c
Hung
Q. Ngo, Computer Science and Engineering, SUNY at Buffalo
This Week’s Agenda
October 10, 2005
Last Time
• CTMC
Today
• A Primer on Queueing Theory
CSE 620 Lecture Notes
Advanced Networking Concepts
Page 1
c
Hung
Q. Ngo, Computer Science and Engineering, SUNY at Buffalo
Basic Model and Kendall Notation
October 10, 2005
B
B
A
B
Denote the system by A/B/k/c − S, where
• A is the inter-arrival time distribution
• B is the service time distribution at each server
• k is the number of servers
• c is the queue capacity (omit if ∞)
• S is the service discipline (omit if FIFO)
Some common distributions of inter-arrival and service times:
• M : Markov, i.e. exponential; e.g. M/M/1, M/M/k
• D: Deterministic; e.g. M/D/k
• Em : Erlang-m (common with phone call arrivals at toll offices)
• Hm : Hyper-m
• G: a general distribution
CSE 620 Lecture Notes
Advanced Networking Concepts
Page 2
c
Hung
Q. Ngo, Computer Science and Engineering, SUNY at Buffalo
October 10, 2005
Basic Parameters and Performance Measures
Input Parameters
• Packet arrival processes (from the outside)
• Distributions of service times at servers
• Service disciplines at servers: FIFO, LIFO, Random,
Priorities (rush order first, shortest processing time first, ...)
• Service capacity (nodes have single or multiple servers)
• Queue capacity (finite or infinite)
Output Performance Measures
• Distributions of waiting times at nodes
• Distributions of sojourn times (wait + service times)
• Distributions of number of packets at nodes
• Distributions of busy times at servers
• Packet loss rates
• Throughput
• ...
CSE 620 Lecture Notes
Advanced Networking Concepts
Page 3
c
Hung
Q. Ngo, Computer Science and Engineering, SUNY at Buffalo
October 10, 2005
Common Parameters and Measures for A/B/k/c
Systems
The averages
• λ: mean arrival rate
• µ: mean service time at each server
• ρ = λ/µ: traffic intensity
• N̄ : mean number of packets in the system
• N̄q : mean number of packets in the queue
• W̄ : mean waiting time overall, also called response time
• W̄q : mean waiting time in the queue
• r: throughput (departure rate)
• u: utilization
The random variables
• N (t): number of packets in the system at time t
• N : number of packets in the system at steady state
• τ : invariant distribution (if any) of {N (t)}t≥0
CSE 620 Lecture Notes
Advanced Networking Concepts
Page 4
c
Hung
Q. Ngo, Computer Science and Engineering, SUNY at Buffalo
Little’s Formulas
October 10, 2005
Think: each packet pays money to wait and/or to be serviced.
Mean rate system earns = λ× Mean amount a packet pays
The following are called Little’s Formulas
• Each packet pays 1$ per unit time in the system, then
N̄ = λ · W̄
• Each packet pays 1$ per unit time in the queue, then
N̄q = λ · W̄q
CSE 620 Lecture Notes
Advanced Networking Concepts
Page 5
c
Hung
Q. Ngo, Computer Science and Engineering, SUNY at Buffalo
October 10, 2005
The M/M/1 Queue
• A BDP with constant birth rate λ, constant death rate µ
• Stability condition
C=
∞
X
n=0
ρn < ∞ iff ρ < 1 iffλ < µ
• Steady state probabilities
1 n
τn = ρ = (1 − ρ)ρn n ≥ 0
C
• Fraction of time system has n packets is τn (λ + µ)
• It then follows that
Pr[N ≥ m]
N̄
W̄
W̄q
r
u
CSE 620 Lecture Notes
=
1−
=
X
m−1
X
τn = ρ m
n=0
nτn =
ρ
1−ρ
N̄
1
=
=
λ
µ−λ
1
ρ
= W̄ − =
µ
µ−λ
= (1 − τ0 )µ = λ
=
1 − τ0 = ρ
Advanced Networking Concepts
Page 6
c
Hung
Q. Ngo, Computer Science and Engineering, SUNY at Buffalo
October 10, 2005
M/M/1: Some Performance Graphing
20
x/(1-x)
18
Mean number of packets in the system
16
14
12
10
8
6
4
2
0
0
0.1
0.2
0.3
0.4
0.5
Utilization
0.6
0.7
20
0.8
0.9
1/(1-x)
18
16
Mean Response Time
14
12
10
8
6
4
2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Utilization
CSE 620 Lecture Notes
Advanced Networking Concepts
Page 7
c
Hung
Q. Ngo, Computer Science and Engineering, SUNY at Buffalo
October 10, 2005
The M/M/k Queue
• This is a birth and death process with constant birth rate λ
and death rates

iµ 1 ≤ i ≤ k
µi =
.
kµ i > k
• The stability condition is
∞
ρk X ρ i
ρn
+
< ∞,
C=
n!
k!
k
n=0
i=0
k−1
X
which holds iff ρ < k.
• At equilibrium,
τ0
τn
=
=
"k−1
X ρn
+
n!
n=0

 τ ρn
0 n!
n
τ0 ρn−k
k!k
CSE 620 Lecture Notes
k
ρ
1
k! 1 − ρ/k
#−1
1≤n≤k
n>k
Advanced Networking Concepts
Page 8
c
Hung
Q. Ngo, Computer Science and Engineering, SUNY at Buffalo
October 10, 2005
The M/M/1/c Queue
• This is a birth and death process with constant birth rate λ
and constant death rate µ, state space {0, 1, . . . , c}
• If ρ 6= 1,
τn =
• If ρ = 1,

 ρn
0
τn =
• Thus,
N̄ =
• Loss probability


ρ
1−ρ
c/2
1
c+1
0
−
n≤c
n>c


ploss = τc =
CSE 620 Lecture Notes
(1−ρ)
1−ρc+1
n≤c
n>c
c+1
c+1
1−ρc+1 ρ
ρ=1

 ρc

ρ 6= 1
(1−ρ)
1−ρc+1
1
c+1
Advanced Networking Concepts
ρ 6= 1
ρ=1
Page 9
c
Hung
Q. Ngo, Computer Science and Engineering, SUNY at Buffalo
October 10, 2005
M/M/1/30: Some Performance Graphing
16
(x/(1-x) - x**31 * 31/(1-x**31))
Mean number of packets in the system
14
12
10
8
6
4
2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Traffic intensity - M/M/1/30
1
0.7
0.8
0.9
x**30*(1-x)/(1-x**31)
0.9
Mean number of packets in the system
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
20
Traffic intensity - M/M/1/30
CSE 620 Lecture Notes
Advanced Networking Concepts
Page 10
c
Hung
Q. Ngo, Computer Science and Engineering, SUNY at Buffalo
October 10, 2005
Simple Comparisons of Queueing Systems
Question
Intuitively, is it better (in terms of response time) to have
an M/M/1 with service rate 10µ or
an M/M/10 with service rate µ,
CSE 620 Lecture Notes
Advanced Networking Concepts
Page 11
c
Hung
Q. Ngo, Computer Science and Engineering, SUNY at Buffalo
October 10, 2005
Simple Comparisons of Queueing Systems
1.4
Wa(x)
Wb(x)
1.2
Response Time
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
Lambda
12
14
16
18
Figure 1: Wa(x): waiting time of the M/M/10 queue
Wb(x): waiting time of the M/M/1 queue
CSE 620 Lecture Notes
Advanced Networking Concepts
Page 12