Question

part 2

Duque Vergere manages a Do or Die Theater complex called Cinema I, II, III, and IV. Each of the four auditoriums plays a different film; the schedule staggers starting times to avoid the large crowds that would occur if all four movies started at the same time. The theater has a single ticket booth and a cashier who can maintain an average service rate of 280 patrons per hour. Service times are assumed to follow an exponential distribution. Arrivals on a normally active day are Poisson distributed and average 210 per hour. To determine the efficiency of the current ticket operation, Duque Vergere wishes to examine several queue-operating characteristics.

d.) What is the average time spent waiting in line to get to the ticket window?
e.) What is the probability that there are more than two people in the system? More than three people? More than four?

Answer 1

We apply the single-server waiting line model:

= Mean Arrival rate of patrons

Here, = 210 per hour

= Mean service rate

Here, = 280 per hour

Average utilization of the system p = /

p = 210 / 280 = 0.75

The average time spent in the system

W = 1 / (280 - 210)

W = 0.014 hour = 0.014 * 60 minutes = 0.86 minutes

Average time spent in the line WQ = p * W

Average time spent in the line WQ = 0.75 * 0.014

Average time spent in the line WQ = 0.0107 hours = 0.0107 * 60 = 0.643 minutes

The probability that there are more than two people in the system = 1 - Probability of less than or equal to two people in the system

Probability of n customers in the system Pn = (1-p) * pn

Probability of 0 customers in the system = (1 - 0.75) * 0.750 = 0.25

Probability of 1 customer in the system = (1 - 0.75) * 0.751 = 0.188

Probability of 2 customers in the system = (1 - 0.75) * 0.750 = 0.141

Probability of 3 customers in the system = (1 - 0.75) * 0.750 = 0.105

Probability of 4 customers in the system = (1 - 0.75) * 0.750 = 0.079

The probability that there are more than two people in the system = 1 - Probability of less than or equal to two people in the system

Probability of less than or equal to two people in the system = Probability of (0 + 1 + 2) customers in the system

Probability of less than or equal to two people in the system = 0.25 + 0.188 + 0.141 = 0.579

The probability that there are more than two people in the system = 1 - 0.579

The probability that there are more than two people in the system = 0.421

The probability that there are more than three people in the system = 1 - Probability of less than or equal to three people in the system

Probability of less than or equal to three people in the system = Probability of (0 + 1 + 2 + 3) customers in the system

Probability of less than or equal to three people in the system = 0.25 + 0.188 + 0.141 + 0.105 = 0.684

The probability that there are more than three people in the system = 1 - 0.684

The probability that there are more than three people in the system = 0.316

The probability that there are more than four people in the system = 1 - Probability of less than or equal to four people in the system

Probability of less than or equal to four people in the system = Probability of (0 + 1 + 2 + 3 + 4) customers in the system

Probability of less than or equal to four people in the system = 0.25 + 0.188 + 0.141 + 0.105 + 0.079 = 0.764

The probability that there are more than four people in the system = 1 - 0.764

The probability that there are more than four people in the system = 0.236

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