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Network Performance and
Quality of Service
5. Queuing Analysis
Introduction


How to analyze changes in network
workloads? (i.e., a helpful tool to use)
Analysis of system (network) load and
performance characteristics
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response time
throughput
Performance tradeoffs are often not intuitive
Queuing theory, although mathematically
complex, often makes analysis very
straightforward
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Queuing Theory

Queuing theory provides a very
general framework for modeling
systems in which customers must line
up (queue) for service (use of
resource)
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Banks (tellers)
Restaurants (tables and seats)
Computer systems (CPU, disk I/O)
Networks (Web server, router, WLAN)
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Queue-based Models
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Queuing model represents:
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Arrival of jobs (customers) into system
Service time requirements of jobs
Waiting of jobs for service
Departures of jobs from the system
Typical diagram:
Customer
Arrivals
Departures
Buffer
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Server
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Parameters for Single-Server
Queuing System
Assuming queue has infinite capacity:
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At  = 1, server is working 100% of the time (saturated), so items are
queued (delayed) until they can be served. Departure rate remains
constant, no matter how great the arrival rate becomes.
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Thus, the theoretical maximum input rate that the system can handle is
max = 1 / TS
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The Fundamental Task of
Queuing Analysis
Given:
• Arrival rate, 
• Service time, Ts
• Number of servers, N
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Determine:
• Items waiting, w
• Waiting time, Tw
• Items queued, r
• Residence time, Tr
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Queuing Process - Example
General Expression:
TRn+1 = TSn+1 + MAX[0, Dn – An+1]
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General Characteristics of
Network Queuing Models
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Item population
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Queue size
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generally assumed to be infinite therefore,
arrival rate is persistent
infinite, therefore no loss
finite, more practical, but often immaterial
Dispatching discipline
 FIFO, typical
 LIFO
 Relative/Preferential, based on QoS
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Multiserver Queuing System
Comments:
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Assuming N identical servers, and  is the utilization of each server.
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Then, N is the utilization of the entire system (aka traffic intensity u )
and the maximum utilization is N x 100%.
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Therefore, the maximum supportable arrival rate that the system can
handle is: max = N / TS
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Multiple Single-Server
Queuing Systems
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Basic Queuing Relationships
General
Single
Server
Multiserver
r = Tr
Little’s Formula
 = Ts
 = Ts
N
w = Tw
Little’s Formula
r=w+
u = Ts = N
Tr = Tw + Ts
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r = w + N
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Queue Notation
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Queues are concisely described using
the Kendall notation, which specifies:
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Arrival process for jobs {M, D, G, …}
Service time distribution {M, D, G, …}
Number of servers {1, n}
Storage capacity (buffers) {B, infinite}
Service discipline {FIFO, PS, SRPT, …}
Examples: M/M/1, M/G/1 etc
Kendall’s notation
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Notation is X/Y/N, where:
X is distribution of interarrival times
Y is distribution of service times
N is the number of servers
M/M/1? M/D/1?
Common distributions
G
= general distribution if interarrival times or
service times
 GI = general distribution of interarrival time with the
restriction that they are independent
 M = exponential distribution of interarrival times
(Poisson arrivals) and service times
 D = deterministic arrivals or fixed length service
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Important Formulas for SingleServer Queuing Systems
Note Coefficient of variation:
if Ts = Ts => exponential
if Ts = 0 => constant
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Mean Number of Items in System (r)Single-Server Queuing
Ts/Ts = Coefficient of variation
M/M/1
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Mean Residence Time – (Tr)
Single-Server Queuing
M/M/1
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Multiple Server Queuing Systems
Multiserver
Queuing
System
Multiple SingleServer Queuing
System
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Important Formulas for Multiserver
Queuing
Note:
Useful only in
M/M/N case,
with equal
service times
at all N
servers.
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Multiple Server Queuing Example
Single server
M/M/1 (2nd Floor)
Multiserver
M/M/? (2nd Floor)
Multiple
Single server
M/M/1 (1st Floor)
M/M/1 (2nd Floor)
M/M/1 (3rd Floor)
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MultiServer vs Multiple Single-Server
Queuing System Comparison
Single server case (M/M/1):
Single server utilization: 
= 10 engineers x 0.5 hours each / 8 hour work day
= 5/8 = .625
Average time waiting: Tw
= Ts / 1 -  = 0.625 x 30 / .375 = 50 minutes
Arrival rate:  = 10 engineers per 8 hours = 10/480 = 0.021 engineers/minute
90th percentile waiting time: mTw(90) = Tw/ x ln(10) = 146.6 minutes
Average number of engineers waiting: w = Tw = 0.021 x 50 = 1.0416 engineers
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Example: Router Queuing
Internet
9600
bps
 = 5 packets/sec
L = 144 octets
…
From data provided:
• Ts = L/R = (144x8)/9600 = .12sec
•  = Ts = 5 packets/sec x .12sec = .6
Determine:
1. Tr= Ts / (1-) = .12sec/.4 = .3 sec
2. r =  / (1-) = .6/.4 = 1.5 packets
ln(1-.90)
- 1 = 3.5 packets
ln (.6)
ln(1-.95)
4. mr(95) =
- 1 = 4.8 packets
ln (.6)
3. mr(90) =
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For 3 & 4, use:
mr(y) =
ln(1 – y/100)
-1
ln 
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Priorities in Queues –
Two priority classes
r
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Priorities in Queues – Example
 
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
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Tr
Router queue services two packet sizes:
• Long = 800 octets
• Short = 80 octets
• Lengths exponentially distributed
• Arrival rates are equal, 8packets/sec
• Link transmission rate is 64Kbps
• Short packets are priority 1,
• Longer packets are priority 2
From data above, calculate:
Ts 1 = Lshort/R = (80 x 8) / 64000 = .01 sec
Ts 2 = Llong/R = (800 x 8) / 64000 = .1 sec
1 =  Ts 1 = 8 x 0.01 = 0.08
2 =  Ts 2 = 8 x 0.1 = 0.8
 = 1 + 2 = 0.88
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
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
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64Kbps
Find the average Queuing Delay (Tr)
through the router:
1 Ts 1 + 2 Ts 2
1 - 1
.08 x .01 + .8 x .1
= .01 +
= 0.098 sec
1-.08
Tr1 = Ts1 +
Tr2 = Ts2 +
= .1 +
Tr =
Tr 1 - Ts 1
1-
.098 - .01
1 - .88
= 0. 833 sec
1
2
T
+
 r1
 Tr2
= .5 x .098 + .5 x .833 = 0.4655 sec
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Network of Queues
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Elements of Queuing Networks
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Queuing Networks
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Jackson’s Theorem and
Queuing Networks
Assumptions:

–
–
–
the queuing network has m nodes, each providing
exponential service
items arriving from outside the system at any node arrive
with a Poisson rate
once served at a node, an item moves immediately to
another with a fixed probability, or leaves the network
Jackson’s Theorem states:

–
–
–
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each node is an independent queuing system with Poisson
inputs determined by partitioning, merging and tandem
queuing principles
each node can be analyzed separately using the M/M/1 or
M/M/N models
mean delays at each node can be added to determine
mean system (network) delays
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Jackson’s Theorem - Application in
Packet Switched Networks
Internal load:
L
Packet Switched
Network
External load, offered to network:
N
N
 =   jk
j=1 k=2
where:
 = total workload in packets/sec
jk = workload between source j
and destination k
N = total number of (external)
sources and destinations
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 =  i
i=1
where:
 = total on all links in network
i = load on link i
L = total number of links
Note:
• Internal > offered load
• Average length for all paths:
E[number of links in path] = /
• Average number of item waiting
and being served in link i: ri = i Tri
• Average delay of packets sent
through the network is:
1 L
Mi
T=

 i=1 Ri - Mi
where: M is average packet length and
Ri is the data rate on link i
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Estimating Model Parameters
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To enable queuing analysis using
these models, we must estimate
certain parameters:
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Mean and standard deviation of arrival
rate
Mean and standard deviation of service
time (or, packet size)
Typically, these estimates use sample
measurements taken from an existing
system
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Sample Means for Exponential
Distribution
Sampling:
• The mean is generally
the most important
quantity to estimate:
N
1
( ) = N Xi
i=1
• Sample mean is itself a
random variable
• Central Limit Theorem:
the probability
distribution tends to
normal as sample size,
N, increases for
virtually all underlying
distributions
• The mean and variance
of X can be calculated
as:
E[ ]= E[X] = 
Var[ ]= 2x/N

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