Previously
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Optimization
Probability Review
Inventory Models
Markov Decision Processes
Agenda
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Hwk
Projects
Additional Topics
Finish queues
Start simulation
Projects
• 10% of grade
• Comparing optimization algorithms
• Diet problem
• Vehicle routing
– Safe-Ride
– Limos
• Airplane ticket pricing
– Over time
– Different fare classes / demands
Additional Topics?
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Case studies
Pricing options
Utility theory (ch 9-10)
Game theory (ch 16)
Queues
service rate µ
departures
arrivals
rate 
queue
c servers
system
• M/M/s (arrivals / service / # servers)
M=exponential dist., G=general
• W = E[T], Wq = E[Tq]
waiting time in system (queue)
• L = E[N], Lq = E[Nq]
#customers in system (queue)
•  = /(cµ)
utilization
(fraction of time servers are busy)
Networks of Queues (14.10)
• Look at flow rates
– Outflow =  when  < 1
• What is the distribution between
arrivals?
– Not independent, formulas fail.
• Special case: all queues are M/M/s
“Jackson Network”
Lq just as if normal M/M/s queue
Queueing Resources
• M/M/s
– Online http://www.usm.maine.edu/math/JPQ/
– Lpc(rho,c) function from textbook (fails on excel 2007,2008)
• G/G/s
– QTP (fails on mac excel)
http://www.business.ualberta.ca/aingolfsson/QTP/
• G/G/s + Networks
– Online http://staff.um.edu.mt/jskl1/simweb
– ORMM book queue.xla at
http://www.me.utexas.edu/~jensen/ORMM/frontpage/jensen.lib
Distribution of Queue Length
• Why care?
– service guarantees
emergency response, missed flights
• M/M/1 case
– N+1 ~ Geometric(1-)
• Otherwise,
– ORMM add-in “state probabilities” P(N=k)
ER Example (p508)
12/hr
1/6
Surgery
5.3/hr
c=3
µ=2/hr
2/hr
1/3
5/6
10/hr
3.3/hr
Diagnosis
c=4
µ=4/hr
2/3
Other units