## Publication

Here, the information related to our paper on modeling VaR and CVaR in communication networks is given. The paper entitled “Upper Bound for Failure Risk in Networks” and authored by Piotr Chołda, Krzysztof Rusek, and Piotr Guzik was presented during 2nd International Workshop on Understanding the inter-play between Sustainability, Resilience, and Robustness in networks (USRR 2014 Fall) and was published in Electronic Notes on Discrete Mathematics (by Elsevier), vol. 51, Mar. 2016.

When using or inspiring by the given information (either changed or not), please cite the abovementioned publication.

## Details of the simulation

The following assumptions were taken in relation to the simulations:

- The network topologies used are retrieved from the SNDlib library and model two large networks: the compact and dense German Research Network (nobel-germany.xml), and the very broad yet sparse US Network (nobel-us.xml).
- The network is modelled as an undirected weighted graph. The weights are non-negative and additive and they are proportional to the physical lengths of the links between the cities. Connections supporting the services are simply routed on the basis of the Dijkstra algorithm.
- For each node and link in a network, the interchanging failure and recovery process is modeled. According to the most commonly assumed conditions, both distributions for link/node failure/repair times are exponential. Their rates were taken from the excellent paper by P. Kuusela and I. Norros,
*On/Off Process Modeling of IP Network Failures*, Proc. DSN 2010, Chicago, IL, 2010. We use the following function to find the failure/repair rates for links: -
if param.l < 25 lambda=lambdar; beta=betam; else lambda = lambdar + 9*lambdar*param.l/param.lmax; beta = betam + 9*betam*param.l/param.lmax; end

- Where param.l is the link length, lambdar and betam are the basic parameters.
- Each service has its own parameters necessary to find the exact value of the penalty: the weights are proportional to the demand volumes provided with the network models.
- Each service’s connection is routed with the shortest path routing found by the Dijkstra algorithm (our networks are represented by weighted digraphs, where the weights representing lengths of the links are non-negative).
- For each scenario, we held 100 000 simulations, where each simulation time was one year of network operation; this is the interval for which penalties due to the assumed compensation policies are estimated.
- After calculating the parametrised normal distribution of penalties for all the services, we are able to calculate both risk measures (where is either VaR or CVaR) for each of them for the following quantiles = 0.9; 0.95; 0.99; 0.999.

However, as we are interested in the estimation of the risk measure related to the total penalty that should be paid by the operator, we do not present the values of the individual risk measures. We only focus on showing that the proposed upper bound is much more eective than the upper bound obtained with the summation of risk measures calculated for separate services.

## Results

Here, we present the results of the simulations compared to the theoretically obtained boundes.