## 2 Literature Review
A transition matrix is used in their solutions with the limitation that this analytical model can be applied only to a few operational scenarios. In the general category of the problem of a queue with unreliable service stations, there is a specific type of problems called repairable queues with degradation. In this case, it can be assumed that the service station can fail during the production activities either because of a fault or due to a degradation of its workload capacity. Such type of scenariois very common and difficult to analyse with a pure analytical approach. The research papers [8] adopt an analytical explicit solution for an M/M/1 of a repairable queueing model, and [24] shows an approach to model the failure rate of the service station that depends on the age of the components, without considering when the system is in idle or in a busy mode. Arrival rate of the jobs and occurrence of failures follow a Poisson process; service time and maintenance time are exponentially distributed. The attention over the working conditions which can affect the system performance is another important setting for this type of modelling. In [25], the authors examined an M/M/1 retrial queue with an unreliable server in an exogenous random environment and obtained the approximate orbit size distribution and mean queueing performance measures using matrix-analytic methods (level dependent QBD). In addition, [26] considered a variable failure rate for a M/M/1 queueing system with repairable service station with different parameters in idle time and busy time. They used transform method (QBD) to obtain the steady-state availability and the generating function method to obtain steady-state mean queueing length. Moreover, [7; 27; 28] focus on a single component system: [7] assumed time-varying operational conditions that are modelled by means of a continuous-time Markov process, whereas [27] combined a continuous-time Markov chain with queueing theory to obtain a model for the optimization of the repairing costs. In another work, [29] studied the M/M/2 repairable queueing system with varying failure rate at different times and used transform method (QBD) in steady-state period to obtain the probability of the states, the steady-state availability and the steady-state queueing length. In [30], authors presented and used an analytical method based on a decomposition approach that applies to a multi-stage manufacturing system. The queueing system was modelled with a Markov process and the degradation with a fixed decreasing yield that does not depend on the physics of the system and of the manufacturing process. As it can be noticed, the previous papers narrow the scenarios to the assumption that the distribution of failure, repair, interarrival and service are random and can be modelled with the Poisson distribution. This represents an important limitation for models which aim to analyse real production environment. Moving towards a more realistic modelling, an evolution model for the failure rate is proposed in [31] to evaluate the optimal integrated predictive maintenance strategy on the manufacturing system with a gamma process to characterize the performance degradation. Although this kind of modelling can be defined as hybrid because it combines at once mechanics and stochastic behaviours of the system, the main limitation of the previous manuscript is that the degradation follows a specific distribution, set as input of the model. A different viewpoint is presented in a recent work [32] that focuses on a joint model of production, quality control and preventive maintenance for a serial-parallel multistage production system. The complexity of the problem is higher than single stage queue and, as in [33], the novelty proposed is to evaluate the machine structure importance measure and productivity when selecting the machine to put in maintenance.