Queuing Home work #1 (Please comment, thanks! Ted)

[Ted_Li]

Homework problems for Queuing 1

1. Your business served 116,800 customers last year. Your business
hours are 9:00 a.m. to 5:00
p.m. 365 days a year. What is the lambda per hour?

Lambda = 116800/365/8 = 40 customers/hour


2. Assuming your arrivals of customers are totally random, explain how
you would interpret 1/lambda
for the business in question one.

1/lambda = 1/40 (hour/customer) = 60/40 (minutes/customer) = 1.5
(minutes/customer)
The average interval between 2 arrival customers is 1.5 minutes.

3. Assume the arrival rate of your customers is lambda = 20, and the
service rate of your production
facilities is mu = 30. Explain why there will sometimes be queues.

Because the arrivals are coming at random time, some time points there
will be more arrivals than the facilities can serve timely, so the
queue will form.

4. Suppose the arrival rate of a system is lambda = 3/hour and the
average service time is 15 minutes.
If the assumptions for a M/M/1 system are met, how many people would
you expect to be in the
system on average? If you could invest in some new equipment that
would speed up service
time to 12 minutes, how many people would you expect to be in the
system on average? What if
you could invest in some new equipment that would speed up service
time to 6 minutes, how
many people would you expect to be in the system on average?

Lambda = 3
Service time = 15 minutes, mu = 60/15 = 4
L = P0*0 + P1*1 + P2*2 + … = (lambda/mu)/(1-lambda/mu) = (3/4)/(1-3/4)
= 3 (in the system on average)
When service time = 12 minutes, mu = 60/12 =5
L = (3/5) / (1-3/5) = 1.5 (in the system on average)
When service time = 6 minutes, mu = 60/6 = 10
L = (3/10) / (1- 3/10) = 3/7 = 0.43 (in the system on average)

5. Suppose the arrival rate is lambda = 36/hour and the average wait
in line before service is 10
minutes. On the average, how many people will be in line?

Lambda = 36
Wq = 10 minutes = 10/60 hours = 1/6 hours
Lq = Lambda * Wq = 36 * 1/6 = 6 (average people stay in line)


6. Suppose the arrival rate of a system is lambda = 4 and mu = 3. How
many people would you expect
to be in the system on average?

Since lambda > mu, in average, there will be more arrivals than the
system can serve, over the time, the queue will increase to infinity
theriotically.

[Kartik Ramachandran]

I got the same.

Thanks,

Kartik

[Nicholas Woo]

I got the same answers for 1-5 but am having some trouble conceptualizing the answer of "infinite" for # 6. The question is asking how many people I would expect to be in the system on average if the arrival rate is greater than the service rate. With that said, it's a given that there will always be "some" number of people in the system.

Now here's where I'm not 100% clear. I agree that the number of people in the system could "grow" infinitely high (or to the max capacity of the real estate) over an extended period of time, but that doesn't necessarily answer the question of "how many people do we expect to be in the system on average," right?   

On Tue, Apr 19, 2011 at 12:41 AM, Ted_Li <xiu...@gmail.com> wrote:

[Nicholas Woo]

Does anyone know the practical interpretation of a negative L value?

--Nick

[alex smirnov]

I got the same answers as Ted

Nick, if lamda > mu, then the state is not in equilibrium, that's why I think infinity is the correct answer for #6

I don't think L can be negative.  How could there be a negative # of customers in the system or in the queue? 

[Nicholas Woo]

Right, a negative solution is clearly impossible. This leads me to believe that the correct answer to #6 is that the model is invalid when lambda exceed mu. We would know from logic that the number of customers in the system could grow infinitely large over an unspecified period of time, but we would not be able to conclude the exact average number of customers in the system.

[alex smirnov]

right, the model only works if lamda/mu<1

as far as average, if the # of customers continues to increase, what does "average" really mean?