Understanding Effective Bandwidths in Circuit-Switched Models and Admission Control

1. CMPE 548Effective Bandwidths CMPE 548 Fall 2005

2. 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

3. Example: M/G/1 FIFO • Consider the constraint Wq≤d, E[Sj]=mj, Var(Sj)=σj2 PK formula CMPE 548 Fall 2005

4. 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

5. 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

6. 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

7. 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

8. 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

9. Probability review Note: Moment-generating function (mgf) of X is given by CMPE 548 Fall 2005

10. 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

11. 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

12. Gaussian source example • Suppose that X[0,t]=λt+Z(t) where Z(t)~N(0,Var(Z(t)). Then, CMPE 548 Fall 2005

13. 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