ignored integer constraints. In the final step, we substituted the AoI bound $\Delta_{\varepsilon}$ (9). This implies that the update rate that achieves the minimal AoI also minimizes the DoI in this case. As can be observed in Fig. 6(a), the minimal AoI bound $\Delta_{\varepsilon}=88$ and the minimal DoI bound $\Phi_{\varepsilon}=44$, corresponding to $\lambda=0.5$, are achieved for the same network utilization of about 0.3.

The direct correspondence of AoI and DoI $\Phi_{\varepsilon}=\lambda\Delta_{\varepsilon}$ observed in Fig. 6(a) is, however, not given in case of a random sensor signal. In Fig. 6(b) we show results for exponential instead of deterministic inter-event times. All other parameters are unchanged. The same set of parameters has also been used for Fig 4(b).

For the time-triggered system, that is signal-agnostic, the AoI is generally unaffected by the choice of the sensor model. Consequently, the AoI in Fig. 6(b) is identical to Fig. 6(a). The DoI increases, however, since a varying number of sensor events may occur during any update interval.

In case of the event-triggered system, the AoI in Fig. 6(b) is larger than in Fig. 6(a) since the arrivals to the network are now a random process. Due to the randomness, the AoI of the event-triggered system is generally larger than the AoI of the time-triggered system, as also observed in Fig. 4.

Regarding the DoI, the event-triggered system has the advantage that it is signal-aware and sends update messages only if needed. Interestingly, both systems, time-triggered and event-triggered, show comparable minimal DoI. For an intuitive explanation consider a burst of sensor events. In this case, the event-triggered system samples the sensor more frequently with the goal to improve the DoI. The increased rate of update messages may, however, cause network congestion and queueing delays that are detrimental to the DoI and outweigh their advantage. Overall this appears to cause similar minimal DoI, however, at a lower average network utilization for the event-triggered system. Concluding, the u-shaped DoI curves in Fig. 6(b) show that both systems are feasible and robust to variations of the network utilization. Configured optimally, the event-triggered system uses less network resources. It generates, however, more bursty network traffic.

A related finding in [5, 7] is that the problem of minimizing the mean-square norm of the state error at the monitor is equivalent to a signal-agnostic AoI minimization problem. In case of our event-triggered and hence signal-aware system, Fig. 6(b) does not confirm a similar result. Here, the network utilization that achieves the minimal tail bounds is different for the AoI and DoI, respectively.

Fig. 6(c) shows results for the same system as in Fig. 6(b) but with deterministic service times $1/\mu=4$ as also used in Fig. 5. In this case the time-triggered system is purely deterministic and achieves a very small AoI that is determined as the sum of the network service time and the width of the update interval. Hence, the AoI is minimal in case of full network utilization. The same applies for the DoI bound.

The event-triggered system shows a much larger AoI that is due to the randomness of the update messages. For low utilization, corresponding to a large threshold $\alpha$, the AoI is large due to infrequent updates if the sensor signal does not change much. In case of high utilization, small $\alpha$, queueing delays start to dominate and the AoI bends sharply upwards.

Despite the large AoI, the event-triggered system achieves a similarly good minimal DoI bound as the time-triggered system. Specifically at low utilization, the DoI bound of the event-triggered system is much smaller. This is a consequence of the deterministic network service, where the delivery of an update message within one message service time $1/\mu=4$ is almost guaranteed, given the utilization is low and queueing delays are avoided. This is particularly favorable for the event-triggered system since once the sensor signal changes by more than the threshold $\alpha$, an update message can be delivered with high probability within short time.

In Fig. 6(b) the minimal DoI bound of the time-triggered system is achieved for $w\approx 13$ and of the event-triggered system for $\alpha=8$, corresponding to utilizations of $0.325$ and $0.25$, respectively. We investigate these parameters in more detail in Fig. 7 where we show AoI and DoI bounds as well as empirical quantiles from $10^{8}$ samples of the AoI and DoI obtained by simulation. While the minimal DoI bounds of the time-triggered and event-triggered systems in Fig. 6(b) are about the same for $\varepsilon=10^{-6}$, we see in Fig. 7(a) that the DoI quantiles differ if $\varepsilon$ is not small. Particularly, the DoI approaches $\alpha$ for $\varepsilon\rightarrow 1$ in case of the event-triggered system and zero in case of the time-triggered system. Conversely, the AoI approaches $w$ for $\varepsilon\rightarrow 1$ in case of the time-triggered system and zero in case of the event-triggered system. We do not display tail bounds for the range of $\varepsilon$ in Fig. 7(a). We include the bounds in Fig. 7(b) and Fig. 7(c) where we show the tail decay. It can be noticed that the DoI bounds and the empirical DoI quantiles of the time-triggered and the event-triggered system exhibit the same speed of tail decay. This dominates the DoI if $\varepsilon$ is small causing similar DoI performance for both systems.

In Fig. 8 we include simulation results for non-optimal parameters $w$ and $\alpha$. For smaller $w$ and $\alpha$ we see an improvement of the AoI and DoI if $\varepsilon$ is not small. This is due to more frequent update messages. At the same time this causes increased network utilization and a smaller speed of tail decay. This consumes the initial advantage when $\varepsilon$ becomes small and leads to worse tail performance. In case of larger than optimal $w$ and $\alpha$, update messages are sent less frequently so that the AoI and DoI increase. This also brings about a reduction of the network utilization that can, however, only achieve a small improvement of the speed of the tail decay which is not relevant for $\varepsilon=10^{-6}$.