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Use of a Computer. Many computer packages are available for performing least-squares regression, such as BMDP, Minitab, SAS, and SPSS. Or, in a computer language such as APL, GAUSS, or SASjIML, in which matrix manipulation is straightforward, it is not hard to write your own program for least-squares regression, using formulas (3.8) and (3.13). For calculating SSR full , note that e = y - XjJ and SSR fuli = e'e. This also works for calculating SSRreduced if X is reduced to Xreduced by omitting the columns corresponding to the coefficients set equal to 0 in the null hypothesis. New vectors jJ and e are calculated based on Xreduced and then SSRreduced = e'e. In the case q = 0, SSRreduced is simply E(y; - ji)2. Test Case. To try out a computational procedure for least-squares regression on your calculator or computer, the data set in Table 3.2 can serve as a quick test case. For simple regression, let X = Xl. The estimated regression line is Y = 33.16 + 1.480X. The estimate of 0' is a = 3.552 and the test statistic for testing f3 = 0 is It I = 2.946. For multiple regression, using both Xl and X 2 as explanatory variables, the estimated regression equation is Y = 51.30 + 0.6025X l - 0.6180X 2 The estimate of 0' is a = 3.328 and the test statistic for testing f3l = f32 = 0 is F = 5.722.

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of k tellers is available Customer arrival is governed by a probability distribution function, as is the service time (the amount of time to be served once a teller becomes available) We are interested in statistics such as how long on average a customer has to wait and what percentage of the time tellers are actually servicing requests (If there are too many tellers, some will not do anything for long periods)

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With certain probability distributions and values of k, we can compute these answers exactly However, as k gets larger the analysis becomes considerably more difficult and the use of a computer to simulate the operation of the bank is extremely helpful In this way, bank officers can determine how many tellers are needed to ensure reasonably smooth service Most simulations require a thorough knowledge of probability, statistics, and queueing theory

--3dE'dw'mw'(m - 2w'). 2'17

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A discrete event simulation consists of processing events Here, the two events are (1 ) a customer arriving and (2) a customer departing, thus freeing up a teller We can use a probability function to generate an input stream consisting of ordered pairs of arrival and service time for each customer, sorted by arrival time' We do not need to use the exact time of day Rather, we can use a quantum unit, referred to as a tick In a discrete time-driven simulation we might start a simulation clock at zero ticks and advance the clock one tick at a time, checking to see whether an event occurs If so, we process the event(s) and compile statistics When no customers are left in the input stream and all the tellers are free, the simulation is over The problem with this simulation strategy is that its running time does not depend on the number of customers or events (there are two events per customer in this case) Rather, it depends on the number of ticks, which is not really part of the input To show why this condition is important, let us change the clock units to microticks and multiply all the times in the input by 1,000,000 The simulation would then take 1,000,000 times longer The key to avoiding this problem is to advance the clock to the next event time at each stage, called an event-driven simulation, which is conceptually easy to do At any point, the next event that can occur is either the arrival of the next customer in the input stream or the departure of one of the customers from a teller's station All the times at which the events will happen are available, so we just need to find the event that happens soonest and process that event (setting the current time to the time that the event occurs) If the event is a departure, processing includes gathering statistics for the departing customer and checking the line (queue) to determine whether another customer is waiting If so, we add that customer, process whatever.

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A discrete time-driven simulation processes each unit of time consecutively It is inappropriate if the interval between successive events is large

3 The probability function generates interarrival times (times between arrivals), thus guaranteeing that arrivals are generated chronologically

6 6 9

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