Queuing with future information

We study an admissions control problem, where a queue with service rate $1-p$ receives incoming jobs at rate $\lambda\in(1-p,1)$, and the decision maker is allowed to redirect away jobs up to a rate of $p$, with the objective of minimizing the time-average queue length. We show that the amount of information about the future has a significant impact on system performance, in the heavy-traffic regime. When the future is unknown, the optimal average queue length diverges at rate $\sim\log_{1/(1-p)}\frac{1}{1-\lambda}$, as $\lambda\to 1$. In sharp contrast, when all future arrival and service times are revealed beforehand, the optimal average queue length converges to a finite constant, $(1-p)/p$, as $\lambda\to1$. We further show that the finite limit of $(1-p)/p$ can be achieved using only a finite lookahead window starting from the current time frame, whose length scales as $\mathcal{O}(\log\frac{1}{1-\lambda})$, as $\lambda\to1$. This leads to the conjecture of an interesting duality between queuing delay and the amount of information about the future.Comment: Published in at http://dx.doi.org/10.1214/13-AAP973 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Universality of Load Balancing Schemes on Diffusion Scale

We consider a system of $N$ parallel queues with identical exponential service rates and a single dispatcher where tasks arrive as a Poisson process. When a task arrives, the dispatcher always assigns it to an idle server, if there is any, and to a server with the shortest queue among $d$ randomly selected servers otherwise $(1 \leq d \leq N)$. This load balancing scheme subsumes the so-called Join-the-Idle Queue (JIQ) policy $(d = 1)$ and the celebrated Join-the-Shortest Queue (JSQ) policy $(d = N)$ as two crucial special cases. We develop a stochastic coupling construction to obtain the diffusion limit of the queue process in the Halfin-Whitt heavy-traffic regime, and establish that it does not depend on the value of $d$, implying that assigning tasks to idle servers is sufficient for diffusion level optimality

Large deviations sum-queue optimality of a radial sum-rate monotone opportunistic scheduler

A centralized wireless system is considered that is serving a fixed set of users with time varying channel capacities. An opportunistic scheduling rule in this context selects a user (or users) to serve based on the current channel state and user queues. Unless the user traffic is symmetric and/or the underlying capacity region a polymatroid, little is known concerning how performance optimal schedulers should tradeoff "maximizing current service rate" (being opportunistic) versus "balancing unequal queues" (enhancing user-diversity to enable future high service rate opportunities). By contrast with currently proposed opportunistic schedulers, e.g., MaxWeight and Exp Rule, a radial sum-rate monotone (RSM) scheduler de-emphasizes queue-balancing in favor of greedily maximizing the system service rate as the queue-lengths are scaled up linearly. In this paper it is shown that an RSM opportunistic scheduler, p-Log Rule, is not only throughput-optimal, but also maximizes the asymptotic exponential decay rate of the sum-queue distribution for a two-queue system. The result complements existing optimality results for opportunistic scheduling and point to RSM schedulers as a good design choice given the need for robustness in wireless systems with both heterogeneity and high degree of uncertainty.Comment: Revised version. Major changes include addition of details/intermediate steps in various proofs, a summary of technical steps in Table 1, and correction of typos