System analysis has been widely used, by ecologists, to help describe the behaviour of various organisms under different environmental conditions such as an increase or a decrease in food supply [153]. Typically, scientists start by isolating the ‘the entities or parts which compose the system’, then define the relationship between these components. Usually, the target is to model the mechanism that controls the dynamics of the ecosystem. Definitely, this requires a clear description of what a state is, and what constitutes a change it. For example, with the derived models, scientists can predict the impact of different events on the population of various species of animals.
Specifically, population dynamics is a branch of mathematical ecology that models the change in the population of different species as well as the processes that influence the changes, e.g. the availability of resources and the presence of the competing species [154].
To build these population growth models, some assumptions are typically made [155]. First, the size of the population, at any given time, represents the state of the system. Secondly, the state is affected by the exogenous variables, such as the
availability of food or predators, and the previous state. For instance, the initial size of the population. Thirdly, the habitat can only support population up to its maximum carrying capacity (K). Fourthly, there is always a seed population that kicks start the process. The rate at which offspring are added is called thebirth rate
and the rate at which the animals die is referred to as the death rate. A plethora of these growth models exit in literature [156, 157, 158].
The rate at which population increases (r) is dependent of the birth rate (b) and the death rate (d). Therefore, r = b−d. Let assume the seed population is N0, so
that rate of change of the population is:
dN
dt =rN0
.
However, the third assumption tells us that population growth is density depen- dent. In other words, the density of a population regulates its growth. This is basically because of the competition in the scarce resource. So the density depen- dence must reflect this fact. For example, (K−NN), which measure the ration of the current population to the maximum carrying capacity of the habitat, or (K −N) that measure the difference between the current capacity and the maximum carrying capacity, respectively, giving us either
dN dt =rN0 K−N N (3.14) or dN dt =rN0[K−N] (3.15)
The solutions to these differential equations give ecologists predictive models that help tackle problems, such as over fishing and saving animals that are on the verge ex- tinction. But a look at these equations (Equation 3.14 and 3.15) shows a remarkable similarity with Equation 3.11. In fact, Equation 3.11 and 3.15 are exactly the same
equation with different variables. WhereN0, K and r represent minimum, maximum
and population evolution constant in Equation 3.15, qmin, qmax, α represent the min-
imum, maximum and video rate evolution constant This inspired us to reformulate the problem of video rate adaptation.
In this context, let us assume that the video rate is the species whose growth we are interested in. Furthermore, we suppose that the playback buffer is its habitat. Next, the rate at which the video rate of the incoming chunks changes is considered the birth rate, and the rate at which a player consumes content from the playback buffer is assumed to be the death rate. Recall from Equation (3.3) that the rate of content arrival is c(ti)q , which is considered to the birth rate. Furthermore, a player consumes content at a constant rate, precisely, one second of content is consumed every wall-clock second. Hence, a constant death rate. Assuming like in the natural habitat there are enough resources to sustain video rate up to the maximum video rate, that is c(ti) > qmax. Therefore, the rate at which the buffer is filled (α) will
be α = c(ti)q −1. However, like in anatural context, there is a seed video rate that
reproduces at the rate of α to kick-start the growth. Hence,
dq
db =αq0.
Increase in video rate, just like the population is density dependent, the higher the video rate, the less we are inclined to increase, because throughput is not unlimited, and users are keen on an increase when video rate is relativity high. Therefore, a density dependence factor is needed, which will force the growth rate in the video rate to decrease as the maximum buffer level is approached. For example, (qmaxq−q), which measure the ration of the current video rate to the maximum available video rate, or (qmax −q) that measures the difference between the current video rate and
the maximum video rate can be used. These respectively give us either
dq db =αq0 qmax−q q (3.16) or
dq
db =αq0[qmax−q] (3.17)
Clearly, Equation 3.11 and 3.17 are exactly alike. Just as biologists solve the differential equation (Equation 3.14 or 3.15) to get the predictive model used for various ecological studies, we can solve either Equation 3.16 or 3.17 to get a predictive model that can be used to predict the video rate to request, given any buffer size less than the maximum value. As in a natural habitat, the buffer size will determine a limit of the maximum video rate (qmax) a player can download. In this case, unlike in
the wild, the maximum video rate is given (as defined in the MPD). Therefore, the task is mainly focused on finding the amount of buffer space required to guarantee the maximum video rate.
As seen in Equation 3.16 and 3.15, one of the advantages of the bio-inspired for- mulation is that some predictive models can be constructed with the same behaviour without much overhead. However, the one that exactly matches Equation (3.11 ) is the Verhulst-Pearl equation [156], perhaps the most well-known population growth model. Furthermore, if the streaming context changes, such that we are forced to modify the some of the assumptions made in deriving Equation 3.11, with the bio- inspired formulation there is no need to start modelling from the beginning all over again. For example, imagine we intend to develop an ABR that strictly serves users with a stable network and capacity the is a least twice the maximum video rate, and using the smart TV. In this scenario, we may what to speed up the convergence, with our current model that can only be done by using a large value of α, but recall this decreases the amount the buffer needed. And since buffer size is not an issue here, this may not be the most optimal solution. A better solution will be to change the point of inflexion of the curve, that is, the point which the rate of video rate increase begins to slow down. With bio-inspired formulation it is easy to try other growth curves, such as Gompertz model [158] with the point of inflexion at qmaxe .