Question

An automatic roller coaster ride takes a constant time of 2.5 minutes to complete a ride. Customers arrive at the facility every 2.75 minutes (exponentially distributed). Report answers to 4 decimal places.

a. Determine the average waiting time in the queue (in minutes).

b. The local bank drive-through teller window can serve a customer at an average of 4.5 minutes per customer (exponentially distributed). Customers arrive in their cars at a rate of 12 per hour (poisson distributed). Determine the average waiting time in the system (in minutes)

c. The bank decides to open a second drive-thru teller window that has its own dedicated queue. The service rate of the second window is the same as the first one. Determine the average waiting time in the system (in minutes).

Answer 1

a) Arrival rate:  λ = 1/2.75 = 0.3636 minutes

Service rate: µ = 2.5 minutes

Average waiting time in queue: Wq = λ/µ(µ-λ) = 0.36/2.5(2.5-0.36) = 0.0673 minutes

b) Service rate: µ = 4.5 minutes

Arrival rate: λ = 12 per hour = 12/60 = 0.2 minutes

Average waiting time in system: Ws = 1/(µ-λ) = 1/(4.5 -0.2) = 0.2326 minutes

c) No. of servers: c=2

Service rate of each teller: µ = 4.5 minutes

Arrival rate: λ = 12 per hour = 12/60 = 0.2 minutes

p = λ/cµ = 0.2/(4.5*2) = 0.0222

m = 0, c= 2

P0 = 0.956

Lq = 0.00002

Wq =Lq/λ = 0.00002/.2 = 0.001

Waiting time in system: Ws = Wq + 1/µ = .001 + 1/4.5 = 0.2232 minutes