Waiting time paradox
Simulation
Bus example: Waiting time paradox
Stochastic variable T:
Lecture
time between two subsequent bus departures
average waiting time = 12 (1 + cT2 ) E (T )
Queuing theory
`mathematics of waiting’
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Simulation, queueing theory
cT =
2
Queuing system
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σ (T )
E (T )
variation coefficien t
Simulation, queueing theory
Queuing systems: examples
Network of servers and queues
Uncertainty in arrival patterns of customers
Production line with buffers
Luggage handling system at Schiphol
and in service times
Dynamic bus station
Terminal layout at airport
Example question: how do waiting times
Planning trucks at gates at HEMA DC
depend on system design?
Design of a the lay-out of a warehouse
Easy cases: formulas from queuing theory
Hard cases: discrete-event simulation
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Planning of a distribution network
Inventory management
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Simulation, queueing theory
Psychological approach
Queue notation (Kendall)
a|b|c|d
Perception of waiting
a: arrival pattern
b: service times
c: number of
Expectation of waiting
“Appreciation= Perception - Expectation”
servers
d: capacity of
queue
M: exponential
G: general
distribution
D: deterministic
Examples: M|M|1,
M|G|1, M|M|c
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Little’s formula
PASTA
assume infinite waiting capacity
Poisson arrivals see time averages
L = λW
W = Wq + E ( B )
On the long run:
Lq = λWq
fraction of customers seeing a certain state
=
fraction of time that system is in that state
L : average number of items in system
W : average throughput time
Lq : average queue length
Wq : average waiting time
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M|M|1 queue (2)
M|M|1 queue
λ=
E( B) λ
1
1
=
, ρ=
, µ=
E( A ) µ
E( B)
E( A )
1 queue, 1 server
A, S exponential distribution (=high variance)
p j = (1 − ρ )ρ j
L=
Long-term average number of clients in the
ρ
1− ρ
W=
system
1
µ(1 − ρ)
L=
ρ
Wq =
µ(1 − ρ)
ρ2
Lq =
1− ρ
ρ
1− ρ
High variation:
L
Plan more careful !!
If FIFO discipline : Wq ( x ) = 1 − ρe −µ (1−ρ ) x
ρ
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Simulation, queueing theory
British study:
Why is the queue before the gentlemen’s toilet so
much longer than the queue before the ladies
toilet?
M|G|1 queue
ρ = λE (B)
Pollaczek - Khintchine
ρ2
L q = (1 + c )
(1 − ρ )
ρE (B)
W q = 12 (1 + c B2 )
(1 − ρ )
1
2
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2
B
Simulation, queueing theory
cB =
σ (B)
E (B)
12
Arrival: Poisson process
Arrival: Poisson process
with rate λ
Average length of stay:
mg = 89 sec
Variation coefficient
cg ≥ cl
with rate λ
Average length of stay:
ml = 39 sec
Variation coefficient cl
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Simulation, queueing theory