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Multiplexing VBR Traffic on a Multiplexer (Switch)

Multiplexing VBR Traffic on a Multiplexer (Switch). VBR : Constant Bit Rate source of traffic. Configuration: a single multiplexer single output link (C) single buffer (B). C. N. B. References.

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Multiplexing VBR Traffic on a Multiplexer (Switch)

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  1. Multiplexing VBR Traffic on a Multiplexer (Switch) • VBR: Constant Bit Rate source of traffic. • Configuration: a single multiplexer single output link (C) single buffer (B) C N B H.Levy, Advanced Comm, CS, TAU, VBR

  2. References • A. Elwalid, D. Mitra, R. H. Wentworth, “A New Approach for Allocating Buffers and Bandwidth to Heterogeneous, Regulated Traffic in an ATM Node”. • L. He and A.K. Wong, “Connection Admission Control Design for GlobView -2000 ATM Core Switches”, Bell Labs Technical Journal, Jan-Mar 98, pp. 94-.. H.Levy, Advanced Comm, CS, TAU, VBR

  3. Characteristics of a VBR source • Peak Cell Rate (PCR=P) (cells/sec or bits/sec) • Sustained Cell Rate (SCR=r) “ “ • Burst Tolerance ( )= size of leaky bucket mechanism that controls traffic (cells) [in ATM standards == measure in time] Permit entrance rate r Transmission Permits BT P Maximal permit exit rate H.Levy, Advanced Comm, CS, TAU, VBR

  4. Characteristics of VBR:Leaky Bucket (cont) • Transmission allowed when bucket is not empty • Worst case scenario: - Source transmits at rate P, for time - Source stops for • Time to fill bucket: = /(P-r) • Maximal Burst Size (MBS): MBS= P* = P* /(P-r) H.Levy, Advanced Comm, CS, TAU, VBR

  5. Performance Requirements • Cell Loss Rate (CLR): 0 < L < 1, fraction of cells that get lost due to buffer overflow. • Cell Delay Variation (CDV): = Delay (cell1) - Delay (cell2) < Delay (cell1). • Claim: Both measures can be estimated via Pr [ # cells in buffer > B] H.Levy, Advanced Comm, CS, TAU, VBR

  6. Performance Requir. (cont) • Reasoning: - CLR : good approximation (infinite buffer approximates finite buffer) - Delay: MAX Delay = B/C Pr[ Delay > B/C] = Pr[ # cells > B] H.Levy, Advanced Comm, CS, TAU, VBR

  7. Problem Formulation • Given:- Source characteristics (P, r, (MBS))- Number of connections N- MUX characteristics: B, C • Question: What is the probability that the loss exceeds L? • Other words: If probability of loss is a performance requirement: what is the multiplexer “size” (# connections that can be handled) H.Levy, Advanced Comm, CS, TAU, VBR

  8. Simplifying Assumption • Assume: “Worst Case Traffic” (WCT) [worst case in terms of buffer requirement is somewat different] • Simplifies analysis • Provides lower bound on capacity (conservative) H.Levy, Advanced Comm, CS, TAU, VBR

  9. Source properties • Data generated at on-period is maximal: • On time: token pool of regulator fills up: • Amount of data in on-time • ==> (time to fill up bucket) H.Levy, Advanced Comm, CS, TAU, VBR

  10. c Analyze Single source in Isolation • connection gets capacity c dedicated to it • v(t) = buffer occupancy • u(t) = link occupancy H.Levy, Advanced Comm, CS, TAU, VBR

  11. Buffer and Link Occupancy Ton Toff b V(t) c u(t) Don Doff H.Levy, Advanced Comm, CS, TAU, VBR

  12. Buffer and link occupancy (cont) • Don = Ton + b/c, Doff = Toff-b/c • w = fraction of time that virtual link/buffer is occupied • w= Don / (Don + Doff) • (token pool increases at P-r) • (virtual link at P-c) • ==> H.Levy, Advanced Comm, CS, TAU, VBR

  13. Lossless Multiplexing • How many sources can multiplex without having any loss ( no statistical multiplexing). • = allocations for buffer and link • One linear relationship is dictated • Also: “balanced system”: • 2 equations ==> solution for • yields lossless capacity, . H.Levy, Advanced Comm, CS, TAU, VBR

  14. Solution H.Levy, Advanced Comm, CS, TAU, VBR

  15. Solution (cont.) • w = fraction of time source is on: H.Levy, Advanced Comm, CS, TAU, VBR

  16. Lossy System: Statistical Multiplexing • Consider system with losses • Want small likelihood of loss. • Each source on-off • Sources differ in their phase over the period (whose duration is Ton+Toff). H.Levy, Advanced Comm, CS, TAU, VBR

  17. Graphical • w Overflow(t) C c0 t w H.Levy, Advanced Comm, CS, TAU, VBR

  18. Probability of loss (overflow) • Let • K = number of sources operated • Loss event : connections or more are active at t. • Probability of loss at t: H.Levy, Advanced Comm, CS, TAU, VBR

  19. Solution (cont.) • Chernoff’s approximation • Approximates the tail of the sum of random variables • M(s) is E[exp(su)] H.Levy, Advanced Comm, CS, TAU, VBR

  20. E[exp(su)] Exp(su) f(u) 1 0 u 1 H.Levy, Advanced Comm, CS, TAU, VBR

  21. Analysis (cont.) • Specific distribution -- Bernoulli: • u=1 with prob w • u=0 with prob 1-w • where H.Levy, Advanced Comm, CS, TAU, VBR

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