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public class Server { private Client client; private int id; private int stopTime = -1; private double meanServiceTime; private ExpRandom random; public Server(int id, double meanServiceTime) { this.id = id; this.meanServiceTime = meanServiceTime; this.random = new ExpRandom(meanServiceTime); } public double getMeanServiceTime() { return meanServiceTime; } public int getStopTime() { return stopTime; } public boolean isIdle() { return client == null; } public void startServing(Client client, int time) { this.client = client; this.client.setStartTime(time); this.stopTime = time + random.nextInt(); System.out.printf("%s started serving client %s at time %d.%n", this, client, time); }

CHAP. 6]

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public void stopServing(int time) { System.out.printf("%s stopped serving client %s at time %d.%n", this, client, time); client = null; } public String toString() { return "Server " + "ABCDEFGHIJKLMNOPQRSTUVWXYZ".charAt(id); } }

The startServing() method (lines 26 32) assigns a new client to the server, stores the start time in the Client object, computes and stores the stop time in its own stopTime field, and prints a report of those actions. The stopServing() method (lines 34 38) stores the stop time in the Client object and prints another report. For a simulation to be realistic, it must use randomly generated numbers to simulate the natural uncertainty of the real word. Those random numbers should have the same distribution as the natural uncertainties that they represent. Service times and time between client arrivals both tend to be distributed exponentially. That means that the probability that the time t is less than a number x is p = 1 e- x. But the Math.random() method returns numbers that are uniformly distributed in the range 0 p < 1. So to convert the random number p to the exponentially distributed random variable x, we solve the equation, obtaining x = (1/ ) ln(1 p). The constant 1/ is the mean of the distribution. Thus we code the nextDouble() method as shown at line 9:

productive capacity and there is no need to expand productive capacity when utilization of existing capacity is declining. c. Investment demand should shift downward. Increased vacancy rates for existing commercial buildings indicate that there will be dif culty selling newly constructed commercial real estate. Thus, commercial real estate construction will decline. Solved Problem 4.4 What is the difference between a lump-sum tax and an income tax Solution: A lump-sum tax is a xed-sum tax that is unrelated to income. An income tax is related directly to earned income. In the case of a proportional income tax the government collects a xed percent of income earned, while for a progressive income tax the rate of taxation increases with the level of income. Lump-sum taxes and proportional and progressive income taxes are illustrated in Figure 4-3. Note that lump-sum taxes remain at $1,000 as income increases from $10,000 to $11,000. When there is a 10 percent proportional income tax rate, tax payments increase from $1,000 to $1,100 as income increases from $10,000 to $11,000. When the tax rate is 10 percent on the rst $10,000 earned and 20 percent on income greater than $10,000, tax payments increase from $1,000 to $1,200 when income increases from $10,000 to $11,000.

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public class ExpRandom extends java.util.Random { private double mean; public ExpRandom(double mean) { this.mean = mean; } public double nextDouble() { return -mean*Math.log(1 - Math.random()); } public int nextInt() { return (int)Math.ceil(nextDouble()); } }

The actual simulation is performed by the main class shown below. It sets four constants for the simulation at lines 2 5: the number of servers, the number of clients arriving for service, the mean service time among the servers, and the mean time between arrivals for the clients. The queue is to hold the clients that have arrived for service and are waiting for an unoccupied server. The simulation instantiates two random exponentially distributed number generators (lines 7 8) and separate arrays for the Server and Client objects (lines 9 10):

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