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Analytics Magazine

Voting Queues: To Queue or Not to Queue?

Spring 2008


© Photographer: Ali Mazraie shadi | Agency:

© Photographer: Ali Mazraie shadi | Agency:

In a U.S. presidential election, that should NOT be a question!

by Alexander S. Belenky and Richard C. Larson

Will the 2008 U.S. presidential election be decided by election queues?

We don’t know, but operations research could be used to make every vote count and help secure the integrity of the voting process.

Queues in U.S. presidential elections first drew national attention in the 2000 election, when voters queued for more than two hours to cast their votes in some counties in Florida [1], and the election hours were extended due to the queues in some battleground states [2]. In the 2004 election, Ohio voters queued for as many as 10 hours in some precincts in Columbus, Cincinnati and Toledo [3], and long queues were reported in other states as well [4, 5, 6].

Besides the inconvenience of waiting, why is this important? Think of two words from queueing theory: balking and reneging. Potential voters who see a long queue may balk from entering it. Others may enter the queue but leave it later, renege, due to frustration at the slow pace of the line. Unwillingness to wait may be due to time personal constraints,such as job or family obligations,or maybe just impatience. If those who balk or renege are differentially more from one political party than another, then the party most affected will complain, saying that some of “its voters” did not vote due to long lines.

Did election queues decide the 2000 and 2004 U.S. presidential elections?

They well might have. In each of the two elections, one state decided the fate of the election: Florida in 2000 and Ohio in 2004. In 2000, more than 5,801,000 votes were cast in Florida, and the official vote tally favored the winner by only 537 votes. With widely reported long queues in the precincts, nobody knows how many of those who came to vote balked or reneged. If there were at least 538 such voters, less than 0.01 percent of all the counted votes, one cannot be sure which candidate could have won Florida — in the absence of balking and reneging. In reviewing the 2000 election, one news analyst summed it up this way: “There were a variety of reasons for these votes not being cast or counted.” After recounting technological problems, he states, “Finally at some voting stations the lines of voters were simply too long and many voters abandoned their attempt to vote because they were not willing to queue for hours” [7]. Abandonment = reneging.

In the 2004 election, with the Ohio winner’s margin less than 119,000 votes, 4-plus hour queue delays [3, 8] could have affected the outcome. Consider the math: 12 of the 88 Ohio counties account for 6,560 of the 11,360 Ohio precincts, and John Kerry won a majority of the votes in these 12 counties [9]. With 72 percent voter turnout of registered voters, 119,000 voters constitute less than 1.5 percent of the registered voters. Anecdotal evidence is compelling: In Columbus, which has 472 precincts, between 5,000 and 15,000 voters were reported to be frustrated and thus balked or reneged [3]. If only 19 potential voters who came to vote in each precinct in these 12 counties balked or reneged, one cannot be sure that the 2004 election outcome in Ohio would have been the same.

Result: the queues might have decided or at least substantially affected the 2000 and 2004 election outcomes.

The political controversy created by misallocation of voting machines is illustrated in this quote by Sen. Barbara Boxer, a Democratic from California: “So, it seems to me that under the Constitution of the United States of America, which guarantees the right to vote, we must ask ourselves several questions today. Why did voters in Ohio wait hours and hours and hours in the rain to vote? Why were voters at one precinct, for example, made to wait in line until nearly 4 a.m., four in the morning, to vote because there were only two machines? At Kenyon College, there were 1,300 voters. They needed 13 machines; they had two. Why did poor and predominantly African-American communities have disproportionately long waits? Why in Franklin County did election officials use only 2,798 machines, when they said they needed 5,000? Why did they hold back 68 machines in warehouses that were perfectly good? Why were 42 of those machines in predominantly African-American districts?”[10].

We take no sides in this debate. But we point out that the substance of the debate is balking, reneging, long waits and the deployment of voting machines — just the types of issues made for O.R. analyses.

© Photographer: Bob Denelzen | Agency:© Photographer: Bob Denelzen | Agency:

Can election queues decide the 2008 U.S. presidential election?

Yes, perhaps — if nothing is done about managing the voter queues, and if the 2008 election is as close as were the last two elections.

In the course of the 2000 election, the slogan “Every vote counts, and every vote must be counted” became popular, and soon afterwards, the Carter-Ford Commission on Federal Election Reform was formed to improve the system of electing a U.S. president. However, due to “hanging chads” and related iconic images of the Florida election, most of attention has been paid to the second part of this slogan, i.e., to voting machines and technologies to audit votes already cast. Only after the 2004 election in Ohio were two bills initiated jointly by two groups of congressmen and U.S. senators [11]. These bills for the first time address the problem of establishing certain voting standards, including the maximum time in queue to cast a vote. Though both bills are pending, their chances to reach the floor of the U.S. Congress seem slim, and the attention of the media to the problem of queues in U.S. presidential elections seems to have faded away.

In the absence of queueing standards and related decision-aiding tools, the allocation of voting machines and personnel to precincts will be at the discretion of local election officials. Queues in the 2008 election may then appear either as a result of poor judgment or intentional manipulation of voting resources.And, with the deployment by both political parties of information technologies for developing “digital states” [12], it becomes even easier to design situations in which voters favoring a particular political party are discouraged to vote by long queues caused by insufficient numbers of voting machines.

Top nine reasons for queueing in U.S. presidential elections.

  1. Underestimated voter turnout. A bigger-than-expected voter turnout may catch off guard election officials who are supposed to respond quickly to such an event if/when it occurs.
  2. The election budget. An insufficient election budget may imply that rho >1.0 during all or part of Election Day. Here rho = p is the queue utilization factor. As we all know, p >1 implies queues that grow.
  3. Types of voting machines. Different types of voting machines have different service times and thus affect queueing differently. As far as we know, no research showing how particular types of voting machines may affect queues on Election Day has so far been conducted.
  4. Activities aimed at mobilizing voters. The 2004 election demonstrated that political parties can increase the participation of low-propensity voters [13]. Success at bringing new voters or reluctant voters to the polling precincts can increase the queueing lambda, or (�), the arrival rates to the queues, thereby making the queues longer.
  5. The demography of the electorate. Changes of the number and types of voters in a county affect service times and thus election-day queues, even if the voter turnout is unchanged compared to the previous election year.
  6. Early voting options. Many states now allow voting in federal elections several days before Election Day in order not to lose voters who are unwilling to stand in queues on Election Day. This decreases Election-Day (�).
  7. Training voters to operate voting machines. Some types of voters — for instance, elderly people, undereducated voters, new immigrants and first-time young voters — generally require more assistance in the precincts than, say, educated 35-year-olds. Out-of-queue training of these voters may reduce service times and thus queues on Election Day.
  8. Weather. Voter turnout is dependent on weather. Good weather usually implies higher turnout; bad weather the reverse. This makes even more difficult the prior estimation of queue parameters.
  9. Unscientific deployments of voting machines and poll workers. The deliberate or accidental misallocation of voting machines among the precincts seems to remain the major cause for long queues in national elections.


A strategy to deal with queues in U.S. presidential elections.

To start to use O.R. in a serious way, we need queue wait standards to be set and agreed upon by stakeholders in the voting process. A standard may be stated in the form, “No more than 5 percent of the voters should wait more than 30 minutes in the voter queue.” Of course, the 5 percent may be 1 percent or 10 percent or any other percentage, just as the 30 minutes could be 10 or even 60 minutes. Whatever the standard, it must be stated in a way that acknowledges that less than 100 percent of the voters will enjoy the standard — due to unavoidable random fluctuations in the queue arrival and service processes.

Standards can be agreed on even in the public sector. The U.S. Postal Service (USPS) has a five-minute standard to serve retail customers. And, to motivate the approach we are proposing, their goal is met in large part by”…using the retail analysis staffing and scheduling model to determine how many employees should be on duty during all periods to maintain service levels within these guidelines” [14]. That’s right, the USPS uses queueing models to staff their retail postal lobbies. If it’s good enough for the U.S. Postal Service, it should be good enough for ensuring equity of access to America’s most precious right: exercising the right to vote.

Suppose the standard is established by the state law and accepted by most if not all stakeholders. Solving the problem of allocating voting machines among county precincts now becomes an O.R. problem in queue design and management. First there is a feasibility test: Are there enough voting machines in the county to meet the standard? If not, how many more do we need? The way we do this, once we have a validated queue model, is to find the minimum number of voting machines needed at each precinct to meet the standard, and then sum over all precincts. Comparing this to the total number currently available determines the overage or underage that we face.

© Photographer: Christian Bernfeld | Agency:© Photographer: Christian Bernfeld | Agency:

The details of a valid and useful queueing model will of course depend on the realities on the ground. Is real-time information available to county managers who can communicate with leadership in each of the precincts? If information is available, can voters who appear at a congested precinct be transported to a nearby uncongested precinct? Or, can voting machines at an uncongested precinct be moved during the voting day to more congested precincts, thereby balancing the load and reducing queue delays at the congested precincts? Would special mobile voting units, which could service voters in their neighborhoods early in the morning and be moved to congested precincts during peak hours, be a comparable or even a better solution? Can voters be informed by radio and television about hours of anticipated congestion and hours of small or zero-length queues, thereby redirecting those who have vote time flexibility to appear during off-hours? In measurement tests of potential voters with long waits, what are the state-dependent probabilities of balking? What are the dynamics of reneging? Does steady-state analysis make sense during the voting day, or must the entire model be transient in nature?

All of these questions and more must be settled by careful research. If we are to bring O.R. to bear on America’s voting system, we must do it with the utmost respect and care. Erlang’s famous equations of 1915 — created for analysis of centralized telephone switching systems — will not be up to the job. More than likely, new queueing models will have to be created.

Politicians and their constituents are crying out for proven, tested methods for deploying fairly and equitably voting machines and related resources to voting places. This is illustrated by a December 2004 letter from Rep. John Conyers Jr., Rep. Melvin Watt, Rep. Jerrold Nadler and Rep. Tammy Baldwin to the Honorable J. Kenneth Blackwell, Ohio secretary of state:

  1. How much funding did Ohio receive from the federal government for voting machines?
  2. What criteria were used to distribute those new machines?
  3. Were counties given estimates or assurances as to how many new voting machines they would receive? How does this number compare to how many machines were actually received?
  4. What procedures were in place to ensure that the voting machines were properly allocated throughout Franklin and other counties? What changes would you recommend be made to insure there is a more equitable allocation of machines in the future? [15].

On engineering systems for servicing U.S. presidential elections.

U.S. presidential elections have traditionally been considered a subject of politics and political science. While establishing voting standards may involve a political process, scientifically handling the voter queues — once the standard is set — should have nothing to do with politics. Developing decision-making systems for servicing American voters according to their constitutional rights is an important step in insuring the integrity and fairness of the process. Training election officials in the use of these systems and informing the voters about the capabilities of these systems should accompany the implementation of such decision-support systems.

We hope to develop a prototype of the first system of such a kind with support of interested organizations that care about the integrity of the election process.


From an O.R. point of view, a voting precinct is a multi-server queueing system. Problems of allocating available voting machines, and perhaps finding the required number of additional voting machines, can be formulated as mathematical programming problems. All of these problems are non-routine due to complications arising in practice: uncertain voter turnout, the ability of the election authorities to deal with long queues, types of machines available, experience of voters, use of early voting options, etc. Whatever the difficulties, we are unaware of any method superior to O.R. for sorting out all the issues and creating a valid and unbiased way to deploy scarce voting resources on Election Day. May O.R. save the day!

Alexander S. Belenky is principle scientist at the MIT Center for Engineering Systems Fundamentals. He is the author of books and scientific articles in the field of optimization and game theory and their applications. His recent books include “Extreme Outcomes of U.S. Presidential Elections” (2003) and “Winning the U.S. Presidency: Rules of the Game and Playing by the Rules” (2004). He was an invited guest on radio and TV shows throughout the country in the course of the 2004 election campaign. His forthcoming book is “How America Chooses Its Presidents.” Belenky holds a Ph.D. in systems analysis and mathematics and a D.Sc. in applications of mathematical methods.

Richard C. Larson is Mitsui Professor of Engineering Systems and of Civil and Environmental Engineering at MIT, where he serves as director of the MIT Center for Engineering Systems Fundamentals. A past president of ORSA (1994-95) and INFORMS (2005), he is a member of National Academy of Engineering, an INFORMS Founding Fellow and recipient of the INFORMS President’s Award and Kimball Medal. He has published various papers on queueing, including “QIE (Queue Inference Engine),” the “Hypercube Queueing Model” and papers on the psychology of queueing.


1. “News Hour with Jim Lehrer,” transcript, Nov. 10, 2000.

2. C. Orman, “State Sen. Scott Addresses Pachyderm Club,” Sedalia Democrat, March 11, 2006.

3. M. Powell, P. Slevin, “Several Factors Contributed to ‘Lost’ Voters in Ohio,” The Washington Post, Dec. 15, 2004.

4. H. McGee, “My Opponent Looks Like Saddam’s Son,” Irish Examiner, Oct. 27, 2004.

5. P. Donovan, “It was About Poor People’s Right to Vote,” The Irish World, Nov. 12, 2004.

6. J. Jones (reviewed by), “U.S. Election 2004,” BBC 1, Nov. 2, 2004.

7. R. Bloor, “Electronic Voting and the U.S. Election,” Oct. 27, 2004, www.

8. Rev. Jesse Jackson, “Something Fishy in Ohio,” Chicago Sunday Times, Nov. 30, 2004.

9. “State results. President by county,”, Nov. 4, 2004. 1

0. “Contesting Ohio Electoral Votes,” transcript of press conference: Sen. Barbara Boxer and Rep. Stephanie Tubbs Jones, Federal News Service, Jan. 6, 2005.

11. Testimony of Norman J. Ornstein, Commission on Federal Election Reform, June 30, 2005, James Baker III Institute for Public Policy, Rice University, Houston, Texas.

12. T. Edswall, “Democrats’ Data Mining Stirs an Intra-party Battle,” The Washington Post, March 8, 2006.

13. D. King, “Youth Came Through with Big Voter Turnout,” The Boston Globe, Nov. 4, 2004.

14. Postal Operations Manual, Sec. 125.21: Service%20Levels%20-%20POM%20125.2.pdf; and

15. (letter dated Dec. 2, 2004).


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