October, 2011


20
Oct 11

Queueing up in R, continued

Shown above is a queueing simulation. Each diamond represents a person. The vertical line up is the queue; at the bottom are 5 slots where the people are attended to. The size of each diamond is proportional to the log of the time it will take them to be attended. Color is used to tell one person from another and doesn’t have any other meaning. Code for this simulation, written in R, is here. This is my second post about queueing simulation, you can find the first one, including an earlier version of the code, here. Thanks as always to commenters for their suggestions.

A few notes about the simulation:

  • Creating an animation to go along with your simulation can take a while to program (unless, perhaps, you are coding in Flash), and it may seem like an extra, unnecessary step. But you can often learn a lot just by “watching”, and animations can help you spot bugs in the code. I noticed that sometimes smaller diamonds hung around for much longer then I expected, which led me to track down a tricky little error in the code.
  • As usual, I’ve put all of the configuration options at the beginning of the code. Try experimenting with different numbers of intervals and tellers/slots, or change the mean service time.
  • If you want to run the code, you’ll need to have ImageMagick installed. If you are on a PC, make sure to include the full path to “convert”, since Windows has a built-in convert tool might take precedence. Also, note how the files that represent the individual animation cells are named. That’s so that they are added in the animation in the right order, naming them sequentially without zeros at the beginning failed.
  • I used Photoshop to interlace the animated GIF and resave. This reduced the file size by over 90%
  • The code is still a work in progress, it needs cleanup and I still have some questions I want to “ask” of the simulation.

13
Oct 11

Waiting in line, waiting on R

I should state right away that I know almost nothing about queuing theory. That’s one of the reasons I wanted to do some queuing simulations. Another reason: when I’m waiting in line at the bank, I tend to do mental calculations for how long it should take me to get served. I look at the number of tellers attending, pick an average teller session length (say one or two minutes), then come up with an average wait per person in line. For example, if there are 4 tellers and the average person takes 2 minutes to do her transaction, then new tellers should become available every 30 seconds. If I’m the 6th person in line, I should expect to wait 3 minutes before I’m attended.

However, based on my experience (the much maligned anecdotal), it often takes much longer than expected. My suspicion is that over time the teller’s get “clogged up” with the slowest people, so that even though an average person might take only 2 minutes, the people you actually see being attended right now are much more likely to be those who take a long time.

To explore this possibility, I set up a simulation in R (as usual, full source code provided at end of post). The first graph, at the beginning of this post, shows the length of queues over time for 4 runs of the simulator, all with the same configuration parameters. Often this graph was completely flat. Note though that when things get out of hand in terms of queue length, they can get way out of hand. To get a feel for how long you would have to wait, given that there is a line in front of you, I tracked how long the first person in line had to wait to be served. Given my configuration settings, this wait would be expected to last 5 intervals. It does seem to take longer than 5 intervals, though I want to tweak the model some and do more testing before I’m ready to quantify that wait.

There are, as with any models, things that make this one unrealistic. The biggest may be that people get in line with the same probability no matter how long the line is. What I need is some kind of tendency to abandon the line if it’s too long. That shouldn’t shorten the wait times for those already in line. I could make those worse. If you assume that slower people are prepared to wait longer in line, then the line is more likely to have slow people. Grandpa Jones is willing to spend an hour in line so he can chat for a while with the pretty young teller; but if the line is too long, that 50-year-old business guy will come back later to deposit his check. I would imagine that, from the bank’s perspective, this presents a tricky dilemma. The people whose time is worth the least are probably the most likely to be clogging up your tellers, upsetting the customers you care the most about (I know, I know, Bank of America cares about all of us equally, right?).

Code so far, note that run times can be very long for high intervals if the queue length gets long:

#### Code by Matt Asher. Published at StatisticsBlog.com ####
#### CONFIG ####
# Number of slots to fill
numbSlots = 5
 
# Total time to track
intervals = 1000
 
# Probability that a new person will show up during an interval
# Note, a maximum of one new person can show up during an interval
p = 0.1
 
# Average time each person takes at the teller, discretized exponential distribution assumed
# Times will be augmented by one, so that everyone takes at least 1 interval to serve
meanServiceTime = 25
 
#### INITIALIZATION ####
queueLengths = rep(0, intervals)
slots = rep(0, numbSlots)
waitTimes = c()
leavingTimes = c()
queue = list()
arrivalTimes = c()
frontOfLineWaits = c()
 
 
#### Libraries ####
# Use the proto library to treat people like objects in traditional oop
library("proto")
 
#### Functions ####
# R is missing a nice way to do ++, so we use this
inc <- function(x) {
  eval.parent(substitute(x <- x + 1))
}
 
# Main object, really a "proto" function
person <- proto(
	intervalArrived = 0,
	intervalAttended = NULL,
 
	# How much teller time will this person demand?
	intervalsNeeded = floor(rexp(1, 1/meanServiceTime)) + 1,
	intervalsWaited = 0,
	intervalsWaitedAtHeadOfQueue = 0,
)
 
#### Main loop ####
for(i in 1:intervals) {
	# Check if anyone is leaving the slots
	for(j in 1:numbSlots) {
		if(slots[j] == i) {
			# They are leaving the queue, slot to 0
			slots[j] = 0
			leavingTimes = c(leavingTimes, i)
		}
	}
 
	# See if a new person is to be added to the queue
	if(runif(1) < p) {
		newPerson = as.proto(person$as.list())
		newPerson$intervalArrived = i
		queue = c(queue, newPerson)
		arrivalTimes  = c(arrivalTimes, i)
	}
 
	# Can we place someone into a slot?
	for(j in 1:numbSlots) {
		# Is this slot free
		if(!slots[j]) {
			if(length(queue) > 0) {
				placedPerson = queue[[1]]
				slots[j] = i + placedPerson$intervalsNeeded
				waitTimes = c(waitTimes, placedPerson$intervalsWaited)
				# Only interested in these if person waited 1 or more intevals at front of line
				if(placedPerson$intervalsWaitedAtHeadOfQueue) {
					frontOfLineWaits = c(frontOfLineWaits, placedPerson$intervalsWaitedAtHeadOfQueue)
				}
 
				# Remove placed person from queue
				queue[[1]] = NULL
			}
		}
	}
 
	# Everyone left in the queue has now waited one more interval to be served
	if(length(queue)) {
		for(j in 1:length(queue)) {
			inc(queue[[j]]$intervalsWaited) # = queue[[j]]$intervalsWaited + 1
		}
 
		# The (possibley new) person at the front of the queue has had to wait there one more interval.
		inc(queue[[1]]$intervalsWaitedAtHeadOfQueue) # = queue[[1]]$intervalsWaitedAtHeadOfQueue + 1
	}
 
	# End of the interval, what is the state of things
	queueLengths[i] = length(queue);
}
 
#### Output ####
plot(queueLengths, type="o", col="blue", pch=20, main="Queue lengths over time", xlab="Interval", ylab="Queue length")
# plot(waitTimes, type="o", col="blue", pch=20, main="Wait times", xlab="Person", ylab="Wait time")

10
Oct 11

Drop in confidence

I’ve been paying a lot of attention lately to how statistics are released to the public. In particular, when are confidence intervals used, and when are they dropped? When are numbers presented as fact, and when are they acknowledged to be fuzzy?

The only time you consistently see confidence intervals reported, in the general press, is for poll results. As in: 66% of respondents believed this poll to be self-reflexive, with a margin of error of plus or minus 5%.

Weather reports often have percentages involved, but it would be a stretch to call these confidence intervals. For example, when the attractive meteorologist on Channel 7 tells you that there is an 80% chance of rain tomorrow, they are presenting that as a fact. Behind that number is a computer simulation that may or may not be able to estimate a confidence interval around that 80% number.

Government statistics and estimates, no matter how bad or biased, almost never come with confidence intervals attached. Gross Domestic Product of the U.S. in 2010? Estimated at $14.64 trillion. Confidence interval for that estimate? Probably so bad you don’t even want to know. Or maybe you do?

Are there times when you’ve been surprised to see a confidence interval reported or missing?