# Stalling in Queuing Systems with Heterogeneous Channels
## 2 QS with Heterogeneous Channels and Stalling Buffer
### Calculation of Steady-State Probabilities  
Assume that the state of a QS with \(m+n\) heterogeneous channels, a stalling buffer of capacity \(K\), and waiting line length \(M\), be defined by numbers: \(\textbf{{i}},\textbf{{j}},\textbf{{k}}\), where \(i\) and \(j\) are the numbers of queries served in fast and slow channels, respectively, and \(k\) is the number of stalled or waiting queries--\(0\leq\textbf{{i}}\leq\textbf{{m}},\textbf{{0}}\leq\textbf{{j}}\leq\textbf{{n}}, \textbf{{0}}\leq\textbf{{k}}\leq\textbf{{K}}+\textbf{{M}},\textbf{{m}}\geq 1, \textbf{{n}}\geq 0\), \(K\geq 0,\textbf{{M}}\geq 0\). Denote the corresponding steady-state probabilities by \(P_{i,j,k}\). Let us draw the state graph of such a QS inn Figure 2; here, the vertexes denote the states connected by arrows representing transitions with non-null probability from one state to the other [1]. The states on the graph are specified as well:  
\(P_{0,0,0}\)--all channels and stalling buffers are free;  
\(P_{i,0,0}\)--in the fast channels are the \(i\) queries, the slow channels and stalling buffer are free;  
Figure 1: Scheme of QS with stalling buffer. Figure 2: Graph of QS state. \(P_{m,0,k}\)--the fast channels are occupied with \(m\) queries, the slow channels are free, the k queries are stalled in the stalling buffer;\(P_{m,0,K}\)--the fast channels are occupied with \(m\) queries, the slow channels are free, the stalling buffer is full; \(P_{0,j,0}\)--in the slow channels are \(j\) queries, the fast channels and the stalling buffer are free; \(P_{i,j,0}\)--in the fast channels are \(i\) queries, in the slow channels are \(j\) queries, the stalling buffer is free; \(P_{m,j,k}\)--the fast channels are occupied with \(m\) queries, in the slow channels are \(j\) queries, \(k\) queries are stalled in the stalling buffer; \(P_{m,j,k}\)--the fast channels are occupied with \(m\) queries, in the slow channels are \(j\) queries, the stalling buffer is full; \(P_{0,n,0}\)--only the slow channels are occupied with \(n\) queries; \(P_{i,n,0}\)--in the fast channels are \(i\) queries, in the slow channels are \(n\) queries, the stalling buffer is free; \(P_{m,n,k}\)--the fast channels are occupied with \(m\) queries, the slow channels are occupied with \(n\) queries, k queries are stalled in the stalling buffer; \(P_{m,n,k+k}\)--the fast channels are occupied with \(m\) queries, the slow channels are occupied with \(n\) queries, the stalling buffer is full, \(k\) queries are waiting in line; \(P_{m,n,k+k}\)--the fast channels are occupied with \(m\) queries, in the slow channels are \(n\) queries, the stalling buffer and the queue are full, and the next arrived query will be lost;  
Typically, the probabilities of these states are calculated using a linear equation system whose coefficients are equal to the rates of transition while the order of the system is equal to the number of states [1]. Thus, one can derive steady-state equations according to the steady-state graph in Figure 2. They are given in the following system (1):  
\[\begin{split} P_{0,j,0}\cdot(\lambda+j\cdot\mu_{2})=P_{1,j,0}\cdot \mu_{1}+P_{0,j+1,0}\cdot(j+1)\cdot\mu_{2},\ \ \ 0\leq j<n,\\ P_{i,j,0}\cdot(\lambda+i\cdot\mu_{1}+j  
* Coefficient of utilization of fast servers with switched-out slow channels \(\mathbf{q}=\frac{\mathbf{\lambda}}{\mathbf{m}\cdot\mathbf{\mu}_{1}}\);
* Rate of capacity of fast and slow servers \(r=\frac{\mu_{1}}{\mu_{2}}\);
* Stalling buffer length \(K\);
* Waiting queue length \(M\).  
Now, the coefficient of QS utilization is as follows:  
\[\rho=\frac{\lambda}{m\cdot\mu_{1}+n\cdot\mu_{2}}=\frac{q}{1+\frac{n}{m\cdot \tau}}.\]  
Let us denote the polynomial functions and ratios that can be used to explicitly derive the steady-state probabilities from systems (1) and (2):  
\[R_{i,j}=\sum_{s=j}^{n}\frac{s!}{j!\cdot(s-j)!}P_{\min(i,m),s,\max(i-m,0)} \tag{3}\]  
\[A_{i,j}=\frac{R_{i,j}}{R_{0,j}} \tag{4}\]  
\[h_{j}=\frac{R_{m+K,j}}{R_{m+K,n}} \tag{5}\]  
The explicit expressions of QS steady-state probabilities are given by Theorem A1. **Theorem A1**.