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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