9 votes
Accepted

Convexity/Concavity of Average Number of Jobs in M/M/1 Queue?

Your calculations (factoring and simplification) are incorrect. $L$ is neither convex nor concave as a function of $\lambda$ and $\mu$. This can be concluded by examining the eigenvalues of the ...
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9 votes
Accepted

Significant bias introduced into simple simulation

You have fallen victim to the renewal paradox, a.k.a. inspection paradox, a.k.a. length-biased sampling. $F_{\Delta}$ is the distribution of service time for the kth customer, but it is NOT the ...
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8 votes
Accepted

MPX Queuing Software for Manufacturing with the GTHUBS Case

Sure thing, you can download from archive.org. You need to run it in DOSBOX.
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  • 96
8 votes

Queuing Theory with Learning Perspective

Certainly, and to take the problem's structure into account, one could model the queue knowing - or guessing - prior information about the queue's structure and/or parameter distribution and use ...
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  • 1,577
7 votes

Current Issues of Interest

This is an attempt at an answer, based on my current understanding: Background: Operations Research is an interdisciplinary field. You go from business administration and economics to theoretical ...
7 votes

Good textbook for queueing theory and performance modeling

Unfortunately, much of the performance analysis and transient approximations for time-varying systems with non-Markovian (non-exponential) properties are not easily obtained in book form (see note at ...
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7 votes

Queuing models in R, $\lambda$ Little

Not directly answering your question of how to code it manually but for discrete simulation of queues in R I would strongly recommend the simmer package. The ...
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  • 1,313
4 votes

Good textbook for queueing theory and performance modeling

I enjoyed Performance Modeling and Design of Computer Systems: Queueing Theory in Action (Amazon link) by Mor Harchol-Balter, which sounds like it fits your bill pretty well. I have it on my desk. ...
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  • 141
4 votes

Good textbook for queueing theory and performance modeling

I have used Stochastic Modeling: Analysis and Simulation by Barry Nelson and found it to be a pretty gentle introduction. It covers stochastic processes, queuing, and simulation.
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4 votes
Accepted

The departure process of an $\rm M/M/\infty$ queue

The following proof approach for Bruke's Theorem was given in this Lecture by Richard Clegg: Definition: A chain is called time-reversible if $p_{ij} = p^*_{ij}$ for all $i$ and $j$. This occurs if ...
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  • 8,185
4 votes

Model or State Uncertainty in Queueing Model due to uncertain arrival rate

After having read Chapter 5.3 of Decision Making Under Uncertainty by Mykel J. Kochenderfer, I have come to some conclusions. We are dealing with model uncertainty, in which case we can formulate a ...
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4 votes
Accepted

How would I model the following problem concerning customer service via FCFS vs Appointment Scheduling

Since you are unfamiliar with OR, I would recommend using discrete event simulation, which I think is the easiest approach for a newcomer (although it may require some programming chops, depending on ...
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  • 28.7k
4 votes

Current Issues of Interest

On the solving side, some hot topics include: presolving techniques GPU-powered algorithms algorithms designed for problems that consume a lot of memory algorithms for distributed architectures ...
4 votes

Significant bias introduced into simple simulation

Note: This answer is intended to show what I have learned from the valuable answer provided by @Mark L.Stone. His post answered my question of why the simulation is biased. Hence, this post provides ...
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3 votes
Accepted

Waiting time in M/M/n queue

Dyer and Proll (1977)1 showed that for an M/M/c queue, the mean waiting time is a strictly decreasing and convex function of c. Reference [1] Dyer, M. E., Proll, L. G. (1977). On the Validity of ...
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  • 5,000
3 votes
Accepted

Average time between two dispatches in a taxi fleet (probably a batch processing queuing system)

Let $\tau$ represent the average time between dispatches. I'm skeptical about $\frac{\lambda T}{t}$ as a possible value of $\tau$ because it fails dimensional analysis. $\tau$ is in units of time. $T$ ...
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  • 28.7k
3 votes

Elevator Traffic (Queueing Theory) Papers

I am not an expert in elevator traffic system simulation, but by googling you could find lots of related papers and literature. Some of them are as follows: Modelling of Elevator Traffic Systems ...
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  • 5,705
3 votes
Accepted

The impact of utilization rate of a queueing system on its average queue time

I do not agree with the assertion that "both high and low utilization rate mean we are going to have a long average queue time". If $\rho$ is low (close to 0), $E[QT]$ is close to 0. Low ...
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  • 28.7k
3 votes

Question about a queueing problem

This is a $M/M/1$ queue with Poisson arrival distribution with $\lambda =1/10$ and Exponential service distribution with $\mu=1/3$. The proportion of the time that the system is busy can be calculated ...
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  • 8,185
2 votes

Queuing Theoretic Model with Memory

Since you are assuming infinite capacity, this sounds like an $M/M/\infty$ queueing system.
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  • 28.7k
2 votes

Good textbook for queueing theory and performance modeling

I learned from Quantitative System Performance Computer System Analysis Using Queueing Network Models by Lazowska, et.al. Unfortunately, it is no longer published, but it is available for free online....
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2 votes

Good textbook for queueing theory and performance modeling

Introduction to queueing theory and stochastic teletraffic models$^1$. The aim of this textbook is to provide students with basic knowledge of stochastic models that may apply to ...
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  • 8,185
2 votes
Accepted

Service probability for M/M/1 queue with reneging

The steady-state probability of being served for an M/M/1 queue with exponential reneging times and no balking is $$ p_s=\frac{1+z}{1+r(1+z)} $$ where $r=\lambda/\mu$ is the service intensity, and $$ ...
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  • 1,957
2 votes
Accepted

If $x=\min\{f(\mathbf{a}),1-\epsilon\}$, how can we model and partition $x$?

Start with the constraints $$x \le f(\mathbf{a})$$ and $$x \le 1-\epsilon.$$ If the nature of the problem is that larger values of $x$ are always preferable in objective terms to smaller values of $x$,...
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  • 28.7k
2 votes

MPX Queuing Software for Manufacturing with the GTHUBS Case

Edit: Grunchy delivers the perfect answer to your question. However, if you want to try out something else, see my answer below: ==== I recommend SIMIO if you just want to play around and visualize ...
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1 vote

Waiting time in M/M/n queue

You can derive them from the balance equations. If you check Taha's or Lieberman's Introduction to OR books, you can find the proofs.
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1 vote

How do derive the steady state probabilities M/M/1/k queueing system?

There are a couple of ways to derive the steady state probabilities for a $M/M/1/k$ queuing system with Markovian* arrivals (the first $M$), exponential service time distribution (the second $M$), a ...
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1 vote

Queuing models in R, $\lambda$ Little

So far this is what I have come up with lambda <- 2 interarrivals <- rexp(5000,lambda) ## (2 items per minute) Provided the $\mu$ we expect that the ...
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