Multiplexing VBR Traffic on a Multiplexer (Switch) by eon20304

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									     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)

N


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                               VBR
                   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-..




3/10/2010         H.Levy, Advanced Comm, CS, TAU,      2
                                VBR
    Characteristics of a VBR source
• Peak Cell Rate (PCR=P) (cells/sec or bits/sec)
• Sustained Cell Rate (SCR=r) “        “
• Burst Tolerance (BT )= size of leaky bucket
  mechanism that controls traffic (cells) [in ATM
  standards == measure in time]

                              Permit entrance rate




                  Maximal permit exit rate
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                               VBR
    Characteristics of VBR:Leaky
            Bucket (cont)
• Transmission allowed when bucket is not
  empty
• Worst case scenario:
   - Source transmits at rate P, for time Ton
   - Source stops for Toff
• Time to fill bucket: Ton = BT/(P-r)
• Maximal Burst Size (MBS):
     MBS= P* Ton = P* BT /(P-r)
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                             VBR
            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]

3/10/2010         H.Levy, Advanced Comm, CS, TAU,   5
                                VBR
        Performance Requir. (cont)
• Reasoning:
    - CLR : good approximation (infinite
       buffer approximates finite buffer)

            - Delay: MAX Delay = B/C
              Pr[ Delay > B/C] = Pr[ # cells > B]


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                                   VBR
            Problem Formulation
• Given:
  - Source characteristics (P, r, BT (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)
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                              VBR
                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)


           Ton      Toff             Ton
(t )
          MBS       P              MBS

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                                      VBR
              Source properties
• Data generated at on-period is maximal:
• On time: token pool of regulator fills up:
                       rP
                        TB
                            NOT

                                      P
• Amount of data in on-time MBS  BT
                                     Pr
• ==>                BT             (time to fill up bucket)
            Toff   
                      r
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                                  VBR
Analyze Single source in Isolation

            (t )                                         u (t )
                                 v (t )               c

• connection gets capacity c dedicated to it
• v(t) = buffer occupancy
• u(t) = link occupancy

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                                  VBR
                    Buffer and Link Occupancy
        Ton              Toff


 b
V(t)

        c
u(t)
                   Don      Doff


       3/10/2010                H.Levy, Advanced Comm, CS, TAU,   11
                                              VBR
     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)
• b  BT ( P  c)  (token pool increases at P-r)
        Pr
•                  (virtual link at P-c)
• ==> b  BT
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                             VBR
            Lossless Multiplexing
• How many sources can multiplex without having
  any loss ( no statistical multiplexing).
• b0 , c0 = allocations for buffer and link
•   One linear relationship is dictated
•   Also: “balanced system”: N B  B / b0  N C  C / c0
•   2 equations ==> solution for b0 , c0
•   yields lossless capacity,        .   N  CN   B




3/10/2010         H.Levy, Advanced Comm, CS, TAU,     13
                                VBR
                                Solution

                   P                                    BT 
             B/C                               if r 
                                                       B/C
            1       (P  r)                               
       c0      BT                                         
                                                 BT        
                   r                                 r  P
                                                B/C        

            b0  c0 B / C

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                                          VBR
            Solution (cont.)

• w = fraction of time source is on:
                w  r / c0




3/10/2010      H.Levy, Advanced Comm, CS, TAU,   15
                             VBR
    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).



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                               VBR
                       Graphical

•w               Overflow(t)

 C



            c0
                 t            w
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                                   VBR
     Probability of loss (overflow)
• Let N C  C / c0
• K = number of sources operated
• Loss event : N C connections or more are
  active at t.
• Probability of loss at t:
•                 K  N    K N
            Ploss   w C (1  w)
                    N 
                                             C


                     C

3/10/2010      H.Levy, Advanced Comm, CS, TAU,   18
                             VBR
            Solution (cont.)
• Chernoff’s approximation
• Approximates the tail of the sum of random
            Ploss  exp sup{sC  K log M ( s)
  variables
                                              
                        s 0                  
• M(s) is E[exp(su)]



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                             VBR
                 E[exp(su)]


                Exp(su)



      1                      f(u)

            0                      1              u
3/10/2010       H.Levy, Advanced Comm, CS, TAU,       20
                              VBR
                    Analysis (cont.)
  • Specific distribution -- Bernoulli:
  • u=1 with prob w
  • u=0 with prob 1-w
log Ploss  K a log(a / w)  (1  a ) log((1  a ) /(1  w)) 
  • where
              a  (C / c0 ) / K


  3/10/2010             H.Levy, Advanced Comm, CS, TAU,   21
                                      VBR

								
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