Prelim 1 – Module 4 Flashcards
Models for calculating the average wait time
Little’s law
Queueing models with 1 server (M/M/1) or c servers (M/M/c)
Little’s law
- analyze existing systems
-works for any system regardless of distribution - sometimes hard to get data
Queueing models with 1 server
- 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
Why do customers have to wait?
Variations in arrival rates and service rates
What is the managerial trade-off?
Utilization of servers vs. customer wait time
Ideally we could avoid the drawbacks of waiting by
- changing prices so that supply equals demand
- scheduling using reservations
- making waiting fun
Maister’s first law of service
Customers compare expectations with perceptions
Maister’s second law of service
It is hard to play catch-up, first impressions are critical
Strategies for waiting line management
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
Ws =
Average time in the system
Ls =
Average # of customers in the system
λ =
Average customer arrival rate
Ls =
λWs
50 emails a day, around 150 unanswered emails in inbox, how long does it take to reply to an email she receives?
𝝀=𝟓𝟎 𝒆𝒎𝒂𝒊𝒍𝒔/𝒅𝒂𝒚
𝑳𝒔=𝟏𝟓𝟎 𝒆𝒎𝒂𝒊𝒍𝒔
𝑾𝒔=? days
𝑳𝒔=𝝀𝑾𝒔
𝑾𝒔=𝑳𝒔/𝝀=(𝟏𝟓𝟎 𝒆𝒎𝒂𝒊𝒍𝒔)/(𝟓𝟎 𝒆𝒎𝒂𝒊𝒍𝒔/𝒅𝒂𝒚)
=𝟑 𝒅𝒂𝒚𝒔
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?
𝑳𝒔=𝟐𝟓 𝒉𝒐𝒖𝒔𝒆𝒔
𝑾𝒔=𝟏𝟐𝟎 𝒅𝒂𝒚𝒔
𝝀=?
𝑳𝒔=𝝀𝑾𝒔
𝝀=𝑳𝒔/𝑾𝒔 =(𝟐𝟓 𝒉𝒐𝒖𝒔𝒆𝒔)/(𝟏𝟐𝟎 𝒅𝒂𝒚𝒔)
=𝟎.𝟐𝟏 𝒉𝒐𝒖𝒔𝒆𝒔/𝒅𝒂𝒚
=𝟕𝟔 𝒉𝒐𝒖𝒔𝒆𝒔/𝒚𝒆𝒂𝒓
240 bottles, usually 2/3 full. Buy on about 8 bottles of wine/month. On average, how long have you stored your wine?
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
The cumulative distribution function for the exponential distribution
𝐹(𝑡)=1−𝑒^(−𝜆𝑡) =EXPON.DIST(𝑡,𝜆,TRUE) (𝑒=2.71828)
What is the probability that the time between consecutive arrivals is less than 30 seconds? More than 2 minutes?
𝐹(0.5)=1−𝑒^(−(1)(0.5))=39.3%
1−𝐹(2)=1−(1−𝑒^(−(1)(2) ) )=13.5%
The Poisson Distribution: A Benefit of the Exponential Distribution
Interarrival times are exponentially distributed <-> # of arrivals follows a poisson distribution
𝑓(𝑛)=((𝜆𝑡)^𝑛 𝑒^(−𝜆𝑡))/𝑛!
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?
𝑓(7)=((1∗5)^7 𝑒^(−1∗5))/7!=10.4%
𝑃𝑂𝐼𝑆𝑆𝑂𝑁.𝐷𝐼𝑆𝑇(3,5∗1,TRUE)=26.5%
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?
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%
If the game goes into overtime, what is the probability that no goal is scored in the next 20 minutes?
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%
What is the probability that Cornell scores the first goal in overtime?
𝜆𝐶𝑂𝑅𝑁𝐸𝐿𝐿/(𝜆𝐶𝑂𝑅𝑁𝐸𝐿𝐿+𝜆𝐻𝐴𝑅𝑉𝐴𝑅𝐷)=2.7/(2.7+3.1)=46.6%
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?
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%
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?
𝜆=0.75∗(50%)=0.375;𝑡=24;𝑥=10
𝑓(𝑛)=((𝜆𝑡)^𝑛 𝑒^(−𝜆𝑡))/𝑛!
= 41.3%
𝜆=
Arrival rate (lambda)
µ=
Service rate (mu)
c=
of servers
ρ = 𝜆/µ
Mean number in service (traffic intensity)
U
Average server utilization = 𝜆/cµ
Lq
Average length of the queue (# of customers)
Wq
Average time waiting before service
Ls
Average # of people in the system (waiting + in service)
Ws
Average time in system
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?
- 𝜆=40 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠/ℎ𝑜𝑢𝑟
𝜇=60 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠/ℎ𝑜𝑢𝑟 - 𝐿𝑠 = 𝐿_𝑞+𝜆/𝜇=1.33+40/60=2
- 𝑊𝑠 = 𝑊𝑞 +1/𝜇=2+1/1=3 𝑚𝑖𝑛𝑢𝑡𝑒𝑠
- 𝐿𝑞= 𝜆2/(𝜇(𝜇− 𝜆))=402/(60(60−40))=1.33
- 𝑊𝑞 =𝐿𝑞/𝜆=1.33/40=0.03 ℎ𝑜𝑢𝑟𝑠=2 𝑚𝑖𝑛𝑢𝑡𝑒𝑠
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
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
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.
Assume customers join a line randomly, then repeat the analysis 𝜆=(2/3)/2=1/3 for each server.
𝑊𝑞=0.5 minutes
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.
Repeat original analysis with
𝜇=(1 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟)/(0.6 𝑚𝑖𝑛𝑢𝑡𝑒𝑠)=1.67 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠/𝑚𝑖𝑛𝑢𝑡𝑒
𝑊𝑞 =0.40 𝑚𝑖𝑛𝑢𝑡𝑒𝑠
