Infinitesimal perturbation analysis of a single-stage fluid queue with loss feedback and non-responsive competing traffic

Abstract

In this paper we perform Infinitesimal Perturbation Analysis (IPA) for a single-stage stochastic fluid queue that is shared between two competing sources, one that employs additive loss-feedback congestion control and the other that employs no congestion-control (i.e., it is unresponsive). This scenario is applicable within the realm of computer communication networks particularly at bottleneck router queues where multiple and diverse flows compete for bandwidth. We optimize the tradeoff between total loss volume and queue workload (a measure for queueing delay). Although a sound knowledge of the system’s dynamics is required to derive the IPA gradient estimators, no knowledge of the underlying probability distributions governing the system is required. What results are fairly simple counting processes, whose values can be computed directly from an ongoing live stream of traffic.

Appendix: Derivation of α 1

During a full-period:

$$ \alpha_{1} = \sigma_{1} - c\gamma\left(\frac{\alpha_{1}}{\alpha_{1} + \sigma_{2}} \right) $$

(30)

$$ \gamma = \sigma_{2} + \alpha_{1} - \beta $$

(31)

Substitute Eq. 31 into Eq. 30

$$\begin{array}{@{}rcl@{}} \alpha_{1} & = & \sigma_{1} - c(\sigma_{2} + \alpha_{1} - \beta)\left(\frac{\alpha_{1}}{\alpha_{1} + \sigma_{2}} \right) \\ & \Rightarrow & \alpha_{1}(\alpha_{1} + \sigma_{2}) = \sigma_{1}(\alpha_{1} + \sigma_{2}) - c(\sigma_{2} + \alpha_{1} - \beta)\alpha_{1} \\ & \Rightarrow & {\alpha_{1}^{2}} + \alpha_{1}\sigma_{2} = \sigma_{1}\alpha_{1} + \sigma_{1}\sigma_{2} - c\alpha_{1}\sigma_{2} - c{\alpha_{1}^{2}} + c\alpha_{1}\beta \\ & \Rightarrow & (1+c){\alpha_{1}^{2}} + (\sigma_{2}-\sigma_{1} + c\sigma_{2} - c\beta)\alpha_{1} - \sigma_{1}\sigma_{2} = 0 \\ & \Rightarrow & (1+c){\alpha_{1}^{2}} + ((1+c)\sigma_{2} - (\sigma_{1}+c\beta))\alpha_{1} - \sigma_{1}\sigma_{2} = 0 \\ & \Rightarrow & \alpha_{1} = \frac{-((1+c)\sigma_{2} - (\sigma_{1}+c\beta))\pm \sqrt{((1+c)\sigma_{2} - (\sigma_{1}+c\beta))^{2} + 4(1+c)\sigma_{1}\sigma_{2}}}{2(1+c)} \\ & \Rightarrow & \alpha_{1} = -\frac{\sigma_{2}(t)}{2} + \frac{\sigma_{1}(t) + c\beta(t)}{2(1+c)} \pm \sqrt{\left(\frac{\sigma_{2}(t)}{2} - \frac{\sigma_{1}(t) + c\beta(t)}{2(1+c)} \right)^{2} + \sigma_{1}(t)\sigma_{2}(t) } \end{array} $$

Now, for all σ ₁(t),σ ₂(t),β(t) and c:

$$\begin{array}{@{}rcl@{}} \sqrt{\left(\frac{\sigma_{2}(t)}{2} - \frac{\sigma_{1}(t) + c\beta(t)}{2(1+c)} \right)^{2} + \sigma_{1}(t)\sigma_{2}(t) } & > & \left|\frac{\sigma_{2}(t)}{2} - \frac{\sigma_{1}(t) + c\beta(t)}{2(1+c)}\right| \end{array} $$

so that

$$\begin{array}{@{}rcl@{}} -\frac{\sigma_{2}(t)}{2} + \frac{\sigma_{1}(t) + c\beta(t)}{2(1+c)} - \sqrt{\left(\frac{\sigma_{2}(t)}{2} - \frac{\sigma_{1}(t) + c\beta(t)}{2(1+c)} \right)^{2} + \sigma_{1}(t)\sigma_{2}(t) } & < & 0 \\ -\frac{\sigma_{2}(t)}{2} + \frac{\sigma_{1}(t) + c\beta(t)}{2(1+c)} + \sqrt{\left(\frac{\sigma_{2}(t)}{2} - \frac{\sigma_{1}(t) + c\beta(t)}{2(1+c)} \right)^{2} + \sigma_{1}(t)\sigma_{2}(t) } & > & 0 \end{array} $$

Therefore

$$\begin{array}{@{}rcl@{}} \Rightarrow \alpha_{1} & = & -\frac{\sigma_{2}(t)}{2} + \frac{\sigma_{1}(t) + c\beta(t)}{2(1+c)} + \sqrt{\left(\frac{\sigma_{2}(t)}{2} - \frac{\sigma_{1}(t) + c\beta(t)}{2(1+c)} \right)^{2} + \sigma_{1}(t)\sigma_{2}(t) } \end{array} $$

Denote

$$\begin{array}{@{}rcl@{}} f(\sigma_{1}(t), \sigma_{2}(t), \beta(t), c) &\equiv& -\frac{\sigma_{2}(t)}{2} + \frac{\sigma_{1}(t) + c\beta(t)}{2(1+c)} \\ &+& \sqrt{\left(\frac{\sigma_{2}(t)}{2} - \frac{\sigma_{1}(t) + c\beta(t)}{2(1+c)} \right)^{2} + \sigma_{1}(t)\sigma_{2}(t) }\end{array} $$