Probability of queuing

We are students and we developing the project of a music festival that takes place in 3 venues. We have identified some risks; one of those is related to the capacity of the venues, the tickets sold and how people will behave during the festival.
The total capacity of the venues is 6000, however we just sell 5800 tickets to have some spare capacity (200). The tickets allow the audience to access the three venues if there is free space on them.
Venue A Venue B Venue C
Capacity (people) 3000 2000 1000

In order to reduce people queuing outside because the venues are full the following measures have been taking:
The concerts in the three venues start at the same time; this is 18.00h, 19.00h and so forth and last 45minutes each.
The line up has been done in such way that bands playing at the same time have a similar fan-base on the social media.
The venues are located in the same area but walking time between them takes approx 5 minutes between venues is required.

The question are:
What is the probability at any given moment of time to have people queuing outside the venues A, B, C?
Probability of having queues in more than one venues simultaneously (e.g A and C)
It is possible to know an estimate, using statistics, of the queuing time? What time of data would be required to carry such analysis?

We appreciate your time and help.


TS Contributor
Your problems seems quite interesting but there are quite a number of follow-up questions need to be asked.

1. Do you mean there are a total of 5800 people (inside the venue + queuing outside) at any given moment of time?

2. If yes, "have a similar fan-base on the social media" means that you want to model the number of people supporting the band in three different venues follows

\( \text{Trinomial}\left(5800; \frac {1} {3}, \frac {1} {3}, \frac {1} {3}\right) \)

3. Why do people queuing outside? Do you mean the people buying the ticket will only watch their most supported band without considering the other two?

4. You have mentioned the show started on time 1800 1900 etc. Do you mean there are different people in each hour? Or people enter freely at anytime? Also do you mean the band will rotate the venue at each time?

5. Not sure how the walking time between each venue will be related to the estimates (unless you are taking somehow complicated model considerations that the people are moved freely between the venues based on several factors). So that's why I am not very sure how "queuing time" plays the role here. As mentioned above unless you consider the people to be very dynamic, can moving around freely throughout the 45 minutes show time; but this backs to the question 3 why do people queue outside? why do not they just go to the empty seat?
Hi, thanks for your time.

The total tickets = people is 5800 at any given moment. It could happen that one band playing in Venue A is quite popular (say 3250 want to attend), because the capacity of the venue A is 3000, 250 will need to wait outside the venue (queue) for someone that is inside to free space. This means that other venues will have 250 people less whatever is the distribution between those venues.
To minimize the impact of queuing, when developing the band line-up we have decided to have bands with more or less same fan base playing at the same time to avoid an 'instability' of people distribution. As a example we have two big bands: say U2 and Coldplay... playing at the same time and have a even distribution of people, because if U2 is playing at the same time that XYZ is likely that most of the people will go to see U2, therefore will need to queue as capacity at venues is limited.
People will need to queue outside because capacity of venues is limited. tickets allow people to move from one venues to another as many times as they wish.

Queuing is important if you consider customer satisfaction. Imagine you pay £100 for the tickets... and everyone want to go to the same concerts, but the capacity of the venues is limited. Once the capacity of the venue has been completed based on first come first served basis, you cannot go inside. The you have two options: 1. you queue and wait for someone else to leave the venue. 2 you move to a different venue.

I hope this clarify your questions.

Looking forward to hearing from you.