## 3 Proportional Lumpability  
The notion of proportional lumpability made its debut in [20]. Much like the notion of quasi-lumpability [15], which is also known as near-lumpability in [13], proportional lumpability builds upon the initial concept of strong lumpability. Furthermore, in contrast to the broader scope of quasi-lumpability, proportional lumpability allows for the precise derivation of a solution for the original process. **Definition 2** (Proportional 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 proportionally lumpable with respect to \(\sim\) (or, alternatively, \(\sim\) is termed a proportional 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 \(s,s^{\prime}\in S_{i}\),_  
\[\frac{q(s,S_{j})}{\kappa(s)}=\frac{q(s^{\prime},S_{j})}{\kappa(s^{\prime})}.\]  
_where \(\kappa\) is a function from \(\mathcal{S}\) to \(\mathbb{R}^{+}\). We say that \(X(t)\) is \(\kappa\)-proportionally lumpable with respect to \(\sim\) (or, alternatively, \(\sim\) is a \(\kappa\)-proportional lumpability for \(X(t)\)) if \(X(t)\) is proportionally lumpable with respect to \(\sim\) and function \(\kappa\)._  
The next theorem proven in [20] establishes that proportional lumpability facilitates the calculation of a precise solution for the original model. **Theorem 1** (Aggregated process for proportional lumpability). : _Consider a CTMC \(X(t)\) with a state space \(\mathcal{S}\), infinitesimal generator \(\mathbf{Q}\), and equilibrium distribution \(\pi\). Let \(\kappa\) denote a function from \(\mathcal{S}\) to \(\mathbb{R}^{+}\), \(\sim\) be a \(\kappa\)-proportional lumpability for \(X(t)\), and \(\widetilde{X}(t)\) be the aggregated process over the state space \(\mathcal{S}/\sim\) and with infinitesimal generator \(\mathbf{Q}\) defined by, for any equivalence classes \(S_{i},S_{j}\in\mathcal{S}/\sim\) where \(S_{i}\
eq S_{j}\), and for any state \(s_{\prime}\in S_{i}\),_  
\[\widetilde{q}(S_{i},S_{j})=\frac{q(s,S_{j})}{\kappa(s)}\,.\]  
_Then, \(\widetilde{\mu}\) such that for any equivalence class \(S\in\mathcal{S}/\sim\),_  
\[\widetilde{\mu}(S)=\sum_{s\in S}\pi(s)\kappa(s) \tag{1}\]  
_is an invariant measure of \(\widetilde{X}(t)\)._  
Definition 3 elucidates a technique for perturbing a CTMC that is proportionally lumpable, transforming it into one that is strongly lumpable. Unlike earlier perturbation approaches, Theorem 2 furnishes a method to calculate the stationary probabilities of a proportionally lumpable chain by leveraging those of the perturbed lumpable chain. The proof of Theorem 2 is available in [20]. **Definition 3** (Perturbed Markov chains). : _Consider a CTMC \(X(t)\) with a state space \(\mathcal{S}\) and infinitesimal generator \(\mathbf{Q}\). Let \(\kappa\) denote a function from \(\mathcal{S}\) to \(\mathbb{R}^{+}\). We say that a CTMC \(X^{\prime}(t)\) with infinitesimal generator \(\mathbf{Q}^{\prime}\) is a perturbation of \(X(t)\) with respect to \(\kappa\) if \(X^{\prime}(t)\) is obtained from \(X(t)\) by perturbing its rates such that for all \(s,s^{\prime}\in\mathcal{S}\) with \(s\
eq s^{\prime}\)._\[q^{\prime}(s,s^{\prime})=\frac{q(s,s^{\prime})}{\kappa(s)}\,.\]  
**Theorem 2** (Equilibrium distribution for proportional lumpability). : _Let \(X(t)\) be a CTMC with state space \(\mathcal{S}\), infinitesimal generator \(\mathbf{Q}\), and equilibrium distribution \(\boldsymbol{\pi}\). Let \(\kappa\) be a function from \(\mathcal{S}\) to \(\mathbb{R}^{+}\).