# Throughput of Buffer with Dependent Service Times
## 2 Related Works  
Based on the author's information, the results shared here are novel, including all the theorems and corollaries listed in points (a)-(f) of the previous section, as well as all the numerical examples in Section 6. The previous studies, mentioned in the next paragraph, focus solely on the stationary analysis and do not address the throughput of the buffer.  
The first article in which dependent service times of a queueing system are modeled in the Markovian service process is [14]. The interarrival times constitute a renewal process in [14], meaning they are independent and identically distributed, with an arbitrary interarrival distribution of finite mean. The buffer capacity is finite. For such a model, the stationary queue size distribution is obtained in [14]. To achieve this, the classic method of using a supplementary variable and calculating the stationary distribution of the embedded Markov chain is utilized. In [15], the tail of this distribution, but for a model with an infinite buffer, was investigated. In [16], other characteristics were studied, including the queue size at arrival epochs and the response time, for both finite and infinite buffer models. Ref. [17] is devoted to a multi-server model with a finite buffer, where the queue length and busy period distributions are derived. In [18], the queue size and queuing delay distributions are investigated under a slightly different service process, in which after an idle period, service always begins with a given phase distribution. In [19], a model with a finite buffer and group arrivals is considered, with two possible group acceptance protocols (total group rejection and partial group rejection). The queue size distribution, queuing delay distribution, and blocking probability of the last and first jobs in a group are obtained. Ref. [20] deals with an infinite-buffer model involving group services, where the queue length distribution and response time are computed. Ref. [21] demonstrates a different method of solving a model with an infinite buffer by exploiting the characteristic equation and its roots. In [22], the response time and inter-departure interval correlation are investigated in a model with an infinite buffer. Refs. [23; 24] analyze a model with an infinite buffer and group arrivals. Specifically, Ref. [23] focuses on the queue length, busy period, and idle period, while [24] is devoted to the queuing delay distribution. Reference [25] also focuses on a group-arrival model but with a finite buffer. In [26], a model with a finite buffer and group service is revisited, with the analysis focusing on optimizing the system's profit by maximizing a utility function composed of several performance metrics. Lastly, in [27], two models of group servicing are compared: in one model, the service phase remains unchanged during the system's idle period, while in the other, the service phase evolves continuously, including during idle periods.  
It should also be stressed that a different method is used in this paper than in [14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27]. The method here is based on solving the system of integral equations in the Laplace transform domain using a special recurrent sequence. The great advantage of this method is that it provides both the time-dependent and stationary solutions simultaneously. The classic approach used in the previous works is based on the analysis of embedded Markov chains and allows for obtaining only the stationary characteristics.  
There is only one recently published study that provides a full, time-dependent analysis of a model with a finite buffer [28]. However, it is devoted solely to queue size, with no investigation of throughput-related characteristics.