Question

On average, 90 patrons arrive per hour at a hotel lobby (interarrival times are exponential) waiting to check in. At present there are five clerks, and patrons wait in a single line for the first available clerk. The average time for a clerk to service a patron is three minutes (exponentially distributed). Clerks earn $10 per hour, and the hotel assesses a waiting time cost of $20 for each hour that a patron waits in line. If needed, round your answers to the nearest cent.

a. Compute the expected cost per hour of the current system $___?

b. The hotel is considering replacing one clerk with an Automatic Clerk Machine (ACM). Management estimates that 20% of all patrons will use an ACM. An ACM takes an average of one minute to service a patron. It costs $48 per day (one day equals eight hours) to operate an ACM. Should the hotel install the ACM? Assume that all customers who are willing to use the ACM wait in a separate queue.

The hotel Select: should - should not replace the clerk with an ACM, as the total hourly cost would Select increase - decrease to $___?

Answer 1

a)

M/M/5 system with λ = 90 customers/hr

μ = 20 customers/hour.

P(j ≥ 5) = .76.

Then, Wq = P(j ≥ 5)/(100-90) = 0.076 hours.

Expected Cost Per Hour = 10(5) + 20(90)Wq

= 50 + 1800(.076)

= $186.80.

b)

With ACM we have an M/M/1 (the ACM) having λ = 18 customers per hour

μ = 60 customers per hour

an M/M/4 having λ = 72 customers/hour

μ = 20 customers/hour.

For ACM Wq =18/(60(60-18)) = 0.0071 hours.

Expected ACM Cost Per Hour = 6 + 20(18)Wq

= $8.57.

For M/M/4 system P(j≥4) = .79

so Wq = 0.79/8 = 0.0988.

Expected Cost Per Hour = 10(4) + 20(72)(0.0988)

= $182.27

Thus with ACM total hourly expected cost is 182.27 + 8.57 = 190.83.

Thus do not use ACM.

The hotel should not replace the clerk with an ACM, as the total hourly cost would decrease.