计算机代写|密码学与网络安全代写cryptography and network security代考|Stationary Data Flow

Several properties related to stationary flow packets or messages can be derived regardless of the distributions of the parameters that define the flow. In the analysis of the data flow, it is assumed that the network conserves the messages, in the sense that they cannot be created, destroyed, or modified by the network.

Messages can only flow into or out of outside the perimeter of the network, or can remain stored for a certain time on the network. The time spent in the network corresponds to the sum of the processing times of the packets by the servers. On each server, the message needs to be opened, eventually corrected, and then routed to the next destination.

If the average rate of entry into the network exceeds the rate of exit, the number of stored messages is constantly increasing. On the other hand, if the average rate is higher than the input rate, the number of stored messages decreases to zero.

From these considerations, one concluded that, for a stable operation of network in the steady state, the input and output rates must be equivalent.
Consider $\alpha(t)$ as the number of incoming packages and $\delta(t)$ the number of network outgoing packets in a certain time interval $(0, t)$. The difference between these quantities, $N(t)$, represents the increase in the number of messages stored on the network in the given interval
$$N(t)=\alpha(t)-\delta(t) .$$
The input rate, in a time interval $t$, is defined as
$$\lambda_t=\frac{\alpha(t)}{t} .$$
Another measure of interest is the total time that all messages spend on the network
$$\gamma(t)=\int_0^t N(x) d x=\int_0^t \alpha(x)-\delta(x) d x .$$
Proceeding in this way, it is possible to find the average number of messages, $N_t$, on the network in the range $(0, t)$
$$N_t=\frac{1}{t} \int_0^t N(x) d x=\frac{\gamma(t)}{t} .$$

计算机代写|密码学与网络安全代写cryptography and network security代考|Markov Model

The Markov model shown in Figure $6.5$ serves as a basis for the analysis traffic on computer networks. A discrete-time Markov chain is used to model a process stochastic set in honor of mathematician Andrei Andreyevich Markov. For this chain, the states prior to the current one are not relevant to the prediction of future states, as long as the current state is known.

Known as the model of birth and death, transitions occur only between states adjacent. For example, from the state $k$, you can go only to $k+1$ or $k-1$ with some probability. This reflects the fact that the likelihood of more than one user entering the system at the same time is negligible. Using the model, it is possible to calculate the steady state probabilities (Kleinrock, 1975).

The transition matrix probabilities $\mathbf{P}=\left{p_{i j}\right}=\left{p\left(y_j \mid x_i\right)\right}$ defines the dynamics of the model. The transition probabilities are obtained from the Markov model, in which $\lambda_k$ and $\mu_k$ are the birth and death parameters.

The Markov chain reaches a steady state after a certain number of iterations. The probabilities of the steady state, $\Pi=\left{p i_k \mid k=1,2,3 \ldots\right}$, can be calculated using one of the known techniques (Kleinrock, 1975; Adke and Manjunath, 1984). Each $k$ state defines the number of users, packages, or other objects in the system.

Two cases of application of the Markov chain are presented. Initially, the problem arises when the birth and death parameters are constant for any state. It can be determined that $\lambda_k=\lambda$ and $\mu_k=\mu$.

The first case, which illustrates the operation of a traditional computer network, connected by wires or cables, which operates with defined flow parameters, produces a geometric distribution for the probabilities
$$p_k=(1-\rho) \rho^k k=0,1,2, \ldots, \text { for } \rho<1,$$
in which $\rho=\lambda / \mu$ is usually called the use of the system. Figure $6.6$ illustrates the geometric probability distribution as a function of the state of the $k$ system.

For the geometric distribution, the statistical average is given by $\rho /(1-\rho)$ and the variance by $\rho /(1-\rho)^2$. The probability of finding more than $L$ users at a given time in the system is $\rho^{L+1}$.

密码学与网络安全代考

计算机代写|密码学与网络安全代写cryptography and network security代考|Stationary Data Flow

$$N(t)=\alpha(t)-\delta(t) .$$

$$\lambda_t=\frac{\alpha(t)}{t} .$$

$$\gamma(t)=\int_0^t N(x) d x=\int_0^t \alpha(x)-\delta(x) d x .$$

$$N_t=\frac{1}{t} \int_0^t N(x) d x=\frac{\gamma(t)}{t} .$$ 范围内的网络上

计算机代写|密码学与网络安全代写密码与网络安全代考|马尔可夫模型

$$p_k=(1-\rho) \rho^k k=0,1,2, \ldots, \text { for } \rho<1,$$

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