## 4 Analysis of the Model with the _MAP_, Simultaneous Service Process and Retrials  
Let us now assume that the described above system does not have a buffer. A customer, that arrives when the number of customers processing in the system is equal to \(N\), cannot be stored for waiting. Instead, he/she leaves the system (it is said that he/she moves to some virtual place called orbit) and tries to enter service after the random time interval. Duration of the intervals between retrials of customers from the orbit has an exponential distribution with the parameters \(\alpha_{i}\) if the number of customers in the orbit is \(i,\ i\geq 1\). By default, \(\alpha_{0}=0\). If the system is full at the retrial epoch, the customer will make more attempts. The number of retrials is assumed to be unlimited. As it was mentioned above, the phenomenon of retrials is typical in many real-world systems, including wireless communication networks. This made the queues with retrials a very popular subject of research. The behavior of the described system with retrials is described by the four-dimensional process  
\[\chi_{t}=\{j_{t},n_{t},v_{t},\eta_{t}\},\ j_{t}\geq 0,\ n_{t}=\overline{0,N},\ \
u_{t}=\overline{1,W},\ \eta_{t}=\overline{1,K_{n_{t}}},\ t\geq 0,\]  
where \(j_{t}\) means the number of customers in the orbit, \(n_{t}\) is the number of customers in service at moment \(t,\ t\geq 0\), and the meaning of the components \(\{v_{t},\eta_{t}\}\) is the same as above. It follows from definitions of arrival, service, and retrial processes that the random process \(\chi_{t}\) is also the continuous-time \(MC\). Let us arrange the states of the \(MC\)\(\chi_{t}\) in the lexicographic order. The set of the states having value \(i\) of the component \(j_{t}\) is called level \(i\), and the set of the states having value \((i,n)\) of the components \((j_{t},n_{t})\) is called sub-level \((i,n)\). Because the important distinction of the retrial systems from the systems with a buffer is that the servers may stay idle while there are customers that would like to enter service, it is clear that all levels \(i\geq 0\) consist of \((N+1)\) sub-levels \((i,n),\ n=\overline{0,N}\). **Lemma 2**. : _The non-zero blocks of the generator \(\hat{\mathbf{Q}}=(\hat{\mathbf{Q}}_{i,j})_{i,j\geq 0}\) of the \(MC\)\(\chi_{t}\) are defined as follows:_  
* _The diagonal blocks_ \(\hat{\mathbf{Q}}_{i,i}\) _are given by the formula:_ \[\hat{\mathbf{Q}}_{i,i}=\mathbf{\Omega}-\alpha_{i}\hat{I},\ i\geq 0,\] _where the block_ \(\mathbf{\Omega}\) _was defined above and_ \[\hat{I}=I_{W\hat{K}_{N}}-\mathrm{diag}\{O_{W\hat{K}_{N-1}},I_{WK_{N}}\};\]
* _The sub-diagonal blocks_ \(\hat{\mathbf{Q}}_{i,i-1}\) _are given by the formula:_ \[\hat{\mathbf{Q}}_{i,i-1}=\alpha_{i}\mathrm{diag}^{+}\{I_{W}\otimes H_{n}^{+},\ n=\overline{0,N-1}\},\ i\geq 1;\]
* _The up-diagonal blocks_ \(\hat{\mathbf{Q}}_{i,i+1}\) _are given by the formula:_ \[\hat{\mathbf{Q}}_{i,i+1}=\mathrm{diag}\{O_{W\hat{K}_{N-1}},D_{1}\otimes I_{K_ {N}}\},\ i\geq 0.\]  
Proof. : It is clear that transitions inside the level \(i\) occur in the retrial system in the same manner as in the corresponding system with the buffer when the buffer is empty. Only the exit from a given state can additionally occur due to a retrial from the orbit during the epoch when the number of customers receiving service is less than \(N\). If it is less than \(N\), the retrying customer is admitted for service. The customer departs from the orbit and a new service begins, i.e., the component \(j_{t}\) of the \(MC\) decreases by one, the component \(n_{t}\) increases by one, and the state of the underlying process \(\eta_{t}\) makes the corresponding transition. This explains the form of the block \(\tilde{\mathbf{Q}}_{i,i-1}\). The form of the block \(\tilde{\mathbf{Q}}_{i,i+1}\) is evident because the increase in the number of customers in the orbit can happen only at a customer arrival moment when the number of customers in service is equal to \(N\). Lemma 2 is proven.