[MUSIC] For a digital network, we will like to have in addition to model which continues and discreet behavior that captures information being transmitted and restore. We may also like to have the property that the events are not necessarily happening periodically. In such a case, unless we know when the events are going to occur beforehand. Coming up with a model, we might end up with some nondeterminism. Because, again, we don't know when the events are going to occur. We don't know the separation like in the ADC or the AC model. So how can we handle that? The idea is to have a model now of a network Where the output of the network. Following the mechanism or the states that we used before. This will be now a of network VN. Let's say that there will be memory that is towards the information that is being transmitted and that's MN. Now, these events are probably going to be occurring at times, t1, t2, t3 where there's no necessarily relationship domestic relationship between, the time between each event. So this is the information and this is the digital network. And this continues. So our goal will be the following. Given. A sequence of. Communication events, And we are going to label that as ti, where this i belong to the naturals. Assume the existence of basically two numbers, one of the numbers will tell me how frequent these events could occur in the best situation. And the other numbers are going to tell me how long they need to wait for the events in the worst situation. So, these numbers will be two parameters Will be TN*, for network, and we're going to call this minimum and this maximum. And these numbers are having this relationship, where this one is larger than cells. So the minimum time is bounded away from itself. So given a sequence of communication events, assume the existence of this numbers, such that the time in between the event t2 And the event t1 or the time between the event at t3, and the event at t2, and in general the time between the event i plus 1, and the event i are bounded by these numbers in the following way. This is saying that, the time elapsed between two consecutive ribbons can not be larger than TN*max, and cannot be smaller than TN*min. So this are particular construction that allows to have some freedom. On this time that will elapse in between events. Notice that if TN*min = TN*max This is saying that the time elapse between events is equal to a constant, therefore we're back. To the model of a very C, ADC, or BAC models. But we won't insist on that being the case even though this is possible. But we will typically have a situation where my network will, in some cases, will send information very fast. In some cases because of bottlenecks or congestion, they will actually send it very late. But we need to know those times okay? So one way to model this type of behavior is to think about what we want to capture as a function of a decreasing timer, okay? So let me try to explain that here. Imagine that I grab this two times, so this is t1 and this is t2, okay? And then someone gives me these two numbers and if this times t2 and t1 satisfies these conditions, then it satisfies in particulary this. And if from these time t1 this is the value of T and *min. The length corresponds to its value and this other Correspond to TN*max. Then I can say from the picture that this inequality is being satisfied, okay? So let's say that these are the numbers that we are given. And let's say we're going to have a timer tau N that will trigger the events. I would like the events again to allow me to do this kind of secret conditions. Okay. So when tau N is equal to zero, again it's arbitrary, but it will become clear when we come up with a model. When tau n is equal 0, I will say that a communication event. Occurs. Okay? [COUGH] All right, so if tao n equal to zero equals to a communication event and this particular value here is an event time, therefore the timer will set zero here. And this was another event the timer will zero here. Similarly, if the minimum event time event occur I will have a tau N equal to zero here. Anything maximum tau N possible, which corresponds to waiting the longest occurs, I will have a communication event here, okay? So since we are looking at tau N equal to zero as the events, and I was saying that I would like to model this with a decreasing timer if my timer is satisfying the following dynamics. Instead of one equals to minus one, but you can see is that in order to reach this point, you should have initialized your timer at this particular value. And if you want to have a tao N equal to zero because of this different equation, you should have initialized your timer at this particular value right here. Since this builds a triangle and the slope of these is 45 degrees, now you have the 45 degrees. Then what happens is that this length is equal to this length. And this length is equal to this length. In a similar vein, if I have an event here that means that my timer has started somewhere like that, okay? So what is that we're arriving to? What we're arriving to is to the at theta for tau n. Essentially, what I need to do is in between events I need to count negative with time, and when tau n is equal to zero, a communication event occurs, and what I need to do is to reset my timer tau n to any value within this range. If the value that I pick is equal to the value of tau N*min, then the next event is going to occur according to this lower mount. If a value that I pick is equal to this other value up here, then the next event is going to occur in the worst case waiting time of tau N TN*max. So how do we capture this? The way that I capture this is by saying that at those event, tau N is reset to the range TN*min and TN*max. Okay? This now captures the mechanisms for the event. Similarly, what we can do now is in between events have the memory state and then to have zero derivtive and at every event have the memory state to be reset to the input of the network so these are my discreet dynamics. This here is my continuous dynamics. This occurs, there is great occurence when the timer is equal to zero while this other one corresponds to the other situation. Notice that I put here the close zero. I could have had put the open zero. But what's going to happen is when I reach zero the timer is trying to push tau n towards a negative value so it doesn't really matter if I close this inner value or not. But if you're more comfortable with the open for now you can keep it open. This difference equation with this constraint, and this differential equation with this constraint, build for us a model of a digital network that captures any sequence. So now this could be changed into any sequence that satisfies these conditions here for these particular numbers. And as we saw for the particular case of an ADC, the initial value of a timer will only affect the initial interval until the first event. And after that, it will be reset according to this mechanism and we will have events that occur according to this flow, okay? So initially you can have an event that occurs sooner than this if you actually initialize it very close to zero or even at zero. [SOUND]