Announcements
• PS #3 out this afternoon
Queueing Systems: Lecture 1
Amedeo R. Odoni
October 4, 2006
• Due: October 19 (graded by 10/23)
• Office hours – Odoni: Mon. 2:30-4:30
- Wed. 2:30-4:30 on Oct. 18 (No office hrs 10/16)
_ Or send me a message
• Quiz #1: October 25, open book, in class
• Old quiz problems and solutions:
posted on 10/19
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Topics in Queueing Theory
Lecture Outline
Introduction to Queues
Little’s Law
Markovian Birth-and-Death Queues
The M/M/1 and Other Markovian Variations
The M/G/1 Queue and Extensions
Priority Queues
Some Useful Bounds
Congestion Pricing
Queueing Networks; State Representations
Dynamic Behavior of Queues
• Introduction to queueing systems
• Conceptual representation of queueing
systems
• Codes for queueing models
• Terminology and notation
• Little’s Law and basic relationships
• Birth-and-death models
• The M/M/1 queueing system
Reference: Chapter 4, pp. 182-203
Queues
A Generic Queueing System
• Queueing theory is the branch of operations research
concerned with waiting lines (delays/congestion)
• A queueing system consists of a user source, a queue
and a service facility with one or more identical parallel
servers
• A queueing network is a set of interconnected queueing
systems
• Fundamental parameters of a queueing system:
- Demand rate
- Capacity (service rate)
- Demand inter-arrival times - Service times
- Queue capacity and discipline (finite vs. infinite;
FIFO/FCFS, SIRO, LIFO, priorities)
- Myriad details (feedback effects, “balking”, “jockeying”, etc.)
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
4
+
Queueing
system
5
Out
Servers
C
Arrival point
at the system
Queue
Source
of users/
customers
Departure point
from the system
C
C C CC CC
C
C
C
C
C
Arrivals
process
Size of
user source
Queue discipline and
Queue capacity
Service process
Number of servers
Applications of Queueing Theory
• Some familiar queues:
_ Airport check-in; aircraft in a holding pattern
_ Automated Teller Machines (ATMs)
_ Fast food restaurants
_ Phone center’s lines
_ Urban intersection
_ Toll booths
_ Spatially distributed urban systems and services
• Level-of-service (LOS) standards
• Economic analyses involving trade-offs among
operating costs, capital investments and LOS
• Congestion pricing
The Airside as a Queueing Network
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
Airport Capacity
“Optimal” capacity
Strengths and Weaknesses of Queueing Theory
• Queueing models necessarily involve approximations
•
•
•
•
and simplification of reality
Results give a sense of order of magnitude, changes
relative to a baseline, promising directions in which to
move
Closed-form results essentially limited to “steady
state” conditions and derived primarily (but not solely)
for birth-and-death systems and “phase” systems
Some useful bounds for more general systems at
steady state
Numerical solutions increasingly viable for dynamic
systems
• Huge number of important applications
A Code for Queueing Models: A/B/m
Distribution of
service time
Queueing System
Number of servers
–/–/–
Distribution of
interarrival time
Customers
Queue
CCCCCC
C
C
C
C
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
Terminology and Notation
• Number in system: number of customers in
•
•
•
•
•
•
queueing system
Number in queue or “Queue length”: number of
customers waiting for service
Total time in system and waiting time
N(t) = number of customers in queueing system
at time t
Pn(t) = probability that N(t) is equal to n at time t
λn: mean arrival rate of new customers when
N(t) = n
μn: mean (total) service rate when N(t) = n
Terminology and Notation (2)
• Transient state: state of system at t is
influenced by the state of the system at t = 0
• Steady state: state of the system is independent
of initial state of the system
• m: number of servers (parallel service
channels)
• If λn and the service rate per busy server are
constants λ and μ, respectively, then λn=λ, μn=
min (nμ, mμ); in that case:
_ Expected inter-arrival time = 1/λ
_ Expected service time = 1/μ
Some Expected Values of Interest
at Steady State
• Given:
_
_
Little’s Law
Number of
users
A(t): cumulative arrivals to the system
C(t): cumulative service completions in the system
λ = arrival rate μ = service rate per service channel A(t)
• 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
N(t)
_
C(t)
t
T
∫
LT = 0
N (t ) dt
T
T
=
T
A(T ) ∫0
⋅
T
A(T )
N (t ) dt
= λT ⋅ WT
Time
Relationships among L, Lq, W, Wq
• Four unknowns: L, W, Lq, Wq
• Need 4 equations. We have the following 3 equations:
_
L = λW (Little’s law)
Lq = λWq
_
W = Wq +
_
1
μ
• If we can find any one of the four expected values, we
can determine the three others
• The determination of L (or other) may be hard or easy
depending on the type of queueing system at hand
∞
• L = ∑ nPn (Pn : probability that n customers are in the system)
n=0
The Fundamental Relationship
queue plus in service) arrivals are
Poisson at rate of λn per unit of time.
4. Whenever n users are in system,
service completions are Poisson at
rate of μn per unit of time.
5. FCFS discipline (for now).
The differential equations that
determine the state probabilities
After a simple manipulation:
λnΔt
1-(λn+ μn)Δt
1. m parallel, identical servers.
2. Infinite queue capacity (for now).
3. Whenever n users are in system (in
Pn (t + Δt) = Pn +1 (t) ⋅ μ n +1 ⋅ Δt + Pn −1 (t) ⋅ λn −1 ⋅ Δt + Pn (t) ⋅[1 − ( μ n + λn ) ⋅ Δt]
Time: t+Δt
n+1 users
Time: t
Birth-and-Death Queueing Systems
Pn(t) = Prob [n users
in system at time t]
n users
n users
dPn (t)
= −(λn + μ n ) ⋅ Pn (t) + λn −1 ⋅ Pn −1 (t) + μ n +1 ⋅ Pn +1 (t)
dt
(1) applies when n = 1, 2, 3,….; when n = 0, we have:
μnΔt
dP0 (t)
= −λ0 ⋅ P0 (t) + μ1 ⋅ P1 (t)
dt
n-1 users
Pn (t + Δt) = Pn +1 (t) ⋅ μ n +1 ⋅ Δt + Pn −1 (t) ⋅ λn −1 ⋅ Δt + Pn (t) ⋅[1 − ( μ n + λn ) ⋅ Δt]
(2)
• The system of equations (1) and (2) is known as the
Chapman-Kolmogorov equations for a birth-and-death
system
(1)
Birth-and-Death System: State
Transition Diagram
The “state balance” equations
• We now consider the situation in which the queueing
λ0
system has reached “steady state”, i.e., t is large
enough to have Pn (t) = Pn , independent of t, or dPn (t) = 0
dt
0
n=0
(3)
n = 1, 2, 3,.. (4)
• The state balance equations can also be written directly
from the state transition diagram
λ2
1
2
μ1
• Then, (1) and (2) provide the state balance equations:
λ0 ⋅ P0 = μ1 ⋅ P1
(λn + μ n ) ⋅ Pn = λ n −1 ⋅ Pn −1 + μ n +1 ⋅ Pn +1
λ1
λm-1
λm+1
m
……
m+1
μm
μ3
μ2
λm
μm+1
……
μm+2
• We are interested in the characteristics of the system
under equilibrium conditions (“steady state”), i.e., when
the state probabilities Pn(t) are independent of t for
large values of t
• Can write system balance equations and obtain
closed form expressions for Pn, L, W, Lq, Wq
M/M/1: Observing State Transition
Diagram from Two Points
Solving…..
Solving (3) and (4), we have:
•
λ
λ ⋅λ
λ
P1 = 0 ⋅ P0 ; P2 = 1 ⋅ P1 = 1 0 ⋅ P0
μ1
μ2
μ 2 ⋅ μ1
and, in general,
λ
⋅λ
⋅.....⋅ λ1 ⋅ λ0
⋅ P0 = K n ⋅ P0
Pn = n −1 n − 2
μ n ⋅ μ n −1 ⋅.....⋅ μ 2 ⋅ μ1
1=
But, we also have:
Giving,
1
P0 =
1+
∞
∑ Kn
n =1
etc.
From point 1:
λP0 = μP1 (λ + μ )P1 = λP0 + μP2
λ
λ
0
μ
∞
∞
n =0
n =1
Condition for steady state:
∞
∑ Kn < ∞
n=1
•
λ
…
2
1
μ
∑ Pn = P0 ⋅ (1 + ∑ K n )
(λ + μ )Pn = λPn−1 + μPn+1
λ
n-1
n+1
n
μ
μ
μ
λ
λ
λ
λ
λ
λ
μ
μ
From point 2:
λP0 = μP1
0
λ
…
2
1
μ
λPn = μPn+1
λP1 = μP2
λ
μ
μ
n-1
μ
n+1
n
μ
λ
λ
μ
μ
M/M/1: Derivation of P0 and Pn
P1 =
Step 1:
∞
∑P
Step 2:
n
n=0
Step 3:
ρ=
Step 4:
⎛λ⎞
⎛λ⎞
λ
P0 , P2 = ⎜⎜ ⎟⎟ P0 , L, Pn = ⎜⎜ ⎟⎟ P0
μ
⎝μ⎠
⎝ μ ⎠
n
∞
⎛λ⎞
= 1, ⇒ P0 ∑ ⎜⎜ ⎟⎟ = 1 ⇒ P0 =
n=0 ⎝ μ ⎠
λ
, then
μ
P0 =
⎛λ⎞
∞
n
∞
∑ ⎜⎜ μ ⎟⎟ = ∑ ρ
n=0
⎠
⎝
1
∞
∑ρ
n=0
= 1− ρ
n
n
=
∞
∞
∞
n=0
n=0
n=0
n=1
d ⎛ ∞ n⎞
d ⎛ 1 ⎞
⎜
⎟
= (1− ρ ) ρ
⎜ ∑ ρ ⎟ = (1− ρ ) ρ
dρ ⎝ n=0 ⎠
dρ ⎜⎝ 1− ρ ⎟⎠
λ
1
⎛λ⎞
∑
⎜⎜ ⎟⎟
n=0 ⎝ μ ⎠
∞
∞
• L = ∑ nPn =∑ nρ n (1− ρ ) = (1− ρ )∑ nρ n = (1− ρ ) ρ ∑ nρ n−1
n
2
M/M/1: Derivation of L, W, Wq, and Lq
⎛ 1 ⎞
ρ
λ
μ
⎟=
=
=
= (1− ρ ) ρ ⎜⎜
2 ⎟
⎝ (1− ρ ) ⎠ (1− ρ ) 1− λ μ μ − λ
L
1
λ 1
•W= =
⋅ =
λ μ −λ λ μ −λ
n
1− ρ ∞
1
=
(Q ρ < 1)
1− ρ 1− ρ
1
• Wq = W − =
μ
and Pn = ρ n (1− ρ )
• Lq = λ Wq = λ ⋅
n=0
High Sensitivity of Delay at High Levels of Utilization
1
μ −λ
−
1
μ
=
λ
μ (μ − λ )
λ
λ2
=
μ (μ − λ ) μ (μ − λ )
M/M/1: An alternative, direct derivation
of L and W
• For an M/M/1 system, with FCFS discipline:
Expected delay
∞
W =∑
(n +1)
n=0
μ
⋅ Pn = E[
N +1
μ
]=
E[N ] +1
μ
=
L +1
μ
• But from Little’s theorem we also have:
L = λ ⋅W
(2)
• It follows from (1) and (2) that, as before:
Demand
Capacity
ρ=1
L=
λ
μ −λ
;
W =
1
μ −λ
Does the queueing discipline matter?
(1)
Additional important M/M/1 results
• The pdf for the total time in the system, w, can
be computed for a M/M/1 system (and FCFS):
M/M/1: E[B], the expected length of a busy period
N
B = busy period
I = idle period
f w (w) = (1 − ρ ) μe −(1− ρ ) μw = ( μ − λ )e −( μ − λ )w for w≥ 0
Thus, as already shown, W = 1/(μ -λ) = 1/[μ (1-ρ)]
• The standard deviation of N, w, Nq, wq are all
proportional to 1/(1-ρ), just like their expected
values (L, W, Lq, Wq, respectively)
• The expected length of the “busy period”, E[B],
is equal to 1/(μ -λ)
t
I
P0 =
But,
B
I
B
I
B
E[length Idle period ]
E[length Busy period ] + E[length Idle period ]
P0 = 1− ρ
Therefore,
E[length Idle period ] = 1
E[B] = E[length Busy period ] =
1
⋅
λ
1
μ (1− ρ )
=
1
μ −λ