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The Multiserver-Job Stochastic Recurrence Equation for Cloud Computing Performance Evaluation

Francois Baccelli, Diletta Olliaro, Marco Ajmone Marsan, Andrea Marin

TL;DR

This paper develops a general analytical framework for performance evaluation of Multiserver-Job Queuing Models (MJQM) under FCFS with arbitrary arrival and service-time distributions. By formulating a Multiserver-Job Stochastic Recurrence Equation (MJSRE) and proving monotonicity and separability within the Monotone-Separable networks paradigm, the authors obtain a threshold-type stability condition governed by $\lambda_c = 1/\gamma$, where $\gamma$ is the growth rate of a pile $H_n$; they also introduce Sub-Perfect Sampling (SPS) to generate tight lower bounds to the steady-state workload $W$, enabling rigorous confidence intervals for metrics like waiting times and wasted servers. The work provides two practical algorithms: (i) SPS for workload sampling and (ii) a procedure to compute the stability threshold, both amenable to massively parallel GPU acceleration. Validation includes two-class and five-class MJQMs and a Google Borg Cell B-style scenario, demonstrating accurate results and substantial speedups over traditional discrete-event simulation (DES). The approach generalizes to multiple resource types and resources with specific identities, offering scalable, rigorous tools for cloud-data-center performance analysis and resource-utilization optimization. The combination of theoretical guarantees, scalable computation, and applicability to realistic, heterogeneous resource setups significantly advances quantitative data-center performance analysis and design.

Abstract

We study the Multiserver-Job Queuing Model (MJQM) with general independent arrivals and service times under FCFS scheduling, using stochastic recurrence equations (SREs) and ergodic theory. We prove the monotonicity and separability properties of the MJQM SRE, enabling the application of the monotone-separable extension of Loynes' theorem and the formal definition of the MJQM stability condition. Based on these results, we introduce and implement two algorithms: one for drawing sub-perfect samples (SPS) of the system's workload and the second one to estimate the system's stability condition given the statistics of the jobs' input stream. The SPS algorithm allows for a massive GPU parallelization, greatly improving the efficiency of performance metrics evaluation. We also show that this approach extends to more complex systems, including MJQMs with typed resources.

The Multiserver-Job Stochastic Recurrence Equation for Cloud Computing Performance Evaluation

TL;DR

This paper develops a general analytical framework for performance evaluation of Multiserver-Job Queuing Models (MJQM) under FCFS with arbitrary arrival and service-time distributions. By formulating a Multiserver-Job Stochastic Recurrence Equation (MJSRE) and proving monotonicity and separability within the Monotone-Separable networks paradigm, the authors obtain a threshold-type stability condition governed by , where is the growth rate of a pile ; they also introduce Sub-Perfect Sampling (SPS) to generate tight lower bounds to the steady-state workload , enabling rigorous confidence intervals for metrics like waiting times and wasted servers. The work provides two practical algorithms: (i) SPS for workload sampling and (ii) a procedure to compute the stability threshold, both amenable to massively parallel GPU acceleration. Validation includes two-class and five-class MJQMs and a Google Borg Cell B-style scenario, demonstrating accurate results and substantial speedups over traditional discrete-event simulation (DES). The approach generalizes to multiple resource types and resources with specific identities, offering scalable, rigorous tools for cloud-data-center performance analysis and resource-utilization optimization. The combination of theoretical guarantees, scalable computation, and applicability to realistic, heterogeneous resource setups significantly advances quantitative data-center performance analysis and design.

Abstract

We study the Multiserver-Job Queuing Model (MJQM) with general independent arrivals and service times under FCFS scheduling, using stochastic recurrence equations (SREs) and ergodic theory. We prove the monotonicity and separability properties of the MJQM SRE, enabling the application of the monotone-separable extension of Loynes' theorem and the formal definition of the MJQM stability condition. Based on these results, we introduce and implement two algorithms: one for drawing sub-perfect samples (SPS) of the system's workload and the second one to estimate the system's stability condition given the statistics of the jobs' input stream. The SPS algorithm allows for a massive GPU parallelization, greatly improving the efficiency of performance metrics evaluation. We also show that this approach extends to more complex systems, including MJQMs with typed resources.
Paper Structure (37 sections, 5 theorems, 22 equations, 7 figures, 3 tables, 2 algorithms)

This paper contains 37 sections, 5 theorems, 22 equations, 7 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Prior Work
  4. Background
  5. Stochastic Recurrence Equations
  6. Loynes' Theorem
  7. Monotone-Separable Stochastic Networks
  8. The Multiserver-Job Stochastic Recurrence Equation
  9. Model Description
  10. The Kiefer and Wolfowitz Stochastic Recurrence Equation
  11. The Multiserver-Job Stochastic Recurrence Equation
  12. Definition of the stochastic recurrence equation and monotonicity
  13. The associated Loynes' sequence
  14. The sub-perfect sampling algorithm
  15. Stability Analysis
  16. ...and 22 more sections

Key Result

Lemma 1

For all fixed $\alpha\in [1,\ldots,s]$, $\sigma$ and $\tau$ in $\mathbb R^+$, the map: is coordinatewise non decreasing. That is, for all ordered workload vectors $W$ and $W'$ of $\;\mathbb{R}_{\geq 0}^s$ such that coordinatewise $W\leq W'$, it holds that, coordinatewise:

Figures (7)

  • Figure 1: Example of HOL blocking in Multiserver-job systems.
  • Figure 2: Left and center: average number of servers wasted due to HOL, in a system with Poisson arrivals and exponential service times. Left: $s=200$, two job classes, comparison of literature and MJSRE. Center: $s=20$, five job classes as in Table \ref{['tab:synth']}. Right: Probability density function of waiting times per class in a MJQM with five classes of jobs and bounded Pareto job service times.
  • Figure 3: Average, variance and 90-th percentile of per-class waiting times versus job arrival rate in the case of a MJQM with 20 servers and 5 job classes, with Poisson job arrivals and exponential job service times. The vertical bar in the left plot indicates the limit of the stability region.
  • Figure 4: Same as Fig. \ref{['fig:synth_exp']} with hyperexponential service times.
  • Figure 5: Average, variance and 90-th percentile of waiting times versus job arrival rate for jobs requiring just 1 server in the case of a MJQM with 20 servers and 5 job classes. This is for Poisson job arrivals and job service times with exponential, hyperexponential, Erlang-3 or bounded Pareto distribution, as in Table \ref{['tab:synth']}. The vertical bars in the left plot indicate the limit of the stability region with the different service time distributions.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Remark 1: Why is FCFS scheduling important in cloud computing data center performance analysis?
  • Lemma 1
  • theorem 1
  • Lemma 2
  • Lemma 3
  • Remark 2: Subsequences
  • Remark 3: Comparison with a forward approach to MJSRE and ordinary DES
  • theorem 2
  • Remark 4: Applicability of Algorithm \ref{['alg:limit']}