# Stalling in Queuing Systems with Heterogeneous Channels
## 4 Applications
### Modelling of Harvesters and Chainsaws Work Productivity  
Coordination of technologies, processes, or devices with different capacities arises very often. Let us consider the problem of coordinated usage of harvesters (Timberjack 1270D) and chainsaws in spruce stands using data of comparative work productivity analysis [39].  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Denotation** & **Formula** & **Value** \\ \hline \multicolumn{4}{c}{Basic characteristics of clusters with stalling} \\ \hline Intensity of interarrival & \(\Lambda\) & - & 0.58 (Gbps) \\ \hline Capacity of fast server & \(\mu_{1}\) & - & 0.146 (Gbps) \\ \hline Capacity of slow server & \(\mu_{2}\) & - & 0.045 (Gbps) \\ \hline Number of fast servers & \(m\) & - & 5 \\ \hline Number of slow servers & \(n\) & - & 2 \\ \hline Coefficient of QS utilization & \(\rho\) & \(\frac{\lambda}{\pi\mu_{1}+\mu_{2}}\) & 0.71 \\ \hline Coefficient of utilization of fast server & \(Q\) & \(\frac{\lambda}{\pi\mu_{1}}=\rho\cdot(1+\frac{\pi}{\pi\sigma})\) & 0.751 \\ \hline Rate of capacity of fast and slow servers & \(R\) & \(r=\frac{\mu_{1}}{\mu_{2}}\) & 6.959 \\ \hline \multicolumn{4}{c}{Calculated characteristics of clusters with stalling} \\ \hline Characteristics & Denotation & Value \\ \hline Occupancy probability of fast channel in QS without slow channel and stalling & \(P_{0}\) & 0.464 \\ \hline The occupancy probability of the fast channels and stalling buffer and free slow channels & \(P_{m,\mu,K}\) & 0.981 \\ \hline Downtime probability & \(P_{0,0,0}\) & 0.0185 \\ \hline Stock probability & \(P_{stack}\) & 0 \\ \hline Number of stuck queries & \(\overline{N}_{stack}\) & 0 \\ \hline Occupancy probability of all slow channel with free fast channel with stalling buffer & \(P_{0,0,0}\) & 0 \\ \hline Probability of queries in slow channels & \(P_{slow}\) & 0 \\ \hline Number of queries in slow channels & \(\overline{N}_{slow}\) & 0 \\ \hline Probability of queries in fast channels & \(\sum\limits_{i=1}^{n+K}\sum\limits_{s=0}^{n}p_{i,s}+p_{m+K,s}\cdot\frac{c-p^{M+1} }{1-\rho}\) & 0.9814 \\ \hline Number of queries in fast channels & \(P_{fast}\) & 3.7561 \\ \hline Probability of stalling & \(P_{stalling}\) & 0.3486 \\ \hline Number of queries in stalling buffer & \(\overline{N}_{stalling}\) & 1.4014 \\ \hline Probability of queue & \(P_{w}\) & 0 \\ \hline Number of queries in waiting line & \(\overline{N}_{w}\) & 0 \\ \hline Probability of loss queries & \(P_{m+K,s}\cdot p^{M}\) & 0 \\ \hline Average number of queries & \(\overline{N}\) & 5.1575 \\ \hline \end{tabular}
\end{table}
Table 2: Characteristics of the server’s cluster with stalling.  
Figure 4: Server components by queuing model.  
Harvester (\(\mu_{1}\)) and chainsaw (\(\mu_{2}\)) work productivity have been compared in Table 3, taking into account the volume of the tree trunk.  
Figure 5 shows how many harvesters and chainsaws will be needed for optimal systems work when the scope of forest harvesting is changed. We can see that, when using one harvester (m = 1) and a system load of 50 m\({}^{3}\) for optimal systems work, 10 chainsaws are needed (Figure 5, graph on the left--see the yellow point on the graph and on the chainsaw axis is showing a 10, and on load axis we can see a 50). In contrast, when using two harvesters (m = 2) and a system load of 100 m\({}^{3}\) for optimal systems work, 20 chainsaws will be needed (Figure 5, graph on the right--see the yellow point on the graph and on chainsaw axis is showing a 20, and on the load axis we can see a 100).