Chapter 8 Homework: Queues

(however, much of this homework depends on lecture notes rather than the textbook)


Your homework should be typed in Word or something similar, with
appropriate spreadsheet segments copied and pasted in.  

1. a) For an M/M/1 queue with lambda=9 jobs/hour and mu=10 jobs/hour,
compute L, W, Wq, and Lq using the formulas given in class.  While you may
_check_ your work using a queueing package like QTS Plus, you should do it
_by hand_ to start with.  Type up not just the answers but also the
calculations that get you there.

b) Repeat (a) after doubling the arrival and service rates
(lambda = 18 jobs/hour and mu=20 jobs/hour); comment on similarities
and differences with part (a).


2.  We said in class that you can't take a multiserver system and
analyze it with one really fast server.  How about splitting it
into many independent single-server systems?
For example, consider an M/M/c system with  mu=1 and lambda=50
with 57 servers. 
a) Compute L, Lq, W, Wq using software like QTS Plus or a web page like
http://www.cs.vu.nl/~koole/ccmath/ErlangC/index.php

b) Instead, consider 57 separate single-server systems, each still
with mu=1 but with lambda=50/57; compute L, Lq, W, Wq for each and in
total (multiply by 57 when appropriate).  Compare to part (a).

c) Explain to me how I decided to have 57 servers in part (a).  Hint: I
didn't use any software (or even a handheld calculator) to determine it.



3. Compute W and L for the large toll bridge example done in class:
Our formulas were:
lambda(t) = 5000*(1 - 0.8cos(2 pi t / 24))
for t measured in hours between 0 and 24.
The maximum departure rate was 500 cars/hour per lane, with 15 lanes,
for a total maximum of 7500 cars/hour.



4.  For the spreadsheet we did that simulated a single-server
First-Come-First-Serve (FCFS) queue using Lindley's Law:
Enter the formula =-LN(rand()) for each inter-arrival time (IATn).
Enter the formula =-0.9*LN(rand()) for each service time (Sn)
Simulate at least 1000 arrivals.
a) Find the mean and standard deviation of the IATs and the Sn.
b) Do a histogram of the IATs and the Sn.
c) Graph the resulting Wqn values.
d) Compute Wq from the simulation.
e) Compare Wq to Question 1, above.
f) Change the service times to be just =0.9 
(that is, all of them are just 0.9), and recompute Wq.
Comment on the results.

You should include a copy of the first part of your spreadsheet
in your report, but don't include all 1000 or so rows, please!

5. The League of American Bicyclists reported, on June 5th 2009,
"A 3 percent reduction in vehicle miles traveled (VMT) in 2008 resulted
in a 30 percent reduction in congestion in metropolitan areas around
the country, suggesting that demand management strategies such as 
increasing bicycling and walking are extremely effective in addressing
congestion and other traffic-related issues."
http://www.bikeleague.org/programs/bikeadvocacy/
a) Using M/M/1 thinking, figure out the relative utilization
(lambda/mu) before and after the 3 percent reduction in VMT.
b) Is M/M/1 an appropriate approximation here?  Explain.
c) Discuss the part of their sentence "suggesting that..."
using what you know from queueing theory.

6. In the government-health-care debate, according to cnn.com,
http://transcripts.cnn.com/TRANSCRIPTS/0907/21/cnr.07.html
SEN. JOHN BARRASSO (R), WYOMING
said "a ten-week wait for an MRI or CAT scan in Canada. 
It's something you can get done the same week here in the United States. "
a) Using M/M/1 thinking, figure out the relative utilization
for MRI or CAT scan machines in Canada.  Remember to account for
"business hours"--don't just do 10 weeks * 7 days/week * 24 hours/day.
Also interesting:
http://secure.cihi.ca/cihiweb/en/media_08feb2006_tab3_e.html
b) Is M/M/1 an appropriate approximation here?  Explain.