A Solving Queueing Problem: Population to Queue to Server

[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.

 

[FactChecker]

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.

 

[mechlite]

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!