# Solving Queueing Problem: Population to Queue to Server

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• mechlite
In summary, the problem is trying to pick a sequence of customer orders that will minimize the server idle time.
mechlite
First post and I'm wondering if I could get some help. I'm new to queueing theory so I'm not sure how to solve this problem.

Population-------------> Queue--------------> Server

I have a calling population that is infinite or a vast amount. The queue capacity is limited at 13. There is 1 server in the process. The population time with the server varies from 10s - 1.5 hrs. The time the population spends in queue before going to the server is 2 hrs. The population chosen MUST go in the queue before the server. So I have a big selection that have to be served, but a smaller capacity. How would I pick the sequence (Queue discipline?) from the calling population that will keep the server running for the most amount of time? If the population sequence selected has short service times, eventually, the server will have to wait 2 hrs for the next selected population. So I need a sequence that will stagger them so while the server is taking x hr to finish with current job, the queue will be able to prepare the next one with minimal wait time and repeat. Appreciate any help with this

EDIT -- More details: there is a long list of customer orders (Population) and before these orders are assembled they have to be prepped. The prepping queue has a max capacity of 13 orders at a time. So I am only able to select 13 customer orders at a time before assembly. Prepping queue time remains pretty much the same (2-3hrs). Assembly (server) time can range from 10 seconds to 2 hrs. I want to pick a sequence out of the customer order list that will minimize the server idle time.

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I have a general comment on this. For practical analysis of these problems, you should apply simulation and not rely on the analytical queueing theory results. The analytical results only apply to very simple, standard situations that are not very realistic or practical. Most queueing problems of anything but the simplest networks require simulations to analyse. There are several software packages available to help build queueing simulations. See https://en.wikipedia.org/wiki/List_of_discrete_event_simulation_software . Once you have a simulation, you can determine steady-state conditions and transient behavior under a variety of conditions. Furthermore, the simulations allow you to collect all sorts of statistics that are very difficult to get analytically.

Here is a recent example showing the difficulty of analyzing a fairly simple queueing problem: https://www.physicsforums.com/threads/waiting-time-in-a-queue-using-poisson-arrival.902175
And the only way I would feel confident of their results is if there was a simulation that supported them.

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berkeman and Buzz Bloom
FactChecker said:
I have a general comment on this. For practical analysis of these problems, you should apply simulation and not rely on the analytical queueing theory results. The analytical results only apply to very simple, standard situations that are not very realistic or practical. Most queueing problems of anything but the simplest networks require simulations to analyse. There are several software packages available to help build queueing simulations. See https://en.wikipedia.org/wiki/List_of_discrete_event_simulation_software . Once you have a simulation, you can determine steady-state conditions and transient behavior under a variety of conditions. Furthermore, the simulations allow you to collect all sorts of statistics that are very difficult to get analytically.

Here is a recent example showing the difficulty of analyzing a fairly simple queueing problem: https://www.physicsforums.com/threads/waiting-time-in-a-queue-using-poisson-arrival.902175
And the only way I would feel confident of their results is if there was a simulation that supported them.
I see what you mean. The analysis on that was in depth. Solving my problem would take way too much time. I will look into the software packages. That might be the fastest and easiest way. Thanks for your help!

## 1. What is a queueing problem?

A queueing problem is a situation where there is a limited number of resources or servers available to serve a large number of customers or individuals. This leads to a buildup of a queue, or line, as customers wait for their turn to be served.

## 2. How do you solve a queueing problem?

There are various approaches to solving a queueing problem, but one common method is by using mathematical models and simulations. These models take into account factors such as arrival rates, service times, and number of servers to find an optimal solution.

## 3. What is the role of population in a queueing problem?

The population in a queueing problem refers to the total number of customers or individuals who require service. This number is important in determining the arrival rate and can affect the queue length and wait time.

## 4. What are the key factors that affect queueing problems?

The key factors that affect queueing problems include arrival rate, service time, number of servers, and the queue discipline. Arrival rate refers to how often customers arrive for service, while service time is the amount of time it takes to serve each customer. The number of servers and the queue discipline, which determines the order in which customers are served, also play a significant role in queueing problems.

## 5. How can queueing problems be applied in real-life situations?

Queueing problems can be applied in various real-life situations, such as in retail stores, call centers, and transportation systems. By understanding and solving queueing problems, businesses and organizations can improve efficiency, reduce wait times for customers, and optimize resource allocation.

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