1
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1answer
56 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
46 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
0answers
56 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{ in the system when they ...
0
votes
0answers
20 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
60 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 ...
1
vote
2answers
67 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 + ...
3
votes
1answer
51 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)} ...
1
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1answer
64 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 ...
0
votes
1answer
253 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 ...
2
votes
1answer
58 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 ...
1
vote
1answer
50 views

Mean number of particle present in the system: birth-death process, $E(X_t|X_0=i)$, $b_i=\frac{b}{i+1}$, $d_i=d$

Let $\{X_t\}$ be a birth–and–death process with birth rate $$ b_i = \frac{b}{i+1}, $$ when $i$ particle are in the system, and a constant death rate $$ d_i=d. $$ Find the expected number of particle ...
2
votes
0answers
158 views

Boundedness of expected reward Markov chain (may be related to discret $M/M/\infty$ queue)

[EDIT]: I read a bit on $M/M/\infty$ queue and it may not be the right comparison and my notation may be confusing (I'm in discrete time and $\lambda,\mu$ look likes rates when they are probability). ...
3
votes
2answers
1k views

One vs multiple servers - problem

Consider the following problem: We have a simple queueing system with $\lambda%$ - probabilistic intensity of queries per some predefined time interval. Now, we can arrange the system as a single ...