Method Based on Discarding Observations Gath ered During the Initial Transient Period

Since observations gathered during the initial transient periods do not char- acterise the steady state, a natural idea is to discard all such observations before further analysis. This requires an estimation of the eective length of the initial transient period. Ignoring the existence of this period can lead to a signicant bias of the nal results. Although, the removal of any observa- tions increases the variance of estimates, which in turn can increase the value of the mean-square error (cf. [DONN81], [TURN77], [WILS78] ), the bias of estimates is generally regarded as the more important factor when consider- ing the quality of estimates, and it justies our decision of implementing an ecient procedure for detecting the length of initial transient periods in the Control Module of AKAROA.

Basic problems related with the existence of initial transient periods and the detection of their lengths are discussed in [PAWL90]. There have been two methods for automatic detection of the length of initial transient period proposed: one proposed by Pawlikowski [PAWL90], and another one pro- posed by Jackway and deSilva [JACK92]. The later is a modied version of the former. No comparative studies of these two methodologies have been done yet. Basing on our own experience, the former has been selected to be implemented in the Control module of AKAROA. It follows the following steps:

A rough, rst approximation of the number nd(0) of initial observations

that should be discarded is obtained applying one of heuristic rules of thumb surveyed in [PAWL90]; namely the rule R5 of that survey:

the initial transient period is over after no observations if the series of observationsx1,x2, ...,xn

0 crosses the meanXn0 k times

.

This rule is sensitive to the value of k, see [GAFA78]. Too large a value will usually lead to an overestimated value of n0 regardless of system utilization,

while too small a value can result in an underestimated n0 in more heavily

loaded systems. Results of our studies conrmed the justness of selecting k=25, recommended in [GAFA78].

Following the rst rough selection of the truncation point for initial data, the length of the initial transient period is more precisely determined se- quentially, by applying one of statistical tests proposed in [SCRU82] and [SCRU83] for testing (un)stationarity of collected observations. These tests are based on the high sensitivity of the sequence of partial sums

Sk =X(n),X(k) ; (2.35)

(k= 0, 1, 2,..., and S0=Sn=0) to the presence of initialization bias in Xn ;

whereXnandXk are means overn and k rst observations, respectively, see

Eqn. 2.1. Following this phenomenon, the tests analyse the convergence of a standardized sequencefT(t)g, 0t1, to the Brownian bridge process with

zero mean and variance equal 1. The sequence fT(t)g is the standardized

sequence of the partial sums Sk, namely

T(t) =bntcS bntc

1

p

x^[X(n)] (2.36)

for 0< t1 (wherebxc denotes the greatest integer not greater than x, and

T(0) = 0. Rejection or acceptance of the hypothesis that a given subsequence of observations is stationary, or equivalently, that the initial transient period is not included in the observations, depends on the probability characteriz- ing the scalar value calculated from the considered sequence. Despite the

sophisticated theory hidden behind these tests they appear to be quite sim- ple numerically, and can be applied to a wide class of simulated processes. A practical problem faced when implementing one of these tests is that they require a priori knowledge of the steady-state variance [X(n)] of the simu- lated process. As proposed in [PAWL90], it has been solved in AKAROA by estimating this variance from a sequence of observations collected some time after the assumed supposed-to-be truncation point, assuming that then the process is at least closer to its steady state.

A logical description of the sequential version of the stationarity test ap- plied in AKAROA is given in below in its pseudo-code form. It should be noted that the actual stationarity test algorithm implemented in the Detec- tInitialTransient object (C++ class) of AKAROA has been inverted from the description given in Fig. 2.1 and reassembled, so that the "data genera- tion process", and the "detect initial transient" are separated, and mutually hidden. Thus the "data generation process" interacts with the "detect initial transient" process only through a well dened interface, and the implemen- tation of either can be changed without aecting the other, so long as they perform their respective tasks.

Figure 2.1: A logical description of the sequential version of the stationar- ity test applied in AKAROA. Note that to separate the "data generation process", from the "detect initial transient", the actual stationarity test al- gorithm implemented in the DetectInitialTransient object (C++ class) of AKAROA has been inverted from the procedure specied here

The structure of the objects for steady state point and interval estima- tion based on the NOBM, OBM, IR, and SA/HW methodologies selected in this Chapter as candidates for implementation in the Control module of AKAROA, and the object for detecting when the eects of the initial tran- sient are insignicant, as well as a comparison of their performance, will be considered in Chapter 3.

Chapter 3