The tag has no wiki summary.

learn more… | top users | synonyms

1
vote
0answers
11 views

Need guidance on a Queuing problem

I can't really go into specifics, I'm more just looking for terms that I can research to get on the right track. Classes of model/processes etc. A close analogy to my problem: I need to optimally ...
0
votes
0answers
15 views

The manager of the cafeteria wishes to calculate the average number of customers in the cafeteria? [closed]

Self-service at a university cafeteria, at an average rate of 7 minutes per customer, is slower than attendant service, which has a rate of 6 minutes per student. The manager of the cafeteria wishes ...
-4
votes
0answers
35 views

a little help please [closed]

In a small convenience store there’s room for only 4 customers. The owner himself deals with all the customers - he likes chatting a bit. On average it takes a customer 4 minutes to pay for his/her ...
0
votes
1answer
19 views

meaning of stationary distribution, null-recurrence and transience in queueing theory

I have 3 question about the meaning of the mathematical term in the reality. Let the queue $Q$ have the stationary distribution $\pi_n=(1-\rho)\rho^n$ for $n\geq0$. Does it mean, when $t$ is large, ...
0
votes
1answer
32 views

Explicit extinction probability of busy period

Let $Q$ be a M/G/1-queue. We denote by $B$ the busy period of the queue, that is defined as follows: $$B:=\inf \{ t>0: Q(t+T_1)=0 \},$$ where $T_1$ is the arriving time of the first costumer. ...
2
votes
1answer
76 views

Removing some arrivals from a Poisson Process

The inter-arrival time of a Poisson Process, $t$, conforms to the exponential distribution, so the probability density function for $t$ is $f(t)=\lambda e^{-\lambda t}$, $t>0$. ($\lambda$ is the ...
0
votes
1answer
24 views

Poisson Modeling/ Queue Theory - Reference Material

Can anyone reccomend some practical reference material related to building and implementing queueing theory models. using stochastic (prefferably Poisson) processes? We are looking to build out a few ...
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 ...
0
votes
1answer
54 views

M/G/1 queue has embedded Markov chain

I tried to prove that the M/G/1 queue has an embedded discrete-time Markov chain. But I'm not sure if I have done it right and properly. Specially I'm not 100% sure if i calculated right the ...
0
votes
3answers
100 views

Transition Matrix of M/M/1 Queue

We know that for an M/M/1 queue the state space is $S=\{0,1,2,... \}$. Further the probability to go from state $i$ to $i+1$ is $\lambda$ for all $i$ in $S$. Moreover, to go from $i$ to $i-1$ is the ...
1
vote
2answers
44 views

Questions about the functions of the operations of a queue

I am looking at the following two functions of two operations of a queue: ...
1
vote
1answer
43 views

How would I find the PDF of one poisson variable in terms of another?

I am looking for the PDF of a Poisson variable. The setup is that pit crews are working on racecars on the track after the drivers reports trouble. The pit crew's expected work time is exponentially ...
1
vote
1answer
46 views

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 ...
1
vote
0answers
46 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 ...
1
vote
1answer
29 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 ...
0
votes
1answer
37 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 ...
2
votes
0answers
57 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 ...
1
vote
1answer
41 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 ...
0
votes
1answer
60 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? ...
0
votes
0answers
31 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 ...
1
vote
1answer
42 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 ...
0
votes
2answers
52 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 ...
0
votes
0answers
11 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 ...
1
vote
0answers
68 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( ...
0
votes
0answers
17 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 ...
1
vote
1answer
58 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 ...
1
vote
1answer
53 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 ...
0
votes
0answers
24 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)} ...
0
votes
0answers
32 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 ...
2
votes
1answer
82 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 ...
4
votes
1answer
48 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 ...
1
vote
1answer
22 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 ...
0
votes
0answers
24 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 ...
0
votes
1answer
72 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 ...
1
vote
1answer
38 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 ...
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
63 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 = ...
0
votes
1answer
20 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 ...
0
votes
1answer
138 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 ...
1
vote
0answers
97 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 ...
2
votes
1answer
213 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
1answer
58 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. ...
0
votes
1answer
33 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 ...
0
votes
1answer
104 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 ...
0
votes
1answer
62 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 ...
0
votes
1answer
124 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" ...
0
votes
1answer
44 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 ...
0
votes
1answer
38 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 ...
1
vote
1answer
146 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 ...
0
votes
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 ...