Here is some mathematical formula to be shared, taken from “Five-minute mathematics” by Ehrhard Behrends. I found it quite useful in estimating the number of customers or visitors.


One of the chapters talks about the Queuing Theory, which is the branch of probability theory.

Have you ever had the feeling that the other lines at the post office or supermarket always move faster than the one you are standing in? Imagine there are 5 lines in post office, every line has approximately the same length and you have to pick one to queue. The probability that you will randomly choose the one that can move fastest is ⅕, which is 20%. Oppositely, you have 80% to wait at the slower line.

This concept of waiting in lines has been under investigation for a long time, “queuing theory” is one of the classical subfields of probability theory. Once you can understand completely about the theory, that understanding can be applied to describe a typical result. Imagine a business that serves the general public: customers arrive, are served, and then leave, it can be a restaurant, shop, we can even think of the visitors to a museum or tourists spot that attract tourists.

(Lots of us in this project in are dealing with customers and visitors, so I think the theory might be helpful in designing the space according to number of customers.)

Few assumptions have been made:
1.The customers arrive at random and individually.
The “random” means that one will not know when the next customer will arrive precisely, only there is a certain average interval between arrivals (technical terminology= exponentially distributed arrival times). The “individually” means no customer arrived in group. So, we assume, on average, a customer arrives every K seconds.

2.When a customer enters the “store”, he or she will be served immediately.
The customer will be served immediately so we can deduct the queuing time of customer. So, the time that the customer spends in the store, we can denote here by L= the average number of seconds that it takes to serve a customer.

The parameters K and L can be change, depend on the situation.
For example, an exhibition hall, we can denote
K= the numbers of visitors
L= the time that visitors spend on average viewing the exhibition
(can also be a measure of the attractiveness of the site)

The problem is to predict the number of customers present at any one time. A large K and small L indicate that on the average there will be a few customers present at any given time.

Such predictions are possible using probability theory.

Let
K = the average time for customer arrives in seconds
L = the time customer stay at the store
λ = L/K
= the number of customers present in the store on average at any given time.

The probability that at a particular moment exactly n number of customers is present is given by the number where n! = the abbreviation for the product 1, 2 … n, n factorial
e = 2.718…, Euler’s number

Example:
Suppose on average, a customer arrives every 60seconds and stay an average of 2 minutes.

K = 60
L = 120
λ = 120/60 = 2

We can calculate the probability that at any given time there will be exactly n customers in the store, as the table follow:



The probability of at most 4 customers is 0.135+0.271+0.271+0.180+0.090= 0.947, or more than 94%. And so the probability of 5 or more is 1-0.947= 0.053, or over 5%.

Through the calculation, we can estimate that there will be four customers stay in the store most of the time. If the store can provide seating for four customers, then it will seldom happen that a customer has to stand.


(actually i don't know whether this important or not, useful or lebih.. i am glad if it helps..^^)




edited from Chapter 27:” Why Am I Always Standing in the Wrong Line?”; Five-Minute Mathematics by Ehrhard Behrends