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18 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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24 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
17 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
13 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
33 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
60 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
65 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
34 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
20 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
39 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
56 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
64 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
31 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
31 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
78 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
28 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
43 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
53 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
53 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
26 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
69 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
64 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
60 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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42 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
45 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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37 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 ...
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2answers
120 views

What property of queuing systems does 'first-come-first-served' maximize?

'First come, first served' is a very very common way of organizing access to a limited resource or service in the real world. It can be explained by saying that whenever the resource is available the ...
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1answer
61 views

Queueing model - expected outflow

Can anybody please help me how to tackle this question? We have one server. The service time is random with mean 1 minute The arrival rate is constant with 3 customers/minute, but they leave if the ...
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1answer
114 views

Two different M/G/Infinity queues

I need some help with this problem. We have a service system where customers arrive randomly, following a Poisson process (intensity λ). The times that the customers spend in service are independent, ...
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0answers
52 views

Queueing theory, meaning of Beta in a Weibulldistribution

I'm having some issues with a service system where customers arrive randomly, following a Poissonprocess. The lengths in time that customers spend in service are independent and Weibull distributed ...
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1answer
158 views

How to prove difference between two independent poisson process is not a poisson process?

It will come under properties of poisson process in some books. The sum of two independent poisson process can be proved as a poisson process using its memoryless property but how to prove difference ...
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0answers
32 views

Statistics & queues

I'm having some issues with a statistics problem. We have a service system where customers arrive randomly, following a Poisson process (intensity λ). The times that the customers spend in service ...
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0answers
97 views

Problem with M/M/2 queueing theory with different service rate $\mu$

I have a problem using M/M/2 queue theory. In this case, the service rate $\mu$ varies with constant $\lambda$. For example : between state 0 and 1, I use service rate $\mu$ but for other states I use ...
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0answers
45 views

M/M/c Queue determine $\Pi_0$

I'm trying to determine how to calculate the probabilities of going from one state to another. In a M/M/4 queue. The information I initially have is the following: $\lambda$ and ...
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71 views

A problem on queues.

"Consider a queueing process with two servers in which the inter-arrival times and service times are exponential with parameters $\lambda$ and $\mu$ respectively. Let $X_n$ be the queue length at the ...
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1answer
53 views

Probability of voting

One hundred students are divided into two equal groups. Both groups vote (yes or no). Find the probability that the groups 1 and 2 both reach a majority. Students vote independently and the ...
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39 views

Statistics - how would I answer this question?

I'm not too sure where to start or how to approach this question? Would anyone have any ideas or be able to help me? Thank you. A security-screening operation for packages sent through the mail has ...
0
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1answer
65 views

Mean sojourn time for a non-Markovian chain? e.g., an M/G/C/C queueing system?

For an M/M/c/c queueing system (when it's at equilibrium) the mean sojourn time of each state can be calculated using the diagonal entries of its transition rate matrix (or the infinitesimal generator ...
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2answers
130 views

Martians and Jovians

In how many ways can five distinct Martians and eight distinct Jovians wait in line if no two Martians stand together?
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63 views

Formal Theory Regarding M/M/s Queue

I have some difficulties with formally deducing the Q-matrix or infinitesimal generator for M/M/s Queues. Although I undestrand the intuitive idea I would like to know the real formal definition of ...
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1answer
83 views

How to model a client / server interaction with queuing theory

I'm interested in modeling a server application where the normal flow of data is as follows: Server A -> Server B -> Server C -> Server B -> Server A That is to say, a job originating from A makes a ...
0
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1answer
76 views

Product form Solution of Jackson Network

Consider a Jackson network with nodes $\{i:1\leq i\leq n\}$ which have the arrival rates $\{\lambda_i\}_{i=1}^n$ from outside and service rates at each node $\{\mu\}_{i=1}^n$. Define ...
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1answer
102 views

Birth-death process: Calculate number of clients that can really enter the system for each unit of time

For a birth-death process that is M/M/1/4/$\infty$/G (as a queue system), how can I calculate the number of clients that can enter the system for each unit of time. Is this solving for $\lambda$ in $4 ...
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0answers
55 views

Stability condition for queueing model with 2 servers, non-identical hypoexponetial-2 and FCFS

We consider a queueing model with 2 servers and a shared queue, let us label the servers as $A$ and $B$. Jobs arrive in batches of size $N$ with rate $\lambda$. Each server has a hypoexponential ...
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1answer
1k views

Utilization difference between a multiple server, single queue and a multiple server, multiple queue system

I've been studying Queue theory for a while and I'm interested in finding out the solution to the following problem. Suppose that we have a system with 4 processors where the arrival rate(λ) is 0.2 ...
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1answer
204 views

Infinite average queueing delay for M/M/1 queues

According to queueing theory, the average queueing delay for an M/M/1 queue can be calculated as $\frac{1}{\mu-\lambda}$, where $\mu$ is the average service rate, and $\lambda$ is the average ...
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1answer
554 views

utilization of m/m/c queue

What is the utilization of m/m/c queue ? in some texts such as http://www.amazon.com/Queueing-Networks-Markov-Chains-Applications/dp/0471565253 said: individual server utilization $\rho ...
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1answer
110 views

m/m/2 question in queueing theory

Customers arrive at a serving-system according to a Poisson process with rate 1. In the system there are two serving stations, A and B, which only take care of one customer at a time. If a customer ...
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1answer
180 views

markov chain application

Two workers handle three machines(i.e. we can at most repair two machines at a time). The time until the machine breaks down is exponentially distributed with expectation value 1/2 and independent of ...
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1answer
167 views

Question on M/M/s queue

costumers arrive to a service station according to a poisson prossees and on average 2 during an hour.the service times and independent of the arrivals and internally independent with mean 45 minuts ...