lab #5 worksheet
----------------

- make a new folder for week5

- files for this week:

  w5exp.m
  w5queue.mws

- start matlab & change to the
  week 5 folder


warm-up 
-------

- run "w5exp", it shows 2 ways to 
  generate transition times that 
  obey the exponential transition
  time process of section 6.2.  it
  also generates the associated 
  probabilities by making a histogram
  of how often a transition time 
  falls into an interval (t->t+dt).

  note:  i used both "hist" and "histc"
  as was most appropriate in each case.  WHY?

- figure No.1 is made by direct simulation
  of figure 6.1.

- figure No.2 is directly made by a single 
  call to "rand"!

- conclusion:  there is a direct method to 
  simulate exponential transition processes.

- in-class challenge:  what is the mean 
  and variance of the two methods?

* be sure that you understand why the direct
  method works before you leave the lab.

- superbonus:  "w5exp.m" seems to have a 
  statistical glitch at t=3+3/8 -- why is this?
  (i've got no clue right now.)


study #1
--------

- use the direct method of generating
  exponential transition times to simulate
  the arrival times for the queueing process
  of section 6.3.

- you DON'T want to use figure 6.2!  we
  will use that approach in lecture tomorrow.

- build a matlab script that does a simulation
  loop of the queueing system of section 6.3.
  at t=0 there is nobody in line.  then arrivals
  happen "randomly" at intervals distributed
  by an exponential transition time with mu=1.

- build a simulation loop to collect the
  following data.  (mine is an easy rip-off
  from the w5exp loop!)

  a)  average queue size at t=25,
  b)  variance of the queue size at t=25,
  c)  probability histogram of queue sizes.

$ how many simulations one needs, seems to be
  a good topic for the discussion forum.

$ use Maple to calculate the p_j(Ndt) probabilities
  using the combinatorial formula given in lecture
  for the center of the distribution.  it is valid
  for the limit as dt->0.  how well did your 
  simulation fare?  to what extent can you tell?


study #2
--------

- use the direct method of generating
  exponential transition times to simulate
  the car arrival time for the street 
  crossing process of section 6.4.

- build a matlab script that does a simulation
  loop of the pedestrian system of section 6.4.
  pedestrian arrives at t=0.  then cars pass by
  "randomly" at intervals distributed by an 
  exponential transition time with mu=1.

- build a simulation loop to collect the
  following data, as a function of how large
  a gap time between cars that the pedestrian 
  wants.  

  a)  average # of cars which pass (assuming 
  pedestrian doesn't wish to use medical 
  coverage & SUV drivers don't stop to let 
  people cross),
  b)  variance of the # of cars,
  c)  probability histogram of # of cars.

  collect similar data on waiting times.

$ confirm as many of your results as you can 
  using the material of the lectures & text.
  (discussion here will likely generate hints
  & ideas.)  hopefully wednesday's lecture 
  will add some ideas.