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11 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 ...
Mark L. Stone's user avatar
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 ...
Mark L. Stone's user avatar
9 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.
Grunchy's user avatar
  • 106
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 ...
dhasson's user avatar
  • 1,687
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 ...
SecretAgentMan's user avatar
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 ...
CMichael's user avatar
  • 1,333
6 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 ...
Oguz Toragay's user avatar
  • 8,652
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. ...
wchargin's user avatar
  • 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.
LarrySnyder610's user avatar
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 ...
Dylan Solms's user avatar
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 ...
prubin's user avatar
  • 39.3k
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
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 ...
TheSimpliFire's user avatar
  • 5,412
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 ...
Dylan Solms's user avatar
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 ...
A.Omidi's user avatar
  • 8,950
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 ...
prubin's user avatar
  • 39.3k
3 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 $x$,...
prubin's user avatar
  • 39.3k
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. $T$ ...
prubin's user avatar
  • 39.3k
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 ...
Oguz Toragay's user avatar
  • 8,652
3 votes
Accepted

Confusion with Expected Queue Length (Lq) Formulas for M/M/s Model in Queuing theory

They are the same, with the caveat that the two sources are using slightly different notation. All symbols other than $\rho$ are the same in both places, but Hillier and Lieberman use $\rho = \frac{\...
prubin's user avatar
  • 39.3k
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 $$ ...
David M.'s user avatar
  • 2,107
2 votes

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 ...
SecretAgentMan's user avatar
2 votes

Queuing Theoretic Model with Memory

Since you are assuming infinite capacity, this sounds like an $M/M/\infty$ queueing system.
prubin's user avatar
  • 39.3k
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....
RBarryYoung's user avatar
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 ...
Oguz Toragay's user avatar
  • 8,652
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 ...
zeybringtbei's user avatar
2 votes

Departure process of M/G/1 queue with hyper-exponential service times

It is not true. You can refer to the following two papers, which provide a general method to analyze the departure process. The Departure Process of the GI/G/1 Queue and Its MacLaurin Series. ...
Weimin Dai's user avatar
1 vote

Modeling an assembly line as a batch processing $M/D^{(b,b)}/r$ queue

t= start of service to end of service. The t for return will be part of average wait time for the queue.
Sutanu Majumdar's user avatar
1 vote

Queueing model without steady state

Most of the formulas I have come across in queueing theory assume steady state, and I don't know of a purely mathematical approach for all cases. So let me show you a simple simulation approach. Using ...
Galen's user avatar
  • 141

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