Page 200 - DCAP601_SIMULATION_AND_MODELING

Simulation and Modelling

Notes In Table 11.1 and Table 11.2 we compare for N = 10 and F(x) = 1 – e the estimators and
–x
confidence intervals for E(C SPTF  C LPTF ) when we do not, resp. do, use common random numbers.
max max
We conclude that in this example the use of common random numbers reduces the standard
deviation of the estimator and hence also the length of the confidence interval with a factor 5.

Table 11.1: Estimation of E(C SPTF  C LPTF ) without using common random numbers
max max

Table 11.2: Estimation of E(CSPTF ) using common random numbers

When we want to use common random numbers, the problem of synchronization can arise:
How can we achieve that the same random numbers are used for the generation of the same
random variables in the two systems?
In the previous example, this synchronization problem did not arise. However, to illustrate this
problem, consider the following situation. In a G/G/1 queueing system the server can work at
two different speeds, v and v . Aim of the simulation is to obtain an estimator for the difference
1 2
of the waiting times in the two situations. We want to use the same realizations of the interarrival
times and the sizes of the service requests in both systems (the service time is then given by the
sizes of the service request divided by the speed of the server). If we use the program of the
discrete event simulation of Section 3 of the G/G/1 queue, then we get the synchronization
problem because the order in which departure and arrival events take place depends on the
speed of the server. Hence, also the order in which interarrival times and sizes of service
requests are generated depend on the speed of the server.
The synchronization problem can be solved by one of the following two approaches:
1. Use separate random number streams for the different sequences of random variables
needed in the simulation.
2. Assure that the random variables are generated in exactly the same order in the two
systems.

Example: The G/G/1 queue, the first approach can be realized by using a separate
random number stream for the interarrival times and for the service requests. The second
approach can be realized by generating the service request of a customer already at the arrival
instant of the customer.

11.3.2 Antithetic Variables

The method of antithetic variables makes use of the fact that if U is uniformly distributed on
(0, 1) then so is 1 – U and furthermore U and 1 – U are negatively correlated.

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