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 replace the clerk with an ACM, as the total hourly cost would to $ .

Answer 1

Answer:

a)

Given,

To compute the expected cost per hour of the current system

λ = 90 clients/hr

μ = 20 clients/hour.

P(j ≥ 5) = .76.

At that point, Wq = P(j ≥ 5)/(100-90)

= 0.076 hours.

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

= 50 + 1800(.076)

Expected Cost Per Hour = $186.80.

b)

Given that,

λ = 18 clients for every hour

μ = 60 clients for every hour

a M/M/4 having λ = 72 clients/hour

μ = 20 clients/hour.

Now ,

let us consider,

ACM Wq = 18 / (60 (60 - 18))

= 0.0071 hours.

Expected ACM Cost Per Hour is given as

= 6 + 20(18)Wq

= $8.57.

For M/M/4 framework P(j≥4) = .79

Wq = 0.79/8

= 0.0988.

Expected Cost Per Hour is given as below'

= 10(4) + 20(72)(0.0988)

= $182.27

In this manner with ACM all out hourly expected expense is give as follows

i.e.,

= 182.27 + 8.57

= 190.83.

Accordingly don't utilize ACM.

The inn ought not supplant the agent with an ACM, as the complete hourly expense would diminish.i.e., reduces.