It is common in banks for bank tellers to have to access a single central cash w

Question

It is common in banks for bank tellers to have to access a single central cash window in order to get cash for customers. Consider such a bank, which has seven tellers, each requiring service from the cash window approximately 19 times per hour (assume Poisson arrivals) and an average time being served at the window (assume negative exponential distribution) of 40 seconds. How busy will the server at the central cash window be, on average (i.e., what percentage of the time should they expect to be serving someone)?

50%

60%

70%

80%

90%

50%

60%

70%

80%

90%

Explanation / Answer

There are seven tellers and each serves 19 customers per hour.

Hence the total number of customers is 7*19 = 133 customers per hour. Hence the arrival rate is 1 customer arrives in 3600/133 seconds. Hence the arrival rate = 27.0677 seconds

Arrival rate () = 27.0677 sec

The server can serve one customer in 40 seconds

Hence the service rate(µ) = 40 seconds

Utilization rate = Arrival rate/ service rate

Utilization rate = 27.0677/40 = 0.67669 = 67.669%

Hence we can apporximate the answer to 70% and state that 70% of the times the machine is expected to be serving someone. Hence answer choice (c ) is the correct answer.

Hence we can find that the server serves customers all the time

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