# Priority Queues with Fractional Service for Tiered Delay QoS
## 4 Performance Analysis  
In this section, we perform queueing analysis for the FSB presented in Section 2. Specifically, we use diffusion approximations and a queue aggregation technique to deduce the probability of overflow for each delay class under heavy traffic conditions. Our analytical results demonstrate how the delay limit for each delay class can be enforced by using proper values of FSB parameters: the unit fractional credit and the buffer capacity. To illustrate our analysis, consider the highest-priority buffer of a two buffer case: Buffer-0 with threshold \(K_{0}\). Queueing analysis of Buffer-0 is performed under the assumption that the traffic input rate is sufficiently high. Packets arriving at Buffer-0 from the flow scheduler are assumed to follow a stationary arrival pattern with known mean and variance. Under heavy traffic conditions, we are making an approximate assumption that Buffer-1 always has packets ready to be promoted to Buffer-0 via the FSA. All packet sizes are assumed to follow the same distribution with known mean and variance. Figure 6 shows five sequential Class-0 packet arrivals with \(\eta_{1}=1/2\). In this example, _every other_ Class-0 packet arrival from the flow scheduler is accompanied by a Class-1 packet promoted to Class-0 from Buffer-1. Thus, the traffic served by Buffer-0 includes Class-0 packets coming straight from the flow scheduler and promoted packets coming from Buffer-1. Essentially, Buffer-0 is a \(G/G/1/K_{0}\) queueing system where inter-arrival times should account for Class-0 traffic from the flow scheduler and traffic promoted from Buffer-1. We approximate the occupancy of Buffer-0 as a diffusion process of a \(G/G/1/K_{0}\) queueing system where \(K_{0}\) is the buffer threshold for Buffer-0. Diffusion process is often used by researchers to model packet schedulers, under heavy load conditions [10], because it leads to closed-form results for the behavior of the scheduler. The key notion is, under heavy traffic conditions, the discontinuities in time from the arrival and departure processes are much smaller than queueing delay. Thus, these processes may be treated as continuous fluid flows, which allow the queue length to be approximated by a diffusion process. For finite buffer cases, the overflow probability can be obtained by imposing boundary conditions on the diffusion process. A comprehensive survey article [17] compared different diffusion approximation techniques and concluded that the diffusion formula by [PERSON] [18] for \(G/G/1/K\) queue is the most accurate and robust. It solved the diffusion equation with jump boundaries and instantaneous returns at 0 and \(K\). A diffusion process is characterized by first and second order statistics of the arrival/departure processes. Let the mean and the variance of packet inter-arrival time be \((\ \overline{t},\ \ \sigma_{t}^{2}\ )\). For a packet of random size \(X\) bits, the mean and the variance of packet service times are denoted \((\overline{X}/C\ \,\ \ \sigma_{X}^{2}/C^{2}\ \ )\), where \(C\) represents the service rate in bits/s. Then the probability of the diffusion process touching the boundary \(K\), denoted \(\delta\), is given by [18]:  
\[\delta=\beta(1-\rho)\exp(\gamma(K-1)) \tag{1}\]  
where:  
\[\rho=\overline{X}/(C\overline{t}) \tag{2}\]  
Figure 6: An example of packets arriving at Buffer-0 with \(\eta_{1}=1/2\). \[\gamma=-2(1-\rho)/(\rho\sigma_{1}^{2}/\tilde{t}^{2}+\sigma_{X}^{2}/\overline{X}^{ 2}) \tag{3}\]  
and:  
\[\beta=\rho(1-\rho^{2}\mathrm{exp}(\gamma(K-1)))^{-1} \tag{4}\]  
Let the mean and the variance of packet inter-arrival time for Buffer-0 from the flow scheduler be \((\;\overline{t}_{0},\;\;\sigma_{t_{0}}^{2}\;)\). Also, assume that Buffer-1 has an unlimited supply of packets to be promoted. Since Buffer-0 has to deal with packets from the flow scheduler and packets promoted from Buffer-1, we define the _effective packet inter-arrival time_, \(t_{0}\)*, as the time interval between two successive packets entering Buffer-0, regardless of the packets' origins.