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Determine wait time in queue without arrival rate

I'm interested in determining wait time in a queue. For example, I'm at the grocery store, there's a single line leading to a set of 5 cashiers, 10 people are in front of me. I know on average it ...
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0answers
25 views

Queueing Delay(W) for M/D/1 queue with different value of service times

I have a problem of calculate the queueing delay of M/D/1 queue. There are two different types of packets with different size. Such that, the arrival rate $\lambda$ and service times $\mu$ will not be ...
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1answer
29 views

Generating functions in a proof

I'm working at the moment with queues. In a proof of a theorem, there is the following statement: Suppose $G(s)$ is a generating function and we have the following conditions: ...
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1answer
17 views

Queuing theory-Multiple server (reducing simple recurrence formulas)

The equations given in 6.3 have been reduced which really eases the computation in further studies. But I tried to find the method of reducing these but I could not find a way at all. Any hints will ...
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1answer
35 views

How do you find the probability of a certain state in Markov Chain?

This question appears without answer in an old exam I found (not a homework question) Suppose messages that enter a system need to be processed by two servers. They arrive at the system at a ...
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0answers
49 views

Taking Laplace-Stieltjes transform to find virtual idle time in G/M/1 queue

I am reviewing some queueing problems from Gross and Harris, and had a question on problem 5.40 part b. The problem is stated as follows: Part B: Show that the stationary output of a $ G/M/1 $ queue ...
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33 views

Poisson Queuing Theory question

I am an undergraduate student and I have studying Queuing Theory in this summer semester. I am stuck on the single server queue model questions and I am not sure how to solve this - At a small ...
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1answer
33 views

Poisson processes and queues

I am trying to understand Poisson processes and queues. I have this exercise: Consider a fuel station with two fuel pumps and one park. Each car that comes to the fuel station when the pumps and the ...
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1answer
55 views

what is difference between open and mixed queue

please consider this image: In this picture that I got it from Here write said that Network C is Open,B is Mixed and A is Closed. I want to know why C isn't mixed? ...
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23 views

G/G/1 Queues - Book with Discrete Time Markov Chain examples

Need some book recommendation or links which have examples how to solve G/G/1 queues with detailed Discrete Time Markov Chain drawn and how to get the steady state distribution, the average number of ...
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1answer
39 views

Solution of a system of differential equations for a continuous time Markov chain.

The equations arise as the Laplace transforms of the forward equations of a continuous time Markov chain for a three-state system, with the following transition rates: Transition , rate $0 ...
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2answers
42 views

Continuous Markov chains, arriving pairs

I have been trying to sort out this exercise but really stuck on this. Preparing myself for exams and found many exercise on continuous Markov chains but I am always stuck when it comes to transition ...
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10 views

conditional mean of geometric RV

Say, there are three nodes: $S$, $R$, $D$. $S$ transmits to $R$, $R$ stores the packets, and later transmits to $D$. At any time, either $S$ or $R$ is selected to transmit according to some random ...
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0answers
48 views

Solving Probabilities for M/M/1 Queue Waiting Time Generating Function

I "believe" that generator, $\bf Q$, of the waiting time distribution for the $M/M/1$ queue is given by the following (I'm not 100% sure if this is even correct): $\bf Q$ = $\left( ...
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0answers
14 views

Estimating queue overflows

(This come up trying to debug a real system at work, and got me wondering...) We have a server that consumes a queue, one item at time. Processing the item is quite quick, but not instant. There is ...
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1answer
56 views

Which Queue to Join at the Super Market

Last night I started wonder about the fastest way to take a shopping trip with my university flat mates and was wonder about how we should queue for the check out. I have a feeling that queue theory ...
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1answer
46 views

What is the distribution of the service-starting time lag w.r.t. two concurrent customers from two parallel $M/M/1/1$ queues?

Consider two parallel, independent $M/M/1/1$ queues (denoted $Q_i, Q_j$) with identical arrival rate $\lambda$ and service rate $\mu$, using FCFS (First Come First Served) discipline. Note that the ...
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0answers
16 views

Conditional Poisson PMF ~ the joint PMF not independent?

Let X denote the number of customers who arrive during a service time and Y the first service time. Customers arrive in a Poisson process with rate $\lambda$. Then: $P(X=x|Y=y)= e^{-(\lambda y)} ...
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20 views

Steady state queue-length distribution of M/G/$1$ queue

The steady state system-length distribution of a FCFS $M/G/1$ queue is well known (see here for reference). I am trying to find it from a different approach, using Little's law, but I am not sure if I ...
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0answers
56 views

What does a customer see when it begins to be served in $M/M/1$ queue?

In queueing theory, the PASTA (Poisson Arrivals See Time Averages) principle [wiki] justifies $a_n = P_n$ where $$a_n = \text{proportion of customers that find } n \text{ in the system when they ...
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1answer
45 views

Rouché theorem in queuing theory

I was looking for the uses of Rouché's theorem, and I came across queuing theory. An article stated that it is a workhorse theorem in this field, but as much as I tried to find some examples on the ...
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1answer
21 views

Poisson arrival and selective removal

Users arrive according to Poisson process with rate λ. If every third user is removed, then do the remaining users form a Poisson process with rate 2λ/3? If every other user is removed, then do the ...
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0answers
20 views

Stationary distribution of Waiting Time in a $GI/GI/1$ queue

I am trying to find if there is any literature where I can find formulas for the stationary distribution of a $GI/GI/1$ queue. Specifically, I need to find $P(W=0)$ where $W$ is the steady state ...
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1answer
51 views

M/M/1 Queuing Theory Question

Lets say I have packets arrive to a terminal at Poisson rate $\lambda$ per hour and my terminal has an exponential service rate $\mu$ per hour (so the mean service time is $\frac{1}{\mu}$). So this is ...
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1answer
29 views

Strict proof of markovity of queing system of type $M/M/n/\infty$

I have a queing system of type $M/M/n/\infty$. The service time is exponential, and the arrival process is poisson. I do understand that because of these two facts the future of the system in ...
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39 views

Probabilistic model of parallel web servers

Note: The following probabilistic model of parallel web servers is abstracted from an engineering project. I am not good at probability theory and I am seeking some evaluations and suggestions. ...
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1answer
51 views

M/M/1 queue with probability of new client leaving

I'm looking at a M/M/1 queue system and trying to show that $\{M_t\}_{t\geq}0$, the number of clients in the system, is a birth-death process. In the simplest of cases this is true if $\lambda_i = ...
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1answer
18 views

Closed queueing networks

When applying Norton's theorem to a closed queueing network, most documentation I found assumes same service rates for all queues. How do you calculate the throughput and delay if the service times ...
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1answer
74 views

M/D/c queue with different service times

In formulas on M/D/c queue it is assumed that service time is the same for all servers. Are there formulas for the case when service times differ between servers (in general case there would be as ...
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0answers
74 views

Poisson distribution-Queueing theory

Vehicles arrive at a junction, in order to swing left, create a line queue ( tail) . The number of vehicle follow Poisson distribution. The length of cycle for the traffic light (for left turns ) is 1 ...
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1answer
138 views

Conditional expectation of conditional expectation

I have a question about conditional expectation. I have always problem with that... It is a step of a proof that I just don't get... I appreciate any help! I have the random variable $$B=S+ ...
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1answer
50 views

What is the Deterministic Traffic Generation Model?

I am studying Markov chains and queuing theory. I was curious about traffic generation models and actually happened to see the Deterministic Traffic Model, referred to as $D$ in Kendall's notation. ...
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1answer
29 views

Queue system with queue-triggered input process

I have a queue system, a classic system with an input generator, a queue and a servant. The servant is a $M$-servant with a certain serving rate $\mu$. The queue can contain an infinite number of ...
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1answer
82 views

M/M/3 queue - reducing wait time by adding servers

Full question below: You are the manager of the customer support division in your company. Your division uses 3 telephone lines operated by 3 separate customer service representatives. A customer is ...
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1answer
60 views

Queuing model $M/M/\infty$

I am considering a queuing model of the form $M/M/\infty$, you find properties of this queue here: http://en.wikipedia.org/wiki/M/M/%E2%88%9E_queue I am interested in the average busy period of this ...
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1answer
118 views

M/H2/1 Queue - Explicit Expression for Response Time Distribution

I am looking for a reference to an explicit expression to the $M/H_2/1$ queue's response time distribution. I.e., when you invert the PK-Formula, I am looking for a reference that gives a "nice" ...
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1answer
41 views

Statisitics for queueing models

I am studying various parameters related to queueing models. Does M/M/1 have lesser delay compared to M/G/1? I think yes. Can anyone verify this? How does this compare to G/M/1? Anyone have any ...
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1answer
37 views

M/M/1/N queue packet

If service rate is the same as the arrival rate for M/M/1/N queue, then intuitively I think that no packets will be dropped. However, using the formula for waiting time, I get a waiting time of ...
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1answer
120 views

Queuing Theory with Poisson Distribution

Suppose customers arrive in a one-server queue according to a Poisson distribution with rate lambda=1 (in hours). Suppose that the service times equal 1/4 hour, 1/2 hour, or one hour each with ...
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1answer
35 views

Std Dev of large set

Lets assume we have a router that transmits packets at 24,000 bytes/msec and that packet lengths are uniformly distributed between 100 and 1,500 bytes. Packets are incoming to the router at a rate of ...
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0answers
49 views

Does there exist a nontrivial cumulative distribution function $F$ on $\mathbb{R}_+$ for which $(F^2)'= (F')^{\ast k}$ for some $k > 0$?

This question arises from the context of computing the distribution of total execution time with an underlying graph of tree. For instance, we can model the execution times of all the nodes on the ...
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1answer
100 views

M/M/s Queue - Gas Station

I have been trying to wrap my head around an M/M/s queue problem but I can't seem to understand what's going on. The problem is queuing at a gas station and I have done some research about the M/M/s ...
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1answer
64 views

Application of queueing theory

Jake's Machine Shop contains a grinder for sharpening the machine cutting tools. A decision must now be made on the speed at which to set the grinder. The grinding time required by a machine operator ...
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1answer
37 views

Queueing Theory: Router question

I have this question I am struggling with: Suppose that packet interarrival time and packet service times at a router follow an exponential probability density function and the router will be ...
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2answers
71 views

Calculate average time to empty the router

Consider a buffer, in which every second the number of packets increases by 1 with probability. Currently there are n packets in the router. Calculate the required to empty. Any help?
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0answers
118 views

How to count discarded entities in a FIFO queue using Simulink?

I'm trying to model a single queue, single server simulation using Simulink in MATLAB, I've recently installed it and I'm pretty new. I've created a Time-Based Entity Generator (with an exponential ...
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2answers
68 views

Random interarrival times (poisson process)

In their monograph "Queues", Cox and Smith state (paraphrased - this is p5): In interval $(t, \Delta t)$ the probability of no arrivals in a completely random process is $1 - \alpha \Delta t + ...
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46 views

Finding the inverse of Laplace–Stieltjes transformation and Convolution related to Probability

I would like to ask you something I do not understand from my book. If I have G following exponential distribution with $G(t)=1-e^{-\lambda t},t\geq 0$ then ...
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1answer
51 views

Inequalities for the tail of the normal distribution (Halfin-Whitt paper)

I am reading the famous paper by Halfin and Whitt, [1]. I'd like to prove remark (1) on page 575. The authors state \begin{align} \frac{\beta \alpha}{(1-\alpha)} = \frac{\phi(\beta)}{\Phi(\beta)} ...
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43 views

Is this two dimensional Markov chain correct for this queueing system?

The problem that I have two single server station with no queuing space a customer goes to station 1 if it is available else it goes to station 2 if it is available or it will be lost output from ...