The city of Los Angeles is currently working on the subway rapid-transit system

Question

The city of Los Angeles is currently working on the subway rapid-transit system between downtown and the San Fernando Valley. The system is designed to carry 50,000 passengers per day. The preliminary design calls for ten stations along the route. The ticketing process will be completely automated. Arriving passengers first must buy a ticket and then proceed to the train platform. The projected passenger arrival rate at the downtown station during the evening rush period (i.e., between 4:00pm and 6:00pm) is 80 customers per minute. Each automated ticketing machine is designed to process four tickets per minute. The traffic analyst has concluded that service time for the ticketing machines follows a negative exponential distribution. Passengers wishing to purchase a ticket wait in a single line and go to the first empty ticket machine. Each train can carry 1000 passengers and arrives at the downtown station every 10 minutes during the rush hour period and departs 3 minutes later. One hour is required to make one complete loop of the subway system. During the rush hour period approximately 400 passengers arrive per train at the downtown station. Of that number approximately 25 percent remain on the train.

The subway system’s Board of Governors has defined performance objectives of mean passenger ticket-buying waiting time to be less than 2 minutes and a mean of no more than 20 customers waiting in line. How many ticket vending machines should be installed at the downtown station to meet the Board’s objectives?

Explanation / Answer

Rush period is 4pm to 6pm. If ticket machines can handle during rush hour they can handle other wise too.

Projected Rush time rate = 80/min

Each automated machine processing rate =4/min

Naturally, waiting time = 80 people/min / 4 tickets/min = 20 mins during rush time.

Capacity of train = 1000; However during rush time approx 400 passengers/train arrive at downtown.

25 % of 400 remain on the train implying only 400-0.25*400 book fresh tickets

Hence, 300 passengers book and this happens every 10 mins.

Waiting time cannot be more than 3 mins as it departs in 3 min

Therefore, for each train 300/3= 100 passengers should be able to book ticket per minute.

Now, performnce objective is to reduce waiting to less than 2 min and less than 20 waiting in the line

Lets consider the 1st thing. Presently, processing rate of ticket mc = 4/min and we have 100 passengers/min in addition to 80 passengers/min who directly arrive at the downtown station

Waiting time = 180/4 =45 mins and to reduce it to less than 2 min; we would need to have >45/2 ticket machines= 23 machines. Now if we see if the second objctive is met.As we have 23 machines and 180 passengers at the peak time/minute; no. of customers waiting in line = 180/23= 8 approximately.

Hence, we need 23 ticketing machines.

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