Question

Skyline pizza is a famous restaurant operating a number of outlets. The restaurant uses a toll-free telephone number to book pizzas at any of its outlets. If the clerk is occupied on one line, incoming phone calls to the restaurant are answered automatically by an answering machine and asked to wait. As soon as the clerk is free, the party that has waited the longest is transferred and answered first. Calls come in at a rate of about 15 per hour. The clerk is capable of taking an order in an average of 3 minutes. Calls tend to follow a Poisson distribution, and service times tend to be exponential. The clerk is paid $20 per hour, but because of lost goodwill and sales, CWD loses about $50 per hour of customer time spent waiting for the clerk to take an order.

Part A Answer the following questions:

a. What is the probability that no customers are in the system (Po)? 2 marks

b. What is the average number of customers waiting for service ( Lq)? 3 marks

c. What is the average number of customers in the system (L)?-2 marks

d. What is the average time a customer waits for service(Wq)? 3 marks

e. What is the average time in the system (W) ?-3 marks

f. What is the probability that a customer will have to wait for service (Pw)?-2 marks

g What is the probability that there is exactly 2 customers in the system- 2 marks

h) What is the probability that there are more than 3 customers in the system-3 mark

Part B

Skyline is considering adding a second clerk to take calls. The store would pay that person the same $20 per hour.

Using appropriate formula for the multiple channel model, answer the following questions:

a. What is the probability that no customers are in the system (Po)? 3 marks

b. What is the average number of customers waiting for service (Lq)? 2 marks

c. What is the average number of customers in the system (L)? 3 mark

s d. What is the average time a customer waits for service (Wq)? 2 marks

e. What is the average time in the system (W)? 2 marks

f. What is the probability that a customer will have to wait for service (Pw)? 2 marks

g. What is the probability that there is exactly 2 customers in the system 2 marks

h. Should it hire another clerk? Explain by showing the cost savings - 4 marks

Answer 1

Answer:

λ = 15 per hour (calls)

µ = 20 per hour ( 3 per minute)

a. Average number of customers waiting to receive service = Lq = pLs

p = λ / µ = 15/20 = 0.75

= 0.75 * 3 = 2.25 customers

b. Probability that there are no customers are in the system = P0 = (1 - p)p^0

p = λ / µ = 15/20 = 0.75

P (0) = (1 - 15/20) * 0.75^0 = 0.25

c. Average number of customers in the system = Ls = λ/µ- λ

= 15 / 20 - 15

= 3 customers

d. Average time a customer waits for service = Wq = λ /µ (µ- λ)

= 15 / 20* (20 - 15)

= 3/20 hours or 9 minutes

e. Average time a customer is in the system = Ws = 1 / µ- λ

= 1/20 - 15

= 0.2 hours or 12 minutes

f. Probability that customer needs to wait= p = λ / µ

= 15 / 20

= 75%

g. Probability that there are exactly 2 customer in the system = P2 = (1 - p)p^2

= (1 - 15/20)*(15/20)^2

= 0.140625

h. Probability that there are 3 or more customers in the ssytem =1 - P2 = 1- (1 - p)p^2

= 1 - (1 - 15/20)*(15/20)^2

= 0.859375

λ = 15 per hour (calls)

µ = 40 per hour ( 3 per minute * 2)

a. Average number of customers waiting to receive service = Lq = pLs

p = λ / µ = 15/40 = 0.375

= 0.375 * 3 = 1.125 customers

b. Probability that there are no customers are in the system = P0 = (1 - p)p^0

p = λ / µ = 15/40 = 0.375

P (0) = (1 - 15/40) * 0.375^0 = 0.625

c. Average number of customers in the system = Ls = λ/µ- λ

= 15 / 40 - 15

= 0.6 customers

d. Average time a customer waits for service = Wq = λ /µ (µ- λ)

= 15 / 40* (40 - 15)

= 15/1000 hours or 0.9 minutes

e. Average time a customer is in the system = Ws = 1 / µ- λ

= 1/40 - 15

= 0.04 hours or 2.4 minutes

f. Probability that customer needs to wait= p = λ / µ

= 15 / 40

= 37.5%

g. Probability that there are exactly 2 customer in the system = P2 = (1 - p)p^2

= (1 - 15/40)*(15/40)^2

= 0.08789

h. Calls per hour = 15

Total Waiting Time = 15 *Wq = 15 * 9 = 135 minutes

New waiting time with 2 clerks = 15 *0.9 = 13.5 minutes

Hours saved = 135 - 13.5 = 121.5 minutes

Cost of Clerk per hour = $20

Less: Loss incurred for 121.5 minutes of waiting time = 121.5 / 60 * 50 = $101.25

Benfit = $81.25. Hence it should hire another clerk.

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