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Questions tagged [queueing-theory]

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

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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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What is the transition rate matrix of two M/M/1 queues in parallel?

Is it, $\begin{pmatrix} -\lambda & \lambda \\ \mu & -(\mu+\lambda) & \lambda \\ &\mu & -(\mu+\lambda) & \lambda \\ &&\mu & -(\mu+\lambda) & \lambda &\\ &...
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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 ...
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Arrival and departure probabilities in an Erlang Loss system

Consider an M/G/c/c system; that is, an Erlang loss system with a general service time. If the system is in the steady state situation, what is the probability that a randomly observed event is an ...
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Ensure equal work distribution in a group whose “composition” changes daily.

I want to ensure that incoming work is equally distributed as much as possible across a group of 100 people. I am not concerned about the complexity of each piece of work. I just want to ensure that ...
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Split Poisson arrival according to output of Optimization problem

I have 3 nodes with Poisson arrival rates $\lambda_1, \lambda_2, \lambda_3$. As shown. The arrivals (packets) are sent to either of the two nodes according to the binary optimization varible $x_{ij}$. ...
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Why does $E[R] = E[B]$ for an $M/M/1$ queue?

I am preparing for an exam and understand why the mean service time $E[B]$ equals $\frac{1}{\mu}$ for an $M/M/1$ queue but I fail to intuitively understand why the mean remaining service time is the ...
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Transient Analysis

Consider a M/G/∞ queue that can be modeled as continuous time markov chain with birth and death process (erlang and poisson distributed respectively in my case). I have to perform transient analysis ...
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M/M/1 and M/M/2 in the same Queue based on number of customers

Suppose you have a queue that starts as M/M/1 with an initial birth rate and initial death rate but the birth rate is larger than the death rate. Once you have n customers in the Queue it switches to ...
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Recovering components of a Poisson process

Consider a scenario with two independent input Poisson processes with rates $P_1 \sim Pois(\lambda_1)$ and $P_2 \sim Pois(\lambda_2)$. These enter a queueing system to form a combined process of rate ...
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What is the difference between “total service time” and “service time”?

In http://www.win.tue.nl/~iadan/queueing.pdf the beginning of section 7.7 (page 70), they write about the total service time of the customer in service, $X$. But isn't this exactly the same as $B$, ...
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arrival rate in single server with general service time distribution.

Customers arrive at a single-server station with Poisson rate $\lambda$. A customer enters the bank if the server is available; otherwise, the customer leaves. The service times of successive ...
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Probability distribution of $W_{q}$ in an $M/M/2$ queuing system

For an $M/M/1$ queuing system, a probability distribution of the waiting time in the queue can be written as $$\mathbb{P}(W_{q}> t)= \rho e^{-\mu t(1-\rho)}$$ where $t$ is some time and $\rho$, as ...
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How would I determine how large an M/M/c queue grows after a certain amount of time?

Most of the reading on queueing theory I've done focuses on when the arrival rate is less than the service rate so that the queue doesn't explode. However, if I have a queueing system (either M/M/1 or ...
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Stationary distribution of the Workload of G/G/1

I dont understand one integration in the proof given in "Applied popability and Queues" Asmussen. Page 275 Theorem 3.4. You dont need the Book to answer. The author shows how the stationary ...
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Probability that one M/M/1 queue empties before another queue with no external arrivals?

Suppose there are 2 independent queues A and B having m and n customers respectively. The service time of each of the queues are Exponentially distributed with service rate as $\mu_{A}$ and $\mu_{B}$. ...
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Help on cancelling out or substituting for n in the equation for P0 of an M/M/c system

I am currently trying to derive the queueing equations for an M/M/c queueing system. The method I am using is to consider to take the Markov chain and use the rate in = rate out principle to first ...
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Expected Waiting Time in a Poisson Process when Events have different Probabilities

Assume I have service times distributed according to a poisson process. There are 3 servers with service rates $\lambda_1$, $\lambda_2$ and $\lambda_3$ and each customer needs to visit them all in the ...
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Probability bound for max queue length M/D/1

I have searched for and not found an analytic upper bound $f$ $P(Q(t) \geq a) \leq f(t,a)$ where $Q(t) = \max_{s \leq t} q(s)$, $q(s)$ is the queue length at time $s$, and $t,a$ are finite. This is ...