Towards Optimal Placement of Monitoring Units in Time-Varying Networks Under Centralized Control

Abstract

The increasing penetration of software-defined communication networks with centralized control has made network management a highly demanding task. Common monitoring approaches in the context of such convoluted high-speed networks have become a serious challenge in terms of complexity and resource management. Management functions rely on monitoring information such as the flow size distribution (FSD), to perform crucial activities such as load balancing and resource provisioning. In this paper, we propose a solution as to how one can utilize limited monitoring resources to estimate the FSD for distinct flows characterized by origin-destination pairs. We provide a method to dynamically adapt placement of monitoring units with some extracted knowledge about the change in FSD’s with time.

Appendix

Lemma 1

Wald’s Identity [20]: Suppose for a sequence of real-valued random variables $\{X_n\}$ and nonnegative integer-valued random variable N following conditions hold,

• 1. $E[\mid X_n \mid ] < \infty$,

• 2. $E[X_n \mathbbm {1}(N \ge n)] = E[X_n] \cdot P(N\ge n), \forall n$ and

• 3. $\sum _{n=1}^{\infty }{E[X_n \mathbbm {1}(N \ge n)]}<\infty$.

Then, the random sums $S_N :=\sum _{n=1}^{N}X_{n}$ and $T_N :=\sum _{n=1}^{N} E[X_n]$ are integrable and $E[S_N]=E[T_N]$. Additionally, if

• 4. $E[X_n]=E[X_1] \forall n$ and

• 5. $E[N]<\infty$

$E[S_N]=E[N] \cdot E[X_1]$.

In our case, $X_i's$ are i.i.d according to a distribution F with finite mean and N is the time to observe $p-th$ percentile of F. Hence, $E[X_n \cdot \mathbbm {1}(N \ge n)] = E[X_n \cdot \mathbbm {1}(X_1< F^{-1}(p),...,X_{n-1}< F^{-1}(p))] = E[X_n] \cdot E[\mathbbm {1}(X_1< F^{-1}(p),...,X_{n-1}< F^{-1}(p))] = E[X_n] \cdot P(N\ge n)$, owing to independence of $X_i's$. Condition (3) is satisfied as the summands form a geometric series with $n{-}th$ term being $E[X_1] \cdot p^n$. Thus, $E[S_N] = E[X_1] \cdot E[N] = E[X_1]/(1-p)$.

Definition 4

Bhattacharyya Distance: Bhattacharyya distance between two probability mass functions (pmf) p and q over same domain X, is defined as $D(p,q) = - \log {(\sum _{x \in X}\sqrt{p_x \cdot q_x})}$.

Compared to the popular metric Kullback-Leibler divergence, this distance is defined even when empirical pmf assigns zero mass to a point that belongs to the support of actual or hypothesized distribution.