Next I would like to provide some basic elements of the so-called Queuing theory. As I already mentioned, for a homogeneous Poisson process the following three equalities are fulfilled. These equalities tell us that the probability that only one, more than one, or zero events occur during some infinitely small interval from t to t+h are equal to lambda h, zero, and one minus lambda h, plus some factors which tend to zero when we divide them by h. Actually, the same formulas hold also for non-homogeneous Poisson process. In fact, in this case, this lambda should be changed to the intensity function lambda(t), and actually, all other elements of these formulae remain the same. Moreover, we can define a non-homogeneous Poisson process as a process which starts from zero, has independent increments and these properties are fulfilled. Homogeneous and non-homogeneous Poisson processes are widely used for describing the queuing systems. The most popular notation for different types of queue is the following. So this notation is based on three characters which are separated by slashes. The first character means the arrival process. It can be of the following types. First of all, it can be memoryless, and in this case we will write a capital letter M. This basically means that the arrival process is Poisson. Secondly, it can be deterministic, and in this case we will write a capital letter D. And thirdly, it can be from any distribution, and in this case we will write a letter G, which stands for "general". We will assume that inter-arrival times are independent, identically distributed, and generally, this process can be only of these three types. The second character is used for the service times. And it can be also either M or D or G. And as for the third character, this is a number of servers. It can be either a positive integer value or infinity. Let me consider more precisely the system M/G/infinity. According to our definition, this is a system where arrival processes are modelled by Poisson process. Also we know that service times are modelled by some distribution which I will also denote by G. And infinity means that the amount of servers is infinite, that is, any arrival is starting to be served immediately after arrival, so, no elements are ever queued. You may think that you will have a call centre, and when you get a new call, then one operator is starting to work with this call. Of course, this system is a bit unrealistic, but it is very useful for showing the most important issues of the queuing theory. You may have the following picture. So the calls are arriving according to some Poisson process N(t), and then let me fix some time moment tau and consider two processes. The first process N_1(t). This process indicates the amount of arrivals who are still being served at time moment tau. And secondly, N_2(t) is an amount of arrivals who are already completed before, or by, tau. Let me consider more precisely this process N_1(t). If we consider the increments of this process, that is, N_1(t+delta) minus N_1(t), this increment in fact indicates how many new arrivals come into the system between t and t+tau, and how many arrivals from this amount are still being served. Therefore, the probability that N_1(t+delta) minus N_1(t) is equal to some number, let me write here one, is in fact equal to the product of two probabilities. The first probability is that one customer arrives between t and t+delta, that is, N(t+delta) minus N(t) is equal to one, multiplied by the probability that this customer is still being served at time tau, that is, the probability that the random variable Y which indicates the service time for this customer is larger than tau - t. It can be some other terms which are in fact of order small o(tau). The first term is equal to delta lambda plus some terms which have smaller rate of convergence to zero, and the second item is equal to one minus G, and the argument is tau - t, plus maybe some terms which have smaller order of convergence to zero. To sum up, we get that this probability is equal to delta lambda (1-G(tau - t)) plus small o(delta). This formula is rather important, and if you consider the probability that this difference is equal to zero or larger than two, you will get very similar expressions. Finally, you may think about this in the following way. In fact, we have a homogeneous Poisson process which is splitted into two subprocesses. The first subprocess N_1 is in fact a non-homogeneous Poisson, because for this process all formulas of this spirit are fulfilled, I mean all formulas with here one, zero, and larger or equal than two. And moreover, the process N_1 is zero at zero, and it possesses the property of having independent increments. Therefore, N_1 is a non-homogeneous process with intensity function equal to lambda multiplied by (1-G(tau - t)). And similar arguments work well also with the second process. And we get that this process is also non-homogeneous Poisson, and the intensity function is lambda multiplied by G(tau - t). This is a very interesting observation, but what is more important and what is a little bit unlogic is that these N_1 and N_2 are in fact independent. This seems to be very surprising because if we consider these two processes, conditioned on the process N_t, then one completely determines the second. But nevertheless, this is true, and I would like to show this now.
