Question

The price of tickets varies depending on the seat location in the Stadium. These seats are color-coded on a diagram so that the single cashier can show purchasers the diagram, have them decide on their seat, & pay for their ticket. The cashier can complete a transaction (selling a ticket) in 2 minutes. 4 Ticket purchasers arrive at the single cashier booth every 10 minutes. Arrivals follow a Poisson distribution while service time follows an exponential distribution.

1. What is the Utilization rate of the ticket sale process?
2. Determine the average number of persons in the line.
3. How long does it take from the time a person joins the line to when they get their ticket?
4. What is the average time a person spends in the line?
5. What is the probability that there is no line?
6. What is the probability that there is 1 person at the cashier & 2 persons waiting in line?

Answer 1

1.

mean arrival rate:

$\lambda=4/10=0.4$ min^-1

mean service rate:

$\mu=1/2=0.5$ min^-1

Utilization rate:

$\rho=\lambda/\mu=0.4/0.5=0.8$

2.

$L_Q=\frac{\rho\lambda}{\mu-\lambda}=\frac{0.8\cdot0.4}{0.5-0.4}=3.2$

3.

$W=\frac{1}{\mu-\lambda}=\frac{1}{0.5-0.4}=10$ min

4.

$W_Q=\rho W=0.8\cdot10=8$ min

5.

$P_0=1-\lambda/\mu=1-0.4/0.5=0.2$

6.

$P_3=(1-\rho)\rho^n=(1-0.8)0.8^3=0.1024$