Question

In: Operations Management

A cinema has a single ticket counter that is manned by a cashier. The cashier is...

A cinema has a single ticket counter that is manned by a cashier. The cashier is capable to handle 280 customers in an hour. Customers arrive at the counter at the rate of four customers per minute. Daniel, the owner who studied queuing models feels that all the seven assumptions for a single-channel model are met. By assuming Exponential service times and Poisson arrival rate, answer the following questions.

a) State three assumptions mentioned above.

b) Determine the average number of customers waiting to buy ticket.

c) Determine the percentage of time the cashier is free.

d) Calculate the average time spent by each customer in the system.

e) Calculate the average time each customer needs to spend waiting to buy ticket.

f) The management of the theatre is planning to increase the number of counters if the probability of the system is busy is higher than 0.5. Is it necessary to increase the number of counters?

Solutions

Expert Solution

(A) Assumptions :

(i) Service time follows the exponential distribution

(ii) Arrival Rate follows follows the poisson distribution

(iii) The average inter-arrival time is larger than average process time

(B)

Arrival Rate () = 4 customers per minute

Arrival Rate () = 240 customers per hour

Service Rate () = 280 customers per hour

Average number of customers waiting to buy a ticket, Lq

Lq = 5.1429 customers

Therefore, average number of customers waiting to buy a ticket = 5.14286 customers

(C)

Percentage of time cashier is free = 1 - (Arrival Rate / Service Rate)

Percentage of time cashier is free = 1 - (240/280)

Percentage of time cashier is free = 0.14286

Percentage of time cashier is free = 14.286%

(D)

Average time a customer spends in the system, W

W = 1/(280-240)

W = 0.025 hours

Wq = 1.5 minutes

Average time a customer spends in system = 1.5 minutes

(E)

Average time a customer spends waiting to buy ticket, Wq

Wq = 5.14286/240

Wq = 0.02143 hours

Wq = 1.2858 minutes

Average time a customer spends waiting to buy ticket = 1.2858 minutes

(F)

Probability that system is busy = Arrival Rate / Service Rate

Probability that system is busy = 240/280

Probability that system is busy = 0.857

Since the Probability of system being busy is more than 0.5, therefore they must increase the number of counters.

keosha answered 11 months ago

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