Bernoulli RP - Poisson RP - Markov RP
Functions that depend on time as well as on one or more random variables serve as models for signals that convey information in various situations. Because of this, it is necessary in many design studies to develop mathematical and computational models of these processes. This allows us to develop tractable analytical problems and to pursue simulation studies of system performance. Such studies span the majority of subdisciplines within Electrical and Computer engineering ranging from the design of automatic control systems to the design of server networks in advanced computer systems.
Our goal for the next few weeks will be study three processes; the Bernoulli Random Process, the Poisson Random Process and the Markov Random Process. If time permits, we will also look at the simulation of the Gaussian or normal random process although we will focus on this process more extensively as we study the next chapter of the textbook.
The Bernoulli Random Process (a.k.a. Binary White Noise) provides a relatively simple introduction to our current topic and we shall use it to "wet our appetite" for the complexities to follow. This type of model is utilized to represent certain types of data in telecommunication systems such as Pulse Code Modulated signals where the transition betwween pulses occurs as discrete, uniformly spaced time intervals.
This random process falls into the class of discrete random sequences. The member functions of the random process, X(nT) assume only one of two values, typically +1 or -1. The probability of +1 is denoted as p, and the probability if -1 is denoted as q=1-p. (i.e. P(1)=p, P(-1)=1-p) When p=q=0.5, the process is known as binary white noise.
From a computational perspective it is easy to model. If we can generate a uniformly distributed random variable, then we can "switch" a random signal between +1 or -1 according to the magnitude of the uniform RV. If the RV is less than q, we make the signal values -1, if it is greater than q, we make the signal value +1.
The MATLAB code below (available on the ftp site as bernproc.m) accomplishes this task.
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The first trace here shows a uniformly distributed random variable over the interval [0,1]. The second trace shows a shifted version of this data so that P(X<0)=q and the P(X>0)=p.
The third trace is a member function of the Bernoulli random process with p=0.4 in this case. This means that, if we generate many more member functions for the process and sample the functions at any time t = t1, then the probability that the value of the member functions is +1 will be 0.4, and the probability that the value of the member functions will be -1 is 0.6. If we select a different time t = t2, this will still be the case. Since we would expect this to be true for all t1 and t2, the random process here is stationary. Note that we can generate another realization of this process simply by running the m-file again. The randomness of the simulation is developed by using the unfrm random number generator in MATLAB.
The Poisson random variable is one example of a discrete random variable. The random variable takes on integer values greater than or equal to zero. It is related to several other random variables such as the binomial RV and the m-Erlang RV.
Essentially, the Poisson RV models experiments or events for which there are only two possible outcomes. We can think of this in terms of the success or failure of a particular outcome, or in terms of the occurence of a distinct event or the lack thereof. The Poisson probability distribution, the probability density function and the cumulative probability distribution functions are then expressed as,
Modeling the pdf in MATLAB is a relatively straightforward task. We simply select a range of integer values for X beginning with x=0, set the variable parameter, alpha, to some positive real value, and then calculate the discrete probabilities for each value of X. The following m-file (available on ftp as poisrv.m) models the pdf and the cumulative proability density function.
The mean and variance of the Poisson random variable are determined in the usual way. The expectation operation involves an infinite sum, similar to that encountered in the corresponding calculations with the geometric RV we examined earlier. The essential results are,
The Poisson random process, as you might expect, is related to the Poisson random variable. It is typically used to models situations in which we expect a finite number of occurences of a particular event within some specified time interval. The practical examples described in the text include an analysis of a data communication problem for a PC or workstation, and an analysis of a telecommunication problem involving a packet transmission system. We will pursue these problems as our next homework assignment.
A Poisson Random Process, X(t), can de defined as follows. Let X(t) be the number of events in the time interval [0,t] for t > 0. Let i denote the number of events in the interval (the Poisson RV) and X(0) = 0. The number of calls in the interval [0,t] is described by a Poisson probability density function. We let the process be memoryless so that it does not matter when t begins or when we beging to count the number of events.
The use of the Poisson pdf to represent problems of this type is based on the following mathematical arguments.
Let the number of occurences of the event in the time interval [0,t] be the random variable, X. (Here i will denote the specific value of X).
Let t and dt denote two contiguous time intevals and let these intervals be independent. That is the occurnce of X in one interval is jointly independent of the occurence of X in any other interval.
Let the parameter lambda, be the mean rate of occurence of the event of interest. We can then make the following statements about the time interval dt.
We develop the model of the random process now by determining the probability that we obtain i calls in the time interval [0,t+dt]. This must be the algrabric sum of the probabilities of two possible combined events.
We have (i-1) events in the time interval [0,t] and we then have ONE new event in the interval dt. Or,
We already have (i) events in the time interval [0,t] and we have NO events in the interval dt.
Mathematically, this is expressed as,
We can solve this equation by rearranging it into a more recognizable, differential form as,
We can solve this integration problem using integration by parts. Applying the initial conditons that PX(i=0,t=0)=1 and PX(i,t=0)=0 for i>0, we arrive at the following random process model,
Thus, if the mean rate of events (lambda) is known, the the equation above specifies the probability that we have i events in some time interval [0,t].
What do we need to specify?
lambda, the mean rate of occurence of the event
the time interval (or intervals) of interest
the length (total time) of the simulation
Example 5.16. Providing some numbers to work with.
The mean rate of events is fixed at lambda = 2.3
The time intervals are set as follows [0, 2.3], [2.3, 4], [4.0, 6.8] and [6.8, 10]. Note that the time intervals are contiguous and do not overlap. Also note that there is no requirement for the time intervals to be equal.
For illustrative purposes, to generate our first member function, we'll assume that the number of events that occur in each time interval correspond to the value of i that has the maximum probability for this value of lambda and time. This is one possible member function. This would not be the case for every member function of course and we'll generalize the situation later. It does however correspond to the example problem in the text book. These results are as follows:
How do we construct the model?
Define variable parameters
Compute pdf for each interval
Select x-value (or i) corresponding to max_value(pdf)
Plot the results
Let's do it!
How should we interpret the results?
This is ONLY one possible member function
If we fix t = t1, then the #events(t1) for this member function, along with other member functions, would need to possess a Poisson distribution
To exactly match the results in the text we would need to add the "fix" of making the number of events for t < 1 equal to zero
Some questions for you to ponder
What is the effect of a change in the mean rate of events?
What is the effect of making the time intervals uniform?
Is it appropriate to consider this realization of the process as a "worst case" example for a particular group of problems?
Is the process stationary or nonstationary? How would you explain your answer?
How might you construct other member functions of the random process?
How might you verify that the statistics of the process approximate that of a Poisson random variable?