# Queueing Model where inflow has to wait for outflow because of a shared channel

• spikkelvissie
In summary: The problem with this scenario is that there are many variables that a standard queuing model does not account for. For instance, the travel time inside the port, the possibility for more than one ship to use the channel at the same time, and the time it takes for a boat to travel from the berth to the channel. A dynamic model would likely be the best option, but my colleague has asked me to find a solution using a static model. While I am not an expert in queuing theory, I have studied it in university. Including the time it takes for a boat to leave the port in the service time would simplify the problem, but there may be instances where a ship enters the port while the previous boat is still on its way to
spikkelvissie
Hi

I am look for a mathematical queuing model that can help with/solve the following scenario. I believe this scenario can me modeled in a dynamic simulation, but I am in need of a solution for a static model.

Scenario:
You are at a port. There is a steady, equally distributed arrival rate of incoming boats. A single queue of boats form in front of a channel. The channel is narrow and therefore the boats cannot pass each other in the channel. Inside the port are 2 berth with a constant service rate. The boats in the queue can only proceed to a berth if the channel is open and a berth is available. The boats that have been serviced at the berth has to return back through that same channel. If an incoming boat and outgoing boat wants to use the channel at the same time, the outgoing boat gets priority. There are no queues inside the port at the berths or at the channel going outward.

In summary: There is only 1 queue, outside the port at the channel, the queue time depends on the availability of the channel, the availability of the berth, the service time at the berth, etc.

Is there a queuing model that describes this scenario?

All of the network models I have looked at assumes the outflow uses a different channel than the inflow and that there are queues inside the port as well.

spikkelvissie said:
Hi

I am look for a mathematical queuing model that can help with/solve the following scenario. I believe this scenario can me modeled in a dynamic simulation, but I am in need of a solution for a static model.

Scenario:
You are at a port. There is a steady, equally distributed arrival rate of incoming boats. A single queue of boats form in front of a channel. The channel is narrow and therefore the boats cannot pass each other in the channel. Inside the port are 2 berth with a constant service rate. The boats in the queue can only proceed to a berth if the channel is open and a berth is available. The boats that have been serviced at the berth has to return back through that same channel. If an incoming boat and outgoing boat wants to use the channel at the same time, the outgoing boat gets priority. There are no queues inside the port at the berths or at the channel going outward.

As far as I can tell, you have a standard queue. The only difference is that the service time includes the entire time a ship is "in port" including from the time an incoming ship enters the channel until that same ship clears the channel on its way out. Even if there are two berths, the time a boat time spends in the channel is part of the service time for calculating mean waiting time. Having two berths potentially reduces mean service time at the berths, but total service time has to include the bottleneck the narrow channel represents.

EDIT: At the start of the service day you can bring two ships in, one behind the other. However, the variance of the service time at the two berths will cause staggering over time. You can use dynamic simulation to see how the variance of service time at the berths affects mean total service time. Note with the Poisson distribution, which usually describes arrival times, the variance is equal to the mean while the gamma distribution is often used for service times where the variance is $k \theta^2$.

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Thank you. I guess the bigger problem is that there are many variables a standard queuing model doesn't account for. For example the travel time inside the port, more than one ship can use the channel behind each other, the time it takes to get through the channel vs. the time it takes to travel from the berth to the channel. The dynamic model would probably be best, but my colleague asked me to look into a way to solve this statically. I am no expert in queuing theory, just did it in University. Making the service time include the time it takes for the boat to leave the port would make this a much simpler problem, but I believe there will be instances where a ship can enter the port while the boat from the berth is still on it's way to the channel, it is this variability that makes it complicated.

It would be ideal if there existed a queuing model where instead of just having arrival and service rate as an input, one can input the shared buffer, the rate of the buffer and the travel time between the buffer and the service point. I believe this travel time has to be separate from service time, since another boat my enter the port while the previous boat is traveling in the port. Unfortunately I don't have any solid data at the moment, I was looking for a generic model which can be used given different types of ports with the same setup.

Thank you for your help. Regards

spikkelvissie said:
Thank you. I guess the bigger problem is that there are many variables a standard queuing model doesn't account for. For example the travel time inside the port, more than one ship can use the channel behind each other, the time it takes to get through the channel vs. the time it takes to travel from the berth to the channel. The dynamic model would probably be best, but my colleague asked me to look into a way to solve this statically. I am no expert in queuing theory, just did it in University. Making the service time include the time it takes for the boat to leave the port would make this a much simpler problem, but I believe there will be instances where a ship can enter the port while the boat from the berth is still on it's way to the channel, it is this variability that makes it complicated.

It would be ideal if there existed a queuing model where instead of just having arrival and service rate as an input, one can input the shared buffer, the rate of the buffer and the travel time between the buffer and the service point. I believe this travel time has to be separate from service time, since another boat my enter the port while the previous boat is traveling in the port. Unfortunately I don't have any solid data at the moment, I was looking for a generic model which can be used given different types of ports with the same setup.

Thank you for your help. Regards

You're welcome, but I think your main interest is the mean queue outside of the channel. Queuing theory, like any probabilistic model, is for calculating the expectation (mean) and the variability around the mean. The theory is that over extended time, the actual mean queue approaches the calculated mean queue. If you treat the entire port operation as a "black box" your only interest is the time between ships entering the channel vs the time between new arrivals to the queue. Mathematically the black box might be modeled as:

$S= F(t, k, \theta) + C_t$ where t is time, (k=1) is a random service encounter using a test variable where $\theta ^ {-1}$ is the mean joint rate of ships served by the two berths. The test variable has an exponential distribution. C is mean transit time within the port (in and out). You most likely would be using the Erlang distribution for some k once you've established a mean service time.

Even with a simulation to evaluate all kinds of special situations, you need information on mean values and measures of variation (variance). With this model, the variance is calculated from the parameters.

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SW VandeCarr said:
You're welcome, but I think your main interest is the mean queue outside of the channel. Queuing theory, like any probabilistic model, is for calculating the expectation (mean) and the variability around the mean. The theory is that over extended time, the actual mean queue approaches the calculated mean queue. If you treat the entire port operation as a "black box" your only interest is the time between ships entering the channel vs the time between new arrivals to the queue. Mathematically the black box might be modeled as:

$S= F(t, k, \theta) + C_t$ where t is time, (k=1) is a random service encounter using a test variable where $\theta ^ {-1}$ is the mean joint rate of ships served by the two berths. The test variable has an exponential distribution. C is mean transit time within the port (in and out). You most likely would be using the Erlang distribution for some k once you've established a mean service time.

Even with a simulation to evaluate all kinds of special situations, you need information on mean values and measures of variation (variance). With this model, the variance is calculated from the parameters.

Thank you, this helps a lot. I really appreciate it. Regards

## 1. What is a queueing model?

A queueing model is a mathematical representation of a system in which entities (such as customers, data packets, or vehicles) arrive at a certain rate, wait in a queue, and are served or processed according to certain rules. It is used to analyze and optimize the performance of systems that involve waiting, such as transportation systems, telecommunication networks, and customer service processes.

## 2. What is the purpose of a queueing model where inflow has to wait for outflow because of a shared channel?

The purpose of this specific type of queueing model is to understand and optimize the performance of systems where there is a shared channel or resource that must be used by both incoming and outgoing entities. This type of system can be found in many real-world scenarios, such as a single lane road where cars must take turns to enter and exit, or a computer network where data packets must share a limited bandwidth.

## 3. How does a queueing model with shared channel work?

In this type of queueing model, incoming entities arrive at a certain rate, join a queue, and wait for their turn to use the shared channel. Once the channel becomes available, the entity at the front of the queue is served and then leaves the system. The outflow of entities is limited by the capacity of the shared channel, so if the inflow rate exceeds the outflow rate, a queue will form.

## 4. What are the key components of a queueing model where inflow has to wait for outflow because of a shared channel?

The key components of this type of queueing model include the arrival rate of entities, the service rate of the shared channel, the number of servers (if there is more than one shared channel), and the capacity of the shared channel. Other factors such as the size of the queue and the behavior of the entities (such as their patience level or priority) can also be included in the model.

## 5. How can a queueing model with shared channel be used in real-world applications?

A queueing model with shared channel can be used to analyze and optimize the performance of various systems, such as transportation systems, telecommunication networks, computer networks, call centers, and more. By understanding the behavior and interactions of the entities in the system, the model can help identify potential bottlenecks and improve the overall efficiency and effectiveness of the system.

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