arXiv subjects

Jalani Williams

Publications and source records attributed to Jalani Williams.

3 recordsLinked to original sources

arXiv2025

An Upper Bound on the M/M/k Queue With Deterministic Setup Times

In many systems, servers do not turn on instantly; instead, a setup time must pass before a server can begin work. These "setup times" can wreak havoc on a system's queueing; this is especially true in modern systems, where servers are regularly turned on and off as a way to reduce operating costs (energy, labor, CO2, etc.). To design modern systems which are both efficient and performant, we need to understand how setup times affect queues. Unfortunately, despite successes in understanding setup in a single-server system, setup in a multiserver system remains poorly understood. To circumvent the main difficulty in analyzing multiserver setup, all existing results assume that setup times are memoryless, i.e. distributed Exponentially. However, in most practical settings, setup times are close to Deterministic, and the widely used Exponential-setup assumption leads to unrealistic model behavior and a dramatic underestimation of the true harm caused by setup times. This paper provides a comprehensive characterization of the average waiting time in a multiserver system with Deterministic setup times, the M/M/k/Setup-Deterministic. In particular, we derive upper and lower bounds on the average waiting time in this system, and show these bounds are within a multiplicative constant of each other. These bounds are the first closed-form characterization of waiting time in any finite-server system with setup times. Further, we demonstrate how to combine our upper and lower bounds to derive a simple and accurate approximation for the average waiting time. These results are all made possible via a new technique for analyzing random time integrals that we named the Method of Intervening Stopping Times, or MIST.

cs.PF↗

arXiv2021

Probing to Minimize

We develop approximation algorithms for set-selection problems with deterministic constraints, but random objective values, i.e., stochastic probing problems. When the goal is to maximize the objective, approximation algorithms for probing problems are well-studied. On the other hand, few techniques are known for minimizing the objective, especially in the adaptive setting, where information about the random objective is revealed during the set-selection process and allowed to influence it. For minimization problems in particular, incorporating adaptivity can have a considerable effect on performance. In this work, we seek approximation algorithms that compare well to the optimal adaptive policy. We develop new techniques for adaptive minimization, applying them to a few problems of interest. The core technique we develop here is an approximate reduction from an adaptive expectation minimization problem to a set of adaptive probability minimization problems which we call threshold problems. By providing near-optimal solutions to these threshold problems, we obtain bicriteria adaptive policies. We apply this method to obtain an adaptive approximation algorithm for the MIN-ELEMENT problem, where the goal is to adaptively pick random variables to minimize the expected minimum value seen among them, subject to a knapsack constraint. This partially resolves an open problem raised in Goel et. al's "How to probe for an extreme value". We further consider three extensions on the MIN-ELEMENT problem, where our objective is the sum of the smallest k element-weights, or the weight of the min-weight basis of a given matroid, or where the constraint is not given by a knapsack but by a matroid constraint. For all three variations we explore, we develop adaptive approximation algorithms for their corresponding threshold problems, and prove their near-optimality via coupling arguments.

cs.DS↗

arXiv2020

Lower Bounds for Caching with Delayed Hits

Caches are a fundamental component of latency-sensitive computer systems. Recent work of [ASWB20] has initiated the study of delayed hits: a phenomenon in caches that occurs when the latency between the cache and backing store is much larger than the time between new requests. We present two results for the delayed hits caching model. (1) Competitive ratio lower bound. We prove that the competitive ratio of the algorithm in [ASWB20], and more generally of any deterministic online algorithm for delayed hits, is at least Omega(kZ), where k is the cache size and Z is the delay parameter. (2) Antimonotonicity of the delayed hits latency. Antimonotonicity is a naturally desirable property of cache latency: having a cache hit instead of a cache miss should result in lower overall latency. We prove that the latency of the delayed hits model is not antimonotone by exhibiting a scenario where having a cache hit instead of a miss results in an increase in overall latency. We additionally present a modification of the delayed hits model that makes the latency antimonotone.

cs.DS↗