# Priority Queues with Fractional Service for Tiered Delay QoS
## 4 Performance Analysis
Then, it is readily shown that the mean and the variance of \(t_{0}\)* are:  
\[(\overline{t}_{0}\)*,\(\sigma_{t_{0}*}^{2})=\left(\;\;\frac{\overline{t_{0}}}{(1+ \overline{n_{1}})},\;\;\;\left(1-\frac{\overline{n_{1}}}{1+\overline{n_{1}}} \right)^{2}\sigma_{t_{0}}^{2}+\frac{\overline{n_{1}}}{1+\overline{n_{1}}} \overline{t}_{0}\;\;\right) \tag{5}\]  
Under the assumption that the packet arrival process remains stationary, we can substitute Equation (5) into Equations (1)-(4) to obtain \(\delta_{0}\), the probability that Buffer-0 is at its threshold \(K_{0}\) when a new Class-0 packet arrives. This is the probability of delay limit violation for Class-0 and it depends on the effective load of Buffer-0, \(\rho_{0}^{*}=\overline{X}/(C\overline{t_{0}}^{*})\). For \(m=2\), a two-buffer system (\(M=1\)) with exponentially distributed inter-arrival time and exponentially distributed service time, Figure 7 shows the simulation result for \(\delta_{0}\) as a function of \(\rho_{0}^{*}\) almost coincides with our result based on diffusion approximations. This analytical method can be readily extended to \(M+1\) buffers, with \(M>1\). Again, we assume the traffic is heavy and each buffer, except Buffer-0, has traffic ready to be promoted. For each \(i\), we define \(L_{i}\) as the aggregate of packet queues formed in the \(i+1\) highest-priority buffers (_i.e._, Buffer-0, Buffer-\(1\),..., Buffer-\(i\)), and \(H_{i}\) the aggregate of packet queues formed in the \(M\)-\(i\) lowest-priority buffers (_i.e._, Buffer-\((i+1)\), Buffer-\((i+2)\),..., Buffer-\(M\)). Figure 8 shows the aggregate queues \(L_{2}\) and \(H_{2}\) for an \(M=4\) system (5 buffers). The analytical result for the two-buffer case can then be applied to the two aggregate buffers, \(L_{i}\) and \(H_{i}\). Figure 7: Analytical _versus_ simulation results: probability of overflow for Buffer-0 _versus_ its load (2 Buffers, and \(m=2\)). That is, we may treat \(L_{i}\) as a \(G/G/1/K_{i}\) queueing system whose blocking probability corresponds to the probability of overflow for Class-\(i\). From the conservation law of priority queues [10], the distribution of the number of packets in the system is invariant to the order of service, as long as the scheduling discipline selects packets independent of their service times. Since the FSB is packet-size-neutral, work-conserving, and no packets are dropped inside \(L_{i}\), we can apply the conservation law to the lower aggregate buffer \(L_{i}\) and use the diffusion result of \(G/G/1/K_{i}\) to find the probability of \(Q_{i}=K_{i}\) when a new packet arrives at \(L_{i}\). Interested reader may refer to Reference [10] for a more comprehensive description on the conservation law of priority queues for modeling queueing system. Let \(\lambda_{i}\) denote the packet arrival rate at Buffer-\(i\) from the flow scheduler. The aggregate packet arrival rate for \(L_{i}\) coming from the flow scheduler is given by:  
\[\Lambda_{i}=\sum\
olimits_{k=0}^{i}\lambda_{k}=1/\overline{t}_{i} \tag{6}\]  
Then, under the assumption that \(H_{i}\) is never empty, the effective inter-arrival time for \(L_{i}\) is given by:  
\[(\overline{t}_{i}*,\sigma_{t_{i}}^{2}*)=\left( \begin{array}{cc}\overline{t }_{i}\\ (1+\eta_{i+1})\end{array} \right),&\left(1-\frac{\eta_{i+1}}{1+\eta_{i+1}} \right)^{2}\sigma_{t_{i}}^{2}+\frac{\eta_{i+1}}{1+\eta_{i+1}}\overline{t}_{i} \end{array}\right) \tag{7}\]  
where:  
\[\eta_{i+1}=1/(m(i+1)) \tag{8}\]  
By substituting Equation (7) into Equations (1)-(4), we can obtain \(\delta_{i}\), the probability of overflow for Class-\(i\) packets. Finally, we may obtain the overall packet blocking probability \(P_{B}\) of the system by observing that all \(M\) buffers can be aggregated to a single buffer \(L_{M}\) and represented by a \(G/G/1/K_{M}\) queue such that \(P_{B}=\delta_{M}\). The FSA allows each delay class, except Class-0, to advance, thus receiving a delay guarantee.