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Queueing Metrics to Help Optimize Slot Floor Experience

It’s no secret that slot players are often willing to wait to play their favorite game. How long will they wait before they walk away?

Authors’ Note: In this fifth article of our 12 part series on customer centric gaming floors, we will examine the value of availability and queueing mathematics, and how they can be applied to the slot floor. Please note these articles are meant to stimulate thought and that we are using some deliberately provocative metaphors and examples which should be taken with a grain of salt.

It’s no secret that slot players are often willing to wait to play their favorite game. What is more of an unknown is how long will they wait before they walk away from the slot?

In the modern casino, determining the average wait time for specific games is an important factor for operators seeking to optimize slot floor performance. On the surface, this would seem a simple equation to solve; in actuality, it requires availability and queuing theory—a level of mathematics well beyond standard optimization metrics.

The availability theory we explore below is based on well-proven manufacturing models adapted to the gaming world. This approach will develop the intuition and mathematical formulas required to drive this important analysis.  Finally we introduce preference filters into the mix to come up with a true determination of how long the best customers have to wait to play their favorite games.


In part two of this series, we described the importance of looking at the player perspective. To look at the gaming machine from the player perspective, we needed to look into metrics that are felt by the player. Players do not sit at their game and think about the hold percentage or the theoretical win per day of the game. Instead players live in the gaming experience—feeling how much they win or lose, they change their betting behavior based on the outcome of the games; and they feel the impact of a game they want to play being occupied by another player or players. To optimize the games, we need data that can see into the gaming experience and optimize it. This approach is very different to optimizing the outcome of the game—to clarify this, we describe two different kinds of metric: optimization and outcome.

Optimization metrics: These are metrics that measure effects the players can observe. In our previous articles, we have explored how occupancy and the cost of the game play including game speed are the key optimization metrics. In this article we dig deeper into a new and more predictive kind of math—the math around queueing theory. In short, we want to optimize the availability of games to players; for example, we want to make sure that high-value players can most often find the game they want, in the location they would prefer to play it. While the mathematics is complex, the results are remarkable in that it allows us to drive the yield of the game. In summary, we have a new class of metrics called availability, and this class of metrics tells us if a game is available at any point in time.

Outcome metrics: These are metrics that the players do not observe. For example, the theoretical win per unit per day on the gaming machine, which is an average from a number of different players. Quite simply, players do not experience the spending of other players. Another outcome metric is the slot floor hold percentage, or what is often incorrectly termed the “price” of our games.


Game availability is central to the user experience and optimization of that experience. Imagine a floor where we relate customer preference for a specific game and the likelihood of that game being available when they would like to play it. To understand the power of availability modeling, let’s consider the following eight optimization questions:

Should I have a 10 pack or a 12 pack of games in this area? The shape and number of games is clearly important, but something as fine grained as the value of a 10 pack of slots versus a 12 pack of slots is hard to see analytically. When looked at using availability, the direct value of availability to players can be calculated.

In a mixed bank should I run two of any game? As we will show below, the probability of finding a game you like is massively increased when you move from one game to two. Armed with this knowledge, the gaming floor can now be optimized to provide high preference games where needed. Furthermore, we can back into the availability models using hypothesis testing; in other words, change the floor and measure the modeled versus actual outcome of the gaming optimization decisions.

Are there any games that are substitutes for a popular game?One solution for a game availability issue is to bring in substitutes, which provides a low-cost way of finding games that act as “drawcards” for other games.

How many games should I have on the floor? Optimization can help operators determine the ideal number of games there should be on the slot floor. For example, a floor may have 1,000 games, but operates optimally with 950. These small changes balance the number of games with the capital/depreciation and running costs of those games.

Can I close off areas of the gaming floor at certain times of the day?This is a critical question when it comes to reducing staff, power and other operating costs. The trick is to do this in a way that still ensures availability of the right gaming experience in a different area of the floor. Consider the example of a room where there exist players who have a strong preference for that room—closing that room will directly impact this player group.

How does a jackpot impact the availability of a game? On a modern gaming floor, there are levers that can be pulled to adjust the availability. For example, adjusting the jackpot behavior changes the occupancy and so it can be modified to maximize yield.

Are players waiting in line to play this game?
Oftentimes customers will play a game with a line
of sight to the slot they want to play. The availability of a game gives a view into the likelihood that players are waiting.

How does the availability invert to predict demand? The inverse of availability combined with game presence gives us a metric that is representative of demand. Game demand is a measure of its attractiveness and gives critical insight into the necessity of adding or removing games from the floor.


These eight optimization questions illustrate how the availability metrics of a game opens the door to a new approach for customer centric gaming floor management. There are a number of mathematical approaches to calculate availability, ranging from simulation models to various statistical techniques. One powerful method is based on the Poisson distribution.  This method is advantageous since a small amount of key metrics can give accurate predictions.

The Poisson distribution is named after by Siméon Denis Poisson and provides a remarkably simple mathematical model that determines the probability of a number of events occurring in a specific period of time. This simple distribution has been applied in powerful ways to understand wait times in queues. For our purposes, we will explore the likelihood that a customer’s preferred game is unavailable when they want to play it. In this example, there are two separate calculations to consider—customer desire to play a specific machine and the likelihood that this machine is available.

Example question: Given a machine is occupied, what is the likelihood that a player wanting to play this machine finds the machine occupied between the hour of 5:00 pm to 6:00 pm on Friday over a six month period?

Example assumptions: The Poisson distribution requires a set of assumptions; the following is a description of how these assumptions are met in the context of the example question:

  • The event of finding the occupied machine can occur more than once in the hour.
  • Each player finding the occupied machine is independent of other players finding the same machine.
  • For the period of time between 5:00 pm to 6:00 pm on Friday, the rate of the event is the same.
  • Two events cannot occur at exactly the same instant.
  • The probability of an event in an interval is proportional to the length of the time we are examining.
  • The quality with which that if these conditions are true determines the applicability of the Poisson distribution.


Given players can find the occupied game in the time between 5:00 pm to 6:00 pm and that this can occur a number of times, the average number of events in an interval is designated λ(lambda). Lambda is the rate at which people find the occupied game, also called the rate parameter. The event of finding the game occupied has a percentage chance of happening, and the formula for determining this percentage is shown in Figure 1: Poisson Distribution.

To explore the power of this formula, let’s consider the example of the person finding the occupied game 2.5 times per hour (λ = 2.5). This example is simplified to show how the math can be applied and, like all models, the quality of the input data and how it meets the assumptions determines the accuracy of the results. That being said, we can see that the Poisson distribution gives fascinating insight into the player experience, and it can show the impact that an occupied game can have on the player gaming experience. What is remarkable is that the data to calculate this is available, and that it opens the door to optimization of the gaming floor based on the availability of games.

However, to take these calculations to the next level, a special kind of math called queuing theory is needed. Indeed, given this illustration of how the mathematics of availability can be calculated, we need to dig into far more sophisticated models to make this concept directly applicable to gaming and to devise gaming specific models that operators can apply to their casino floor. Queuing theory is common in many business areas, retail checkout models are an example, and it has been proven in a wide variety of applications. Using these calculations, critical questions such as the impact of adding an additional game or the likelihood of finding a customer’s game occupied can be explored. These models are extremely powerful and in future parts to this series we will cover how to create and apply them.

This article was originally published on September 7, 2016 in Casino Journal and is reprinted here with permission of the publisher.

Andrew Cardno & Dr. Ralph Thomas

Andrew Cardno, CTO and founder of VizExplorer, is an internationally acclaimed thought leader in the field of data visualization, advanced analytics and big data. A prolific inventor with more than 60 patent applications, he has published more than 100 industry and academic articles, and is co-author of the 2012 book The Math that Gaming Made. Dr. Ralph Thomas is chief data scientist and general manager gaming division for VizExplorer, a San Diego, Calif.-based provider of software that makes life easier, more productive and more profitable for casino operators.

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