How Is Queueing Theory Applied in a Busy Store's Christmas Sale?

[Erenjaeger]

Homework Statement

Anyone know anything about queueing theory? would really appreciate some help.
the question goes as follows:
The annual S&C Christmas sale is so popular that it is necessary to limit the number of customers who can be inside the store simultaneously; this limit is set at 60 people, with other customers having to queue on the street outside until somebody leaves the store. It is projected that when the store opens at 9am, there will already be 100 customers wanting to enter immediately, with further customers arriving thereafter as a Poisson process at rate 150 per hour until noon, and 50 per hour after that. Once a customer does get inside, the time they spend there is thought to be exponentially distributed with mean 30 minutes.
(a) Describe this queueing system in Kendall notation.

Homework Equations

not really any relevant equations I don't think, not even a precalc question but i didnt know where to post it

The Attempt at a Solution

From the question I can see that λ=150 customers / hour initially after the people waiting get in and λ=50 customers / hour after noon. also μ = 1/0.5 = 2 customers / hour / server
and i can see that the queue is an M/M/something queue C being the number of servers but I am not sure how to work that out.
by the way λ and μ are arrival rate and service rate respectively.

 

[StoneTemplePython]

All inter-arrival periods are exponential, on both the in-processing and out processing -- the one exception to this is the discontiunities -- i.e. when the store starts up and again at noon but setting those aside, they are memoryless. So M/M seems about right.

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What does this tell you with respect to servers / processing capacity?

The annual S&C Christmas sale is so popular that it is necessary to limit the number of customers who can be inside the store simultaneously; this limit is set at 60 people, with other customers having to queue on the street outside until somebody leaves the store.

 

[Erenjaeger]

All inter-arrival periods are exponential, on both the in-processing and out processing -- the one exception to this is the discontiunities -- i.e. when the store starts up and again at noon but setting those aside, they are memoryless. So M/M seems about right.

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What does this tell you with respect to servers / processing capacity?

Is that limit of 60 people including the servers? I can't see how I could derive how many servers there are from the information given unless i just guess and say 60 people in the store / 2 customers served per hour = 30 servers ?

 

[Erenjaeger]

Is that limit of 60 people including the servers? I can't see how I could derive how many servers there are from the information given unless i just guess and say 60 people in the store / 2 customers served per hour = 30 servers ?

or just 60 customers + however many servers since the question says "limit the number of customers"?

 

[Ray Vickson]

Is that limit of 60 people including the servers? I can't see how I could derive how many servers there are from the information given unless i just guess and say 60 people in the store / 2 customers served per hour = 30 servers ?

Is that limit of 60 people including the servers? I can't see how I could derive how many servers there are from the information given unless i just guess and say 60 people in the store / 2 customers served per hour = 30 servers ?

You said that "once a customer gets into the store, the time they spend there is expl(30 min). That is, the "service time" of a customer does not depend in any way on the number of other customers in the store. It is as though there are enough servers to serve however many customers are inside the store --- a 60-server queue (or, alternatively, a self-service system in which each customer serves itself).

So, it looks to me like an M/M/60 queue with no "capacity" restrictions. Up to a total customer content of 60 there is no queue---everybody is in service. For more than 60 customers, a "queue" begins; these are the customers lined up on the sidewalk outside the store. (If you did have a finite capacity restriction, that would not be the "60"; rather, it would be a restriction on the number of people that could line up outside the store.)

 

[Erenjaeger]

You said that "once a customer gets into the store, the time they spend there is expl(30 min). That is, the "service time" of a customer does not depend in any way on the number of other customers in the store. It is as though there are enough servers to serve however many customers are inside the store --- a 60-server queue (or, alternatively, a self-service system in which each customer serves itself).

So, it looks to me like an M/M/60 queue with no "capacity" restrictions. Up to a total customer content of 60 there is no queue---everybody is in service. For more than 60 customers, a "queue" begins; these are the customers lined up on the sidewalk outside the store. (If you did have a finite capacity restriction, that would not be the "60"; rather, it would be a restriction on the number of people that could line up outside the store.)

Right so it is an M/M/60 queue. But just so I understand, I know once a customer gets into the store, the time they spend there is expl(30 min) which is 1/half an hour or 1/0.5 = 2 customers per minute = μ. but you said "that is, the "service time" of a customer does not depend in any way on the number of other customers in the store." Which I can see why that is, because why would the service differ depending on the amount of customers right? so does the fact that its exponentially distributed infer that ? or is that just the case all the time? I am still confused though how you can get from the fact that the service time doesn't depend on the amount of customers in the store to the fact that there are 60 servers? couldn't there be 30 servers or 50 or any number between 1 and 60?

 

[StoneTemplePython]

Right so it is an M/M/60 queue. But just so I understand, I know once a customer gets into the store, the time they spend there is expl(30 min) which is 1/half an hour or 1/0.5 = 2 customers per minute = μ. but you said "that is, the "service time" of a customer does not depend in any way on the number of other customers in the store." Which I can see why that is, because why would the service differ depending on the amount of customers right? so does the fact that its exponentially distributed infer that ?

the fact that when customers are in queue -- they are treated as iid, which implies that.

Im still confused though how you can get from the fact that the service time doesn't depend on the amount of customers in the store to the fact that there are 60 servers? couldn't there be 30 servers or 50 or any number between 1 and 60?

the problem states 60 customers in a store at once, max. That is your capacity.
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edit: I'm a bit concerned I overloaded the word "capacity" here. The problem is set up so that customers (aka packets) get "serviced" when in the store. Once in the store, each servicing is iid (with the exponential distribution). The store never tells people to 'wait' unless it is full. The store gets full at 60 people. This is equivalent to saying there are 60 servers running. If the store told people to wait outside when only 50 customers were inside, then the number of servers would be 50.

you may want to look at example 2.3.1 on page 17, and example 2.4.1 on page 19 of here:

https://ocw.mit.edu/courses/electri...ring-2011/course-notes/MIT6_262S11_chap02.pdf

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In the real world, of course, the more customers in the store, the longer you may spend waiting in line at the checkout counter, for instance.