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Exhaustive Polling Systems

Exhaustive Polling Systems. Alec Zimmer and Joey Kraisler. Setup. Deterministic, exhaustive polling system n nodes, n ≥ 3 J obs queue continuously at nodes, at rate λ i for the i th node Server processes one node at a time, at rate μ i for the i th node

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Exhaustive Polling Systems

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  1. Exhaustive Polling Systems Alec Zimmer and Joey Kraisler

  2. Setup • Deterministic, exhaustive polling system • n nodes, n ≥ 3 • Jobs queue continuously at nodes, at rate λi for the ith node • Server processes one node at a time, at rate μifor the ith node • ρi := λi/μi < 1 for all i • Σρi > 1

  3. Switching Rule • Server switches between nodes based on a generalized greedy rule • If at the ith node, the server processes it until xi = 0. • Zero switching time • Have bi > 0, i = 1,2,…,n • Server then starts processing the jth node, where bjxj ≥ bkxk for all k

  4. The Process • A collection of queue lengths (x1,…,xn) can be thought of as a particle in Rn • We can project the system to the n-simplex x1+…+xn = 1 • The system then consists of a particle moving from face to face of the n-simplex • Let ϕbe the map that sends a point on the boundary of the n-simplex to the next point the particle visits on the boundary

  5. Node Process • We’re concerned about where the particle representing queue lengths travels • MacPhee et. al. (2006) show that when heading to the ith side, the particle goes on a line towards the node where θj:= (1/μj)Σλi − 1

  6. Parameter simplification • Can just consider the case when μi=1 for all i • Scale coordinates by • Project back to the simplex

  7. New nodes • We then get θ := θj =Σ λi − 1 = Σ ρi − 1 and Giving an n-simplex with sides parallel to the original one

  8. Contraction • Let A be our original simplex formed by the ei, VA be the intersection of the boundary of A with the simplex formed by the vi • If z is in VA, so is ϕ(z) • There is some γ < 1 such that |ϕ(z) – ϕ(z’)| < γ|z – z’| for z,z’ on and going to the same sides • Projecting towards a point decreases distance • “Angle shift” inside VA further contracts • Get |ϕ(n)(z) – ϕ(n)(z’)| < γn|z – z’|

  9. Periodicity • A trajectory z, ϕ(z), ϕ(2)(z), … is a periodic orbit if there is some m such that ϕ(k+m)(z) = ϕ(k)(z) for all k ≥ 0 • Theorem All points on a periodic orbit lie in VA for any number of nodes n. After a trajectory visits all n nodes, it is inside VA.

  10. Theorem All points on a periodic orbit lie in VA for any number of nodes n. After a trajectory visits all n nodes, it is inside VA. Proof: • If a trajectory is inside VA, it will stay in VA, so an orbit is either entirely inside or outside VA. Use contradiction. • Suppose orbit involves m nodes, m < n. • Consider the m-1 plane containing the relevant nodes. No point in the orbit lies in this plane • Each time we map a point in the orbit to the next point, the distance to the m-1 plane decreases. Contradiction

  11. Theorem All points on a periodic orbit lie in VA for any number of nodes n. After a trajectory visits all n nodes, it is inside VA. • Now suppose we have a trajectory that visits all n nodes. We use induction on n. • n = 3 covered by MacPhee et. al. (2006) • If true for n-1, consider the first n-1 nodes the trajectory visits. The point will then be in the projection of the first n-1 nodes with respect to the nth node. • The next map sends the point inside VA. QED

  12. Theorem All points on a periodic orbit lie in VA for any number of nodes n. After a trajectory visits all n nodes, it is inside VA. • Corollary For any starting point z, there is some t0 > 0 such that ϕ(t)(z) is in a contracting region for all t ≥ t0

  13. Further Research (I) • For n = 3, decision points di are where bjxj = bkxk • MacPhee et. al. (2006) proves the following: Theorem For almost all decision points di, i = 1,2,3, the triangle process Z has finitely many periodic orbits. For such sets of decision points, all trajectories ϕ(t)(z) are eventually periodic and each converges onto one of these orbits as t → ∞. • This relies on almost all decision points having finitely many preimages

  14. Further Research (II) • But, this doesn’t directly work for n > 3. • Have to consider decision lines/planes also

  15. Acknowledgements Thanks to MishaGuysinsky and Shilpak Banerjee

  16. References Macphee, I. M.; Menshikov, M. V.; Popov, S.; and Volkov, S (2006). Periodicity in the Transient Regime of Exhaustive Polling Systems. Ann. Appl. Probab. Vol. 16, No. 4, 1816-1850.

  17. Questions?

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