Ioannis Lambadaris — Page 2

arXiv2013

Optimal Server Assignment in Multi-Server Queueing Systems with Random Connectivities

We study the problem of assigning $K$ identical servers to a set of $N$ parallel queues in a time-slotted queueing system. The connectivity of each queue to each server is randomly changing with time; each server can serve at most one queue and each queue can be served by at most one server during each time slot. Such a queueing model has been used in addressing resource allocation problems in wireless networks. It has been previously proven that Maximum Weighted Matching (MWM) is a throughput-optimal server assignment policy for such a queueing system. In this paper, we prove that for a system with i.i.d. Bernoulli packet arrivals and connectivities, MWM minimizes, in stochastic ordering sense, a broad range of cost functions of the queue lengths such as total queue occupancy (which implies minimization of average queueing delays). Then, we extend the model by considering imperfect services where it is assumed that the service of a scheduled packet fails randomly with a certain probability. We prove that the same policy is still optimal for the extended model. We finally show that the results are still valid for more general connectivity and arrival processes which follow conditional permutation invariant distributions.

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arXiv2013

Power Strip Packing of Malleable Demands in Smart Grid

We consider a problem of supplying electricity to a set of $\mathcal{N}$ customers in a smart-grid framework. Each customer requires a certain amount of electrical energy which has to be supplied during the time interval $[0,1]$. We assume that each demand has to be supplied without interruption, with possible duration between $\ell$ and $r$, which are given system parameters ($\ell\le r$). At each moment of time, the power of the grid is the sum of all the consumption rates for the demands being supplied at that moment. Our goal is to find an assignment that minimizes the {\it power peak} - maximal power over $[0,1]$ - while satisfying all the demands. To do this first we find the lower bound of optimal power peak. We show that the problem depends on whether or not the pair $\ell, r$ belongs to a "good" region $\mathcal{G}$. If it does - then an optimal assignment almost perfectly "fills" the rectangle $time \times power = [0,1] \times [0, A]$ with $A$ being the sum of all the energy demands - thus achieving an optimal power peak $A$. Conversely, if $\ell, r$ do not belong to $\mathcal{G}$, we identify the lower bound $\bar{A} >A$ on the optimal value of power peak and introduce a simple linear time algorithm that almost perfectly arranges all the demands in a rectangle $[0, A /\bar{A}] \times [0, \bar{A}]$ and show that it is asymptotically optimal.

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arXiv2012

Analysis and Design of Irregular Graphs for Node-Based Verification-Based Recovery Algorithms in Compressed Sensing

In this paper, we present a probabilistic analysis of iterative node-based verification-based (NB-VB) recovery algorithms over irregular graphs in the context of compressed sensing. Verification-based algorithms are particularly interesting due to their low complexity (linear in the signal dimension $n$). The analysis predicts the average fraction of unverified signal elements at each iteration $\ell$ where the average is taken over the ensembles of input signals and sensing matrices. The analysis is asymptotic ($n \rightarrow \infty$) and is similar in nature to the well-known density evolution technique commonly used to analyze iterative decoding algorithms. Compared to the existing technique for the analysis of NB-VB algorithms, which is based on numerically solving a large system of coupled differential equations, the proposed method is much simpler and more accurate. This allows us to design irregular sensing graphs for such recovery algorithms. The designed irregular graphs outperform the corresponding regular graphs substantially. For example, for the same recovery complexity per iteration, we design irregular graphs that can recover up to about 40% more non-zero signal elements compared to the regular graphs. Simulation results are also provided which demonstrate that the proposed asymptotic analysis matches the performance of recovery algorithms for large but finite values of $n$.

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arXiv2011

On the Stability Region of Multi-Queue Multi-Server Queueing Systems with Stationary Channel Distribution

In this paper, we characterize the stability region of multi-queue multi-server (MQMS) queueing systems with stationary channel and packet arrival processes. Toward this, the necessary and sufficient conditions for the stability of the system are derived under general arrival processes with finite first and second moments. We show that when the arrival processes are stationary, the stability region form is a polytope for which we explicitly find the coefficients of the linear inequalities which characterize the stability region polytope.

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arXiv2011

Delay Optimal Server Assignment to Symmetric Parallel Queues with Random Connectivities

In this paper, we investigate the problem of assignment of $K$ identical servers to a set of $N$ parallel queues in a time slotted queueing system. The connectivity of each queue to each server is randomly changing with time; each server can serve at most one queue and each queue can be served by at most one server per time slot. Such queueing systems were widely applied in modeling the scheduling (or resource allocation) problem in wireless networks. It has been previously proven that Maximum Weighted Matching (MWM) is a throughput optimal server assignment policy for such queueing systems. In this paper, we prove that for a symmetric system with i.i.d. Bernoulli packet arrivals and connectivities, MWM minimizes, in stochastic ordering sense, a broad range of cost functions of the queue lengths including total queue occupancy (or equivalently average queueing delay).

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arXiv2011

Explicit Characterization of Stability Region for Stationary Multi-Queue Multi-Server Systems

In this paper, we characterize the network stability region (capacity region) of multi-queue multi-server (MQMS) queueing systems with stationary channel distribution and stationary arrival processes. The stability region is specified by a finite set of linear inequalities. We first show that the stability region is a polytope characterized by the finite set of its facet defining hyperplanes. We explicitly determine the coefficients of the linear inequalities describing the facet defining hyperplanes of the stability region polytope. We further derive the necessary and sufficient conditions for the stability of the system for general arrival processes with finite first and second moments. For the case of stationary arrival processes, the derived conditions characterize the system stability region. Furthermore, we obtain an upper bound for the average queueing delay of Maximum Weight (MW) server allocation policy which has been shown in the literature to be a throughput optimal policy for MQMS systems. Using a similar approach, we can characterize the stability region for a fluid model MQMS system. However, the stability region of the fluid model system is described by an infinite number of linear inequalities since in this case the stability region is a convex surface. We present an example where we show that in some cases depending on the channel distribution, the stability region can be characterized by a finite set of non-linear inequalities instead of an infinite number of linear inequalities.

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arXiv2011

Density Evolution Analysis of Node-Based Verification-Based Algorithms in Compressed Sensing

In this paper, we present a new approach for the analysis of iterative node-based verification-based (NB-VB) recovery algorithms in the context of compressive sensing. These algorithms are particularly interesting due to their low complexity (linear in the signal dimension $n$). The asymptotic analysis predicts the fraction of unverified signal elements at each iteration $\ell$ in the asymptotic regime where $n \rightarrow \infty$. The analysis is similar in nature to the well-known density evolution technique commonly used to analyze iterative decoding algorithms. To perform the analysis, a message-passing interpretation of NB-VB algorithms is provided. This interpretation lacks the extrinsic nature of standard message-passing algorithms to which density evolution is usually applied. This requires a number of non-trivial modifications in the analysis. The analysis tracks the average performance of the recovery algorithms over the ensembles of input signals and sensing matrices as a function of $\ell$. Concentration results are devised to demonstrate that the performance of the recovery algorithms applied to any choice of the input signal over any realization of the sensing matrix follows the deterministic results of the analysis closely. Simulation results are also provided which demonstrate that the proposed asymptotic analysis matches the performance of recovery algorithms for large but finite values of $n$. Compared to the existing technique for the analysis of NB-VB algorithms, which is based on numerically solving a large system of coupled differential equations, the proposed method is much simpler and more accurate.

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arXiv2011

Optimal Multi-Server Allocation to Parallel Queues With Independent Random Queue-Server Connectivity

We investigate an optimal scheduling problem in a discrete-time system of L parallel queues that are served by K identical, randomly connected servers. Each queue may be connected to a subset of the K servers during any given time slot. This model has been widely used in studies of emerging 3G/4G wireless systems. We introduce the class of Most Balancing (MB) policies and provide their mathematical characterization. We prove that MB policies are optimal; we define optimality as minimization, in stochastic ordering sense, of a range of cost functions of the queue lengths, including the process of total number of packets in the system. We use stochastic coupling arguments for our proof. We introduce the Least Connected Server First/Longest Connected Queue (LCSF/LCQ) policy as an easy-to-implement approximation of MB policies. We conduct a simulation study to compare the performance of several policies. The simulation results show that: (a) in all cases, LCSF/LCQ approximations to the MB policies outperform the other policies, (b) randomized policies perform fairly close to the optimal one, and, (c) the performance advantage of the optimal policy over the other simulated policies increases as the channel connectivity probability decreases and as the number of servers in the system increases.

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arXiv2011

Density Evolution Analysis of Node-Based Verification-Based Algorithms in Compressive Sensing

In this paper, we present a new approach for the analysis of iterative node-based verification-based (NB-VB) recovery algorithms in the context of compressive sensing. These algorithms are particularly interesting due to their low complexity (linear in the signal dimension $n$). The asymptotic analysis predicts the fraction of unverified signal elements at each iteration $\ell$ in the asymptotic regime where $n \rightarrow \infty$. The analysis is similar in nature to the well-known density evolution technique commonly used to analyze iterative decoding algorithms. To perform the analysis, a message-passing interpretation of NB-VB algorithms is provided. This interpretation lacks the extrinsic nature of standard message-passing algorithms to which density evolution is usually applied. This requires a number of non-trivial modifications in the analysis. The analysis tracks the average performance of the recovery algorithms over the ensembles of input signals and sensing matrices as a function of $\ell$. Concentration results are devised to demonstrate that the performance of the recovery algorithms applied to any choice of the input signal over any realization of the sensing matrix follows the deterministic results of the analysis closely. Simulation results are also provided which demonstrate that the proposed asymptotic analysis matches the performance of recovery algorithms for large but finite values of $n$. Compared to the existing technique for the analysis of NB-VB algorithms, which is based on numerically solving a large system of coupled differential equations, the proposed method is much simpler and more accurate.

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arXiv2010

Network Capacity Region of Multi-Queue Multi-Server Queueing System with Time Varying Connectivities

Network capacity region of multi-queue multi-server queueing system with random ON-OFF connectivities and stationary arrival processes is derived in this paper. Specifically, the necessary and sufficient conditions for the stability of the system are derived under general arrival processes with finite first and second moments. In the case of stationary arrival processes, these conditions establish the network capacity region of the system. It is also shown that AS/LCQ (Any Server/Longest Connected Queue) policy stabilizes the system when it is stabilizable. Furthermore, an upper bound for the average queue occupancy is derived for this policy.

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arXiv2010

An Efficient Approach Toward the Asymptotic Analysis of Node-Based Recovery Algorithms in Compressed Sensing

In this paper, we propose a general framework for the asymptotic analysis of node-based verification-based algorithms. In our analysis we tend the signal length $n$ to infinity. We also let the number of non-zero elements of the signal $k$ scale linearly with $n$. Using the proposed framework, we study the asymptotic behavior of the recovery algorithms over random sparse matrices (graphs) in the context of compressive sensing. Our analysis shows that there exists a success threshold on the density ratio $k/n$, before which the recovery algorithms are successful, and beyond which they fail. This threshold is a function of both the graph and the recovery algorithm. We also demonstrate that there is a good agreement between the asymptotic behavior of recovery algorithms and finite length simulations for moderately large values of $n$.

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