Question

A typical subway in Manhattan has six turnstiles, each of which can be controlled by the station manager to be used for either entrance or exit, but not both. The manager must decide at different times of the day how many turnstiles to use for entering passengers and how many to use for exiting passengers. At the NYU station stop, passengers enter the station at a rate of about 84 per minute between the hours of 7:00 a.m. – 9:00 a.m. Passengers exiting the station in this time window leave at a rate of around 48 per minute. Each turnstile can allow an average of 30 passengers per minute to enter or exit. Arrival and service times have been thought to follow a Poisson and negative exponential distribution, respectively. Assume riders form a common queue at both entry and exit turnstiles and proceed to the first empty turnstile. The NYU station foreman, Dudley Doright, does not want the average passenger at his station to wait in a turnstile line for more than 6 seconds, nor does he want more than 8 people in any queue during this 7:00 a.m. – 9:00 a.m. time.

a. How many turnstiles should be open in each direction every morning between 7:00 a.m. – 9:00 a.m.?
b. What assumptions are necessary to ensure that Dudley has accurately assessed this waiting-line system?

Answer 1

To prevent queuing times of more than 6 seconds and more than 8 people in line, the service rates for entering and exiting turnstiles must exceed the arrival rates of 84 and 48 passengers per minute, respectively. Using queuing theory, appropriate numbers of turnstiles need to be allocated for each purpose, ensuring that 30 times the number of turnstiles exceeds the arrival rate.

Step-by-step explanation:

Subway Turnstile Allocation

To determine the number of turnstiles that should be open for entering and exiting passengers, we need to calculate the service rate and compare it with the arrival rate of passengers. The arrival rates of entering and exiting passengers are 84 per minute and 48 per minute, respectively. Each turnstile can service 30 passengers per minute. To meet Dudley Doright's requirements (no more than a 6-second wait and no more than 8 people in queue), we would resort to a queuing theory formula to calculate the necessary number of turnstiles for each direction.

Given the arrival rates (λ) and service rates (μ), we can utilize the queuing theory formula L = λ / (μ - λ), where L is the average number of customers in the system. To ensure that the average wait time does not exceed 6 seconds, or 0.1 minutes, the average number of customers (L) must be kept below a certain threshold. By solving for the number of turnstiles needed, we aim for the service rate to be higher than the arrival rate.

For the entering passengers, let E be the number of turnstiles needed for entrance. Hence, the overall service rate for entrance is 30E. Similarly, let X be the number of turnstiles for exiting passengers, then the overall service rate for exit is 30X. Since we need 84 people entering per minute, E must be such that 30E > 84. For exiting, 30X > 48.