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CN Computer Networks Research Group G R Department of Computer - - PowerPoint PPT Presentation

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Closed-Form Expression for the Collision Probability in the IEEE EPON Registration Scheme Swapnil Bhatia (with Dr. Radim Barto s) CN Computer Networks Research Group G R Department of Computer Science University of New Hampshire This

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Closed-Form Expression for the Collision Probability in the IEEE EPON Registration Scheme

Swapnil Bhatia (with Dr. Radim Bartoˇ s) Computer Networks Research Group Department of Computer Science

CN

R

G

University of New Hampshire

This work was supported in part by the Cisco University Research Program.

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Outline

◮ Introduction to Ethernet Passive Optical Networks (EPONs) ◮ IEEE EPON Registration Scheme ⋄ Description and model ◮ Efficiency of the Registration Scheme ⋄ Two devices ⋄ n devices ◮ Summary and discussion

2 of 29 IEEE GLOBECOM 2005

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IEEE 802.3ah Ethernet Passive Optical Network

◮ Shared, passive, optical, access network ◮ Directed tree topology ◮ Central arbitrator: OLT (Optical Line Terminator) ◮ Subscriber Device: ONU (Optical Network Unit) ◮ Data rate of 1 Gbps (bidirectionally) ◮ 20 km reach = ⇒ RTT ≤ 200 µs

OLT ONU ONU ONU ONU ONU ONU ONU ONU ONU ONU ONU ONU ONU Backbone User data Passive Optical OLT: Optical Line Terminator ONU: Optical Network Unit 2 k m Fiber Optical Splitter

◮ Recently standardized by the IEEE 802.3ah group

3 of 29 IEEE GLOBECOM 2005

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IEEE EPON Discovery and Registration Scheme

◮ Allow new ONUs to join the network ⋄ Synchronization ◮ Provide new ONUs with transmission parameters ⋄ Logical link identifiers, security (?) and QoS (?) ⋄ Enable RTT calculation by new ONUs ◮ RTT to new ONUs unavailable to the OLT, initially ⋄ = ⇒ a contention-based scheme

4 of 29 IEEE GLOBECOM 2005

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IEEE EPON Discovery Protocol

◮ OLT reserves a sufficiently large window of time ◮ Broadcasts a “Discovery start” message to all ONUs ⋄ Contains length of window ◮ New ONUs follow Random Wait rule:

Discovery window Random delay REGISTER_REQ GATE GATE REGISTER_ACK REGISTER OLT ONU

⋄ Wait for a random period after receiving message ⋄ Transmit request to join network ⋄ Expect a Register reply before next Discovery cycle ⋄ Finish with an Ack, or retry

5 of 29 IEEE GLOBECOM 2005

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IEEE EPON Discovery Protocol

◮ Discovery window required periodically ⋄ Every few minutes ◮ No subscriber traffic during Discovery window ⋄ Including voice, video etc. ◮ = ⇒ Important to choose window size wisely

Discovery window Random delay REGISTER_REQ GATE GATE REGISTER_ACK REGISTER OLT ONU

◮ Questions: ⋄ How do we choose the most efficient window size? ⋄ What do we mean by the efficiency of the Random Wait scheme?

6 of 29 IEEE GLOBECOM 2005

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Efficiency of the IEEE EPON Discovery Protocol

◮ Observation: Larger the discovery window size (say w), ⋄ larger the number of successful registrations ⋄ but larger the bandwidth wasted ◮ ∴ High efficiency means: ⋄ Maximal number of successful registrations, with ⋄ Minimal window size ◮ ∴ Efficiency of Discovery Window ⋄ Could be defined as: ρ = Number of successful registrations Size of Discovery window ◮ “Number of successful registrations” is a random variable

7 of 29 IEEE GLOBECOM 2005

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Efficiency of the IEEE EPON Discovery Protocol

◮ Suppose: ⋄ there are n ONUs ⋄ Ps(n) is the probability of a successful registration for an ONU, in the presence of n − 1 other ONUs ⋄ T is the duration of the discovery window reserved by the OLT ◮ Efficiency of Discovery Window ρ = n · Ps(n) T

8 of 29 IEEE GLOBECOM 2005

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Probability of a Successful Registration: Ps(n)

◮ For ONU-i, let random variable: Xi

  • its random RTT,

Yi

  • the random wait, and

Zi

  • the time of arrival of the message at the OLT.

Random delay GATE OLT ONU REGISTER_REQ window Reserved

Zi Yi X′

i

X′

i

T = 2p + w

= ⇒ Zi = Xi + Yi where : Xi = 2X′

i

9 of 29 IEEE GLOBECOM 2005

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Efficiency of the IEEE EPON Discovery Protocol

◮ Efficiency of Discovery Window ρ = n · Ps(n) 2p + w where: n is the total number of ONUs in the EPON, Ps(n) is the probability of a successful registration for an ONU, in the presence of n−1 other ONUs, p is the maximum propagation delay to any new ONU, and w is the size of the discovery window ◮ Rule: Choose a window size w that maximizes ρ, given n and p. ◮ Question: How do we find Ps(n)?

10 of 29 IEEE GLOBECOM 2005

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Probability of a Successful Registration: Ps(n)

◮ Suppose OLT chooses window size to be w µs ⋄ = ⇒ Wait Yi is ∈ [0, w] ⋄ distributed uniformly (IEEE standard) ◮ Suppose ONUs are located at a maximum propagation delay p ⋄ = ⇒ RTT Xi ∈ [0, 2p] ⋄ distributed uniformly (reasonable) ◮ Then, density of Zi = Xi + Yi:

fZi(z)

1 M

Zi m M M + m

z m M m + M − z m M

m = min(2p, w), M = max(2p, w)

11 of 29 IEEE GLOBECOM 2005

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Probability of a message collision: 1 − Ps(n)

◮ Consider 1 − Ps(2): the probability of a message collision for an ONU, in the presence of 1 other ONU ◮ Consider arrival times Z1 and Z2 ( = ⇒ independent) ◮ The joint density of Z1, Z2:

14 12 10 14 8 0.002 12 z1 6 10 0.004 8 4 6 0.006 z2 4 2 0.008 2 0.01 M k M M + m Z

1

− Z

2

= k Z

1

− Z

2

= − k Z1 f1(z1) f2(z1) f3(z1) f1(z2) f2(z2) f3(z2) Z2 m k M + m

◮ The collision event: |Z1 − Z2| ≤ k where k message length

12 of 29 IEEE GLOBECOM 2005

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Probability of a message collision: 1 − Ps(2)

k k M m M m M + m m + k m + k m − k M + k M − k M + k m − k M − k M + m Z

1

− Z

2

= − k Z

1

− Z

2

= k M + m − k M + m − k R1 R3 R5 R6 R4 R7 R8 R9 R12 R2 R10 R11 R13 Z1 Z2 f1(z1) f2(z1) f3(z1) f1(z2) f2(z2) f3(z2) m M k k m M M + m m + k M + m − k M + m − k Z

1

− Z

2

= k Z

1

− Z

2

= − k m + k R1 R2 R3 R4 R5 R6 R7 R8 R9 Z1 f1(z1) f2(z1) f3(z1) f1(z2) f2(z2) f3(z2) Z2 M − k M − k M + m

m ≥ k m < k M − m ≥ k

13 of 29 IEEE GLOBECOM 2005

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Probability of a message collision: 1 − Ps(2)

k k m m M M M + m M + m m − k M − k m + k M + m − k M + m − k Z1 Z2 R1 R2 R3 R4 R5 R7 R9 R10 R12 M + k m − k R11 R13 M + k

8 6

M − k m + k f1(z1) f2(z1) f3(z1) f1(z2) f2(z2) f3(z2)

Z

1

− Z

2

= − k Z

1

− Z

2

= k

m m k M M k M − k M − k m + k m + k M + m − k M + m − k M + m Z1 Z2

Z1 − Z2 = −k Z1 − Z2 = k

f3(z1) f2(z1) f1(z1) f1(z2) f2(z2) f3(z2) R2 R4 R6 R8 R9 R5 R7 R1 R3 M + m

m ≥ k m < k M − m < k

14 of 29 IEEE GLOBECOM 2005

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Probability of a message collision: 1 − Ps(2)

m m M M M + m Z1 Z2 f3(z1) f2(z1) f1(z1) f1(z2) f2(z2) f3(z2) R2 M + m M + m − k M + m − k k k

Z1 − Z2 = −k Z1 − Z2 = k

R3 R1 k k

Z1 − Z2 = −k Z1 − Z2 = k

Z2 Z1 M M + m M M + m f3(z2) f2(z2) f1(z2) f1(z1) f2(z1) f3(z1) m m

M + m > k M + m ≤ k m < M < k

15 of 29 IEEE GLOBECOM 2005

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Probability of a message collision: 1 − Ps(2)

M Z1 Z2 R2 M + m M − k k k

Z1 − Z2 = −k Z1 − Z2 = k

R3 R1 m m M M + m M + m − k fX(z1) fX(z2) M − k M + m − k

m = 0, M > k

16 of 29 IEEE GLOBECOM 2005

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Probability of a message collision: 1 − Ps(2)

m = 0 M < k M + m ≤ k m < k m < k M > k k > 0, M > 0 M − m < k

true false true false true true true false false true true false false false true false

P7 P6 P6 P6 P4 P5 P3P2 P1

17 of 29 IEEE GLOBECOM 2005

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Probability of a message collision: 1 − Ps(2)

400 300 200 p 100 0.2 100 200 0.4 300 0.6 w 400 0.8 1

k = 2.528 µs m = min(2p, w), M = max(2p, w)

18 of 29 IEEE GLOBECOM 2005

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How about Ps(n)?

◮ A successful transmission by Z1 can be expressed as:

n

  • i=2
  • |Z1 − Zi| > k
  • ◮ ∴ Ps(n):

P n

  • i=2
  • |Z1 − Zi| > k
  • =

∞

  • −∞

P n

  • i=2
  • |Z1 − Zi| > k|Z1 = t
  • · P(Z1 = t) · dt

=

∞

  • −∞

[1 − FZ2(t + k) + FZ2(t − k)](n−1) · fZ2(t) · dt

19 of 29 IEEE GLOBECOM 2005

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How about Ps(n)?

fZi(z)

1 M

Zi m M M + m t − k t M + m − k t + k M − k M + k m + k m − k

z m M m + M − z m M

P n

  • i=2
  • |Z1 − Zi| > k
  • =

∞

  • −∞

[1 − FZ2(t + k) + FZ2(t − k)](n−1) · fZ2(t) · dt ◮ Must consider relative magnitude of t in addition to m, M, k as before ⋄ Many cases, quite tedious

20 of 29 IEEE GLOBECOM 2005

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How about Ps(n)?

◮ A simple approximation ⋄ An ONU’s transmission was successful in the presence of n − 1

  • ther ONUs (Ps(n))

⋄ = ⇒ it was successful in the presence of each of the n − 1 ONUs, taken one at a time (Ps(2)) Ps(n) ≈ Ps(2)n−1 ⋄ Done!

21 of 29 IEEE GLOBECOM 2005

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Probability of a Successful Registration: Ps(n)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 50 100 150 200 250 300 350 400 Probability of successes in first attempt (99% confidence) Window size (micro sec)

  • No. nodes

2 8 16 30 50 65 90 100 Closed-form

p > 0

22 of 29 IEEE GLOBECOM 2005

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Probability of a Successful Registration: Ps(n)

◮ For ONU-i, let random variable: Xi

  • its random RTT,

Yi

  • the random wait, and

Zi

  • the time of arrival of the message at the OLT.

Random delay GATE OLT ONUs window Reserved

T = 2p + w p p

= ⇒ Zi = Xi + Yi where : Xi = 2X′

i

23 of 29 IEEE GLOBECOM 2005

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Probability of a Successful Registration: Ps(n)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 50 100 150 200 250 300 350 400 Probability of successes in first attempt (99% confidence) Window size (micro sec)

  • No. nodes

2 4 8 16 35 65 100 Closed-form

p = 0

24 of 29 IEEE GLOBECOM 2005

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Probability of a Successful Registration: Ps(n)

200 150 0.2 0.4 0.6 0.8 100 1 100 n w 200 50 300 400 200 0.2 0.4 150 0.6 100 0.8 1 100 200 n w 50 300 400 0

p > 0 (Random RTT) p = 0 (Identical RTT)

25 of 29 IEEE GLOBECOM 2005

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Efficiency of the IEEE Discovery Scheme

400 300 200 0.01 0.02 0.03 150 200 0.04 w 0.05 0.06 100 0.07 100 n 50 400 300 200 0.01 150 0.02 200 0.03 w 100 0.04 100 n 50

p > 0 (Random RTT) p = 0 (Identical RTT) ρ = n · Ps(n) 2p + w

26 of 29 IEEE GLOBECOM 2005

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Choosing an efficient window size

p > 0 (Random RTT) p = 0 (Identical RTT)

50 100 150 200 250 300 350 200 400 600 800 1000 1200 1400 1600 Number of nodes Window size (micro seconds) uniform identical

# devices Most efficient (M = 0) window size (µs) 2 35.82 4 64.63 8 105.20 10 122.39 16 168.43 32 273.77 50 380.49 64 459.65

27 of 29 IEEE GLOBECOM 2005

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Summary

◮ Characterized efficiency of the IEEE Registration scheme ◮ Exact closed-form expression for two ONUs ◮ Approximate closed-form expression for n > 2 ONUs ◮ Performance for variable message lengths ◮ Analysis of multi-step performance of registration scheme possible ◮ Work may be applicable to similar schemes in other technologies

28 of 29 IEEE GLOBECOM 2005

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Discussion and Questions

29 of 29 IEEE GLOBECOM 2005