## 3 Proportional Lumpability
### Comparison with Lumpability of the Embedded Markov Chain  
We contrast proportional lumpability with the lumpability of the embedded Markov chain, as defined in [21]. Examples 2 and 3 offer fresh insights into this comparison. One common method for determining the stationary probability distribution of an ergodic continuous-time Markov chain \(X(t)\) involves examining its embedded Markov chain \(X^{E}(t)\). Technically, the embedded Markov chain is a regular discrete-time Markov chain (DTMC), sometimes referred to as its jump process. For a given \(X(t)\) with state space \(\mathcal{S}\), each entry of the one-step transition probability matrix of the corresponding embedded Markov chain is denoted by \(p(s,s^{\prime})\), signifying the conditional probability of transitioning from state \(s\) to state \(s^{\prime}\). This probability is defined as follows:  
\[p(s,s^{\prime})=\frac{q(s,s^{\prime})}{q(s)}\quad\text{ for }s\
eq s^{\prime}\]  
while \(p(s,s)=0\). Let us assume that \(X^{E}(t)\) is aperiodic, and let \(\boldsymbol{\pi}^{*}\) be its steady-state distribution. Therefore, we can compute the equilibrium distribution \(\boldsymbol{\pi}\) of \(X(t)\) in the following way: let \(W=\sum_{s\in\mathcal{S}}\pi^{*}(s)/q(s)\), then  
\[\pi(s)=\frac{\pi^{*}(s)}{Wq(s)}\,.\]  
It is important to note that our definition of \(q_{\sim}(s)\) differs from that of \(q(s)\) in general. Therefore, the proportionally lumpable property of \(X(t)\) does not necessarily imply that the corresponding embedded Markov chain \(X^{E}(t)\) is lumpable. Conversely, if \(X^{E}(t)\) exhibits lumpability, then \(X(t)\) is proportionally lumpable with respect to the function \(\kappa\) from \(\mathcal{S}\) to \(\mathbb{R}^{+}\), where \(\kappa(s)=q(s)\) for all \(s\in\mathcal{S}\). In summary, we can affirm that if \(X(t)\) possesses a strongly lumpable embedded process, then it is also proportionally lumpable. However, the reverse is not necessarily true. **Example 2**. : _Let us reconsider the reliability problem for a system comprising \(N\) components. Suppose our focus now shifts to determining the number of operational components at any given time. Thus, the state space \(\mathcal{S}=C_{i}:0\leq i\leq N\), where \(C_{i}\) represents the system state with \(i\) functioning components. We posit that in each state \(C_{i}\), the time to failure of a component follows an exponential distribution with rate \(\mu_{i}\). Additionally, each component can undergo restoration with a rate of \(\lambda\). In certain scenarios, system failures may occur due to the simultaneous malfunctioning of components resulting from common underlying factors. These common-cause failures could stem from issues like shared power supply failures, environmental conditions (such as earthquakes, floods, humidity, etc. ), common maintenance issues, and so forth. Simultaneous failures attributed to common causes may happen at a rate of \(\mu_{c}\). The state transition diagram for this system repair model is depicted in Figure 3._  
_We can prove that this model is proportionally lumpable with respect to the relation \(\sim\) over \(\mathcal{S}\) given by the reflexive, symmetric, and transitive closure of \(\{(C_{i},C_{j}):1\leq i,j\leq N\}\), and the function \(\kappa\) such that \(\kappa(C_{i})=q_{\sim}(C_{i})\) for \(i\in\{0,\ldots,N\}\). This relation induces two equivalence classes, named \(S_{0}=\{C_{0}\}\) and \(S_{1}=\{C_{1},\ldots,C_{N}\}\), and the model in Figure 3 results to be proportionally lumpable, whose aggregated chain is depicted in Figure 4._  
_In this case, the model in Figure 3 does not have a strongly lumpable embedded process due to the fact that \(q(C_{i})\
eq q_{\sim}(C_{i})\) for each \(i\in\{0,\ldots,N\}\)._  
**Example 3**. : _Consider the model described in Example 1. We showed that the CTMC depicted in Figure 1 is proportionally lumpable. It is easy to see that this model has also a strongly lumpable embedded process.