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

• Erenjaeger
In summary, the given problem can be described as an M/M/60 queue with exponentially distributed inter-arrival times and service times. This means that the customers arrive at a rate of 150 customers per hour initially, and 50 customers per hour after noon. The service time for each customer is exponentially distributed with a mean of 30 minutes, and there are no restrictions on the number of servers available. This results in a queue beginning when the total customer content exceeds 60, with customers lined up on the sidewalk outside the store. The service time for each customer does not depend on the number of other customers in the store.
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.

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?

Erenjaeger said:
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.

StoneTemplePython said:
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.

- - - -
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 said:
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"?

Erenjaeger said:
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 said:
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.)

Ray Vickson said:
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?

Erenjaeger said:
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.

Erenjaeger said:
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.

Last edited:

## What is Queueing theory problem?

Queueing theory problem is a mathematical study of waiting lines, or queues, and is used to analyze and optimize systems where customers arrive randomly and wait in line for service.

## What are the main components of a queueing system?

The main components of a queueing system are the arrival process, the service process, the queue structure, and the performance measures. The arrival process refers to how customers arrive to the system, the service process refers to how the customers are served, the queue structure refers to how customers are organized in the queue, and the performance measures refer to the metrics used to evaluate the efficiency and effectiveness of the system.

## What are the different types of queueing systems?

There are several types of queueing systems, including single-server, multi-server, and networked systems. Single-server systems have one server serving all customers, while multi-server systems have multiple servers serving customers simultaneously. Networked systems involve multiple queues and servers interconnected in a larger system.

## What is the goal of queueing theory?

The goal of queueing theory is to find the optimal balance between customer waiting time and service capacity. This allows for the design and management of systems that can handle customer demand efficiently and effectively, minimizing wait times and maximizing resource utilization.

## How is queueing theory applied in real-world scenarios?

Queueing theory has many applications in various industries, such as telecommunications, transportation, healthcare, and manufacturing. It is used to improve customer service, reduce waiting times, and optimize resource allocation in systems with unpredictable arrival patterns, such as call centers, airports, hospitals, and production lines.

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