2
votes
0answers
132 views

Changing a queueing processes

Situation Consider a general queueing system $\mathscr{S}$, whose customer arrival times are independent, and whose service times are independent; both of these are allowed to have general ...
1
vote
1answer
52 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 ...
2
votes
1answer
77 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{ customers in the system when ...
0
votes
1answer
68 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 ...
0
votes
0answers
40 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. ...
0
votes
1answer
60 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 = ...
2
votes
1answer
196 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+ ...
0
votes
0answers
48 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 ...
0
votes
1answer
315 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 ...
1
vote
0answers
232 views

Modified M/M/1/2 with 2 possible arrival rates and M/M/1/5 queue

I've been stuck on this question for hours, and could use some help :) "An M/M/1/2 queue has service rate $\mu$ and arrival rate of either $\lambda_1$ or $\lambda_2$. The rate can change only when ...
2
votes
1answer
202 views

Question on M/M/2 queue variation

I have the following question: 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 exponential distributed with expectation ...
0
votes
1answer
245 views

M/M/1 queues And finding equilibrium probability that the shop is empty

Customers arrive at a barbers shop at the incidents of a Poisson process of rate λ. Each person is served in order of arrival (by the single barber), and takes an exponential, rate μ service time. ...
0
votes
1answer
108 views

Birth processes with immigration and catastrophe

On the volcanic island of Montserrat the number of species increases(by immigration from neighbouring islands) at rate α. However, at rate η the volcano explodes, and all life is wiped out, although ...
3
votes
1answer
71 views

How to get transition rates in a $M/M/\infty$ queue

I am told for an $ M/M/\infty$ queue the transition rates $q$ are as follows. $q(n,n+1) = \lambda$ $q(n,n-1) =n\mu$ Can anybody explain the intuition behind $q(n,n-1)$?
4
votes
0answers
157 views

Is Queueing Theory dead? [closed]

I was studying queueing theory for my class and noticed that we are now able to either solve with certainity most queiening problems or simulate them. is queueing a dead research area? I read this ...
1
vote
1answer
132 views

Two stage cyclic queue

Given a cyclic queue of two servers of exponential service rates, if there are N customers at one server at time t, how do i start about showing that N can be modeled as a birth and death process? and ...
2
votes
1answer
236 views

M/G/1/K - evalutate birth and death rates

Within a queue with capacity = K and exponential interarrival times, death rate is μ and birth rate λ. A packet is discarded when the queue is full. When the source is active there's a probability ...
0
votes
1answer
400 views

variable death / birth rate in stochastic process

Within a queue with capacity = K death rate is μ and birth rate λ. A packet is discarded when the queue is full with probability Pk=P(K elements in the queue) Moreover there's a probability $p1 > ...
1
vote
1answer
119 views

Queueing Theory - Probability that all jobs have been served?

Suppose I have M/M/1 system with $\lambda = 4$ per hour and $\mu = 5$ per hour. How can I find out if all jobs have been served after, say, 8 hours? At first I thought about doing $P(n > 40)$ since ...
5
votes
1answer
401 views

Crowded and quiet periods in a $M/M/1$ queue

I'm trying to solve the following exercise (not homework): Consider a $M/M/1$ queue with an arrival rate of 60 customers per hour and a mean service time of 45 seconds. A period during which there ...
3
votes
2answers
436 views

Birth and Death Process Question (Queuing)

A small shop has two people who can each serve one customer at a time. There is also space for two customers to wait. Anyone who arrives and sees that the shop is full will go to another store. ...