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I have been tasked with finding out approximately how many people were in a location at a given time. I know how many people arrived per hour and how many people left per hour and the average time someone was here by arrival hour. So data looks like so:

arr     hourly_arrival  avg_time_here   hourly_leaving
0       1631        7           2575
1       1294        7       2434
2       1135        6       2248
3       930     6       2011
4       878     7       1803
5       856     7       1619
6       1152        6       1603
7       1710        6       1091
8       2354        6       1487
9       3153        5       1605
10      3652        6       1913
11      3873        6       2220
12      3901        6       2642
13      3766        6       2983
14      3623        6       3355
15      3672        6       3515
16      3613        6       3607
17      3644        6       3672
18      3735        6       3599
19      3654        6       3343
20      3423        6       3702
21      3072        6       3832
22      2675        6       3595
23      2124        6       3092

0 being midnight and 23 being 11pm. What is the best way to go about this? What I have done is the following but it just does not seem right.

I took the hourly_arrival column and divided each row by 365.25 as the data encompasses a year and got the average arrival per hour. The avg_time_here column represents the average hours someone is here for that given arrival hour. For hourly leaving, I did the same as hourly_arriving.

avg_hrl_arrival becomes avg_hrl_arr = hourly_arrival / 365.25

avg_hrl_leave becomes avg_hrl_leave = hourly_leaving / 365.25

Then at each hour I said avg_people_here is

avg_people_here = avg_hrl_arr * avg_time_here

rgk
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MCP_infiltrator
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1 Answers1

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As far as I understand this, you could use Queuing theory and specifically the Birth-Death process. If you assume exponential arrival, this could classify as a M/M/1/FCFS queue.

You could estimate the arrival rate per hour $\lambda$ as a simple average of the number of people arriving at each hour in the day and assume the service rate $\mu$ $\geq$ $\lambda$.

You are asked to find the average number of people in the queue, $L$, which is obtained using Little's formula as below. $$L = \lambda \cdot W$$ where $W$ is the average waiting time in queue, which gives, $$L = \frac{\rho}{1-\rho}$$ where $\rho = \frac{\lambda}{\mu}$ is the traffic intensity.

rgk
  • 136