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Response time in a pair of processor sharing queues with Join-the-Shortest-Queue scheduling

Julianna Bor, Peter G Harrison

TL;DR

The paper addresses the problem of characterizing the full response-time distribution in two parallel JSQ-PS queues, a long-standing gap for Kendall-type models with PS discipline. It develops functional equations for the conditional LSTs of response times and solves them numerically by translating to a linear-algebra problem via contour discretization and 2D interpolation, using a known equilibrium queue-length generating function $G(x,y)$ to decondition. The main contributions are a practical numerical method that yields the density and moments (validated against regenerative simulation for both medium and heavy load) and insights into routing choices and performance relative to static scheduling. The approach has potential impact on the design and analysis of JSQ-based load balancing in networks, with further work aiming to reduce numerical pole effects and to explore variant routing rules.

Abstract

Join-the-Shortest-Queue (JSQ) is the scheduling policy of choice for many network providers, cloud servers and traffic management systems, where individual queues are served under processor sharing (PS) queueing discipline. A numerical solution for the response time distribution in two parallel PS queues with JSQ scheduling is derived for the first time. Using the generating function method, two partial differential equations (PDEs) are obtained corresponding to conditional response times, where the conditioning is on a particular traced task joining the first or the second queue. These PDEs are functional equations that contain partial generating functions and their partial derivatives, and therefore cannot be solved by commonly used techniques. We are able to solve these PDEs numerically with good accuracy and perform the deconditioning with respect to the queue-length probabilities by evaluating a certain complex integral. Numerical results for the density and the first four moments compare well against regenerative simulation with 500,000 regeneration cycles.

Response time in a pair of processor sharing queues with Join-the-Shortest-Queue scheduling

TL;DR

The paper addresses the problem of characterizing the full response-time distribution in two parallel JSQ-PS queues, a long-standing gap for Kendall-type models with PS discipline. It develops functional equations for the conditional LSTs of response times and solves them numerically by translating to a linear-algebra problem via contour discretization and 2D interpolation, using a known equilibrium queue-length generating function to decondition. The main contributions are a practical numerical method that yields the density and moments (validated against regenerative simulation for both medium and heavy load) and insights into routing choices and performance relative to static scheduling. The approach has potential impact on the design and analysis of JSQ-based load balancing in networks, with further work aiming to reduce numerical pole effects and to explore variant routing rules.

Abstract

Join-the-Shortest-Queue (JSQ) is the scheduling policy of choice for many network providers, cloud servers and traffic management systems, where individual queues are served under processor sharing (PS) queueing discipline. A numerical solution for the response time distribution in two parallel PS queues with JSQ scheduling is derived for the first time. Using the generating function method, two partial differential equations (PDEs) are obtained corresponding to conditional response times, where the conditioning is on a particular traced task joining the first or the second queue. These PDEs are functional equations that contain partial generating functions and their partial derivatives, and therefore cannot be solved by commonly used techniques. We are able to solve these PDEs numerically with good accuracy and perform the deconditioning with respect to the queue-length probabilities by evaluating a certain complex integral. Numerical results for the density and the first four moments compare well against regenerative simulation with 500,000 regeneration cycles.
Paper Structure (19 sections, 5 theorems, 28 equations, 5 figures, 6 tables)

This paper contains 19 sections, 5 theorems, 28 equations, 5 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Functional equations for the response time of two JSQ-PS queues
  3. Generating function for the queue-length distribution
  4. Generating functions of conditional response time densities
  5. LST of the unconditional response time distribution
  6. Solving the functional equations
  7. Unconditional response time
  8. Moments
  9. Evaluation
  10. Comparison with simulation
  11. Static scheduling models
  12. Finding the optimal routing probabilities
  13. Accuracy and cost
  14. Conclusion
  15. Appendices
  16. ...and 4 more sections

Key Result

Lemma 1

Analogous expressions can be obtained for $\alpha_F,\beta_F,\gamma_F,\delta_F$, replacing $E$ by $F$ on the right hand sides.

Figures (5)

  • Figure 1: Two JSQ-PS queues in parallel where the arriving task joins the shorter queue. The arrival rate is $\lambda$, the service rates are $\mu_1$ and $\mu_2$,
  • Figure 2: Steps to approximate f(x) from 8 known values around the unit circle.
  • Figure 3: Response time density $w(t)$ (vertical axis) against $t$ (horizontal axis) for two JSQ-PS servers with parameters $\lambda=1, \mu_1=0.9, \mu_2=1.1, a_1=0, a_2=1$: comparison with simulation.
  • Figure 4: Response time density $w(t)$ (vertical axis) against $t$ (horizontal axis) for two JSQ-PS servers with parameters $\lambda=1, \mu_1=0.4, \mu_2=0.8, a_1=a_2=0.5$: comparison with simulation.
  • Figure 5: Mathematica code for 2-dimensional linear interpolation.

Theorems & Definitions (9)

  • Definition 1
  • Lemma 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Lemma 2
  • Proof 1
  • Proof 2
  • Proof 3