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Questions tagged [queueing-theory]

Queueing theory is the mathematical study of waiting lines, or queues.

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Expected time for two new customers in a $(\lambda_A=1,\lambda_S=2)-M/M/2$ in steady state.

Two customers enter a $(\lambda_A=1,\lambda_S=2)-M/M/2$ in steady state. Find the expected time that it will take for both to make through queue and have their service done. My attempt: if this is a ...
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Service queue with expiration

from an Internet of Things setting we have the following question. There is an information service. It has a contract with a customer that it would provide information which is at most N seconds old (...
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Queue for Single Server with Capacity constraints

We have single server system and has a max capacity of 25 instances and each customer entering the server can occupy any number of instances(<= 25). Now we want to know if the current capacity is ...
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Are the two random variables independent?

Consider the following two random variables, In first case you record the number of people arriving at a queue, for a random amount of time. Note, here the arrival of people in the queue is random (...
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Queuing theory model for chain of queues and servers.

I want to apply queuing theory for our service which consists of multiple microservices which communicate via message queues. Basically, all events from the inbound queue are processed by one or ...
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Laplace-Stieltjes-Transform Waiting Time M/G/1 Queue

Given is an M/G/1 queue with $A_i$ i.i.d. $\sim$ exp and $B_i$ i.i.d. $\sim$ $F(t)$, $F(t)=1-\frac{1}{2}e^{-2t}-\frac{1}{2}e^{-\frac{1}{2}t}$ and $\rho=\frac{1}{2}$. I am supposed to see the ...
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How can I calculate probability of the waiting time in an M/G/1 queue?

Let $W$ denote the waiting time in an M/G/1 queue with customers arriving according to a Poisson Process and service times i.i.d with distribution $F(t)$. How can I calculate $\mathbb{P}(W>0)$ and ...
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Probability of the waiting time M/G/1 and G/M/1 queue

Task: Given is $F(t)=1-\frac{1}{2}e^{-2t}-\frac{1}{2}e^{-\frac{1}{2}t}$ and $\rho=\frac{1}{2}$. My task is to calculate 1) $\mathbb{E}[W]$, $\mathbb{P}(W>0)$ and $\mathbb{P}(W>1)$ in an M/G/...
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M/M/1 Queuing Process

I have two part question involving an M/M/1 queuing process. The first part I did but the second I'm not sure about. The question involves a gas-station pump where customers arrive at a rate of 20 ...
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Queueing theory analysis

I'm looking for references that analyze $M/M/c/K/\infty/D$ queues as $K$ and $D$ vary and $K$ is finite. My motivation is to compare the availability and statistics of the service times between ...
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m/m/1 queuing theory

I am trying to write a c simulation code for the system M/M/1/K, where K is system capacity that takes different values. I am a beginner and thus have limited knowledge on this.Please if you have any ...
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Deterministic model of throughput in a (single tread) production line

I am looking for a simple and canonical model of throughput (number of batches per unit of time) of a sequential and deterministic production line. For this simple model, I am searching for theorems ...
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Why are the transition probabilities for an $M/M/1$ queue $\lambda$ and $\mu$?

The transition state diagram for an $M/M/1$ queue is shown as The only explanation I have for the transition probabilites is that arrivals occur at rate $\lambda$ according to a Poisson process and ...
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What density distribution could be used for time delays?

I am analyzing a process related with time delays produced in scheduled departures or arrivals of a distribution trucks. The process consists on: The definition of the problem is: Each truck has ...
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Queuing processing and Gamma distribution

I've been trying to solve the following exercise and I was hoping for your input. If $Q$ is a queueing process with arrival rate $\lambda$ and service rate $\mu$, and a customer arrives to find ...
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mean and variance in backlog process

We have a discrete process $(N_k)_{k\in\mathbb{N}}$ defined as follows. At each time $k$ an amount of Poisson$(\nu)$ packets come in, but there is also a probability $1/e$ that exactly $1$ packet ...
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Little's Law to calculate threads in order to service client's requests.

I am totally dummy in math so do't mind if my question is silly, but i need help. According to an Article Little's Law can be used to determine number of threads required in the pool to effectively ...
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Probability of a being at a certain state after some time

A system can be in two states, $a$ and $b$. The waiting time for $a$ to become $b$ is modelled by means of an exponential distribution with parameter $\lambda,$ $1-e^{-\lambda t},$ and the waiting ...
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On distribution of the time between two consecutive events (either arrival or departure) in an M/M/1 queueing system

For an M/M/1 queueing system, distribution of the time ($t_e$) between two consecutive events (either arrival or departure) can be derived as follows with the independent assumption, $$F(t_e\ge t)=F(...
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If the inter-arrival times of customers are i.i.d. exponential distribution, is it necessary that the number of customers is a Poisson process?

Suppose customers arrive with time interval $U_i$ i.i.d. $Exp(\lambda)$, therefore, $$F(U_i\le t)=1-e^{-\lambda t}$$ The arrival time of customer $i$ is $$T_i=\sum^i_{j=1}{U_j}$$ The number of ...
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Queuing Theory m/m/1 system

A fast-food restaurant has one drive-through window. An average of 40 customers per hour arrives at the window. It takes an average of 1 minute to serve a customer. Assume that interarrival and ...
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How to formulate a queueing model where riders and drivers arrive randomly and are matched at every specific interval?

I am going to develop a queueing model in which riders and drivers arrive with inter-arrival time exponentially distributed. All the riders and drivers arriving in the system will wait for some ...
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How to determine the rate of input in a queue M/M/c.

I have the exit rate ($\mu$) and the average waiting time in the queue ($W_q$). I need solve to rate of input ($\lambda$) in a queue. I now: $\rho = \frac{\lambda}{c\mu} < 1$ $\pi_0 = \left[\left(...
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Estimate the shops total revenue per day.

You are a frequent customer at a coffee shop, where you typically wait 3 minutes to be served. Furthermore, on average, you spend 4 dollars per visit. Over many months you estimate that on a ...
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Queuing Model with Different Resource Requirements for Individual Users

Is there a queuing model where different customers have different resource requirements, and the service time depends on these requirements? For example, in a web server, different requests require ...
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The Men's Department.

I was given this question to solve: The Men's Department of a large store employs one tailor for customer's fitting. The number of customers requiring fitting has a mean arrival time rate of 24 per ...
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Solving Exponential Distributions with Preemptive Queueing

A router implements a preemptive priority queueing policy, where high priority packets are served first, and their arrival interrupts low priority packets’ service. If the service is interrupted, the ...
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Queueing Theory Help M/M/3

Comtex plc employs three people in its mail room to sort and despatch mail going through its internal mail system. Letters arrive at an average rate of 150 an hour and each employee can deal with 60 ...
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Multiclass M/M/1 queue which can simultaneously have one customer per class

Suppose there is a queue with exponential service time $\frac{1}{\mu}$ which accepts customers from K classes with Poisson distribution and rate $\lambda_k$ but if a customer from any class arrives at ...
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M/M/S Queue, probability interpretation

In an M/M/s queue, what does this expression mean? : $\sum_{n=0}^{s-1}{(s-n)P_n}$ Furthermore, is it possible that the following equation holds? : $\sum_{n=0}^{s-1}{(s-n)P_n} = (1-\rho)s$ If ...
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Are there general programmatic models for “servers” in queue systems?

Are there general programmatic models for "servers" in queue systems? Particularly, If one feeds clients with some params that are drawn from distribution, e.g. service time. Then once they go into ...
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What does “use exponentially distributed inter-arrival times, but make inter-arrival time 25” mean?

I'm asked to simulate inter-arrival times from exp-distribution and then I'm given that inter-arrival times must be 25, 40, 20, 40. I think these mean "average inter-arrival time", so does that mean ...
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How to analyze this type of queue

The setup is as follows: Families arrive at a taxi stand according to a Poisson process with rate $\lambda$. An arriving family finding $N$ other families waiting for a taxi does not wait. Taxis ...
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Number of people waiting in a M/G/$\infty$ queue at time t

For a M/G/$\infty$ queue (parameter of M is $\lambda=1$), we are given that G ~ U(0,3). My goal is to find the probability that there is no one waiting in the queue at time t = 10. So far, I have ...
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The average service time of the drive-in teller and the inside-bank teller?

A bank has one drive-in teller (who can serve customers without leaving their cars). The drive-in teller has a room (i.e., a queue) for one additional customer to wait. Customers arriving when the ...
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Comparison between mean time spent in M/M/1 and M/M/2

Prove that mean time spent in an M/M/1 system having arrival rate $\lambda$ and service rate $2μ$ is less than the mean time spent in an M/M/2 system with arrival rate $\lambda$ and each service rate $...
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how to plot the curves of the number of vehicles arriving in the line in function of time considering the speed in the queue?

A three lane highway is considered in each direction, with a free speed of $100~\rm km / h$ and a capacity of $2000~\rm veh / h / lane$. The average flow observed in the morning is $4500~\rm veh / h$. ...
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Queuing theory for task 1

A customer who comes to a fast food restaurant will get a $10\%$ discount if he arrived within $4$ minutes after the previous customer. If the time between the arrival is between $4$ and $5$ minutes, ...
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Can I apply queuing model in this process?

My model I want to apply queuing model to my process. There is total N users in a system. Initially, all user in A (in above diagram). The transition probability between each mode is written above ...
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Waiting time in the system using M/D/C queuing model

Can Anyone please tell me formula of expected waiting time in system for M/D/C queuing model? May I get it by replacing mu by c*mu in m/c/1 formula?
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Unused server in 2-server system with priority

The Problem Consider the following system: Customers arrive according to a Poisson process with intensity $\lambda$ over a time period $[0,T]$. There are two servers $X$ and $Y$ that can serve the ...
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Does the shuffling of a sequence of measurements produce an i.i.d. sequence?

Given a stationary sequence of measurements, say, of response times (i.e., sojourn times), of a queue (e.g., M/M/1), if we randomly reshuffle such samples, will we get a sequence of i.i.d. samples? ...
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In what cases is the average time in the system dependent / independent of a service discipline?

My teacher says that it does not depend on the discipline when we have M/G/1 (we did not learn others like G/G/k). I found this statement only for M/M/1 (here[page 8] and here). But I doubt it, even ...
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Adding a probability to a number (probability theory)

In the textbook of stochastic process, it says Suppose there is only one server. Let $L$ be the long-run average number of customers in the system. Let $L_Q$ be the average queue length in ...
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The u* rate for M/M/2 queue to have same “average waiting time in queue of a customer” with M/M/1 system with rate u

Ok here is the question, in a supermarket I have one cashier with u and a single M/M/1 queue, Suppose I want to get 2 cashier with same rate u*.What would should be the u* value for me to have same" ...
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Blocking probability of two non-preemptive priority queues with finite buffer sizes

I am trying to compare the blocking probability of two priority queues with finite buffer sizes. Specifically, the problem setting is described as follows: There are two queues, one is with higher ...
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Queueing theory - finding λ and μ from the text.

I have few tasks to do related to queuing theory which I struggle to understand key points. I have few texts and I'm supposed to find λ and μ which really tells me that I'm doing something wrong. I ...
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Is there is some literature for queue with random arrival/service rate?

I have read literature for queuing theory where we have time-varying rates. I am interested in knowing any literature that concerns itself with the service rate or arrival rate as a random variable ...
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What is the intuitive meaning of the capacity of a timing channel?

I am studying a timing channel. A queue can act as a noise source to such systems as it will disturb the information coded in the incoming time. If the queue has a deterministic service time than we ...
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calculating distribution of wait times for individuals in a queue

I'm attempting to generate output similar to the queueview program offered by ICMI (but can't use that program). For a given number of staff (computed using the Erlang-C probability in iterative ...