Load Balancing via Random Local Search in Closed and Open systems

In this paper, we analyze the performance of random load resampling and migration strategies in parallel server systems. Clients initially attach to an arbitrary server, but may switch server independently at random instants of time in an attempt to improve their service rate. This approach to load balancing contrasts with traditional approaches where clients make smart server selections upon arrival (e.g., Join-the-Shortest-Queue policy and variants thereof). Load resampling is particularly relevant in scenarios where clients cannot predict the load of a server before being actually attached to it. An important example is in wireless spectrum sharing where clients try to share a set of frequency bands in a distributed manner.Comment: Accepted to Sigmetrics 201

Tight Load Balancing via Randomized Local Search

We consider the following balls-into-bins process with $n$ bins and $m$ balls: each ball is equipped with a mutually independent exponential clock of rate 1. Whenever a ball's clock rings, the ball samples a random bin and moves there if the number of balls in the sampled bin is smaller than in its current bin. This simple process models a typical load balancing problem where users (balls) seek a selfish improvement of their assignment to resources (bins). From a game theoretic perspective, this is a randomized approach to the well-known Koutsoupias-Papadimitriou model, while it is known as randomized local search (RLS) in load balancing literature. Up to now, the best bound on the expected time to reach perfect balance was $O\left({(\ln n)}^2+\ln(n)\cdot n^2/m\right)$ due to Ganesh, Lilienthal, Manjunath, Proutiere, and Simatos (Load balancing via random local search in closed and open systems, Queueing Systems, 2012). We improve this to an asymptotically tight $O\left(\ln(n)+n^2/m\right)$. Our analysis is based on the crucial observation that performing "destructive moves" (reversals of RLS moves) cannot decrease the balancing time. This allows us to simplify problem instances and to ignore "inconvenient moves" in the analysis.Comment: 24 pages, 3 figures, preliminary version appeared in proceedings of 2017 IEEE International Parallel and Distributed Processing Symposium (IPDPS'17

A Stochastic Model for Car-Sharing Systems

Vehicle-sharing systems are becoming important for urban transportation. In these systems, users arrive at a station, pick up a vehicle, use it for a while and then return it to another station of their choice. Depending on the type of system, there might be a possibility to book vehicles before picking-up and/or a parking space at the chosen arrival station. Each station has a finite capacity and cannot host more vehicles and reserved parking spaces than its capacity. We propose a stochastic model for an homogeneous car-sharing system with possibility to reserve a parking space at the arrival station when picking-up a car. We compute the performance of the system and the optimal fleet size according to a specific metric. It differs from a similar model for bike-sharing systems because of reservation that induces complexity, especially when traffic increases

Stationary Distribution Analysis of a Queueing Model with Local Choice

The paper deals with load balancing between one-server queues on a circle by a local choice policy. Each one-server queue has a Poissonian arrival of customers. When a customer arrives at a queue, he joins the least loaded queue between this queue and the next one, ties solved at random. Service times have exponential distribution. The system is stable if the arrival-to-service rate ratio called load is less than one. When the load tends to zero, we derive the first terms of the expansion in this parameter for the stationary probabilities that a queue has 0 to 3 customers. We investigate the error, comparing these expansion results to numerical values obtained by simulations. Then we provide the asymptotics, as the load tends to zero, for the stationary probabilities of the queue length, for a fixed number of queues. It quantifies the difference between policies with this local choice, no choice and the choice between two queues chosen at random

On the flow-level stability of data networks without congestion control: the case of linear networks and upstream trees

In this paper, flow models of networks without congestion control are considered. Users generate data transfers according to some Poisson processes and transmit corresponding packet at a fixed rate equal to their access rate until the entire document is received at the destination; some erasure codes are used to make the transmission robust to packet losses. We study the stability of the stochastic process representing the number of active flows in two particular cases: linear networks and upstream trees. For the case of linear networks, we notably use fluid limits and an interesting phenomenon of "time scale separation" occurs. Bounds on the stability region of linear networks are given. For the case of upstream trees, underlying monotonic properties are used. Finally, the asymptotic stability of those processes is analyzed when the access rate of the users decreases to 0. An appropriate scaling is introduced and used to prove that the stability region of those networks is asymptotically maximized

Mean field and propagation of chaos in multi-class heterogeneous loss models

We consider a system consisting of parallel servers, where jobs with different resource requirements arrive and are assigned to the servers for processing. Each server has a finite resource capacity and therefore can serve only a finite number of jobs at a time. We assume that different servers have different resource capacities. A job is accepted for processing only if the resource requested by the job is available at the server to which it is assigned. Otherwise, the job is discarded or blocked. We consider randomized schemes to assign jobs to servers with the aim of reducing the average blocking probability of jobs in the system. In particular, we consider a scheme that assigns an incoming job to the server having maximum available vacancy or unused resource among randomly sampled servers. We consider the system in the limit where both the number of servers and the arrival rates of jobs are scaled by a large factor. This gives rise to a mean field analysis. We show that in the limiting system the servers behave independently—a property termed as propagation of chaos. Stationary tail probabilities of server occupancies are obtained from the stationary solution of the mean field which is shown to be unique and globally attractive. We further characterize the rate of decay of the stationary tail probabilities. Numerical results suggest that the proposed scheme significantly reduces the average blocking probability of jobs as compared to static schemes that probabilistically route jobs to servers independently of their states