Waiting Lines Example-1

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Waiting Lines

Waiting Lines Example-1

1. Students arrive at the Administrative Services Office at an average of one every 15 minutes, and their request take on average 10 minutes to be processed. The service counter is staffed by only one clerk, Judy Gumshoes, who works eight hours per day. Assume Poisson arrivals and exponential service times. M/M/1 System, R = 4 customers/hour, Poisson (Ci =1), Rp = 6 customers/hour, Exponential (Cp =1)

a) What percentage of time is Judy idle?

b) How much time, on average, does a student spend waiting in line?

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Waiting Lines


M/M/1 System, R = 4 customers/hour, Rp = 6 customers/hour

a) What percentage of time is Judy idle?

= R/Rp = 66.67% of time she is busy

1- = 33.33% of time idle

b) How much time, on average, does a student spend waiting in line?

Ti = Ii/R 1.33/4 = 0.33 hours

= 0.33 hours or 20 minutes

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2. You are working at a bank and doing resource requirements planning. You think that there should be six tellers working in the bank. Tellers take 15 minutes per customer with a standard deviation of 5 minutes. On average one customer arrives in every three minutes according to an exponential distribution.

a) On average how many customer would be waiting in line?

b) On average how long would a customer spend in the bank?

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c = 6, R = 20 customers/hour, Rp = 4 customers/hour

= R/Rp = 20 / (4×6) = 0.83

Ci = 1, Cp = 5/15 = 0.33

a) On average how many customer would be waiting in line?

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b) On average how long would a customer spend in the bank?

Ti = Ii/R 1.62/20 = 0.081 hours, or 4.86 minutes

Tp = 1/Rp = 1/4 hours or 15 minutes

T = Ti+Tp = 4.86+15 = 19.86 minutes

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3. Consider a call center that employs 8 agents. Past data collected on the customer inter-arrival times has shown that the mean time between customer arrivals is 1 minute, and has a standard deviation of 1/2 minute. The amount of time in minutes the past 10 callers have spent talking to an agent is as follows: 4.1, 6.2, 5.5, 3.5, 3.2, 7.3, 8.4, 6.3, 2.6, 4.9.

a) What is the coefficient of variation for the inter-arrival times?

b) What is the mean time a caller spends talking to an agent?

c) What is the standard deviation of the time a caller spends talking to an agent?

= 1.88 minutes

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d) What is the coefficient of variation for the times a caller spends talking to an agent?

e) What is the expected number of callers on hold, waiting to talk to an agent?

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a) What is the coefficient of variation for the inter-arrival times?

Ci = Sa/Ta = 0.5/1 = 0.5

b) What is the mean time a caller spends talking to an agent?

= average (4.1, 6.2, 5.5, 3.5, 3.2, 7.3, 8.4, 6.3, 2.6, 4.9) = 5.2 minutes.

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c) What is the standard deviation of the time a caller spends talking to an agent?

= 1.88 minutes

d) What is the coefficient of variation for the times a caller spends talking to an agent?

(standard deviation) / mean = 1.88/5.2 = 0.192

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e) What is the expected number of callers on hold, waiting to talk to an agent?

R= 1 per minute, c = 8,

Average processing rate for 1 agent is = 1/5.2

There are c=8 agents, Rp = 8/5.2 = 1.54

ρ = R/Rp= 1/(1.54) = 0.65

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f) What is the expected number of callers either on hold or talking to an agent?

I = Ii + Ip= 1.62 + 0.65*8 = 6.82

g) What is the expected amount of time a caller must wait to talk to an agent?

Ti = Ii / R = 1.62 minutes

h) What is the expected amount of time between when a caller first arrives to the system, and when that caller finishes talking to an agent?

T = I / R = 6.82 minutes

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4. Wells Fargo operates one ATM machine in a certain Trader Joe’s. There is on average 8 customers that use the ATM every hour, and each customer spends on average 6 minutes at the ATM. Assume customer arrivals follow a Poisson process, and the amount of time each customer spends at the ATM follows as exponential distribution.

a) What is the percentage of time the ATM is in use?

R = 8 per hour

Rp = 10 per hour

ρ = 8/10 = 0.8

=80%

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b) Suppose that the percentage of time the ATM is in use is 0.8.

b2) On average, how many customers are in line waiting to use the ATM?

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c) Suppose that the number of customers in line waiting to use the ATM is 3. (This may or may not be the answer you found in part b). All information remains as originally stated. What is the average time a customer must wait to use the ATM? State your answer in minutes.

Ti = Ii/R = 3/8 [3(60)]/8 = 22.5 minutes

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5. Matador housing office has one customer representative for walk-in students. The arrival rate 10 customers per hour and the average service time is 5 minutes. Both inter-arrival time and service time follow exponential distributions.

a) What is the average waiting time in line?

R = 10 customers/hour, Rp = 12 customers/hour, ρ = 10/12= 0.83

Ti = Ii/R = 4.17/10 = 0.42 hour

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6. Monterey post station has 7 tellers from Monday to Saturday. Customers arrive to the station following a Poisson process with rate 36 customers per hour. The service time is exponentially distributed with mean 10 minutes.

a) What is the utilization rate of the tellers?

R = 36, Rp = 7/10=.7 minute or 42 hour

ρ = R/Rp= 36/42 = 6/7 = 85.7%

b) What is the average number of customers waiting in line?

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On Sunday, instead of tellers, the post station only opens 3 auto-mail machines to provide automatic service. Each machine can weight different size of package, print self-adhesive labels and accept payments. Arrival is Poisson with rate 20 customers per hour. The service time is 4 minutes with probability 0.75 and 20 minutes with probability 0.25.

c) What is the mean service time?

ms = 4*0.75 + 16 * 0.25 = 8 minutes

d) What is the coefficient of variation of service time?

Ss = sqrt (0.75 * 32 + 0.25 * 92 ) = 6.93

Cs = 6.93 / 7 = 0.866

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e) What is the utilization rate?

R = 20 customers/hour, Rp = 3/8=.375 min or 22.5 hour

ρ =R/Rp= 20/(22.5) = 88.9%

f) What is the average number of customers waiting in line?

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7. Bank of San Pedro has only one teller. On average, one customer comes every 6 minutes, and it takes the teller an average of 3 minutes to serve a customer. To improve customer satisfaction, the bank is going to implement a unique policy called, “We Pay While You Wait.” Once implemented, the bank will pay each customer $3 per minute while she or he waits in line. (So the clock starts when a customer comes to the end of the line, and stops when he or she begins to talk to the teller.) Bank of San Pedro hired you as a consultant and you are responsible for estimating how much the “We Pay While You Wait” program will cost. Your preliminary study indicates there are, on average, 0.5 customers waiting in line Assume linear cost. If a customer waits for 20 seconds in line, Bank of San Pedro will pay $1.

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Calculate the capacity of the tellera) 10 customers/hour

b) 3.33 customers/hour

c) 20 customers/hour

d) 30 customers/hour

e) Cannot be determined

It takes the teller an average of 3 minutes to serve a customer

60/3 = 20 customers per hour


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Calculate proportion of the time the teller is busy. a) 100%

b) 80%

c) 62%

d) 50%

e) 40%

R=10, Rp=(1/3)*60= 20

R/Rp= 10/20 = 0.5

= 50%


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Waiting Lines

How long, on average, does a customer wait in line? a) 6 minutes

b) 4.8 minutes

c) 3 minutes

d) 2.6 minutes

e) 2 minutes

Indeed Ii was even given in the problem.

Ti= Ii/R

Ti= 0.5/10 = .05

0.05*60= 3 minutes


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Calculate the expected “hourly” cost of the “We Pay While You Wait” program. a) $9

b) $36

c) $60

d) $90

e) $140

Ii =0.5. Therefore, half a customer is always there. For each hour one customer gets 60*3 = 180. Thus o.5 customer gets 90$.

May be you do not believe me.

Each customer waits, on average, 3 minutes.

So he or she receives, on average, 3*3=$9.

There are 10 customers arriving per hour.

So the overall cost of this program is 9*10=$90.


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American Vending Inc. (AVI) supplies vending food to a large univeristy. Because students often kick the machines out of anger and frustration, management has a constant repair problem. The machines break down on an average of three per hour, and the breakdowns are distributed in a Poisson manner. Downtime costs the company $25/hour per machine, and each maintenance worker gets $4 per hour. One worker can service machines at an average rate of five per hour, distributed exponentially; two workers working together can service seven per hour, distributed exponencially; and a team of three workers can do eight per hour, distributed exponentially.

What is the optimal maintenance crew size for servicing the machines?


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A local grocery store has two cashier stations, and one experienced cashier and one novice cashier. During a typical working day (8 hours), 120 customers will show up. The novice cashier will serve 48 customers and the experienced cashier will serve 72 customers. On average it takes 6 minutes for the novice cashier to serve one customer and 3 minutes for the experienced cashier to serve one customer.

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During the rush hours, approximate 25 customers will show up during an hour. Does the store have enough capacity for the rush hour?

A Yes, the capacity of the store is 25 customers per hour.B Yes, the capacity of the store is 30 customers per hour.C No, the capacity of the store is 10 customers per hour.D No, the capacity of the store is 20 customers per hour.E None of the above.

The manager finds that during a day, both cashier stations have two customers waiting on average. On average, how long does a customer stay in the novice cashier’s waiting line?

A 8 minutesB 11 minutesC 16 minutesD 20 minutesE None of the above

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On average, how long does a customer stay in the experienced cashier’s line?

A 7 minutesB 9.4 minutesC 13.3 minutesD 15 minutesE None of the above

On average, how long does a customer spend in the store?A 14 minutesB 16 minutesC 19 minutesD 24 minutesE None of the above

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On average, how long does a customer stay in the experienced cashier’s line?

A 7 minutesB 9.4 minutesC 13.3 minutesD 15 minutesE None of the above

On average, how long does a customer spend in the store?A 14 minutesB 16 minutesC 19 minutesD 24 minutesE None of the above

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To shorten the waiting-time, the manager does a detailed study and finds that half of the customers purchase 5 items or less, and half of the customers purchase more than 5 items. The manager decides to let the novice cashier only serves the customers purchase 5 items or less. After the change, it turns out that both cashier stations have 1.75 customers waiting on average. Assume that the novice cashier serves all of the customers purchasing 5 items or less and the experienced cashier serves all of the customers purchasing more than 5 items.

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What is the average waiting time in the novice cashier’s line? A 13 minutesB 14 minutesC 17 minutesD 19 minutesE None of the above

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Question 33. What is the average waiting time in the experienced cashier’s line?

A 9 minutesB 11 minutes C 12 minutesD 14 minutesE None of the above

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Waiting Lines Example-1