Questions tagged [queuing-theory]
Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide a service.
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Confusion with Expected Queue Length (Lq) Formulas for M/M/s Model in Queuing theory
I have found two formulas for Expected Queue Length $L_q$ for the M/M/s model regarding Queuing Theory, namely:
Found inside the "Introduction to the Operations Research" book made by ...
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Queueing system with 2 heterogeneous servers with prioritizing customers
This is a special case of a $M/M/c$ queuing system with $c=2$. Here, there is a single customer queue that feeds into two servers, one of which has a higher rate of service than the other. Call them ...
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How does one model weather in Simio and have it change how vehicles operate within the model?
For context, I am using the Simio modeling software for an OR class, and for fun I am making a simulation model of the traffic outside my dorm. I have recreated the layout and distance of the streets, ...
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Service probability for M/M/c queue for reneging where reneging time is a function of service time
There is an M/M/1 (getting a result for M/M/c is ideal) queue with arrival rate 𝜆 and service rate 𝜇 and participants can renege.
However, the difference between normal reneging settings (constant ...
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Understanding queueing theory used to model the vehicle routing problem
I am trying to understand some journal articles on the vehicle routing problem, specifically Vehicle routing with dynamic travel times: A queueing approach and An M/M/c queue model for vehicle routing ...
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Loan Allocation Problem
I need help with the following use case:
Context :
Let's say there is a fintech "F", that helps with aggregating personal loans. Now, "F" can pool in customers who need a personal ...
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m/m/1/k=1 variance
Is there a nice formula for the finite time variance of the m/m/1/k=1 queue in steady state? (Exponential arrivals $\lambda$, exponential service time $\mu$, 1 server and no queue, so if a job arrives ...
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Modeling an assembly line as a batch processing $M/D^{(b,b)}/r$ queue
Consider an assembly line where some parts are produced in a station following a Poisson process with a rate of $\lambda$.
These parts are put directly in a bin (transfer time to the bin is negligible)...
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The impact of utilization rate of a queueing system on its average queue time
For an $M/M/c$ queue where arrivals follow a Poisson distribution with rate $\lambda$ and are iid, service follows a Poisson process with rate $\mu$ and there are $c$ parallel servers, we can estimate ...
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If $x=\min\{f(\mathbf{a}),1-\epsilon\}$, how can we model and partition $x$?
I have been dealing with a problem for sometime and although tried different things and asked some questions before, I think the problem might be somewhere that we didn't look before.
Variables $0\le ...
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Multiserver Queue Theory Optimization problem
I have a design optimization problem where I need to connect a customer with a server via call. The scenario is as follows:
Customer-1 is connected with $N$ servers out of a total pool of $P$ servers....
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Average time between two dispatches in a taxi fleet (probably a batch processing queuing system)
We have a fleet of taxis with $t$ taxis available. All taxis are identical in the sense that they have the same capacity for $p$ passengers and each taxi is dispatched only when its capacity is full.
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Departure process of M/G/1 queue with hyper-exponential service times
I am working on two queues in tandem; the first queue is M/G/1 with hyper-exponential service times and the second queue has exponential service times. I want to know if the second queue can be ...
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Queueing model without steady state
The questions says:
A printer is used to process print jobs. The interarrival times between jobs are exponentially distributed with a mean of 70 seconds. The time that is required to perform a job is ...
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Elevator Traffic (Queueing Theory) Papers
I am about to start my final year thesis. The topic would be related to the queueing theory, specifically for the elevator traffic systems. I would need to do some literature review before going ...
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Significant bias introduced into simple simulation
This question has been asked on Cross Validated
Introduction
Service is allocated to an infinite source of customers i.e. there is always a service in progress. The duration of the $i^{th}$ service ...
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Minimising average holding cost per unit time is equivalent to minimising average delay per customer in queue
This question was originally asked on Math Stack Exchange. It is probably better suited here.
Introduction
When one thinks about a queue, it is natural to want to find a policy that minimises the ...
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What are some references for Fluid and Diffusion models in Queuing?
In queuing theory and stochastic process, there are many papers that have applied fluid and diffusion models to approximate the original models. However, the concepts and basics of these models are ...
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How would I model the following problem concerning customer service via FCFS vs Appointment Scheduling
I'm unfamiliar with OR, and would like to get some advice on how to "think" about this scenario:
I work for an organization that primarily has store fronts. Customer's arrive throughout the ...
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Service probability for M/M/1 queue with reneging
Consider an M/M/1 queue with arrival rate $\lambda$ and service rate $\mu$, where participants renege after a random amount of time which follows an exponential distribution with mean $\tau$. We ...
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How do derive the steady state probabilities M/M/1/k queueing system?
How do I derive the steady state probabilities for the $M/M/1/k$ queueing systems with finite system capacity $k$?
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Two M/M/1 queues working independently
If two M/M/1 queues with arrival rate $\lambda$ and service rate $\mu$ are working independently. Is the system equivalent to a single M/M/1 queue with arrival rate $2\lambda$ and service rate $2\mu$?
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Waiting time in M/M/n queue
I was able to numerically "show" that an M/M/n queue is concave increasing in the number for $n$ servers, which makes intuitive sense. However is there a formal analytical proof for it? Where can I ...
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Approximate average waiting time when a m\m\1 queue shows transient behavior
I have an m/m/1 queue and a set of arrival rate $\{\lambda_{t_1}, \lambda_{t_2} ...., \lambda_{t_n}\}$ and a fixed service rate $\mu$. I want to calculate the average waiting time during each time ...
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Combining Two Different Queues
I am trying to create an optimization model for a problem that involves two different types of queues. Given Poisson demand (for both), there is a queue with constant service time and another queue ...
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MPX Queuing Software for Manufacturing with the GTHUBS Case
I'm teaching an Operations Management Class. A long time ago, there was a simple DOS-based product called MPX. It was based on queuing theory and let you model manufacturing processes. There was ...
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Queuing Theory with Learning Perspective
I am willing to work on queuing models but in classical queuing models, it is assumed the probability distributions of arrival and service are known or at least the rate is known. However, I am ...
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Current Issues of Interest
What are some current issue of interest in Operations Research? I am interested in current topics that experts in the field are interested in researching.
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Convexity/Concavity of Average Number of Jobs in M/M/1 Queue?
I am working on a problem involving the average number of jobs $L$ in an M/M/1 queue with arrival rate $\lambda$, service rate $\mu$. For traffic intensity $\rho = \frac{\lambda}{\mu}$,
$$
L = \frac{\...
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Queuing Theoretic Model with Memory
Consider a telephone company which receives call request at some arrival rate and serves each request with some service rate. This can be modeled using a Poisson Process. However I wish to model the ...
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Question about a queueing problem
Arrivals at a telephone booth are considered to be Poisson with an average time of 10 minutes between one arrival and the next. The length of phone calls is assumed to be distributed exponentially, ...
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The departure process of an $\rm M/M/\infty$ queue
Burke's theorem says that the output process of an $\rm M/M/1$, an $\rm M/M/C$, and a $\rm M/M/\infty$ queue with arrival rate $\lambda$ and service rate $\mu$ follows a Poisson with parameter $\...
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Model or State Uncertainty in Queueing Model due to uncertain arrival rate
Introduction
I am currently modelling a scenario where two queues need to be served by a single server in a non preemptive discipline. I am quite sorted on generating the optimal policy via Value or ...
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Good textbook for queueing theory and performance modeling
Can someone recommend a good self-study textbook for queueing theory and performance modeling? My interest is in applying this to understanding the behavior of some real-world server networks, ...
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Queuing models in R, $\lambda$ Little
It's noted that the number of folks in a stationary system will maintain an average equal to the rate of arrival multiplied by the mean of the service distribution.
The formula $L = \lambda w$ is ...
