Hind Castel-Taleb

Stochastic comparisons: A methodology for the performance evaluation of fixed and mobile networks

We propose to use a mathematical method based on stochastic comparisons of Markov chains in order to derive performance indices bounds. The main objective is to find Markovian bounding models with reduced state spaces, which are easier to solve. We apply the methodology to performance evaluation of complex telecommunication systems modelled by large size Markov chains which cannot be solved by exact methods. This methodology can be applied for continuous- or discrete-time Markov chains. In the first study, we consider an MPLS switch represented by two stages of buffers. Various kinds of traffic with different QoS levels enter the first stage, and transit in the second stage. The goal is to compute packet loss rates in the second stage. In the other study, we define a CAC scheme in a mobile network which gives the priority to the handover over the new calls. Performance evaluation of the CAC scheme consists in the computation of the dropping handover and call blocking probabilities. For the two studies, systems are represented by large state Markov chains whose resolution is difficult. We propose to define intuitively bounding systems in order to compute performance measures bounds. Using stochastic comparisons methods, we prove that the new systems represent bounds for the exact ones. Different methods can be used. For the MPLS switch, we use the coupling equivalent to the sample-path ordering, allowing the comparison of the loss rates. In the case of the CAC scheme, we apply the increasing sets formalism used to define weaker orderings, enabling the comparison of the dropping handovers and blocking probabilities. We validate stochastic comparison method by presenting some numerical results illustrating the interest of the approach.

Stochastic bounds and histograms for network performance analysis

Exact analysis of queueing networks under real traffic histograms becomes quickly intractable due to the state explosion. In this paper, we propose to apply the stochastic comparison method to derive performance measure bounds under histogram-based traffics. We apply an algorithm based on dynamic programming to derive bounding traffic histograms on reduced state spaces. We indeed obtain easier bounding stochastic processes providing stochastic upper and lower bounds on buffer occupancy histograms (queue length distributions) for finite queue models. We evaluate the proposed method under real traffic traces, and we compare the results with those obtained by an approximative method. Numerical results illustrate that the proposed method provides more accurate results with a tradeoff between computation time and accuracy. Moreover, the derived performance bounds are very relevant in network dimensioning.

Performance measure bounds in mobile networks by state space reduction

We present in this paper a mathematical method based on stochastic comparisons of Markov chains in order to compute in a mobile network performance measures that are important Q parameters for users: the dropping handover of voice and the blocking probability of a new voice call. The key idea of this methodology is that given a complex system represented by a multi-dimensional Markov chain which is too large to be solved, we propose to reduce the state space, and so to define a new Markov chain which is a simplified version of the original one. This reduced Markov chain is defined as an aggregated one, which represents a stochastic bound for performance measures written as increasing reward function on the stationary distribution. The main steps of the construction of the aggregated Markov chain applied into mobile networks are presented in this paper. As the number of mobile users with different kinds of applications increases, the associated model is more complex and it can be represented by multi-dimensional continuous time Markov chain with a very large size. Thus, we define an aggregated Markov chain represented by a multi-dimensional birth and death process which is very easy to solve. We have proved that there is a weak ordering between the original Markov chain and the aggregated one using increasing set formalism. We have computed upper bounds of dropping handover and blocking probability for different values of input parameters. Numerical results prove that upper bounds give good results and so the stochastic methodology is an interesting mathematical tool for the performance evaluation of complex systems. Keywords: Mobile Networks, Quality of Service, Stochastic ordering, stochastic comparisons, Continuous Time Markov Chains.

Modeling and performance evaluation of the IEEE 802.15.4e LLDN mechanism designed for industrial applications in WSNs

AbstractThe IEEE 802.15.4 standard has been introduced for low latency and low energy consumption in wireless sensor networks. To better support the requirements of industrial applications, where the use of this standard is limited, the low latency deterministic network (LLDN) mechanism of the IEEE 802.15.4e amendment has been proposed. In this paper, we develop a three dimensional Markov chain model for the IEEE 802.15.4e LLDN mechanism. Then, we estimate the stationary probability distribution of this chain in order to derive theoretical expressions of some performance metrics, as the reliability, energy consumption, throughput, delay and jitter. After that, we conduct a comparative study between the IEEE 802.15.4e LLDN and the IEEE 802.15.4 slotted carrier sense multiple access with collision avoidance (CSMA/CA). Numerical results show that the deterministic behavior of the LLDN mechanism significantly reduces the collision probability providing best performances in terms of reliability, energy consumption, throughput and delay compared to the IEEE 802.15.4 slotted CSMA/CA. Finally, the accuracy of our theoretical analysis is validated by Monte Carlo simulations.

Loss rates bounds for IP switches in MPLS networks

We propose in this paper to evaluate the performance of an IP switch in MPLS networks. We focus our study on an important QoS parameters for users which is the packet loss rates. We use a mathematical method based on stochastic comparisons of Markov chains in order to compute loss rate bounds. The system understudy can be represented by a multi-dimensional Markov chain which is too large to be solved by numerical methods. The key idea of the proposed methodology is to define bounded models with a reduced state space, so easier to solve. These bounded models provide upper and lower bounds on packets loss rates. We give in this paper the proof that the bounded models provide really stochastic bounds on performance measures. Numerical results confirm the proof and show the quality of these bounds.

Stochastic bounds on partial ordering: application to memory overflows due to bursty arrivals

We apply stochastic bounding methods with a partial order on the state space to the analysis of memory overflow in a router. Usually, stochastic bounds are associated to a total order implying useless constraints and decreasing the tightness of bounds. Here we present the basic methodology of sample path comparison with a partial order and some numerical results to show the accuracy of the results. We analyze the probability of a buffer overflow with two types of packets, a Pushout access mechanism and Markov modulated batch arrivals. This problem is strongly related to the memory rejection out a Fiber Delay Loop in an all optical router using deflection routing.

Stochastic Monotonicity in Queueing Networks

Stochastic monotonicity is one of the sufficient conditions for stochastic comparisons of Markov chains. On a partially ordered state space, several stochastic orderings can be defined by means of increasing sets. The most known is the strong stochastic (sample-path) ordering, but weaker orderings (weak and weak*) could be defined by restricting the considered increasing sets. When the strong ordering could not be defined, weaker orderings represent an alternative as they generate less constraints. Also, they may provide more accurate bounds. The main goal of this paper is to provide an intuitive event formalism added to stochastic comparisons methods in order to prove the stochastic monotonicity for multidimensional Continuous Time Markov Chains (CTMC). We use the coupling by events for the strong monotonicity. For weaker monotonicity, we give a theorem based on generator inequalities using increasing sets. We prove this theorem, and we present the event formalism for the definition of the increasing sets. We apply our formalism on queueing networks, in order to establish monotonicity properties.

Multi-services MAC protocol for wireless networks

Summary form only given. Due to random access in wireless networks using CSMA/CA like in Wifi networks, the integration of services with a lot of quality of service needs is impossible. In this paper, we propose to study and to evaluate a new MAC protocol that takes into account different types of traffic (e.g.. voice and data) and for each traffic, different priority levels are considered. To improve the QoS of WIFI MAC protocols, we add a selective reject and push-out mechanisms. To model our protocol, using Markov chain is impossible because it provides Markov chain with a large state-space. This is due to the resource management and user mobility. Thus, we propose to build an aggregated Markov chain with a less state-space that allows to compute easily performance measures. We have used stochastic comparisons of Markov chains to prove that the considered access protocol (with selective reject and push-out mechanisms) gives less loss rates of high priority connections (data and voices) than the traditional one (without selective reject and push-out mechanisms). We give numerical results to confirm mathematical proofs.

Weak Stochastic Comparisons for Performability Verification

The probabilistic model checking provides a precise formalism for the performance and reliability verification of telecommunication systems modeled by Markov chains. We study a queueing system similar to a Jackson network except that queues have a finite capacity. We propose to study in this paper (state and path) formulas from the Continuous Stochastic Logic (CSL), in order to verify performability properties. Unfortunately, transient and stationary analysis is very complex for multidimensional Markov processes. So we propose to use the stochastic comparisons in the sense of weak orderings to define bounding processes. Bounding processes are represented by independent M/M/1 queues for which transient and stationary distributions can be computed as the product of probability distributions of each queue. We use the increasing set method, and we develop an intuitive formalism based on events to establish weak stochastic comparisons.

Bounding Aggregations on Phase-Type Arrivals for Performance Analysis of Clouds

We evaluate the performance of a cloud system using a hysteresis queueing system with phase-type and batch arrivals. To represent the dynamic allocation of the resources, the hysteresis queue activates and deactivates the virtual machines according to the threshold values of the queue length. We suppose by batches, and follow a phase-type process. This system is analyze, especially when the size of the state space increases and the length of batch arrival distribution is large. So, to solve this problem, we propose to use stochastic bounds and the performance measures and compare the proposed bounding models with the exact one. The relevance of our methodology is to offer a trade-off between computational complexity and accuracy of the results and provide very interesting solutions for network dimensioning.