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CMPE 548 Effective Bandwidths

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CMPE 548Effective Bandwidths

CMPE 548 Fall 2005

- 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

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

PK formula

CMPE 548 Fall 2005

- 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

- 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

- 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

- 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

- 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

Note: Moment-generating function

(mgf) of X is given by

CMPE 548 Fall 2005

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

- 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

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

CMPE 548 Fall 2005

- 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