Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Tail Optimality and Performance Analysis of the Nudge-M Scheduling Algorithm

Nils Charlet, Benny Van Houdt

TL;DR

This work introduces Nudge-M, a scheduling policy for two-type job systems where decisions rely only on arrival types and their order, to achieve strong tail performance improvements over FCFS. The authors establish that Nudge-M is asymptotically tail-optimal within a large family of Nudge-like policies by deriving explicit ATIR expressions and proving a unique optimal parameter M_{opt} that matches the Nudge-K optimum from prior work. In heavy traffic, ATIR(M_{opt}) converges to a Gamma-Boost–like limit, and the optimal M is shown to depend only on coarse job-size statistics, not on full distributions. The paper also provides a practical numerical framework using phase-type distributions and Markov-modulated fluid queues to compute mean and distributional performance for both type-1 and type-2 jobs, highlighting significant mean and tail gains with Nudge-M. The results offer a rigorous, broadly applicable approach to tail optimization under limited information, with potential extensions to more types and general arrival processes.

Abstract

Recently it was shown that the response time of First-Come-First-Served (FCFS) scheduling can be stochastically and asymptotically improved upon by the {\it Nudge} scheduling algorithm in case of light-tailed job size distributions. Such improvements are feasible even when the jobs are partitioned into two types and the scheduler only has information about the type of incoming jobs (but not their size). In this paper we introduce Nudge-$M$ scheduling, where basically any incoming type-1 job is allowed to pass any type-2 job that is still waiting in the queue given that it arrived as one of the last $M$ jobs. We prove that Nudge-$M$ has an asymptotically optimal response time within a large family of Nudge scheduling algorithms when job sizes are light-tailed. Simple explicit results for the asymptotic tail improvement ratio (ATIR) of Nudge-$M$ over FCFS are derived as well as explicit results for the optimal parameter $M$. An expression for the ATIR that only depends on the type-1 and type-2 mean job sizes and the fraction of type-1 jobs is presented in the heavy traffic setting. The paper further presents a numerical method to compute the response time distribution and mean response time of Nudge-$M$ scheduling provided that the job size distribution of both job types follows a phase-type distribution (by making use of the framework of Markov modulated fluid queues with jumps).

Tail Optimality and Performance Analysis of the Nudge-M Scheduling Algorithm

TL;DR

This work introduces Nudge-M, a scheduling policy for two-type job systems where decisions rely only on arrival types and their order, to achieve strong tail performance improvements over FCFS. The authors establish that Nudge-M is asymptotically tail-optimal within a large family of Nudge-like policies by deriving explicit ATIR expressions and proving a unique optimal parameter M_{opt} that matches the Nudge-K optimum from prior work. In heavy traffic, ATIR(M_{opt}) converges to a Gamma-Boost–like limit, and the optimal M is shown to depend only on coarse job-size statistics, not on full distributions. The paper also provides a practical numerical framework using phase-type distributions and Markov-modulated fluid queues to compute mean and distributional performance for both type-1 and type-2 jobs, highlighting significant mean and tail gains with Nudge-M. The results offer a rigorous, broadly applicable approach to tail optimization under limited information, with potential extensions to more types and general arrival processes.

Abstract

Recently it was shown that the response time of First-Come-First-Served (FCFS) scheduling can be stochastically and asymptotically improved upon by the {\it Nudge} scheduling algorithm in case of light-tailed job size distributions. Such improvements are feasible even when the jobs are partitioned into two types and the scheduler only has information about the type of incoming jobs (but not their size). In this paper we introduce Nudge- scheduling, where basically any incoming type-1 job is allowed to pass any type-2 job that is still waiting in the queue given that it arrived as one of the last jobs. We prove that Nudge- has an asymptotically optimal response time within a large family of Nudge scheduling algorithms when job sizes are light-tailed. Simple explicit results for the asymptotic tail improvement ratio (ATIR) of Nudge- over FCFS are derived as well as explicit results for the optimal parameter . An expression for the ATIR that only depends on the type-1 and type-2 mean job sizes and the fraction of type-1 jobs is presented in the heavy traffic setting. The paper further presents a numerical method to compute the response time distribution and mean response time of Nudge- scheduling provided that the job size distribution of both job types follows a phase-type distribution (by making use of the framework of Markov modulated fluid queues with jumps).
Paper Structure (14 sections, 15 theorems, 98 equations, 5 figures)

This paper contains 14 sections, 15 theorems, 98 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Model and Algorithms
  3. The Asymptotic Tail Improvement Ratio of Nudge-M
  4. Heavy Traffic Regime
  5. Optimality of Nudge-$M$ within a family of Nudge policies
  6. Nudge-M swap distribution for a tagged type-2 job
  7. Mean response time of Nudge-M
  8. Response time distribution of type-2 jobs
  9. Response time distribution of type-1 jobs
  10. FCFS scheduling:
  11. Nudge-1 scheduling:
  12. Nudge-$M$ scheduling:
  13. Conclusions and future work
  14. Nudge-$K,M$ versus Nudge-$M,L$

Key Result

Theorem 1

Let $c_{W^{(1)}}(M) = \lim_{t \rightarrow \infty} e^{\theta_Z t} P[W_M^{(1)} > t]$, then with $w_1 = p\tilde{S}_1(-\theta_Z) / \tilde{S}(-\theta_Z)$ and $w=(1-p)/\tilde{S}(-\theta_Z)$, which decreases in $M$ as $w_1+w < 1$.

Figures (5)

  • Figure 1: The asymptotic tail improvement ratio of Nudge-$M$ over Nudge-$K$ (left) and $\gamma$-Boost over Nudge-$M$ (right) with optimized parameters for $p=0.9$ with exponential type-1 and type-2 jobs. Nudge-$M$ yields a much higher ATIR than Nudge-$K$, especially for high loads $\lambda$. $\gamma$-Boost improves Nudge-$M$ by exploiting arrival time information, but the gains vanish as the load tends to one. The flat white area in the first plot corresponds to an ATIR of zero which occurs when $M_{opt} \leq 1$.
  • Figure 2: The asymptotic tail improvement ratio of Nudge-$M$ over FCFS with $M=M_{opt}$ and $M=M_{heavy}$ for $p=2/3$ and $E[X_1]/E[X_2]=4$. Setting $M=M_{heavy}$ yields most of the gains even when $\lambda$ is not very close to one.
  • Figure 3: The asymptotic tail improvement ratio of Nudge-$K$, Nudge-$L$, Nudge-$K,L$, Nudge-$M$, and $\gamma$-Boost (with optimized $K,L$, $M$, and boost), over FCFS with exponential job sizes. Nudge-$M$ outperforms the other Nudge algorithms, but the difference with Nudge-$K,L$ is small. $\gamma$-Boost outperforms the different Nudge algorithms by exploiting the arrival time information.
  • Figure 4: The mean response time improvement ratio (MTIR) of Nudge-$M$ with $M = M_{opt}$, Nudge-$M$ with $M = 20$, and a priority queue over FCFS with $p = 2/3$, and $E[X_2]/E[X_1] = 4$. For low loads, Nudge-$M$ with a high parameter coincides with the priority queue. This plot also shows how Nudge-$M$ improves the mean response time without making the tail worse than FCFS.
  • Figure 5: The tail improvement ratio of Nudge-$K$, Nudge-$L$, Nudge-$K,L$ and Nudge-$M$ (with parameters that maximize the ATIR) over FCFS with $E[X_2]/E[X_1]=4$, $p=2/3$, $\lambda=0.7$. Nudge-$M$ outperforms the other Nudge algorithms asymptotically, but the difference with Nudge-$K,L$ is often small. The second plot shows that Nudge-$M$ does not always stochastically improve upon Nudge-$K,L$.

Theorems & Definitions (29)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Corollary 1
  • proof
  • ...and 19 more