Previously

• Optimization

• Probability Review

• Inventory Models

• Markov Decision Processes

Agenda

• 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?

• Case studies

• Pricing options

• Utility theory (ch 9-10)

• Game theory (ch 16)

• 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)

Queues

system

arrivals departures

queue serversrate

service rate µ

c

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– Onlinehttp://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)

Diagnosisc=4

µ=4/hr

Surgeryc=3

µ=2/hr

Other units

12/hr 1/6

5/6

1/3

2/3

5.3/hr

3.3/hr

10/hr

2/hr