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CMPE 548 Effective Bandwidths. Admission control. Simple call admission control for the circuit-switched model: Suppose a collection of sources n j of type j ε J which require a bandwidth α j share a link with capacity C

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Cmpe 548 effective bandwidths

CMPE 548Effective Bandwidths

CMPE 548 Fall 2005

Admission control
Admission control

  • Simple call admission control for the circuit-switched model:

    • Suppose a collection of sources nj of type jεJ which require a bandwidth αj share a link with capacity C

    • One can check if bandwidth is available by considering the linear constraint

  • Admission control with statistical guarantees for statistically multiplexed heterogeneous packetized traffic streams

    • What is αj for an arbitrary source?

CMPE 548 Fall 2005

Example m g 1 fifo
Example: M/G/1 FIFO

  • Consider the constraint Wq≤d, E[Sj]=mj, Var(Sj)=σj2

PK formula

CMPE 548 Fall 2005

M g 1 fifo continued
M/G/1 FIFO continued

  • Let’s define the effective bandwidth αj(d):

    • Note how the effective bandwidth incorporates statistical properties of the source and the QoS requirements!

  • Note that we once again have the linear constraint:

CMPE 548 Fall 2005

Effective bandwidth
Effective bandwidth

  • Problems related to resource sharing can be analyzed using the notion of “effective bandwidth” which is a scalar (or a statistical descriptor) that summarizes resource usage and which depends on the statistical properties and QoS requirements of a source

  • Definition:

    • log E[.] is the log-moment generating function of RV X[0,t]

    • X[0,t] is the load produced by the source in time interval [0,t]

CMPE 548 Fall 2005

Space time parameters s and t
Space-time parameters: s and t

  • In α(s,t), s, t are system parameters defined by the context of the source.

    • The characteristics of multiplexed traffic, QoS requirements, link resources (capacity & buffer)

  • Space parameter s (in kb-1) is an indication of degree of multiplexing and depends, among others, on the size of the peak rates of the multiplexed sources relative to the link capacity

  • Time parameter t corresponds to the most probable duration of buffer busy period prior to overflow

CMPE 548 Fall 2005

Important properties of s t
Important properties of α(s,t)

  • If X[0,t]=ΣXi[0,t] where {Xi[0,t]} are independent then α(s,t)=Σαi(s,t)

  • For any fixed value of t, α(s,t) is increasing in s: Effective bandwidth decreases as degree of multiplexing increases (s→0)

  • For any fixed value of t, α(s,t) lies between the mean and the peak of the arrival rate measured over an interval of length t

  • t→0: a bufferless model; t→∞: large buffers

CMPE 548 Fall 2005

More on space time parameters s t
More on space-time parameters (s, t)

  • In particular, for link capacities much larger than the peak rates of the multiplexed sources, s→0 and α(s,t)→mean rate of the source

  • For link capacities not much larger than the peak rates of the sources s is large and α(s,t)→max value of X[0,t]/t (deterministic multiplexing)

  • Time parameter t identifies the time-scales that are important for buffer overflow

    • Large t implies slow time-scales are responsible for buffer overflow

    • Parameter t increases with buffer size (or link capacity)

CMPE 548 Fall 2005

Probability review
Probability review

Note: Moment-generating function

(mgf) of X is given by

CMPE 548 Fall 2005

Chernoff bound
Chernoff bound

Assume sx>>1, and let’s take β(s)=1 (based on “numerical experience”):

when Ploss<<1)

and E[esQ]≤β(s)

(approximately, could be a bad approximation!)

CMPE 548 Fall 2005

Poisson source example
Poisson source example

  • Consider a Poisson source:

    • Mgf of Poisson(λ) RV X[0,t] is

    • Then,

      Example: Suppose Ploss=10-5. Then, s=11.5/x.

      If we pick x>>11.5 cells so that s<<1, α(s,t)=λ

      For large enough buffer, the effective capacity of a Poisson source is just the average rate of that source.

      Now, let x=10 cells. Then, s=1.15 and α(s,t)=2λ!

      The effective capacity doubles!

CMPE 548 Fall 2005

Gaussian source example
Gaussian source example

  • Suppose that X[0,t]=λt+Z(t) where Z(t)~N(0,Var(Z(t)). Then,

CMPE 548 Fall 2005

Gaussian sources self similarity
Gaussian sources: Self-similarity

  • It has been shown that Ethernet traffic exhibits self-similar behavior, in which case Var(Z(t))=σ2t2H with Hurst parameter 0.5<H<1

  • Then, the effective bandwidth of such source is

  • Note that α(s,t) grows as a fractional power of t

CMPE 548 Fall 2005