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## Section: New Results

### Performance Evaluation of Communication Networks

Participants : Thomas Begin, Philippe Nain, Isabelle Guérin Lassous.

#### First-Come-First-Served Queues with Multiple Servers and Customer Classes

This a joint work with A. Brandwajn [5]. We present a simple approach to the solution of a multi-server FCFS queueing system with several classes of customers and phase-type service time distributions. The proposed solution relies on solving a single two-class model in which we distinguish one of the classes and we aggregate the remaining customer classes. We use a reduced state approximation to solve this two-class model. We propose two types of aggregation: exact, in which we merge the phase-type service time distributions exactly, and approximate, in which we simplify the phase-type distribution for the aggregated class by matching only its first two moments. The proposed approach uses simple mathematics and is highly scalable in terms of the number of servers, the number of classes, as well as the number of phases per class. Our approach applies both to queues with finite and infinite buffer space.

#### A study of systems with multiple operating levels, probabilistic thresholds and hysteresis

This a joint work with A. Brandwajn [6]. Current architecture of many computer systems relies on dynamic allocation of a pool of resources according to workload conditions to meet specific performance objectives while minimizing cost (e.g., energy or billing). In such systems, different levels of operation may be defined, and switching between operating levels occurs at certain thresholds of system congestion. To avoid rapid oscillations between levels of service, "hysteresis" is introduced by using different thresholds for increasing and decreasing workload levels, respectively. We propose a model of such systems with general arrivals, arbitrary number of servers and operating levels where each higher operating level may correspond to an arbitrary number of additional servers and soft (i.e. non-deterministic) thresholds to account for "inertia" in switching between operating levels. In our model, request service times are assumed to be memoryless and server processing rates may be a function of the current operating level and of the number of requests (users) in the system. Additionally, we allow for delays in the activation of additional operating levels. We use simple mathematics to obtain a semi-numerical solution of our model. We illustrate the versatility of our model using several case study examples inspired by features of real systems. In particular, we explore optimal thresholds as a tradeoff between performance and energy consumption.

#### Covert cycle stealing in an M/G/1 queue

Consider an M/G/1 queue where arriving jobs are under control of a party (Willie). There exists a second party, Alice who may or may not want to introduce a sequence of jobs to be serviced. Her goal is to prevent Willie from being able to distinguish between these two cases. The question that we address is: can Alice introduce her stream of jobs covertly, i.e., prevent Willie from distinguishing between the two possibilities, either her introducing the stream or not, and if so, at what rate can she introduce her jobs? We present a square-root law on the amount of service Alice can receive covertly. The covertness criterion is that the probabilities of false alarm and missed detection is arbitrarily close to one. One result we have established is the following: consider exponential service times for Alice's jobs and Willies' jobs with rate ${\mu }_{1}$ and ${\mu }_{2}$, respectively. During $n$ Willie's job busy periods, Alice can submit covertly $O\left(\sqrt{n}\right)$ jobs if ${\mu }_{1}<2{\mu }_{2}$, $O\left(\sqrt{n/logn}\right)$ jobs if ${\mu }_{1}=2{\mu }_{2}$, and $O\left({n}^{{\mu }_{1}/{\mu }_{2}}\right)$ jobs if ${\mu }_{1}>2{\mu }_{2}$. This is the first time that such a phase transition has been observed in this context. This ongoing research, carried out by P. Nain in collaboration with D. Towsley (Univ. Massachusetts) and B. Jiang (Shanghai Jiao Tong Univ.), has various applications in the context of service level agreement.

#### LRU caches

The work on network caches operating under the standard Least-Recently-Used (LRU) management policy, initiated in 2017 (see 2017 Dante Activity Report), has been completed and published  [13]. Under weak statistical assumptions on the content request process, this work establishes the validity of the so-called “Che's approximation” as the cache size and the number of content go to infinity.

#### Stochastic Multilayer Networks

A stochastic multilayer network is the aggregation of $M$ networks (one per layer) where each is a subgraph of a foundational network $G$. Each layer network is the result of probabilistically removing links and nodes from $G$. The resulting network includes any link that appears in at least $K$ layers. This model is an instance of a non-standard site-bond percolation model. Two sets of results are obtained in  [28]: first, we derive the probability distribution that the $M$-layer network is in a given configuration for some particular graph structures (explicit results are provided for a line and an algorithm is provided for a tree), where a configuration is the collective state of all links (each either active or inactive). Next, we show that for appropriate scalings of the node and link selection processes in a layer, links are asymptotically independent as the number of layers goes to infinity, and follow Poisson distributions. Numerical results are provided to highlight the impact of having several layers on some metrics of interest (including expected size of the cluster a node belongs to in the case of the line). This model finds applications in wireless communication networks with multichannel radios, multiple social networks with overlapping memberships, transportation networks, and, more generally, in any scenario where a common set of nodes can be linked via co-existing means of connectivity.