# Question: Antonio runs a shoe repair store by himself Customers arrive

Antonio runs a shoe repair store by himself. Customers arrive to bring a pair of shoes to be repaired according to a Poisson process at a mean rate of 1 per hour. The time Antonio requires to repair each individual shoe has an exponential distribution with a mean of 15 minutes.

(a) Consider the formulation of this queueing system where the individual shoes (not pairs of shoes) are considered to be the customers. For this formulation, construct the rate diagram and develop the balance equations, but do not solve further.

(b) Now consider the formulation of this queueing system where the pairs of shoes are considered to be the customers. Identify the specific queueing model that fits this formulation.

(c) Calculate the expected number of pairs of shoes in the shop.

(d) Calculate the expected amount of time from when a customer drops off a pair of shoes until they are repaired and ready to be picked up.

(e) Use the corresponding Excel template to check your answers in parts (c) and (d).

(a) Consider the formulation of this queueing system where the individual shoes (not pairs of shoes) are considered to be the customers. For this formulation, construct the rate diagram and develop the balance equations, but do not solve further.

(b) Now consider the formulation of this queueing system where the pairs of shoes are considered to be the customers. Identify the specific queueing model that fits this formulation.

(c) Calculate the expected number of pairs of shoes in the shop.

(d) Calculate the expected amount of time from when a customer drops off a pair of shoes until they are repaired and ready to be picked up.

(e) Use the corresponding Excel template to check your answers in parts (c) and (d).

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