## Section: New Results

### Performance Evaluation

Participants : Hamza Ben Ammar, Yann Busnel, Pierre L'Ecuyer, Gerardo Rubino, Yassine Hadjadj-Aoul, Sofiène Jelassi, Patrick Maillé, Yves Mocquard, Bruno Sericola.

**Stream Processing Systems.** Stream processing systems are today gaining momentum as tools to perform analytics on continuous data streams. Their ability to produce analysis results with sub-second latencies, coupled with their scalability, makes them the preferred choice for many big data companies.

A stream processing application is commonly modeled as a direct acyclic graph where data operators, represented by nodes, are interconnected by streams of tuples containing data to be analyzed, the directed edges (the arcs). Scalability is usually attained at the deployment phase where each data operator can be parallelized using multiple instances, each of which will handle a subset of the tuples conveyed by the operators’ ingoing stream. Balancing the load among the instances of a parallel operator is important as it yields to better resource utilization and thus larger throughputs and reduced tuple processing latencies.

*Shuffle grouping* is a technique used by stream processing frameworks to share input load among parallel instances of stateless operators. With shuffle grouping each tuple of a stream can be assigned to any available operator instance, independently from any previous assignment. A common approach to implement shuffle grouping is to adopt a Round-Robin policy, a simple solution that fares well as long as the tuple execution time is almost the same for all the tuples. However, such an assumption rarely holds in real cases where execution time strongly depends on tuple content. As a consequence, parallel stateless operators within stream processing applications may experience unpredictable unbalance that, in the end, causes undesirable increase in tuple completion times. We proposed Online Shuffle Grouping (OSG), a novel approach to shuffle grouping aimed at reducing the overall tuple completion time. OSG estimates the execution time of each tuple, enabling a proactive and online scheduling of input load to the target operator instances. Sketches are used to efficiently store the otherwise large amount of information required to schedule incoming load. We provide a probabilistic analysis and illustrate, through both simulations and a running prototype, its impact on stream processing applications.

We consider recently an application to continuous queries, which are processed by a stream processing engine (SPE) to generate timely results given the ephemeral input data. Variations of input data streams, in terms of both volume and distribution of values, have a large impact on computational resource requirements. Dynamic and Automatic Balanced Scaling for Storm (DABS-Storm) [17] is an original solution for handling dynamic adaptation of continuous queries processing according to evolution of input stream properties, while controlling the system stability. Both fluctuations in data volume and distribution of values within data streams are handled by DABS-Storm to adjust the resources usage that best meets processing needs. To achieve this goal, the DABS-Storm holistic approach combines a proactive auto-parallelization algorithm with a latency-aware load balancing strategy.

*Sampling techniques* is a classical method for detection in large-scale data streams. We have proposed a new algorithm that detects on the fly the $k$ most frequent items in the sliding window model [37]. This algorithm is distributed among the nodes of the system. It is inspired by a recent and innovative approach, which consists in associating a stochastic value correlated with the item's frequency instead of trying to estimate its number of occurrences. This stochastic value corresponds to the number of consecutive heads in coin flipping until the first tail occurs. The original approach was to retain just the maximum of consecutive heads obtained by an item, since an item that often occurs will have a higher probability of having a high value. While effective for very skewed data distributions, the correlation is not tight enough to robustly distinguish items with comparable frequencies. To address this important issue, we propose to combine the stochastic approach with a deterministic counting of items. Specifically, in place of keeping the maximum number of consecutive heads obtained by an item, we count the number of times the coin flipping process of an item has exceeded a given threshold. This threshold is defined by combining theoretical results in leader election and coupon collector problems. Results on simulated data show how impressive is the detection of the top-$k$ items in a large range of distributions [38].

**Throughput Prediction in Cellular Networks.** Downlink data rates can vary significantly in cellular networks, with a potentially non-negligible effect on the user experience. Content providers address this problem by using different representations (e.g., picture resolution, video resolution and rate) of the same content and switch among these based on measurements collected during the connection. Knowing the achievable data rate before the connection establishment should definitely help content providers to choose the most appropriate representation from the very beginning. We have conducted several large measurement campaigns involving a panel of users connected to a production network in France, to determine whether it is possible to predict the achievable data rate using measurements collected, before establishing the connection to the content provider, on the operator's network and on the mobile node. We establish evidence that it is indeed possible to exploit these measurements to predict, with an acceptable accuracy, the achievable data rate. We thus introduce cooperation strategies between the mobile user, the network operator and the content provider to implement such anticipatory solution [23].

**Call Centers.**
In emergency call centers (for police, firemen, ambulances) a single event can sometimes trigger many incoming calls in a short period of time. Several people may call to report the same fire or the same accident, for example. Such a sudden burst of incoming traffic can have a significant impact on the responsiveness of the call center for other events in the same period of time. We examine in [53] data from the SOS Alarm center in Sweden. We also build a stochastic model for the bursts. We show how to estimate the model parameters for each burst by maximum likelihood, how to model the multivariate distribution of those parameters using copulas, and how to simulate the burst process from this model. In our model, certain events trigger an arrival process of calls with a random time-varying rate over a finite period of time of random length.

**Performance of CDNs.**
In order to track the users who illegally re-stream live video streams, one solution is to embed identified watermark sequences in the video segments to distinguish the users. However, since all types of watermarked segments should be prepared, the existing solutions require an extra cost of bandwidth for delivery (at least multiplying by two the required bandwidth). In [66], we study how to reduce the inner delivery (traffic) cost of a Content Delivery Network (CDN). We propose a mechanism that reduces the number of watermarked segments that need to be encoded and delivered. We calculate the best- and worst-case traffics for two different cases: multicast and unicast. The results illustrate that even in the worst cases, the traffic with our approach is much lower than without reducing. Moreover, the watermarked sequences can still maintain uniqueness for each user. Experiments based on a real database are carried out, and illustrate that our mechanism significantly reduces traffic with respect to the current CDN practice.

Beyond CDNs, Network Operators (NOs) are developing caching capabilities within their own network infrastructure, in order to face the rise in data consumption and to avoid the potential congestion at peering links. These factors explain the enthusiasm of industry and academics around the Information-Centric Networking (ICN) concept and its in-network caching feature. Many contributions focused these last years on improving the caching performance of ICN. In [41], we propose a very versatile model capable of modeling the most efficient caching strategies. We first start by representing a single generic cache node. We then extend our model for the case of a network of caches. The obtained results are used to derive, in particular, the cache hit probability of a content in such caching systems. Using a discrete event simulator, we show the accuracy of the proposed model under different network configurations.

**Probabilistic analysis of population protocols.**
In [20], we studied the well-known problem of dissemination of information in large scale distributed networks through pairwise interactions. This problem, originally called rumor mongering, and then rumor spreading has mainly been investigated in the synchronous model. This model relies on the assumption that all the nodes of the network act in synchrony, that is, at each round of the protocol, each node is allowed to contact a random neighbor. In the paper, we drop this assumption under the argument that it is not realistic in large scale systems. We thus consider the asynchronous variant, where at random times, nodes successively interact by pairs exchanging their information on the rumor. In a previous paper, we performed a study of the total number of interactions needed for all the nodes of the network to discover the rumor. While most of the existing results involve huge constants that do not allow us to compare different protocols, we provided a thorough analysis of the distribution of this total number of interactions together with its asymptotic behavior. In this paper we extend this discrete time analysis by solving a conjecture proposed previously and we consider the continuous time case, where a Poisson process is associated to each node to determine the instants at which interactions occur. The rumor spreading time is thus more realistic since it is the real time needed for all the nodes of the network to discover the rumor. Once again, as most of the existing results involve huge constants, we provide tight bound and equivalent of the complementary distribution of the rumor spreading time. We also give the exact asymptotic behavior of the complementary distribution of the rumor spreading time around its expected value when the number of nodes tends to infinity.

The context of [57] is the two-choice paradigm which is deeply used in balanced online resource allocation, priority scheduling, load balancing and more recently in population protocols. The model governing the evolution of these systems consists in throwing balls one by one and independently of each others into $n$ bins, which represent the number of agents in the system. At each discrete instant, a ball is placed in the least filled bin among two bins randomly chosen among the $n$ ones. A natural question is the evaluation of the difference between the number of balls in the most loaded and the one in the least loaded bin. At time $t$, this difference is denoted by Gap(t). A lot of work has been devoted to the derivation of asymptotic approximations of this gap for large values of $n$. In this paper we go a step further by showing that for all $t\ge 0$, $n\ge 2$ and $\sigma \ge 0$, the variable Gap(t) is less than $a(1+\sigma )ln\left(n\right)+b$ with probability greater than $1-1/{n}^{\sigma}$ , where the constants $a$ and $b$, which are independent of $t$, $\sigma $ and $n$, are optimized and given explicitly, which to the best of our knowledge has never been done before.

The work described in [58] focuses on pairwise interaction-based protocols, and proposes an universal mechanism that allows each agent to locally detect that the system has converged to the sought configuration with high probability. To illustrate our mechanism, we use it to detect the instant at which the proportion problem is solved. Specifically, let ${n}_{A}$ (resp. ${n}_{B}$) be the number of agents that initially started in state $A$ (resp. $B$) and ${\gamma}_{A}={n}_{A}/n$, where $n$ is the total number of agents. Our protocol guarantees, with a given precision $\epsilon >0$ and any high probability $1-\delta $, that after $O(nln(n/\delta \left)\right)$ interactions, any queried agent that has set the detection flag will output the correct value of the proportion ${\gamma}_{A}$ of agents which started in state $A$, by maintaining no more than $O(ln(n)/\epsilon )$ integers. We are not aware of any such results. Simulation results illustrate our theoretical analysis.

All these works are part of the thesis [11].

**Fluid Queues.**
Stochastic fluid flow models and in particular those driven by Markov chains have been intensively studied in the last two decades. Not only they have been proven to be efficient tools to mimic Internet traffic flow at a macroscopic level but they are useful tools in many areas of applications such as manufacturing systems or in actuarial sciences to cite but a few. We propose in a forthcoming book, entitled Advanced Trends in Queueing Theory, edited by V. Anisimov and N. Limnios in the Mathematics and Statistics Series, Sciences, Iste & J. Wiley, a chapter which focus on such a model in the context
of performance analysis of a potentially congested system.
The latter is modeled by means of a finite-capacity system whose content is described by a Markov driven stable fluid flow. We step-by-step describe a methodology to compute exactly the loss probability of the system. Our approach
is based on the computation of hitting probabilities jointly with the peak level reached during a busy period, both in the infinite and finite buffer case. Accordingly we end up with differential Riccati equations that can be solved numerically. Moreover we are able to characterize the complete distribution of both the duration of congestion and of the total information lost during such a busy period.

**Organizing both transactions and blocks in a distributed ledger.** We propose in [39] a new way to organize both transactions and blocks in a distributed ledger to address the performance
issues of permissionless ledgers. In contrast to most of the existing solutions in which the ledger is a chain of blocks extracted from a tree or a graph of chains, we present a distributed ledger whose structure is a balanced directed acyclic graph of blocks. We call this specific graph a SYC-DAG. We show that a SYC-DAG allows us to keep all the remarkable properties of the Bitcoin blockchain in terms of security, immutability, and transparency, while enjoying higher throughput and self-adaptivity to transactions demand. To the best of our knowledge, such a design has never been proposed.

**Additional intermittent server.**
We analyzed the performance of a system consisting of a queue with one regular single server supported by an additional intermittent server who, in order to decrease the mean response time, i) leaves the back office to join the first server when the number of customers reaches the threshold $K$, and ii) leaves the front office when it has no more customers to serve.
This study produced a closed-form solution for the steady state probability distribution and for different metrics such as expected response times for customers or expectation of busy periods.
Then, for a given value of $K$, the influence of the intermittent server on the response time is exhibited. The consequences on the primary task of the intermittent server are investigated through metrics such as mean working and pseudo-idle periods. Finally, a cost function is proposed from which an optimal value of the threshold $K$ is obtained. The results were the subject of the book chapter [71].

**Operational availability prediction.**
The evaluation of the operational availability of a fleet of systems on an operational site is far from trivial when the size of the state space of a faithful Markovian model makes this issue unrealistic for many large models. The main difficulty comes from the existence on the site of “on line replaceable units” (lru ) that may be unavailable from time to time when a breakdown occurs. To be more precise, let us say that the intrinsic availability is an upper limit of the operational availability because the considered unavailability corresponds only to the time necessary to proceed with the exchange of the defective element.
This assumes that the repairer and the spare part are always immediately available on the operational site. To reduce the intervention time at an operational site, system repair consists of replacing the defective subset with an identical one in good condition. These exchangeable subsets are called lru s. Restoring a piece of system by exchanging an lru makes it possible to obtain a rapid return to service of the system and does not require the presence on the operational site of several specialists. Thus, we can change an aircraft engine in a few hours thanks to stakeholders who are knowledgeable about the procedures for intervention on fasteners and connections but who are not required to know the techniques to repair a broken engine.
The operational availability (the one that is of interest to the user) will generally be lower than the intrinsic availability because the latter integrates the unavailability of repairer or lru , if any. And if the waiting time for a repairer is generally measured in hours, the unavailability of a lru can be measured in days, in weeks, even in months! It is therefore this last point which a potential customer should worry about in priority. Moreover, the unavailability of the lru is more difficult to model and evaluate than that of the repairer.

In a first study we considered a virtual system with only one type of lru in order to understand the influence of various parameters on the operational availability both of a faithful Markovian model (for which we are able to get the exact answer) and of a proposed approximate method. By doing so, we were able to compute the relative errors induced when using this latter solution. The approximate method showed a quite good accuracy [56]. The generalization to systems with multiple types of line replaceable units was conducted in a second study. The main idea is to consider a non product-form queuing network and to aggregate subsets of it as if they were parts of a product-form queuing network. Note that if the spare lru s were always available on the operational site (which is never the case) we could fairly model the behavior of the support system on the site by means of a product-form network. Also in this second study, the potential lack of repairer is taken into account. The low relative complexity of the new recurrent approximate method allows it to be used for applications encountered in the field of maintenance. Although approximate, the method provides the result with a small relative error, ranging from ${10}^{-2}$ to ${10}^{-8}$ for examples which can be compared with our reference Markovian model. For now, the method only concerns systems consisting of a series of lru s [64]).

**Modeling loss processes in voice traffic.**
Markov models of loss incidents happening during packet voice communications are needful for many engineering tasks, namely network dimensioning and automatic quality assessment. Two very simple ones are Bernoulli and 2-state Markov models, but they carry limited information about incurred loss incidents. On the other hand, a general Markov loss model with ${2}^{k}$ states, where $k$ is the window length used for observing the voice packet arrival process, leads to heavy computations and an excessive lookahead delay. Moreover, legacy Markov loss models concentrate mostly on capturing some physical characteristics of loss incidents, rather than their perceived effects.

In [16], we propose a comprehensive and detailed Markov loss model considering the distinguished perceived effects caused by different loss incidents. Specifically, it explicitly differentiates between (1) isolated 20 msec loss incidents which are inaudible by the human ears, (2) highly and lowly frequent short loss incidents (20-80 msec) that are perceived by humans as bubbles and (3) long loss incidents ($\ge $ 80 msec) inducing interruptions that dramatically decrease speech intelligibility. Our numerical analysis show that our Markov loss model captures subtle characteristics of loss incidents observed in empirical traces sampled over representative network paths.

**Transient analysis of Markovian models.**
Continuing with a research line that we started years ago with colleagues in California, where we addressed the transient state distributions of Markovian queuing models using the concept of pseudo-dual proposed by Anderson, we discovered this year a new way to attack these problems, leading to an approach with a much wider application range. Moreover, this methodology clarifies some phenomena that appeared when Anderson's tools were used. The result is now two new concepts we propose, related in general to arbitrary square matrices (possibly infinite), the *power-dual* and the *exponential-dual*, and the way we can apply them to the analysis of linear systems of difference or differential equations. The first elements of this new theory were discussed in [26].