The rate map represented by Equation 3.11 is expected to be used by an algorithm designer to build an adaptation logic that guarantees the set of requirements earlier enumerated at the Section 3.2. This section presents an analytical analysis of the model, to show that it will indeed help a rate selection algorithm developer achieve his/her aim if used. Recall that a rate map is used to guide an ABR navigate its path from the lowest video rate to the highest video rate. To guarantee high average video rate and network utilisation the rate map must converge at the highest video rate, and remain there for the remaining duration of the streaming session. The next sections will discuss the video rate converges, stability, and the impact ofα on them.
3.4.1
Convergence
We start by having a look at the pictorial representation of the rate map. Figure 3.5 presents three plots of Equation (3.11), with each plot starting from a different min- imum video rate (qmin = 100kbps,2000kbps,12000kbps) but same maximum video
rate (and qmax = 8000kbps). The most important observable characteristic of the
a flatter slope, as the buffer occupancy increases the slope becomes steeper, halfway through the range of the video rates the slope began to flatten again. This pattern is same regardless of the starting video rate. Furthermore, we can observe that all the curves converge at the maximum video rate, though at different buffer positions.
To generalise this, the limit ofR() as buffer tends to infinity is taken, which results in limB→∞R() =qmax . Put differently,qmax is asymptotically reached, independent
of the initial value of the video rate (q0). In summary, regardless of the starting video
rate the maximum value thatR() will return, assuming an infinite buffer size, isqmax.
To find the minimum buffer size that guarantees this convergence, we solve R() =
qmax. It should be noted that any increase in the buffer size above the obtained value
does not result in any rise in video rate. Therefore, barring any other consideration by an algorithm designer, this can be considered asBmax.
3.4.2
Stability
Having seen the rate map will converge at the right video rate, next, we will try to find out if the converged point is stable. You may recall that stability is part of the user requirement. The equilibrium of the model is when dR()dB = 0. In other words when the system does not change its video rate. The result of equating Equation (3.6) to zero gives us two equilibrium points, q∗ = 0 and q∗ =qmax.
It is evident that when a client has not started requesting any video, it will stay in that state forever. However, it is interesting to investigate the behaviour of the model nearq∗ = 0. Since close toq∗ = 0 the buffer level is low. Whenq is very small,
αq2 is small compared to αqq
max. Therefore, equation (3.6) becomes dR()
dB ≈αqqmax.
We can infer from this equation that provided α > 0 any small perturbation in the system state will result in an exponential growth of the video rate away from the current video rate, hence resulting in an equilibrium that is unstable.
The second equilibrium point is q∗ = qmax. Again we are interested in what
happens near this point. Let us assume that
. When we substitute q=qmax+ into Equation 3.6, we get
dR()
dB =−αqmax−
2 (3.12)
However, if q is close to qmax, for all α >0 the 2 will be very small, therefore we
have dR()dB ≈ −αqmax. Thus, small perturbation will decay exponentially, reverting
to theqmax. Hence, the equilibrium q∗ =qmax is asymptotically stable.
3.4.3
Impact of the Evolution Constant on a Buffer
The constantα determines the speed of the video rate evolution. Figure 3.6 shows a plot derived from Equation 3.1 using different values ofα. As can be seen, an increase in the value ofαincreases the amplitude of the path, which represents the maximum rate at which the video will evolve. In other words, the higher the value of α the faster the system converges at the maximum video rate. And since the quicker the convergence, the higher the average video rate, seemingly, an increase in the value of
α is desirable. However, by changing the subject of the formula of Equation 3.10 to
α we get the following equation:
α = ln q(qmax−qmin) qmin(qmax−q) . 1 qmax(Bt−Bt0), (3.13) From Equation 3.13 we can infer that the value ofαonly depends on the allowable buffer size, providedq =qmax−, where the value of is very small compared to the
qmax. This is because all other variables are constants. Therefore, the larger the buffer
size, the smaller the value ofα, hence the longer the convergence time. But since the larger the buffer size, the more stable the system is, there is a trade-off between the average video rate and stability. Equation 3.13 gives the expression that calculates the exact value of α. The derived value optimises both the video rate stability and its average value. Any value above the one computed in Equation 3.13 will reduce video rate stability and increase video rate and vice versa.
Figure 3.6: Derived trajectory of video quality evolution