THE QUEUE PROBLEM

There is queue everywhere. The management must find weather the line is growing or decreasing. There is simple formula.

  P = at
    'a' is the average new persons joining the queue in each hour. 't' is the time taken to serve each person. If p is smaller than 1, everyone in the queue will be served, but if it is greater than 1, the queue will grow without limit. This makes sense: if the average time to serve a customer is longer than the average time it takes for the next customer to arrive, you cannot expect the queue to decrease!
    Let us see an example. Let 10 customers join the line every hour. Let 5 minutes (1/12 of an hour) be the average time required to serve a customer.

Then p = 10*1/12 =5/6 =0.833 meaning that the queue will not grow out of control.
    Also, the average length of queue is p^2/2(1-p)
= (5/6) ^2/2(1-5/6)
= 25/12 =2...
    That means, there will be always 2 people waiting in the line.
    The average wait time is p*t/2(1-p)
                                      = 5/6*1/12/2(1-5/6)
                                      = 5/24 hours or 12.5 minutes.
    On the average, every customer has to wait 12.5 minutes.  The p gives so much information.  Hence it is called 'utilization factor'.
     When you are in a supermarket, it is difficult to decide which checkout to pick.  There are many factors to consider,
1. How many people are in each queue?
2. How much shopping does each person in the queue have?
3. How talkative is the cashier?
4. Will a new checkout be open?
    But there is a simple and elegant solution. One need not worry about so many factors. Rather than having 10 distinct lines, have one big line (snake). A sign will direct the next person in line to the first available cashier. Even though, the ‘snake' looks long, it will move extremely fast. In traditional layout of 10 counters, and 10 lines, if there is a delay at one counter, it holds up everyone behind them slightly. But in 'snake' system, there are still 9 other cashiers getting people through quickly. If you join snake queue before someone named 'x', you will be served before 'x'. 'snake' treats everyone fairly. That is why, this system is followed in airports. A queue system need not irritate the people, it can be pleasant one also.

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LISSAJOUS FIGURES

Definition: "When a particle is subjected to two sine wave motion or two oscillatory motion at right angles, the particle describes lissajous figures". We know sine wave motion and circular motion is basically same. Hence we draw two circles A and B perpendicular to each other. The circle B rotates twice faster than circle A. That is, frequency of circle B is two times than that of A. A particle at the intersection of two circles is subjected to two sine wave motion A and B at 90 degree simultaneously. The particle will describe figures depending on the frequency and phase of A and B . In our case, the ratio of frequency is 1:2 and the two waves are in phase. To draw lissajous figures : A moving point in both the circles are chosen. Here we should remember; during the time taken by the circle A to complete one rotation, circle B completes two. Hence the points are marked on the circles according to their speed. Then straight lines

THE PARABOLA

A jet of water shooting from a hose pipe will follow a parabolic path. What is the so special about parabola. Y= x^2 Draw a graph for the above equation. It will result in a parabola. This parabola is also called unit parabola. Any equation involving square will yield a parabola. Example: Y = 2x^2 +3x+3 (also called quadratic equation) X= 2 and -2, both satisfies the equation 4 = X^2. Parabolic equations always have two solutions. Any motion taking place freely under gravity follows parabolic path. Examples: An object dropped from a moving train, A bomb dropped from flying plane, A ball kicked upwards. If a beam of light rays fall on the parabolic shaped mirror, they will be reflected and brought to focus on a point. This fact is made use of in Dish Antenna, Telescope mirrors, etc. Inverted parabola shape is used in the construction of buildings and bridges. Because the shape is able to bear more weight. A plane

CASINO'S GAME

Let us find out how the casino survives with mathematics. Say, your friend invite you for a game of dice. You must bet (wager) 2 dollars. If you roll 'six' you will get back 8 dollars. The game will go on for 30 rounds. All sounds good. The probability of rolling 'six' is 1/6. Since the game will be played for 30 times, the 'expected win' is 30*1/6 = 5. That is, you are expected to win 5 rounds out of 30. Hence your gain will be 5 * 8 =40 dollars. ok. This also implies that you will loose 25 rounds. Hence your loss will be 25*2 =50 dollars. Your net gain will be gain-less = 40-50 = -10 dollars. For 30 rounds, the loss is -10 dollars, Hence, for one round =-10/30 = -1/3 dollars. There will be a loss of -1/3 or 0.33 dollars per round. It is not a fair game. Let us make a simple formula to calculate 'Pay out per round\. The probability for a win = p The pay-out in case of win = V No. of rounds = n The expect

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