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Rare events in a polling system: Rays and Spirals

Robert D. Foley, David R. McDonald

Abstract

It's a situation everyone dreads. A road is down to one lane for repairs. Traffic is let through one way until the backlog clears and then traffic is let through the other way to clear that backlog and so on. When stuck in a very long queue it is inevitable to wonder how did I get into this mess? We study a polling model with a server having exponential service time with mean $1/μ$ alternating between two queues, emptying one queue before switching to the other. Customers arrive at queue one according to a Poisson process with rate $λ_1$ and at queue two with rate $λ_2$. We discuss how we get at a rare event with a large number of customers in the system. In fact this can happen in two different ways depending on the parameters. In one case one queue simply explodes and runs away without emptying. We call this the ray case. In the other spiral case the queues are successively emptied but in a losing battle as the system zigzags to the rare event. This dichotomy extends to the steady state distribution and leads to quite different asymptotic behaviour in the two cases.

Rare events in a polling system: Rays and Spirals

Abstract

It's a situation everyone dreads. A road is down to one lane for repairs. Traffic is let through one way until the backlog clears and then traffic is let through the other way to clear that backlog and so on. When stuck in a very long queue it is inevitable to wonder how did I get into this mess? We study a polling model with a server having exponential service time with mean alternating between two queues, emptying one queue before switching to the other. Customers arrive at queue one according to a Poisson process with rate and at queue two with rate . We discuss how we get at a rare event with a large number of customers in the system. In fact this can happen in two different ways depending on the parameters. In one case one queue simply explodes and runs away without emptying. We call this the ray case. In the other spiral case the queues are successively emptied but in a losing battle as the system zigzags to the rare event. This dichotomy extends to the steady state distribution and leads to quite different asymptotic behaviour in the two cases.
Paper Structure (29 sections, 26 theorems, 233 equations, 4 figures, 1 table)

This paper contains 29 sections, 26 theorems, 233 equations, 4 figures, 1 table.

Table of Contents

  1. Tandem polling model
  2. State space & macro behavior:
  3. Stability of the tandem polling model
  4. Uniformization
  5. Twisted $Z$
  6. Twisted $M$
  7. Path of the rare event of hitting a high level
  8. The ray case
  9. The ray case: all directions
  10. Computing $\pi(1,y,1)$ and $\pi(0,y,2)$
  11. A bridge to $(1,y,1)$
  12. Extending to $\pi(0,y,2)$
  13. The Cascade case
  14. Extending to $\pi(x,1,2)$ in the Cascade case
  15. Extending to $\pi(x,0,1)$
  16. ...and 14 more sections

Key Result

Theorem 1

If there is a ray on sheet 1; i.e. $\tilde{\lambda}_1>\tilde{\mu}$ then where

Figures (4)

  • Figure 1: Jump rates when queue $1$ is served are on the left sheet and on the right sheet when queue $2$ is served. Open circles denote points not in the state space
  • Figure 2: Ray-Spiral case: $\lambda_1=0.3$, $\lambda_2=0.05$, $\mu=0.65$
  • Figure 3: Ray-Spiral case: $\lambda_1=6/20$, $\lambda_2=1/20$, $\mu=13/20$
  • Figure 4: Spiral-Spiral case: $\lambda_1=0.3$, $\lambda_2=0.15$, $\mu=0.55$

Theorems & Definitions (45)

  • Theorem 1
  • proof
  • Theorem 2
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • Theorem 3
  • proof
  • ...and 35 more