Team 2053 5.2 Predicting Bottlenecks with Queuing Theory One intuition for modeling the airplane boarding problem is to think of it as a stochastic process. A stochastic process is a collection of random variables that must take on a value at every state, where states are indexed by some parameter (in our case time)2. A simple example of using a stochastic process to model the airline boarding problem would be if we considered each entering passen- ger to be associated with a random variable that described their assigned seat Although this may seem simplistic, it is conceivable to assign every potent parameter in the airplane boarding process a random variable that is associated with time. We determined that this level of detail was prohibitive based on the mount of computation that would be required even for just a few variables Despite this, we can still use several tools associated with stochastic processes to learn about the plane boarding problem To analyze this stochastic process formulation, we use queuing Queu- ing theory deals with analyzing the way that random variables processes interact. Traditionally queuing theory is utilized for de ng the .erage hroughput of a system. While the airplane boarding problem does not possess a quantity directly corresponding to throughput, we will show that we can gain a better understanding of bottlenecks and their effects using this approach We place a"processor"at each row. This processor corresponds to each paes The first step in our analysis is to partition the airplane into a series of quer ger making a decision at this point either to keep walking or to stop and enter their row. Each processor has a queue that stores passengers. Queues have a size of 1 and will stop the processor feeding them if they are full. This would represent people backing up if someone stops in the aisle. a diagram for this layout can be seen in figure 2 ( Figure 2: A Queuing Theory Model of an Airplane In the above diagram uk represents the processing rate of the "processor".We can choose this variable to directly correspond to the average walking speed of people. Each Pk represents some probability at which passengers are diverted into the their rows or continue walking in the aisle. In some cases people will take longer to get into their rows depending upon how long it takes for themTeam 2053 9 of 30 5.2 Predicting Bottlenecks with Queuing Theory One intuition for modeling the airplane boarding problem is to think of it as a stochastic process. A stochastic process is a collection of random variables that must take on a value at every state, where states are indexed by some parameter (in our case time) [2]. A simple example of using a stochastic process to model the airline boarding problem would be if we considered each entering passen￾ger to be associated with a random variable that described their assigned seat. Although this may seem simplistic, it is conceivable to assign every potential parameter in the airplane boarding process a random variable that is associated with time. We determined that this level of detail was prohibitive based on the amount of computation that would be required even for just a few variables. Despite this, we can still use several tools associated with stochastic processes to learn about the plane boarding problem. To analyze this stochastic process formulation, we use queuing theory. Queu￾ing theory deals with analyzing the way that random variables in stochastic processes interact. Traditionally queuing theory is utilized for determining the average throughput of a system. While the airplane boarding problem does not possess a quantity directly corresponding to throughput, we will show that we can gain a better understanding of bottlenecks and their effects using this approach. The first step in our analysis is to partition the airplane into a series of queues. We place a “processor” at each row. This processor corresponds to each passen￾ger making a decision at this point either to keep walking or to stop and enter their row. Each processor has a queue that stores passengers. Queues have a size of 1 and will stop the processor feeding them if they are full. This would represent people backing up if someone stops in the aisle. A diagram for this layout can be seen in figure 2. u_0 u_1 u_2 u_n−1 p_1 p_3 p_2n−1 p_0 p_2 p_2(n−1) Figure 2: A Queuing Theory Model of an Airplane In the above diagram uk represents the processing rate of the “processor”. We can choose this variable to directly correspond to the average walking speed of people. Each pk represents some probability at which passengers are diverted into the their rows or continue walking in the aisle. In some cases people will take longer to get into their rows depending upon how long it takes for them 9