## 2 Description of a Simultaneous Service Process
### Description  
The description of the service process is as follows. Let the number of customers in service at the moment \(t\) be \(n_{t},\ n_{t}=\overline{0,N}\). This notation means that \(n_{t}\) admits values in the set \(\{0,1,\ldots,N\}\). Let \(\eta_{t}\) be the auxiliary finite-state process such that the two-dimensional process \(\{n_{t},\eta_{t}\}\) behaves as the quasi-death process between the successive service beginning epochs. The quasi-death process is the case of the well-known Quasi-Birth-and-Death (\(QBD\)) process, see [1].  
Under the fixed state \(n\) of the process \(n_{t}\), \(n=\overline{0,N}\), the process \(\eta_{t}\) admits values from the set of cardinality \(K_{n},\ K_{n}<\infty\). The quasi-death process \(\{n_{t},\eta_{t}\}\) under the fixed value \(n\) of the process \(n_{t}\) is completely defined by two matrices, \(H^{0}_{n}\) and \(H^{-}_{n}\). The non-diagonal entries of the square matrix \(H^{0}_{n},\ n=\overline{0,N}\), of size \(K_{n}\) describe the transition rates of the process \(\eta_{t}\) within its state space which do not lead to the end of customer service. The diagonal entries of the matrix \(H^{0}_{n}\) are negative. The module of such an entry defines the rate of the process \(\eta_{t}\) exit from the corresponding state. The matrix \(H^{-}_{n},\ n=\overline{1,N}\), of size \(K_{n}\times K_{n-1}\) describes transition rates of the process \(\eta_{t}\) when one of \(n\) customers processed in the system departs.  
Note that the matrix \(H^{0}_{n}\) is the sub-generator and \(H^{0}_{n}\mathbf{e}+H^{-}_{n}\mathbf{e}=\mathbf{0}^{T},\ n=\overline{1,N}\), where \(\mathbf{e}\) and \(\mathbf{0}\) are column and row vectors of \(1\)'s and \(0\)'s of a suitable size, respectively, and \({}^{T}\) denotes transposition of the vector.  
Because when \(n=0\) customers are present in the system, it does not make sense to speak about their deaths. Thus, below by default we assume that \(K_{0}=1,\ H^{0}_{0}=0,\ H^{-}_{0}=0\).  
If the number of customers receiving service in the system is \(n,\ n=\overline{0,N-1}\), and a new customer arrives, this customer is admitted to the system. The number of customers in service increases to \(n+1\) and the process \(\eta_{t}\) makes the transition with the probabilities that constitute the stochastic matrix \(H_{n}^{+},\ n=\overline{0,N-1}\), of size \(K_{n}\times K_{n+1}\), \(H_{n}^{+}\mathbf{e}=\mathbf{e},\ n=\overline{0,N-1}\).  
Let us briefly present the examples of the determination of the matrices \(H_{n}^{0}\), \(H_{n}^{-}\), and \(H_{n}^{+}\).  
Let customers' service be provided by \(N\) independent servers. Service time by a server has a \(PH\) distribution with the irreducible representation \((\boldsymbol{\beta},S)\) where \(\boldsymbol{\beta}=(\beta_{1},\beta_{2},\ldots,\beta_{M})\) is the stochastic row vector of size \(M\) and \(S\) is a sub-generator of size \(M\). Denote \(\mathbf{S}_{0}=-S\mathbf{e}\). Such a service time can be interpreted as the time of reaching the single absorbing state \(M+1\) by the underlying \(MC\)\(\zeta_{t}\) that admits the value \(m,\ m=\overline{1,M}\), at the moment of service beginning with the probability \(\beta_{m}\), transits in the set of transient states \(\{1,2,\ldots,M\}\) with the rates defined by the entries of the matrix \(S\) and, finally, transits to the absorbing state with the rates defined by the entries of the column vector \(\mathbf{S}_{0}\). For more detailed information about \(PH\) distribution, see, e.g., [1; 17; 20].  
The most popular in the existing literature two ways to description of the service process of many customers, generate two examples of constructing the matrices \(H_{n}^{0}\), \(H_{n}^{-}\), and \(H_{n}^{+}\) defining the process \(\{n_{t},\eta_{t}\}\) for this system with \(N\) identical independent servers and the \(PH\) distribution of service times.