# Stalling in Queuing Systems with Heterogeneous Channels
## 4 Applications  
### Modelling of Heterogeneous Server Cluster with Stalling Buffer  
Compatibility problems of heterogeneous networks are solved by offering specialized data maintenance solutions, evaluating their combining cases, and analysing errors, links, etc. Optimization of the network node stalling buffer is important for combining networks of different capacities [31]. Web hosting companies are forced to change old slow servers to new fast servers, to serve an increasing flow of service. The optimal solution to this problem is using both groups--fast and slow servers connected to one heterogeneous server cluster using a stalling buffer (Figure 3).  
Performance of servers' W is calculated by using disks and disk arrays IOPS (input/output operations per second) :  
\[IOPS_{disk}=\frac{MeX_{rpn}}{\frac{McX_{rpn}}{1000}+\frac{MeX_{ latency}}{1000}},\]where \(MeX_{rpm}\)--disk rotational speed per minute; \(MeX_{scek}\)--disc seek time in milliseconds; \(MeX_{latency}\)--disc latency in milliseconds. Disk array (RAID) IOPS is calculated as follows:  
\[IOPS_{raid}=IOPS_{disk}\cdot(N_{disks}-D_{rp})\cdot D_{\rm R\%}+\frac{IOPS_{ disk}\cdot(N_{disks}-D_{wp})\cdot D_{w\%}}{C},\]  
where N\({}_{\rm disks}\)--the number of disks in an array; D\({}_{\rm rp}\)--reading parity for disks array; D\({}_{\rm wp}\)--writing parity for disks array; D\({}_{\rm R\%}\)--total percent of reading; D\({}_{\rm w\%}\)--total percent of writing; C--disks array overhead. Disk array RAID5 in slow and fast servers is used. For RAID5: D\({}_{\rm rp}=1\), D\({}_{\rm wp}=0\), D\({}_{\rm R\%}=70\), D\({}_{\rm w\%}=30\), \(C=4\).  
Slow servers are HP ProJiant DL180 G6 [36] width two Intel Xeon L5640@ 2.27 GHz processors (CPU Benchmarks 6358), with 128 GB DDR3 operating memory (read speed--13,887 MB/s; write speed--10,076 MB/s; MiN\({}_{r/\rm{wspeed}}=11,982\), MiN\({}_{\rm type}=8\)); we used five disks (N\({}_{\rm disks}=5\)) HP 3 TB 3.5\" LFF 6 G Dual Port SAS 7.2 K RPM (MeX\({}_{r/\rm{wdata}}=97\) Mb/s, data transferring rate MeX\({}_{\rm rate}=6.0\) Gb/sec, IOPS\({}_{\rm disk}=57\)) connected to RAIDS (IOPS\({}_{\rm raid}=18,098\)).  
Fast servers are HP ProJiant DL380 Gen9 [37] width two Intel Xeon E5-2699 v4 @ 2.20 GHz processors (CPU Benchmarks 23200), with 128 GB DDR4 operating memory (read speed--22,075 MB/s; write speed--16,842 MB/s; MiN\({}_{r/\rm{wspeed}}=19,458.5\), MiN\({}_{\rm type}=16\)); we used seven disks (N\({}_{\rm disks}=7\)) HP 2.4 TB SAS 12 G Enterprise 10 K LFF (MeX\({}_{r/\rm{wdata}}=195\) Mb/s, data transferring rate MeX\({}_{\rm rate}=12\) Gb/s, IOPS\({}_{\rm disk}=133\)) connected to RAIDS (IOPS\({}_{\rm raid}=62843\)). Servers were divided into two synchronized clusters. The interarrival traffic time was distributed under the exponential law with parameter \(\lambda\) and the length of service was also distributed under this law with parameters \(\mu_{1}\) and \(\mu_{2}\). This was the case in systems using M length waiting buffer and K length stalling buffer (see Figure 3).  
Performance of servers in clusters are calculated using the following formula (Figure 4):  
\[W=\frac{U\cdot MeX_{rate}\cdot MeX_{r/\rm{wdata}}\cdot IOPS_{raid}}{MiN_{r/ \rm{wspeed}}\cdot MiN_{type}},\]  
where U--summarized performance processors of server controller (CPU Benchmarks); MeX\({}_{\rm rate}\)--number of input output transferring's to array of disks; MeX\({}_{r/\rm{wdata}}\)--amount of data (bytes) transmitted per input/output operation; MiN\({}_{r/\rm{wspeed}}\)--data transferring rate; MiN\({}_{\rm type}\)--coefficient taking into account the type of memory [38]. Calculated coefficient of performance for fast servers W\({}_{\rm fast}=1190\) and for slow server W\({}_{\rm slow}=171\).  
The characteristics of the QS cluster with five fast and two slow servers are computed, and the results are displayed in Table 2.  
Figure 3: Heterogeneous servers’ clusters with a stalling buffer.