Question

1. On average, 90 patrons per hour arrive at a hotel lobby (interarrival times are exponential) waiting to check in. At present there are five clerks, and the patrons wait in a single line for the first available clerk. The average time for a clerk to service a patron is 3 minutes (exponentially distributed). Clerks earn $10 per hour, and the hotel assesses a waiting time cost of $20 for each hour a patron waits in line. The hotel is considering replacing one clerk with an Automated Clerk Machine (ACM). Management estimates that 20% of all patrons will use an ACM. An ACM takes an average of 1 minute (also exponential) to service a patron. It costs $48 per day (1 day equals 8 hours to operate an ACM. Assume that all customers who are willing to use the ACM wait in a separate queue. Please use excel to solve the problem.

1. Should the hotel use the ACM to reduce costs?
2. What is the break-even cost of the ACM so that the expected hotel costs are the same?

Answer 1

a)

Total operating cost of the current system (five clerks) is determined using EXCEL as follows:

Service rate of each clerk = 60/3 = 20 per hour

EXCEL FORMULAS:

Cell	Formula	Copy to
H3	=($D$4/$D$5)^G2/FACT(G2)+H2	H3:H8
D9	=1/((VLOOKUP(D6+1,$G$2:$H$8,2,0)-VLOOKUP(D6,$G$2:$H$8,2,0))/(1-D4/(D5*D6))+VLOOKUP(D6,$G$2:$H$8,2,0))
D10	=D4/D5/D6
D11	=(VLOOKUP(D6+1,$G$2:$H$8,2,0)-VLOOKUP(D6,$G$2:$H$8,2,0))*D9*D10/(1-D10)^2+D4/D5
D12	=D11-D4/D5
D13	=D11/$D$4
D14	=D12/$D$4
D20	=D6*D17+D12*D18

After replacing one clerk with an ACM, there are two systems:

1) M/M/1 system served by ACM. arrival rate = 90*20% = 18 per hour , service rate = 60/1 = 60 per hour

Lq = 18^2/(60*(60-18)) = 0.129

Total operating cost per hour = Cs+Lq*Cw = 48/8+0.129*20 = $ 8.57

2) M/M/s system with 4 clerks

Arrival rate = 90-18 = 72 per hour

Total operating cost is computed using EXCEL as follows:

Total operating cost / hour of the complete system = Total operating cost of M/M/1 system + Total operating cost of M/M/s system

= 8.57+181.8

= $ 190.37

Total cost of the system after replacing a clerk with ACM is higher.

Therefore, the hotel should NOT use the ACM

b)

Breakeven cost of the ACM = 187.25-181.80-.129*20

= $ 2.87 per hour

= $ 22.96 per day