## 2 Background
### Strong (or Ordinary) Lumpability  
In the field of performance and reliability analysis, the notion of lumpability introduces a method for aggregating models. This method allows for the development of a reduced Markov chain compared to the original one while maintaining the ability to determine precise results for the original process.  
The concept of lumpability is defined by equivalence relations that span the state space of the Markov chain. These relations create a division within the state space of the Markov chain, enabling the grouping of equivalent states into larger macro-states, which ultimately decreases the total state space. When the division fulfills the _strong_ (or _ordinary_) lumpability condition [11; 31], the equilibrium solution of the condensed process can provide a precise solution for the initial one.  
The definition of strong lumpability was originally presented in [11] and has been extensively investigated in subsequent research [13; 24; 32; 33; 34].  
**Definition 1** (Strong lumpability).: _Consider a CTMC \(X(t)\) with a state space \(\mathcal{S}\), and let \(\sim\) denote an equivalence relation over \(\mathcal{S}\). We define \(X(t)\) to be strongly lumpable with respect to \(\sim\) (or, alternatively, \(\sim\) is termed a strong lumpability for \(X(t)\)) if \(\sim\) induces a partition on the state space of \(X(t)\) such that for any equivalence classes \(S_{i},S_{j}\in\mathcal{S}/\sim\) where \(S_{i}\
eq S_{j}\), and for any states \(s,s^{\prime}\in S_{i}\), the transition rates from \(s\) to \(S_{j}\) and from \(s^{\prime}\) to \(S_{j}\) are equal, i.e.,_  
\[q(s,S_{j})=q(s^{\prime},S_{j})\,.\]  
Therefore, an equivalence relation over the state space of a Markov process has a strong lumpability if it divides the space into equivalence classes, ensuring that within each class, the collective transition rates to any other class remain the same. It is important to highlight that every Markov process exhibits strong lumpability with respect to the identity relation, as well as the trivial relation, which comprises only one equivalence class.  
In the paper by [PERSON] et al. [11], it is demonstrated that for an equivalence relation \(\sim\) over the state space of a Markov process \(X(t)\), the aggregated process remains a Markov process for any initial distribution if and only if \(\sim\) represents a strong lumpability for \(X(t)\). Furthermore, the transition rate between two macro-states \(S_{i}\) and \(S_{j}\) in \(\mathcal{S}/\sim\) is equal to \(q(s,S_{j})\) for any \(s\) belonging to \(S_{i}\).  
**Proposition 1** (Aggregated process for strong lumpability).: _Consider a CTMC \(X(t)\) with a state space \(\mathcal{S}\), infinitesimal generator \(\mathbf{Q}\), and equilibrium distribution \(\boldsymbol{\pi}\). Let \(\sim\) be a strong lumpability for \(X(t)\) and \(\widetilde{X}(t)\) be the aggregated process with state space \(\mathcal{S}/\sim\) and infinitesimal generator \(\widetilde{\mathbf{Q}}\) defined by, for any equivalence class \(S_{i},S_{j}\in\mathcal{S}/\sim\),_  
\[\widetilde{q}(S_{i},S_{j})=q(s,S_{j})\]for any \(s\in S_{i}\). Then, the equilibrium distribution \(\widetilde{\pi}\) of \(\widetilde{X}(t)\) is characterized by the property that for any equivalence class \(S\in\mathcal{S}/\sim\),_  
\[\widetilde{\pi}(S)=\sum_{s\in S}\pi(s)\,.\]