OR/MS Today - December 2007

Minimizing Voting Queues

The Right Not to Wait

Forecasting and simulation show promise in reducing waiting times to vote.

By Douglas A. Samuelson, Theodore T. Allen and Mikhail Bernshteyn

A group of O.R. analysts has developed a promising solution to the problem of long waiting lines at polling places. The solution includes improved forecasts of how long people will take to vote, taking the number of offices and issues on the ballot into account, and simulation of different proposed allocations of voting machines to precincts. The results, tested in a few counties in 2006, indicate that improved allocation of machines to precincts, producing much better voter service, can be achieved at minimal cost. The results also indicate, however, that the acquisition of new touch-screen voting machines, in response to the Help America Vote Act (HAVA), may make the problem worse in many instances because of the increased time required to vote using these machines.

The company, Sagata, Ltd., of Columbus, Ohio, markets this solution as a software package, Sagata VotePro. Outputs include expected waiting times at the worst precincts, expected latest precinct closing times and average overall waiting times. The software can show the expected performance of a given voting equipment allocation or indicate a recommended new allocation based on specified goals.

Simulation is necessary for several reasons:

  • Analytical methods generally focus on mean waiting times for the region of interest, averaging both over different polling places and different times of day. The critical metric, however, is the peak waiting times in the few precincts with the longest waits. This is especially important if these precincts happen to correspond predominantly to voters of one party preference of ethnic group. Thus, for example, although average waiting times overall were short (less than five minutes) in Franklin County (Columbus) in 2004, a few precincts had waits exceeding five hours. Since these precincts were largely African American, there is particular interest in whether these voters received their legally guaranteed equal access to voting. In turn, election officials do not look forward to answering media inquiries about the precincts with such problems, even if there are only a few of them.

  • Analytical queueing results apply to a steady-state infinite time horizon, while an election day is limited in duration.

  • According to many election officials and our own experience, it is unrealistic to assume a constant voter arrival rate throughout the day.

  • Additional relevant conditions such as voting equipment breakdowns (down times) need to be considered. A down time can be caused by a machine being out of paper, cartridge out of ink, software problems or other events. We know of no analytical formulas for systems with multiple types and time distributions of server breakdowns, even for a steady-state system.
Are Election Officials "Always Wrong"?

It appears that people generally underestimate how difficult it can be to achieve that all voters wait no more than a specified threshold. For example, New York State has recently introduced a rule that no voter should wait more than 30 minutes before casting a vote. The more voters there are within a jurisdiction that wants to satisfy this goal, the more difficult it is to achieve the goal. The cause of this is what some probability modelers, especially in reliability, call "multiplicity" — the way that the probability of at least one undesirable outcomes grows with the number of independent "trials," Assume, for instance, that one precinct has a chance of 1 percent of having a voter who would experience a wait longer than 30 minutes. Then, if the precincts are probabilistically independent, a county with 200 precincts would face a chance of 1-(1-0.01)200 = 0.86 of having at least one precinct where there would be a voter waiting longer than 30 minutes.

This effect, in turn, implies a larger than expected "safety stock" of machines is needed to avert having even one precinct with long waits, taking unexpected events into account. For demonstration purposes, assume a one-minute average voting time and a 70 percent voter turnout for a county with approximately 200 precincts each servicing about 1,200 people. The relationship between the number of machines and the expected average waiting time looks like Figure 1.

Figure 1: Relationship between number of machines and expected average waiting time.

However, a seemingly good average waiting time of, say, four minutes, in the Figure 1 plot would translate for some (few) voters into waits of about two hours in the plot shown in Figure 2, which shows the expected waiting time for the precinct with the longest waiting time. This suggests that the New York State requirement that no voter wait for more than 30 minutes is more difficult to achieve than acceptable wait on average.

Figure 2: Expected waiting time for the precinct with the longest waiting time.

Also, these plots indicate the consequences of allocating an insufficient number of machines, as was the case in the well-publicized Ohio presidential election in 2004. The performance deteriorates quickly as the number of machines decreases, both in terms of the average waiting time across all vote centers and waiting times at the "worst" precinct.

An additional complication is that, if machine allocation is done in a fair scientific way, it is generally impossible to know ahead of time which precinct is going to be the "worst." The nature of randomness will lead to a few vote centers being less "lucky" in terms of voter arrival pattern throughout the day, voter age/literacy, etc.

How the Software Works

VotePro is a discrete-event simulation program. The program keeps track of each voter's arrival time, time in line and time spent voting on a machine. It also keeps the information on whether a machine broke down and the voter had to rejoin the line. After the simulation has run (for example, with each simulated day having 1 million voters queue, vote and depart for a large county), the resulting information can be assembled to produce any statistics of interest; for example, how late will the precincts close past the official closing time, how many precincts will perform well on average, and how bad will the worst precinct be. In order to obtain the information of how variable the results are, the VotePro user can specify how many times to replicate the simulation. The final results are reported with means and standard deviations as shown in Table 1.

Table 1:

Criterion Description Estimate StDev (mean) StDev (obs)
Max of Avg. Avg. waiting time of the worst location every election 257.9 21.8 119.3
Avg. of Avg. Avg. waiting time of all voters 5.5 0.1 0.7
Avg. of Max Avg. waiting time of the voter who waited longest in each 17.9 0.2 1.2
Max Overtime Expected number of minutes the latest poll closes 520.3 44.3 242.8
Avg. Overtime Avg. number of minutes all polls stay open after 7:30 pm 14.6 0.2 1.1

Figure 3, a typical output from the program, indicates that some locations will experience average waiting times in excess of 30 minutes. The program also produces more detailed results, showing maximum waits in addition to the precinct averages shown, which in this case predicted that some voters might wait in excess of five hours.

Figure 3: Program output shows some locations will experience average waiting times in excess of 30 minutes.

Studies in Ohio in 2006

The 2004 presidential election in Ohio, one of the most closely watched in history, highlighted several problems with the electoral and vote-counting process. While much of the media attention focused on questions about vote counting and discrepancies between exit polls and actual vote counts, the difference in waiting times among precincts appears to have produced a much larger effect, as some reporters estimated that as many as 20,000 voters, mostly African American, might have been turned away or discouraged from voting in Franklin County (Columbus) alone. Regular readers of OR/MS Today may recall a feature article by Alexander Belenky and Richard Larson (June 2006) calling attention to the potential severity of these problems, and a letter from us (August 2006) commenting on the issue and mentioning the VotePro work, then in progress. The mass media carried a number of other stories and commentaries about the potential magnitude of the problem, such as the Washington Post "op-ed" article by former Governors Dick Thornburgh and Richard Celeste in August 2006.

In 2006, Franklin County commissioned Election Science Institute (ESI), which in turn included Sagata in its team, to conduct "what if" analyses and generate recommendations about how to avoid the waiting time problems the county experienced in 2004. This study concluded that a one-minute increase in how long it takes to vote from 5.7 to 6.7 minutes would shift Franklin County from minimal to long waits. Such a shift could occur, for example, because of extra ballot items or poor voter instruction, for example. The study also predicted that certain precincts, mainly in the Gahanna area, would have relatively long waits, and additional resources (if any) should be transferred to these precincts. The county official in charge said that he would use that information to put additional absentee ballots in Gahanna but would not shift the allocation. For 2008, state officials are currently considering using simulation for allocation.

Actual experience in the general election in November 2006 bore out the model's predictions. Again, although the county had purchased many more machines, there were long waits in a few polling places because the allocation was insufficiently tuned to account for variations in how long people took to vote. This is a function of the length of the ballot, the complexity of some of the issues, whether voters have studied the ballot ahead of time, the structure of the voting process and how quickly voters assimilate the information.

An unexpected and unfortunate characteristic of some of the new machines is that they tend to require more time for most voters to get through the ballot. Rather than seeing the whole ballot at once, as on older lever-pull machines, voters are guided through one office or issue at a time. Also, when a voter skips an office or question, the machine prompts to make sure the non-vote was intentional. This feature apparently causes many voters to pause and reflect, substantially increasing the average time voters spend casting their votes.

Also in 2006, Sagata, InfoLogix and a number of other service providers, again as part of an ESI effort, conducted a thorough assessment of how well the new DRE touch-screen voting machines performed in Cuyahoga County (Cincinnati), Ohio. The results raised concerns about a number of issues, including the increased time some voters needed to use the machines, the difficulty of auditing results, and the vulnerability of the machines to both errors in use and deliberate tampering.

Studies in Other Jurisdictions

Sagata studied the 2006 November election in Travis County (Austin and vicinity), Texas as well. In this case, VotePro predicted what officials already believed: lines would not be an important concern. What was interesting, however, points to what might become a national trend. Typically, approximately half of the voters in Travis vote early, and the officials choose not to deploy all of their machines because of the associated cost savings. It is possible that, through simulation, officials all across the country in the future could use simulation-based evidence of overcapacity (particularly for low turnout elections or when many vote early) and avoid the need to deploy all of their machines.

A more recent study in New York State also pointed out the potential problem of longer voting times offsetting the benefits of acquiring new voting machines. Officials in at least one major county are accustomed to voters requiring less than 30 seconds to vote once they are at the "open-face" lever machines they still use. When we told them that we had data from direct recording equipment (DRE) machines that require the user to scroll through multiple pages in Franklin, Ohio, with average voting times longer than seven minutes, they were amazed. They were also very worried because with larger numbers of the more expensive DRE machines being required, they would exceed their budgets. In general, if average voting times increase tenfold, one could easily need to acquire ten times as many machines as the jurisdiction had before just to duplicate the waiting time performance experienced previously. The county officials were particularly interested in the implications of simulations relating to the specific machines to be purchased since previous studies had indicated variations in average voting times. In fact, even linearly proportionate increases in numbers of machines might not suffice, as greater average voting times also tend to have greater variation, and it is the longest times, not the average, that determine the severity of the resulting problem.


In short, detailed study has highlighted a number of concerns about the new machines many jurisdictions throughout the United States have acquired or are acquiring in response to HAVA. In particular, we observed longer times required to vote, and greater sensitivity of these times to the length of the ballot, meaning that even jurisdictions that acquired many more voting machines may still need improved allocation of machines to polling places to avoid having unacceptable waits in some locations.

Douglas A. Samuelson is a principal decision scientist at Serco, a professional services company in Vienna, Va. In 2005-2006, he worked — via InfoLogix, Inc., his own research and development company — on voting access and auditing issues for Votewatch/Election Science Institute.

Theodore T. Allen is associate professor of Industrial and Systems Engineering at Ohio State University and a partner in Sagata, Ltd., a business statistics company headquartered in Columbus, Ohio. Sagata worked for Election Science Institute in Ohio in 2005-2006.

Mikhail Bernshteyn is the director of the Canadian branch of Sagata, Ltd. He received his Ph.D. at the Ohio State University in operations research and has performed several interdisciplinary research projects involving optimization, simulation and statistics.


  1. Allen, Theodore T., Mike Bernshteyn and Douglas A. Samuelson, 2006, "Voting Queues Present Complicated Problems," Letters, OR/MS Today, August 2006.
  2. Allen, Theodore, and Mikhail Bernshteyn, 2007, "Mitigating Voter Waiting Times," Chance, Winter 2007.
  3. Belenky, Alexander and Richard Larson, 2006, "To Queue or Not to Queue?" OR/MS Today, June 2006.
  4. Election Science Institute, 2006, "DRE Analysis for the May 2006 Primary, Cuyahoga County, Ohio," final report, August 2006; published, along with rejoinders and discussion of the implications, as "ESI Memorandum to Ohio Election Officials," on www.electionscience.org/reports/viereports, Aug. 22, 2006.
  5. Kyle, Susan, Douglas A. Samuelson, Fritz Scheuren, and Nicole Vicinanza, 2004, "Explaining Discrepancies Between Official Votes and Exit Polls in the 2004 Presidential Election," Chance, Spring 2007.
  6. Sagata, Ltd. Web site, www.sagata.com.
  7. Thornburgh, Dick, and Richard Celeste, 2006, "Watch Out for Voting Day Bugs," The Washington Post, Aug.28, 2006, page A15.

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