Prelim 1 – Module 4

Question 1

Models for calculating the average wait time

Answer 1

Little’s law
Queueing models with 1 server (M/M/1) or c servers (M/M/c)

Question 2

Little’s law

Answer 2

• analyze existing systems
  -works for any system regardless of distribution
• sometimes hard to get data

Question 3

Queueing models with 1 server

Answer 3

• rough-cut design of a new service process
• can get deep insights without lots of information
• service times and inter-arrival times must follow exponential distribution
• 4 equations
• need table or spreadsheet to calculate Lq for systems with more than 1 server

Question 4

Why do customers have to wait?

Answer 4

Variations in arrival rates and service rates

Question 5

What is the managerial trade-off?

Answer 5

Utilization of servers vs. customer wait time

Question 6

Ideally we could avoid the drawbacks of waiting by

Answer 6

1. changing prices so that supply equals demand
2. scheduling using reservations
3. making waiting fun

Question 7

Maister’s first law of service

Answer 7

Customers compare expectations with perceptions

Question 8

Maister’s second law of service

Answer 8

It is hard to play catch-up, first impressions are critical

Question 9

Strategies for waiting line management

Answer 9

That old empty feeling: unoccupied time goes slowly
A foot in the door: pre-service waits seem longer than in-service waits
The light at the end of the tunnel: reduce anxiety with attention
Excuse me, but I was first: a first-come, first-served (FCFS) queue discipline is often perceived as most fair

Question 10

Ws =

Answer 10

Average time in the system

Question 11

Ls =

Answer 11

Average # of customers in the system

Question 12

λ =

Answer 12

Average customer arrival rate

Question 13

Ls =

Answer 13

λWs

Question 14

50 emails a day, around 150 unanswered emails in inbox, how long does it take to reply to an email she receives?

Answer 14

𝝀=𝟓𝟎 𝒆𝒎𝒂𝒊𝒍𝒔/𝒅𝒂𝒚
𝑳𝒔=𝟏𝟓𝟎 𝒆𝒎𝒂𝒊𝒍𝒔

𝑾𝒔=? days

𝑳𝒔=𝝀𝑾𝒔

𝑾𝒔=𝑳𝒔/𝝀=(𝟏𝟓𝟎 𝒆𝒎𝒂𝒊𝒍𝒔)/(𝟓𝟎 𝒆𝒎𝒂𝒊𝒍𝒔/𝒅𝒂𝒚)
=𝟑 𝒅𝒂𝒚𝒔

Question 15

A real estate agent has noticed that it takes about 120 days to sell a house and that typically there about 25 houses on the market. About how many transactions are there in a year?

Answer 15

𝑳𝒔=𝟐𝟓 𝒉𝒐𝒖𝒔𝒆𝒔

𝑾𝒔=𝟏𝟐𝟎 𝒅𝒂𝒚𝒔

𝝀=?

𝑳𝒔=𝝀𝑾𝒔

𝝀=𝑳𝒔/𝑾𝒔 =(𝟐𝟓 𝒉𝒐𝒖𝒔𝒆𝒔)/(𝟏𝟐𝟎 𝒅𝒂𝒚𝒔)
=𝟎.𝟐𝟏 𝒉𝒐𝒖𝒔𝒆𝒔/𝒅𝒂𝒚
=𝟕𝟔 𝒉𝒐𝒖𝒔𝒆𝒔/𝒚𝒆𝒂𝒓

Question 16

240 bottles, usually 2/3 full. Buy on about 8 bottles of wine/month. On average, how long have you stored your wine?

Answer 16

𝝀=𝟖 𝒃𝒐𝒕𝒕𝒍𝒆𝒔/𝒎𝒐𝒏𝒕𝒉
𝑳𝒔=(𝟐/𝟑)𝟐𝟒𝟎 𝒃𝒐𝒕𝒕𝒍𝒆𝒔=𝟏𝟔𝟎 𝒃𝒐𝒕𝒕𝒍𝒆𝒔

𝑾𝒔=? days

𝑳𝒔=𝝀𝑾𝒔

𝑾𝒔=𝑳𝒔/𝝀=(𝟏𝟔𝟎 𝒃𝒐𝒕𝒕𝒍𝒆𝒔)/(𝟖 𝒃𝒐𝒕𝒕𝒍𝒆𝒔/𝒎𝒐𝒏𝒕𝒉)
=𝟐𝟎 𝒎𝒐𝒏𝒕𝒉𝒔

Question 17

The cumulative distribution function for the exponential distribution

Answer 17

𝐹(𝑡)=1−𝑒^(−𝜆𝑡) =EXPON.DIST(𝑡,𝜆,TRUE) (𝑒=2.71828)

Question 18

What is the probability that the time between consecutive arrivals is less than 30 seconds? More than 2 minutes?

Answer 18

𝐹(0.5)=1−𝑒^(−(1)(0.5))=39.3%
1−𝐹(2)=1−(1−𝑒^(−(1)(2) ) )=13.5%

Question 19

The Poisson Distribution: A Benefit of the Exponential Distribution

Answer 19

Interarrival times are exponentially distributed <-> # of arrivals follows a poisson distribution
𝑓(𝑛)=((𝜆𝑡)^𝑛 𝑒^(−𝜆𝑡))/𝑛!

Question 20

What is probability that 7 arrive in 5 min with arrival rate 𝜆=1 per minute?

What is probability that 3 or fewer arrive in 5 min?

Answer 20

𝑓(7)=((1∗5)^7 𝑒^(−1∗5))/7!=10.4%

𝑃𝑂𝐼𝑆𝑆𝑂𝑁.𝐷𝐼𝑆𝑇(3,5∗1,TRUE)=26.5%

Question 21

The Cornell hockey team scores 2.7 goals per game.
The Harvard hockey team scores 3.1 goals per game.
If Harvard scores two goals, what is the probability that Cornell ties? Wins?

Answer 21

𝑓(𝑛)=((𝜆𝑡)^𝑛 𝑒^(−𝜆𝑡))/𝑛!

TIE: 𝑃𝑂𝐼𝑆𝑆𝑂𝑁.𝐷𝐼𝑆𝑇(2,2.7,FALSE)=24.5%
WIN: 1−𝑃𝑂𝐼𝑆𝑆𝑂𝑁.𝐷𝐼𝑆𝑇(2,2.7,𝑇𝑅𝑈𝐸)=50.6%

Question 22

If the game goes into overtime, what is the probability that no goal is scored in the next 20 minutes?

Answer 22

𝐹(𝑡)=1−𝑒^(−𝜆𝑡)

𝜆=𝜆𝐶𝑂𝑅𝑁𝐸𝐿𝐿+𝜆𝐻𝐴𝑅𝑉𝐴𝑅𝐷=2.7+3.1=5.8 𝑔𝑜𝑎𝑙𝑠 𝑝𝑒𝑟 60 𝑚𝑖𝑛𝑢𝑡𝑒𝑠=0.10 𝑔𝑜𝑎𝑙𝑠 𝑝𝑒𝑟 𝑚𝑖𝑛𝑢𝑡𝑒
1−𝐸𝑋𝑃𝑂𝑁.𝐷𝐼𝑆𝑇(20,0.1,𝑇𝑅𝑈𝐸)=13.5%

Question 23

What is the probability that Cornell scores the first goal in overtime?

Answer 23

𝜆𝐶𝑂𝑅𝑁𝐸𝐿𝐿/(𝜆𝐶𝑂𝑅𝑁𝐸𝐿𝐿+𝜆𝐻𝐴𝑅𝑉𝐴𝑅𝐷)=2.7/(2.7+3.1)=46.6%

Question 24

You have 10 rooms left, and only 24 hours remain to sell those rooms. At the price of $129, customer reservations arrive at a rate of 0.75 per hour. What is the probability that you sell all 10 rooms?

Answer 24

𝜆=0.75;𝑡=24;𝑥=10
Probability that we sell 10 or more = 1- Prob that we sell 9 or fewer
𝑓(𝑛)=((𝜆𝑡)^𝑛 𝑒^(−𝜆𝑡))/𝑛!
= 98.5%

Question 25

If we set the price to $189, only 50% of the customers would decide not to buy. What is the probability now that you sell all 10 rooms?

Answer 25

𝜆=0.75∗(50%)=0.375;𝑡=24;𝑥=10
𝑓(𝑛)=((𝜆𝑡)^𝑛 𝑒^(−𝜆𝑡))/𝑛!
= 41.3%

Question 26

𝜆=

Answer 26

Arrival rate (lambda)

Question 27

µ=

Answer 27

Service rate (mu)

Question 28

c=

Answer 28

of servers

Question 29

ρ = 𝜆/µ

Answer 29

Mean number in service (traffic intensity)

Question 30

U

Answer 30

Average server utilization = 𝜆/cµ

Question 31

Lq

Answer 31

Average length of the queue (# of customers)

Question 32

Wq

Answer 32

Average time waiting before service

Question 33

Ls

Answer 33

Average # of people in the system (waiting + in service)

Question 34

Ws

Answer 34

Average time in system

Question 35

One front desk clerk during busy check in times (4-7pm). 1 min to check a person in. 40 customers arrive per hour. How long do customers wait on average? How long is the line on average?

Answer 35

1. 𝜆=40 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠/ℎ𝑜𝑢𝑟
   𝜇=60 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠/ℎ𝑜𝑢𝑟
2. 𝐿𝑠 = 𝐿_𝑞+𝜆/𝜇=1.33+40/60=2
3. 𝑊𝑠 = 𝑊𝑞 +1/𝜇=2+1/1=3 𝑚𝑖𝑛𝑢𝑡𝑒𝑠
4. 𝐿𝑞= 𝜆2/(𝜇(𝜇− 𝜆))=402/(60(60−40))=1.33
5. 𝑊𝑞 =𝐿𝑞/𝜆=1.33/40=0.03 ℎ𝑜𝑢𝑟𝑠=2 𝑚𝑖𝑛𝑢𝑡𝑒𝑠

Question 36

One front desk clerk during busy check in times (4-7pm). 1 min to check a person in. 40 customers arrive per hour.

Add a second clerk, but keep one line

Answer 36

Change from c=1 to c=2.
Use Appendix C table and select value for 𝜌=0.65 (rounding from 𝜌=0.67) to get
𝐿𝑞=0.077 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠
Or use spreadsheet to get: (more precise)
𝐿𝑞=0.083 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠

𝑊𝑞 =𝐿𝑞/𝜆=(0.083 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠)/(2/3 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠/𝑚𝑖𝑛𝑢𝑡𝑒)=0.125 𝑚𝑖𝑛𝑢𝑡𝑒𝑠

Question 37

One front desk clerk during busy check in times (4-7pm). 1 min to check a person in. 40 customers arrive per hour.

Add a second clerk, but have two lines.

Answer 37

Assume customers join a line randomly, then repeat the analysis 𝜆=(2/3)/2=1/3 for each server.
𝑊𝑞=0.5 minutes

Question 38

One front desk clerk during busy check in times (4-7pm). 1 min to check a person in. 40 customers arrive per hour.

Revise the check-in process so that it only takes 0.6 minutes.

Answer 38

Repeat original analysis with
𝜇=(1 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟)/(0.6 𝑚𝑖𝑛𝑢𝑡𝑒𝑠)=1.67 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠/𝑚𝑖𝑛𝑢𝑡𝑒

𝑊𝑞 =0.40 𝑚𝑖𝑛𝑢𝑡𝑒𝑠