Question 1.

An electronics store has a repair department with one staff member working each day. The repair department is open every day starting from 9AM, and closes 6 hours later at 3PM. Customers bring devices to the store according to a Poisson process, with an average number of 5.5 devices arriving per 6-hour working day. The time to repair a device is exponentially distributed, with an average completion time of 50 minutes. As soon as a repair is completed, the customer can collect their device from the store. Assume that the staff member works continuously throughout the day, and then continues any unfinished jobs at the start of the next working day.

Given these assumptions, the number of devices in the store can be modelled as an M/M/1 queue. Assume that the queue has been running for long enough that its steady state properties apply.

(a) Write down the arrival rate, service rate, and the traffic intensity for this queue. Include relevant units.

(b) On a given day, what is the probability that 2 or fewer customers bring a device to the store before noon?

(c) Once the staff member begins repairing a device, what is the probability that it takes 1 hour or less to repair that device?

(d) What is the average number of devices in the queue (including any which are currently being repaired)?

(e) The store tells customers that they can expect to be able to collect their device by the next working day. Is the store’s claim justified? Explain.

(f) On average, how much time in one 6-hour work day will the staff member be actively repairing devices?

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