[Paper Review] Large Deviations for the Single Server Queue and the Reneging Paradox

Rami Atar, Amarjit Budhiraja+2 more|arXiv (Cornell University)|Mar 16, 2019

Advanced Queuing Theory Analysis16 references4 citations

TL;DR

This paper establishes a sample path large deviations principle for the M/M/1+M queue with reneging, proving that the decay rate of atypical reneging counts—both abnormally high and abnormally low—does not depend on the individual customer reneging rate θ. The key result reveals a large deviations analogue of the 'reneging paradox,' where the LD cost associated with reneging vanishes under optimal measures, despite θ's role in the model's dynamics.

ABSTRACT

For the M/M/1+M model at the law-of-large-numbers scale, the long run reneging count per unit time does not depend on the individual (i.e., per customer) reneging rate. This paradoxical statement has a simple proof. Less obvious is a large deviations analogue of this fact, stated as follows: The decay rate of the probability that the long run reneging count per unit time is atypically large or atypically small does not depend on the individual reneging rate. In this paper, the sample path large deviations principle for the model is proved and the rate function is computed. Next, large time asymptotics for the reneging rate are studied for the case when the arrival rate exceeds the service rate. The key ingredient is a calculus of variations analysis of the variational problem associated with atypical reneging. A characterization of the aforementioned decay rate, given explicitly in terms of the arrival and service rate parameters of the model, is provided yielding a precise mathematical description of this paradoxical behavior.

Motivation & Objective

• To establish a sample path large deviations principle (LDP) for the M/M/1+M queue with reneging under fluid scaling.
• To analyze the decay rate of probabilities for atypically large or small long-run reneging counts when the arrival rate exceeds the service rate.
• To resolve the 'reneging paradox' at the large deviations scale by showing the decay rate is independent of the individual reneging rate θ.
• To characterize the decay rate explicitly in terms of arrival and service rates only, using calculus of variations and Euler-Lagrange equations.
• To demonstrate that under the optimal change of measure, the normalized reneging rate remains at its law of large numbers value, implying vanishing LD cost for reneging.

Proposed method

• Model the system using Poisson random measures (PRM) to describe arrivals, service completions, and reneging events.
• Apply a general variational representation for expectations of functionals of PRM to derive Laplace principle upper and lower bounds.
• Define the rate function I_T(ξ,ζ) as the infimum over control processes φ that satisfy state dynamics equations involving arrival, service, and reneging intensities.
• Use the contraction principle to derive the LDP for the reneging count process Y^n(t) and its normalized limit.
• Apply calculus of variations to the associated variational problem, deriving Euler-Lagrange equations for the minimizer of the rate function.
• Restrict analysis to trajectories where queue length ξ(t) ≥ 1, and construct minimizers via a transformation of fluid trajectories from a reduced model.

Experimental results

Research questions

• RQ1 Does the decay rate of atypical reneging counts in the M/M/1+M queue depend on the individual customer reneging rate θ at the large deviations scale?
• RQ2 Can a sample path large deviations principle be rigorously established for the joint process of normalized queue length and normalized reneging count?
• RQ3 What is the explicit form of the decay rate for rare events involving abnormally high or low reneging counts when λ > μ?
• RQ4 Why does the LD cost associated with reneging vanish under the optimal measure, despite θ being a parameter in the model?
• RQ5 How does the optimal control trajectory behave in the variational problem, and what does it reveal about the structure of atypical reneging behavior?

Key findings

• The decay rate for atypical reneging counts is independent of the individual reneging rate θ, confirming a large deviations analogue of the 'reneging paradox'.
• The decay rate C(γ) is explicitly given in terms of λ and μ only, and is identical for both large and small deviations from the typical reneging rate γ₀ = (λ − μ)+.
• Under the optimal change of measure for atypical reneging, the normalized reneging rate remains equal to its law of large numbers value, implying the LD cost for reneging is zero.
• The minimizer of the rate function corresponds to trajectories where the queue length remains at or above 1, and the optimal control is derived via a transformation of a reduced fluid model.
• The rate function I_T(ξ,ζ) is finite only when the state trajectory (ξ,ζ) satisfies the fluid dynamics equations with appropriate intensity controls, and is infinite otherwise.
• The large deviations principle holds for the pair (X^n, Y^n) in the space of cadlag paths, with the rate function I_T defined via an infimum over control processes satisfying the state dynamics.