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. The 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 plus hash H are equal to lambda H, zero, and one minus lambda H plus some factors which tend to zero once that 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 its intensity function lambda of T. Actually, all other elements of this formula remain the same. Moreover, we can define and 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 annotation 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 we'll write a capital letter M. This basically means that the arrival process is Poisson. Secondly, it can be deterministic, and in this case, we'll write a capital letter D. And thirdly, it can be from any distribution and in this case, we'll write letter G, which stands for general. We will assume that interarrival times are independent, identically distributed. Generally, this process can be only of these three types. The second character is used for the service times. And it can be also as 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 infinitive. Let me consider more precisely the system M, G, and Infinitive. According to our definition, this is a system where arrival processes are modeled by Poisson process. Also, we know that service times are modeled by some distribution which I also denote by G. And the infinitive means that the amount of zeros is infinite. It is any arrival, it's starting to be soft immediately after arrival. So no elements are ever queued. You may think that you have a call center and when you get new your calls that 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 might have the following picture. So the calls are arriving. According to some Poisson process N of T, and then let me fix some time moment torque, and consider two processes. The first process N1 of T. This process indicates the amount of arrivals who are still being served at time moment torque. And secondly, N2 of T is an amount of arrivals who are already completed before by torque. Let me consider more precisely this process N1 of T. If we consider the increment of this process, it is N1 T plus Delta minus N1 T. This increment in fact indicates how many new arrivals come into the system between T and T plus torque and how many arrivals from this amount are still being served. Therefore, the probability that N1 T plus delta minus N1 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 plus Delta, so this is N T plus delta minus N T is equal to one, multiplied by the probability that this customer is still being served at time torque. It is a probability that a certain variable Y which indicates the service time for this customer is larger than torque minus T. It can be some other terms which are the effect of other small O of torque. 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 torque minus 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 one minus G torque minus T plus small O of delta. These formulas are important and if you consider the probabilities of these different cycles 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 N1 is in fact non-homogeneous Poisson, because for this process, all formulas of this period are fulfilled. I mean all formulas was here, one, zero, and larger equals than two. Moreover, the process N1 is zero at zero and it possesses a property of having independent increments. Therefore, N1 is non-homogeneous process whose intensity function equals to lambda multiplied by one minus G torque minus T. And similar arguments work well also with the second process. And I guess this process is also non-homogeneous Poisson and the intensity function is lambda multiplied by G torque minus T. This is a very interesting observation, but what is more important and what is a little bit on logic is that these N1 and N2 are in fact independent. This seems to be very surprising because if we consider these two processes, condition on the process N T as in one completed as legitimizes second. But, nevertheless this is true and I would like to show this now.
