1.225J
1.225J (ESD 225) Transportation Flow Systems
Lecture 8
Delays in Probabilistic Models:
Elements from Queueing Theory
Profs. Ismail Chabini and Amedeo Odoni
Lecture 8 Outline
‰ Introduction to Queueing
‰ Conceptual Representation of Queueing Systems
‰ Codes for Queueing Models
‰ Terminology and Notation
‰ Little’s Law and Basic Relationships
‰ Exponential Distribution for Interarrival and Service times Modeling
‰ State Transition Diagram
‰ Derivation of waiting characteristics for M/M/1
‰ Summary
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Lecture 8, Page 2
1
Applications of Queueing Theory
‰ Some familiar queues:
• Airport check-in
• Automated Teller Machines (ATMs)
• Fast food restaurants
• On hold on an 800 phone line
• Urban intersection
• Toll booths
• Aircraft in a holding pattern
• Calls to the police or to utility companies
‰ Level-of-service (LOS) standards
‰ Economic analyses involving trade-offs among operating costs, capital
investments and LOS
‰ Queueing theory predicts various characteristics of waiting lines (or
queues) such as average waiting time
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Lecture 8, Page 3
Queueing Models Can Be Essential in Analysis of Capital
Investments
Cost
Total cost
Optimal
cost
Cost of building the capacity
Cost of losses due to waiting
“Optimal” capacity
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Capacity
Lecture 8, Page 4
2
Strengths and Weaknesses of Queueing Theory
‰ Queueing models necessarily involve approximations and
simplification of reality
‰ Results give a sense of order of magnitude, of changes relative to
a baseline, of promising directions in which to move
‰ Closed-form results are essentially limited to “steady state”
conditions and derived primarily (but not solely) for birth-anddeath systems and “phase” systems
‰ Some useful bounds for more general systems at steady state
‰ Numerical solutions are increasingly viable for dynamic systems
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Lecture 8, Page 5
Queueing Process and Queueing System
Queueing System
Servers
C
Arrival point
at the system
C
Queue
Source
of users/
customers
Departure point
from the system
C
C C C C C C
C
C
C
C
Arrivals
process
Size of
user source
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Queue discipline and
Queue capacity
Service process
Number of servers
Lecture 8, Page 6
3
Queueing network consisting of five queueing systems
Queueing
system
2
In
Queueing
system
1
Queueing
system
3
Point where
users merge
Point where
users make
a choice
+
Queueing
system
5
Out
Queueing
system
4
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Lecture 8, Page 7
A Code for Queueing Models: A/B/m
Distribution of
service time
Queueing System
Number of servers
–/–/–
Distribution of
interarrival time
Customers
C
C
CCCCCC
C
C
Queue
S
S
S
S
Service
facility
‰ Some standard code letters for A and B:
• M: Negative exponential (M stands for memoryless)
• D: Deterministic
• Ek:kth-order Erlang distribution
• G: General distribution
‰ Model covered in this lecture: M/M/1
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4
Terminology and Notation
‰ State of system: number of customers in queueing system
‰ Queue length: number of customers waiting for service
‰ N(t) = number of customers in queueing system at time t
‰ Pn(t) = probability that N(t) is equal to n
‰ λn: mean arrival rate of new customer when N(t) = n
‰ µn: mean (combined) service rate when N(t) = n
‰ Transient condition: state of system at t depends on the state of the
system at t=0 or on t
‰ Steady state condition: system is independent of initial state and t
‰ s: number of servers (parallel service channels)
‰ If λn and the service rate per busy server are constant, then λn=λ, µn=sµ
‰ Expected interarrival time =
1
1
λ
‰ Expected service time = µ
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Lecture 8, Page 9
Quantities of Interest at Steady State
‰ Given:
• λ = arrival rate
• µ = service rate per service channel (number of servers =1, in this
lecture)
‰ Unknowns:
• L = expected number of users in queueing system
• Lq = expected number of users in queue
• W = expected time in queueing system per user (W = E(w))
• Wq = expected waiting time in queue per user (Wq = E(wq))
‰ 4 unknowns ⇒ We need 4 equations
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Lecture 8, Page 10
5
Little’
Little’s Law
A(t): cumulative arrivals to the system
C(t): cumulative service completions in the system
Number of
users
A(t)
N(t)
C(t)
t
T
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∫
LT = 0
N (t )dt
T
T
Time
T
=
A(T ) ∫0 N (t ) dt
⋅
= λT ⋅ WT
T
A(T )
Lecture 8, Page 11
Relationships between L, Lq, W,
W, and Wq
‰ 4 unknowns: L, W, Lq, Wq
‰ Need 4 equations. We have the following 3 equations:
• L = λW (Little’s law)
• Lq = λWq
1
• W = Wq +
µ
‰ If we know L (or any one of the four expected values), we can determine
the value of the other three
‰ The determination of L may be hard or easy depending on the type of
queueing model at hand (i.e. M/M/1, M/M/s, etc.)
∞
‰ L = ∑ nPn ( Pn : probability that n customers are in the system)
n=0
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6
Modeling Interarrival Time and Service Time
• T : Interarrival (service) time random variable
αe −αt , t ≥ 0
• Density function : fT (t) = 
,t < 0
0
• P{0 ≤ T ≤ t} = 1− e −αt , E (T ) =
1
α
, var(T ) =
1
α2
• For small ∆t, P{0 ≤ T ≤ ∆t} ≈ α ∆t (why?)
∞
• e x = 1+ x + ∑
k =2
xk
k!
(−α ∆t) k
)
k!
k =2
≈ α ∆t (for small ∆t)
∞
• P{0 ≤ T ≤ ∆t} = 1− e −α ∆t = 1− (1− α ∆t + ∑
• Interarrival Time : α = λ ; Service Time : α = µ
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Lecture 8, Page 13
State Transition Diagram for M/M/1
‰ States:
0
…
2
1
n-1
n+1
n
‰ During ∆t:
P1λ∆t
P0 λ∆t
0
P2 λ∆t
Pn +1λ∆t
Pn λ∆t
Pn −1λ∆t
n-1
Pn −1µ∆t
P3 µ∆t
P2 µ∆t
P1µ∆t
…
2
1
Pn − 2 λ∆t
n+1
n
Pn + 2 µ∆t
Pn +1µ∆t
Pn µ ∆t
‰ Another way to represent it: State Transition Diagram
λ
λ
0
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µ
λ
…
2
1
µ
λ
µ
n-1
µ
n+1
n
µ
λ
λ
λ
µ
µ
Lecture 8, Page 14
7
Observing State Transition Diagram from Two Points
‰ From point 1:
λP0 = µP1
(λ + µ ) P1 = λP0 + µP2
λ
λ
0
(λ + µ ) Pn = λPn −1 + µPn +1
λ
λ
…
2
1
n-1
n+1
n
µ
µ
µ
µ
λ
λ
λ
µ
µ
µ
‰ From point 2:
λP0 = µP1
λPn = µPn +1
λP1 = µP2
λ
λ
0
λ
…
2
1
λ
λ
λ
n-1
n+1
n
µ
µ
µ
µ
λ
µ
µ
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µ
Lecture 8, Page 15
Derivation of P0 and Pn
n
2
‰ Putting it all together: P1 =
λ
λ
λ
P0 , P2 =   P0 , L, Pn =   P0
µ
µ
µ
n
∞
λ
‰ Since ∑ Pn = 1, ⇒ P0 ∑   = 1 ⇒ P0 =
n =0
n =0  µ 
∞
‰ Let ρ =
λ
, then
µ
‰ Therefore, P0 =
1
λ
 
∑
n =0  µ 
∞
n
n
∞
λ
1− ρ ∞
1
  = ∑ ρ n =
=
(Q ρ < 1)
∑
1
−
1
−
µ
ρ
ρ
n= 0 
 n =0
∞
1
∞
∑ρ
= 1− ρ
n
and Pn = ρ n (1 − ρ )
n =0
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8
Derivation of L, W, Wq, and Lq
∞
∞
∞
∞
n =0
n =0
n =0
n =1
• L = ∑ nPn =∑ nρ n (1 − ρ ) = (1 − ρ )∑ nρ n = (1 − ρ ) ρ ∑ nρ n −1
d  ∞ n
d  1 


= (1 − ρ ) ρ
 ∑ ρ  = (1 − ρ ) ρ
dρ  n = 0 
dρ  1 − ρ 
λ
 1 
ρ
λ
µ
=
= (1 − ρ ) ρ 
=
=
2 
λ
 (1 − ρ )  (1 − ρ ) 1 − µ µ − λ
•W=
L
λ
=
1
λ 1
⋅ =
µ −λ λ µ −λ
• Wq = W −
1
µ
=
• Lq = λ Wq = λ ⋅
1
µ −λ
−
1
µ
=
λ
µ (µ − λ )
λ
λ2
=
µ (µ − λ ) µ (µ − λ )
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Lecture 8, Page 17
Lecture 8 Summary
‰ Introduction to Queueing
‰ Conceptual Representation of Queueing Systems
‰ Codes for Queueing Models
‰ Terminology and Notation
‰ Little’s Law and Basic Relationships
‰ Exponential Distribution for Interarrival and Service times Modeling
‰ State Transition Diagram
‰ Derivation of waiting characteristics for M/M/1
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Lecture 8, Page 18
9