Queuing with future information

Abstract

We study an admissions control problem, where a queue with service rate 1 − p receives incoming jobs at rate λ ∈ (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 ∼ log 1/(1−p) 1 1−λ , as λ → 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 λ → 1. 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 O(log 1 1−λ ), as λ → 1. This leads to the conjecture of an interesting duality between queuing delay and the amount of information about the future.

Necessity of Future Information in Admission Control

“…A main message of Spencer et al (2014) is that one can drastically reduce queueing delay with a sufficient amount of future information. In particular, there exists c h > 0, such that if the length of the lookahead window satisfies…”

“…Despite having identical modeling assumptions, our proof techniques are quite different from those employed by Spencer et al (2014). The core of our arguments hinges upon a relationship between diversions and future idling of the server, evaluated over certain subset of input sample paths.…”

“…In a recent paper, Spencer et al (2014) initiated an investigation along this direction in a class of queueing admission control problems, illustrated in Figure 1. An overloaded queue with service rate 1 − p receives incoming jobs at rate λ ∈ (1 − p, 1), and the system manager is allowed to divert incoming jobs up to a rate of p, with the objective of minimizing the time-average queueing delay among the admitted jobs.…”

“…They prove that, while delay under an optimal online policy diverges to infinity in the heavy-traffic regime, it can stay bounded by making use of future information. However, the diversion polices of Spencer et al (2014) require the length of the lookahead window to grow to infinity at a non-trivial rate in the heavy-traffic regime, and it remained open whether substantial performance improvement could still be achieved with less future information.We resolve this question to a large extent by establishing an asymptotically tight lower bound on how much future information is necessary to achieve superior performance, which matches the upper bound of Spencer et al (2014) up to a constant multiplicative factor. Our result hence demonstrates that the system's heavy-traffic delay performance is highly sensitive to the amount of future information available.…”

“…We resolve this question to a large extent by establishing an asymptotically tight lower bound on how much future information is necessary to achieve superior performance, which matches the upper bound of Spencer et al (2014) up to a constant multiplicative factor. Our result hence demonstrates that the system's heavy-traffic delay performance is highly sensitive to the amount of future information available.…”