Frank Kelly introduced in [Kel91] a scalar involving the statistical properties of a single source, the statistical properties of traffic being superposed with the traffic source and the capacity and buffer of a multiplexer. This scalar expresses the equivalent bandwidth which estimates the required resources for the source in order to respect its Quality of
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Service (QoS) requirements. For example, consider nj sources of type j, each having
equivalent bandwidth αj, and J the number of source types. The linear constraint to
meet the QoS of all J type sources is defined as follows:
J
X
j=1
nj · αj ≤ C∗ (7.1)
Where C∗ is equivalent capacity of the link which depends on different parameters (Link
capacity, buffer, QoS, and the statistical properties of the traffic mix). The equiva- lent bandwidth estimation is based on the asymptotic analysis. It concerns mainly the way the buffer overflow probability decays as a function of some quantity. Two quan- tities are generally used: The size of the buffer which gives the large buffer asymptotic [EM93, CW95], and the size of the system (link, number of sources, . . . ) which gives the many sources asymptotic [SG95, CW96]. The many sources asymptotic consider that the buffer per source and the capacity per source are constant. However, the definitions of equivalent bandwidth based on the large buffer asymptotic have been found not ac- curate. The main explanation of this refers to the gain we have when superposing many sources together and which does not figure in the large buffer asymptotic. Indeed, Kelly [Kel96] tried to include this information into the definition of the equivalent bandwidth through two parameters: space and time. Briefly, the equivalent bandwidth according to Kelly depends on the link’s operating point through these two parameters which can be calculated using the many sources asymptotic.
7.4.1.1 Many Sources Asymptotic
We consider J independent source types being superposed in a multiplexer. The number of sources of each type j is defined by nj = Nρj, while N represents the total size of
the system and ρ = (ρ1, ρ2, ..., ρJ) is the proportion vector by type of source. To comply
with this notation the buffer size is defined as B = Nb and the link capacity C = Nc with parameters b (resp. c) corresponding to the buffer (resp. capacity) per source.
Consider the time interval [0, t], and note Xj[0, t] the load produced by a j source.
Kelly introduces [Kel96] the equivalent bandwidth of a source of type j, assuming that Xj[0, t] has stationary increments, as:
αj(s, t) =
1
s × tlog E
h
eXj[0,t]i (7.2) s, t are source context parameters defining the system, i.e., the characteristics of the superposed traffic, capacity, buffer. . . The time parameter t is related to the duration of the busy period of the buffer prior to overflow, while the space parameter s expresses the degree of superposing. The space parameter depends on the ratio between the peak rate of the superposed sources and the link capacity. Thus, the physical interpretation of the space parameter shows that when link capacity is larger than the peak rate of the superposed sources, the space parameters tends to zero. As a consequence, the term αj(s, t) tends to the mean rate of the source. On the other hand, when link capacity is not
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much larger than the peak rate of the superposed sources, s is big and consequently the term αj(s, t) tends to the maximum value of Xj[0, t] /t (Note that Xj[0, t] is a random
variable).
Practically, it is important to know the buffer overflow probability in presence of superposed traffics. Based on Kelly’s definition of the equivalent bandwidth, Courcou- betis and Weber [CW96] demonstrated the sup inf formula that gives the many sources asymptotic for buffer overflow expressed usually as P (overf low) = e−NI+o(N) ≈
N →∞e −NI
derived from the general form:
lim
N →∞
1
N log(P (overf low)) = supt infs
s.t J X j=1 ρjαj(s, t) − s(ct + b) = −I (7.3)
Usually P (overf low) is denoted as Q(Nc, Nb, Nρ) expressing the probability that in an infinite buffer where Nρ = (Nρ1, Nρ2, ..., NρJ) sources are superposed and served at
C = Nc rate, the queue length exceeds the threshold B = Nb. I is generally called the asymptotic rate function.
The QoS constraint on the overflow probability is expressed as P (overf low) ≤ e−γ.
However, the effective bandwidth αj(s, t) provides a measure of resource usage for a par-
ticular operating point of the link, expressed through parameters s and t. For example, if a source of type j1 has twice as much equivalent bandwidth as a source of type j2, then
for this particular operating point of the link, one source of the first type uses twice as much resources than a source of the second type. The asymptotics underlying the above results assume only stationarity of sources.
7.4.1.2 Large Buffer Asymptotic
The many sources asymptotic definition of the equivalent bandwidth takes into account the effects of statistical superposition of traffic sources. Meanwhile, the definition of the equivalent bandwidth based on the large buffer asymptotic considers only the character- istics of the source as well as the QoS constraint. Thus, if we consider the QoS constraint P (overf low) ≤ e−δB [CW95, EM93, dVW95], where B is the total buffer. Then, the
equivalent bandwidth of a source of type j is given by:
αj(s) = 1 st→∞lim 1 t log E h esXj[0,t]i (7.4) In fact, the last equation is the same as (7.2) when t → ∞. Thus, the last equation is only accurate for large buffer sizes as the time parameter t becomes large. That is why for finite buffer sizes, a significant miss utilization of link capacity could occur.
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Figure 7.5: The Evolution of ON-OFF VoIP Source Equivalent Bandwidth
7.4.1.3 Equivalent Bandwidth of ON-OFF Sources by Kelly’s Formula The effective bandwidth of an ON-OFF source model (for an individual application) is given by the following equation [Kel96]:
α(s, t) = 1
s.tlog [1 + p.(exp(s.t.α
∗(s, t)) − 1)] (7.5)
Where α∗(s, t) is the effective bandwidth of the ON state and p is the proportion of time
spent in the ON state. The mean and peak of the source are represented by M and h respectively, so α∗(s, t) = h and p = M
h. Then the available bandwidth is represented by:
αM,h(s, t) = 1 s × tlog 1 + M h .(e s.t.h − 1) (7.6)
As an example we show on Figure 7.5 the effective bandwidth for a VoIP traffic source based on G711 codec and classical communication parameters for ON and OFF periods (See Chapter 5).
The two dimension graph shows how the equivalent bandwidth varies with time and scale parameters between average and peak rate values. However, determining the time and scale parameter (or the operating point) according to network configuration requires the resolution of the sup inf formula. This may be problematic when such decisions must be taken by the SIP proxy server for bandwidth reservation in real time. Algorithms proposed for the resolution of the sup inf formula do not allow the estimation of the operating point (t, s) analytically.
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