Monte Carlo Simulation

The most basic simulation methodology is known as Monte Carlo simulation. The Monte Carlo method is a game of chance (hence the name) in which a random experiment is replicated a number of times. Every Monte Carlo simulation program contains two counters, the first of which is a replication counter that is incremented by one each time the random experiment is replicated. An event of interest is identified and the second counter is incremented by one each time the event of interest is observed.

To place this in the communications context, assume that the simulation is being performed in order to estimate the symbol error probability of a communications system. In this situation the underlying random experiment is the passing of a digital symbol through a communications system. This is a random experiment because channel noise, interference, or other random disturbances may, or may not, cause a transmitted symbol to be received in error. Since we are executing the simulation to estimate the symbol error probability, the event of interest is a symbol being received in error. Thus, each time a symbol is passed through the system in the simulation, the replication counter is incremented by one. Each time a transmitted symbol is received in error the event counter is incremented by one. The simulation is assumed to process N symbols. After N symbols have been passed through the system, we calculate the relative frequency of symbol error. This relative frequency is an estimate of the symbol error probability and is given by

where NE is the number of symbol errors observed in passing the N total symbols through the simulation. The probability of symbol error is given by

Since a simulation must execute in finite time in order to be useful, the underlying random experiment can only be replicated a finite number of times. As a result, the simulation is unable to determine the probability of symbol error. The best we can do is to estimate the probability through the relative frequency.

This immediately gives rise to an important question. How large should N be for the relative frequency to be a valid estimate of a probability? Ultimately the question of required accuracy depends on the application. However, in general, we have interest in the following:

• We wish the relative frequency to be a unbiased estimator. In other words, the statistical estimation of the relative frequency should be equal to the probability. In equation form

• where E denotes statistical expectation. Another way of expressing this is that on average we obtain the correct result.
   

• We wish the estimator to be consistent. For a consistent estimator, the variance of the estimator goes to zero as N goes to infinity.

The advantages of the Monte Carlo method are that it is very flexible. The Monte Carlo method can be easily applied to any system in which the model for each functional block in the system block diagram is known in sufficient detail. The fundamental disadvantage of the Monte Carlo method is that the required simulation run times are often impractically long if statistically valid results are to be obtained.