(Fun) want to chase her? How long are you waiting?

I want to chase her from the fruit shell network? Calculate how long it will take for chunhao to be published on
23:15:45

She is in love, but not for you. How long will it take? Don't worry. Let's estimate how to make yourself feel. We can also estimate the number of people she has been chasing since she entered the age of youth. The spare part must be read!

How many people will a girl chase her from her first pursuit to her marriage? How many times does she talk about love? In fact, we can build a simplified model for estimation.

This is a process of waiting and processing.

Assume that a girl with no bad conditions will always be followed by others (the frequency of being pursued is different ). If you think of the pursuer as a queue, when the pursuer appears, it can be regarded as a "request" and put in this queue. For a girl, to "process" the "request", she has only two options: accept or reject (other complex actions are not considered here ). Once she starts to process a request, there are only two results:

The girl will fall in love with him until she breaks up (or gets married). After her consideration for a short or long time, she will explicitly reject the target.

A breakup or rejection means that the process is completed. This means that he is removed from the queue. The time that a girl has been in love with this person, or the time that she decides to reject is called "Processing Time "; A person's time from becoming her pursuer to breaking up or being rejected is called "waiting for time ". So how many girls are in the queue on average? On average, how long will each person wait in a girl's queue?

Simplified to Mathematical Model

On average, for a girl, the frequency of appearance of the pursuer can be treated as random, and the interval between the two pursuers can be expressed by exponential distribution. As a result, the frequency of appearance of the pursuer follows the Poisson distribution. On the other hand, because of the random appearance of the pursuers, the desired objects of girls are random, which leads to the random time for girls to fall in love, that is, the processing time is subject to the exponential distribution, within a period of time, girls fall in love with each other at a frequency of Poisson distribution.

To simplify complex problems, let's make the following idealized assumptions:

(1) The pursuers are all dead-eyed. Once the pursuers become pursuers, they will not give up and stay in the queue until they break up or are rejected. That is to say, when an object in the queue is not processed, it will queue for processing.

(2) girls do not have to foot on two ships, and each time they can only fall in love with one person (each time they only process one request ).

(3) When a girl is processing a request, she will not be harassed by any other pursuer with a gentleman's demeanor.

(4) When girls process requests in the queue, they should follow the first come first served (FCFS) principle. If they are reliable, they will fall in love and refuse if they are unreliable, deciding to reject someone may take some time.

Since the time interval and processing time of the pursuer follow the exponential distribution, we use a (t) and B (t) To represent the density function of the two respectively ):

According to the characteristics of Poisson distribution, we can know that the average value of the time interval of the target is 1/λ, and the average value of processing time is 1/μ,λ refers to the frequency at which the target user appears, and μ refers to the frequency at which the girl processes the request..

Use queuing theory to calculate the average waiting time of the pursuer

When the number of pursuers exceeds one, a queue is formed. At this time, we can introduce a Markov chain (Markov chain) to analyze this problem:

In, each circle represents a State of the queuing system, where the number represents the length of the queue: the number of current pursuers. In each state, every time a suitor comes, the system goes to the next state (the number of suitors increases by 1). After a girl finishes a relationship or rejects a person, the system is back to the previous state (the number of pursuers is reduced by 1, because a pursuer is rejected or becomes an ex-boyfriend ).

P n indicates the probability that the system is in the state n (there are n pursuers in the girl's Queue). To reach a stable state, the system must satisfy the following recursive equation:

Through iteration, the expression of P n can be obtained:

It is also noted that for this Markov chain, the probability and value of each State should be 1 (because it contains all the conditions): P0 + P 1 +... + P n = 1. Combined with the above formula, the final solution is:

With probability, we can calculate the average length of the chaser queue. The Random Variable N represents the number of pursuers in the queue in a stable state, as expected:

Simplicity:

Knowing the expectations of the number of suitors for this girl, we also want to know how long each suiter will wait on average. To solve this small problem, we have to use the queuing theory (Queueing Theory) of Little's formulas:

W is the average waiting time of the pursuer (from entering the queue to breaking up with the girl or being rejected). Combined with the above results, it is not difficult to conclude:

Estimation of reality with conclusions

The formula has been launched, and the theoretical weapon has been firmly held in hand. The rest is to find a way to understand the girl's history and get two key data: λ and μ. The frequency of the target is determined by the girl's popularity. The frequency of her fall in love is roughly equal to the number of times she talked about the relationship and the time when the girl was chased from the beginning. For example, a girl has a person every month to pursue her, and she talks about love every six months. The half year here refers to the time when the girl broke up in love and fell in love, in this cycle, the average number of requests processed by girls is 7. Then the average waiting time of the target is 1/(7/6-1/1) = 6, and the average waiting time of the target is 6 months. But how many people will spend half a year waiting for a chance to present themselves? Note: here, the requests are processed, that is, the girl's attention to you. Maybe
The result of the 12-month wait is rejected. So here we can see that it is quite large to wait for the cost to be favored by the sweetheart. This may be the cause of the situation that is endangered.

On the other hand, if the frequency of a girl's love affair is the same, the more popular (usually the more beautiful) the girl is, the more frequent the pursuer will appear, therefore, the value of W is larger. This shows that the probability of success is not high if you want to impress "popular" girls with sincerity and waiting. If you really want to catch up, it would be better to "give up and dare to drag the Emperor down the horse", even if she is in love, also take practical actions.

Interestingly, this "research result" is also useful to girls. The girl puts her data in the formula to calculate the average length of the pursuit, and then she can calculate how many pursuers she can meet in a certain period of time. For example, after calculation, the average waiting time for a suseeker is two months. In the ten-year period from the age of 16 to the age of 26, the number of suitors is expected to be 60. If you want to pick a good husband, you have to know what the girl knows. According to the mathematician's suggestion, we should first reject the above.
37% of people will study one by one, waiting for the next person who is better than these people.

Two points about the Model

Although we use this mathematical conclusion to guide love, it should be noted that the reality is not so idealistic. There are two points to note here. First of all, in the model hypothesis, we assume that the pursuer will not leave automatically once it enters the queue. However, after a girl does not respond for a while, most people will know that it is difficult to leave automatically and will not wait so long, especially when a girl is in love.

On the other hand, girls can only process one request at a time. It is common to fall in love with only one person at a time, but it is also normal to investigate and reject several pursuers at the same time (rather than rejecting one person at a time as assumed in the model ). In reality, almost all of the pursuers are ignored directly. From this perspective, the request processing frequency μ is usually greater than the request arrival frequency λ (because many people can be rejected in a short time), which is similar to the model item.

In fact, this model is most suitable for girls to be tempted by the pursuit Queue (such a small number of pursuers, the corresponding Lambda will not be very high ). This queue will not be too long, because it will rarely show many girl-motivated pursuers at the same time. However, to reject any such pursuit, you must carefully consider it for a while before you can make up your mind.

Although it is an idealized model, it is more accurate than asking her in person.

References:

[1] fundamentals of Queueing Theory 4th edition, 2008

[2] Wikipedia: Poisson distribution

[3] Wikipedia: Markov Chain