Stationary analysis of the Shortest Queue First service policy

We analyze the so-called Shortest Queue First (SQF) queueing discipline whereby a unique server addresses queues in parallel by serving at any time that queue with the smallest workload. Considering a stationary system composed of two parallel queues and assuming Poisson arrivals and general service time distributions, we first establish the functional equations satisfied by the Laplace transforms of the workloads in each queue. We further specialize these equations to the so-called "symmetric case", with same arrival rates and identical exponential service time distributions at each queue; we then obtain a functional equation $$M(z) = q(z) \cdot M \circ h(z) + L(z)$$ for unknown function $M$, where given functions $q$, $L$ and $h$ are related to one branch of a cubic polynomial equation. We study the analyticity domain of function $M$ and express it by a series expansion involving all iterates of function $h$. This allows us to determine empty queue probabilities along with the tail of the workload distribution in each queue. This tail appears to be identical to that of the Head-of-Line preemptive priority system, which is the key feature desired for the SQF discipline

Cache Miss Estimation for Non-Stationary Request Processes

The aim of the paper is to evaluate the miss probability of a Least Recently Used (LRU) cache, when it is offered a non-stationary request process given by a Poisson cluster point process. First, we construct a probability space using Palm theory, describing how to consider a tagged document with respect to the rest of the request process. This framework allows us to derive a general integral formula for the expected number of misses of the tagged document. Then, we consider the limit when the cache size and the arrival rate go to infinity proportionally, and use the integral formula to derive an asymptotic expansion of the miss probability in powers of the inverse of the cache size. This enables us to quantify and improve the accuracy of the so-called Che approximation

Entropic Forces in Binary Hard Sphere Mixtures: Theory and Simulation

We perform extensive Monte Carlo simulations of binary hard-sphere mixtures (with diameter ratios of 5 and 10), to determine the entropic force between (1) a macrosphere and a hard wall, and (2) a pair of macrospheres. The microsphere background fluid (at volume fractions ranging from 0.1 to 0.34) induces an entropic force on the macrosphere(s); the latter component is at infinite dilution. We find good overall agreement, in both cases, with the predictions of an HNC-based theory for the entropic force. Our results also argue for the validity of the Derjaguin approximation relating the force between convex bodies to that between planar surfaces. The earlier Asakura-Oosawa theory, based on a simple geometric argument, is only accurate in the low-density limit.Comment: 13 pages, LaTeX, 10 figure