Answer to Question #86117 in Statistics and Probability for Anand

Question

There are two reservation counters for all ticket booking for customers, who arrive in
Poisson fashion at an average rate of 10 per hour. The service time for booking clerks
at both the counters are exponentially distributed, with mean time of 5 minutes.
These counters remain open for 12 hours per day.
i) Find the hours of the day for which all the clerks are busy.
ii) Find the expected waiting time of customers in the queue.

Answer 1

i) Find the hours of the day for which all the clerks are busy.

Two counters can serve:

2 customers / 5 min = 10 customers / 25 min

That is, clerks are busy 25 minutes per hour.

Accordingly, the clerks are busy per day:

25 * 12 = 300 min = 5 hours (300/60)

ii) Find the expected waiting time of customers in the queue.

Arrival rate: 𝜆=10/1hour

Service rate: 𝜇=1/5min=12/1hour

The expected waiting time of customers in the queue:

𝑊𝑞 =𝜆 / (2𝜇(𝜇 − 𝜆))=10/(2*12*(12-10))=10/48=5/24 hours = 12.5 min