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The Amplitude Dynamics of Impulsive Queues

Ruici Gao, Jamol Pender

Abstract

In this paper, we analyze a multiserver Markovian queue with customer abandonment i.e. the Erlang-A queue uder a novel framework, i.e. the random impulsive differential equations (RIDEs). This framework captures systems that evolve continuously while experiencing sudden, discrete interventions. The combination of such framework with Erlang-A queue give rise to multiple real-life applications, such as improving efficiency at call centers and designing optimal timing to apply quantum error correction in trying to shun away from decoherence in quantum computing. We derive closed-form expressions for steady-state amplitude bounds and average queue lengths under impulse, and we identify the impulse timings that optimize system performance.

The Amplitude Dynamics of Impulsive Queues

Abstract

In this paper, we analyze a multiserver Markovian queue with customer abandonment i.e. the Erlang-A queue uder a novel framework, i.e. the random impulsive differential equations (RIDEs). This framework captures systems that evolve continuously while experiencing sudden, discrete interventions. The combination of such framework with Erlang-A queue give rise to multiple real-life applications, such as improving efficiency at call centers and designing optimal timing to apply quantum error correction in trying to shun away from decoherence in quantum computing. We derive closed-form expressions for steady-state amplitude bounds and average queue lengths under impulse, and we identify the impulse timings that optimize system performance.
Paper Structure (31 sections, 11 theorems, 116 equations, 5 figures)

This paper contains 31 sections, 11 theorems, 116 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main Contributions of the Paper
  3. Organization of the Paper
  4. Infinite Server Queue
  5. Review of the Mt/G/infinity Queue
  6. The Erlang-A Queueing Model
  7. Designing the Impulse
  8. Impulse Design in Infinite Server
  9. Impulse Design in Erlang-A
  10. Numerical Experiments
  11. q0 > c, lambda > mu c
  12. $q_0 \leq c, \lambda > \mu c$
  13. $q_0 > c, \lambda \leq \mu c$
  14. $q_0 \leq c, \lambda \leq \mu c$
  15. Conclusion
  16. ...and 16 more sections

Key Result

Lemma 2.1

Let $q(t)$ be the solution to the following differential equation where $q(0) = q_0$. Then, the solution for any value of $t$ is given by and as $t \to \infty$ we have

Figures (5)

  • Figure 1: Linear Impulsive Differential Equation Plots. (a)$\lambda = 10, \mu = 1, b = -1/2, q_0 = 0, \delta = 2$. (b)$\lambda = 10, \mu = 1, b = -1/2, q_0 = 12, \delta = 2$. (c)$\lambda = 10, \mu = 1, b = 1, q_0 = 12, \delta = 2$ . (d)$\lambda = 10, \mu = 1, b = 1, q_0 = 30, \delta = 2$.
  • Figure 2: Fluid Mean of Overload System (starts above capacity). (a).$\lambda = 10, \mu = 1, b = 0.5, T = 5, q_0 = 3, \theta = 2, c = 2$. (b). $\lambda = 10, \mu = 2, b = 0.5, T = 5, q_0 = 6, \theta = 3, c = 4$.
  • Figure 3: Fluid Mean of Overload System (starts below capacity). (a). $\lambda = 10, \mu = 1, b = 0.5, T = 4, q_0 = 1, \theta = 2, c = 2$. (b). $\lambda = 10, \mu = 1, b = 0.5, T = 0.35, q_0 = 1, \theta = 2, c = 2$. (c). $\lambda = 10, \mu = 1, b = 0.5, T = 0.2, q_0 = 1, \theta = 2, c = 2$.
  • Figure 4: Fluid Mean of Underload System (starts above capacity). (a). $\lambda = 9, \mu = 5, b = 0.5, T = 2, q_0 = 5, \theta = 2, c = 2$. (b). $\lambda = 9, \mu = 5, b = 0.5, T = 0.15, q_0 = 5, \theta = 2, c = 2$. (c). $\lambda = 9, \mu = 5, b = 0.5, T = 2, q_0 = 3, \theta = 2, c = 2$.
  • Figure 5: Fluid Mean of Underload System (starts below capacity). (a). $\lambda = 9, \mu = 5, b = 0.5, T = 1.5, q_0 = 1, \theta = 2, c = 2$. (b). $\lambda = 9, \mu = 5, b = 0.5, T = 0.5, q_0 = 1, \theta = 2, c = 2$.

Theorems & Definitions (21)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 3.1
  • proof
  • ...and 11 more