1. CSE 550 Computer Network Design
Dr. Mohammed H. Sqalli
COE, KFUPM
Spring 2007 (Term 062)

2. Outline Queuing Models Application to Networks Traffic Flow Analysis
CSE-550-T062
Lecture Notes - 3

3. Queuing Models - Single-Server Queue -
λ: average number of packets arriving per second [pps]
Utilization, fraction of time the facility is busy: ρ = λTs
Theoretical maximum input rate that can be handled by the system is: λmax = 1/Ts
Queues become very large near system saturation, growing without bound when ρ = 1
Practical considerations limit the input rate for a single server to 70-90% of the theoretical maximum
Little's formula (general relationship) : r = λTr and w = λTw
CSE-550-T062
Lecture Notes - 3

4. Queuing Models - Multiserver Queue -
Utilization: ρ = λTs/N
Theoretical maximum input rate that can be handled by the system is: λmax = N/Ts
Traffic intensity: u = Nρ
CSE-550-T062
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5. Queuing Models - Multiple Single-server queues -
Example of a Network of Queues
Traffic Partitioning
Traffic Merging
Queues in Tandem
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6. Queuing Models - Notation
The notation X/Y/N is used for queuing models
X = distribution of the inter-arrival times
Y = distribution of service times
N = number of servers
The most common distributions are:
G = general independent arrivals or service times
M = negative exponential distribution
D = deterministic arrivals or fixed length service
Example: M/M/1
CSE-550-T062
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7. Queuing Models - Single-server queues -
M/G/1 model:
The arrival rate is Poisson and the service time is general
M/M/1 model:
The standard deviation is equal to the mean, the service time distribution is exponential, i.e., service times are essentially random
M/D/1 model:
The standard deviation of service time is equal to zero, i.e., a constant service time
The poorest performance is exhibited by the exponential service time (M/M/1), and the best by a constant service time (M/D/1)
Usually, the exponential service time can be considered to be the worst case:
An analysis based on this assumption will give conservative results
CSE-550-T062
Lecture Notes - 3

8. Queuing Models - Single-server queues -
Coefficient of variation = σTs/Ts
Zero: Constant service time (M/D/1)
Example: all transmitted messages have the same length
Ratio less than 1: Using M/M/1 model would give answers on the safe side: it will give queue sizes and times that are slightly larger than they should be
Example: a data entry application for a particular form
Ratio close to 1: This is a common occurrence and corresponds to exponential service time (M/M/1)
Example: message sizes varying over the full range, shared LAN, and packet-switching networks
Ratio greater then 1: Need to use the M/G/1 model and not rely on the M/M/1 model
Example: a system that experiences many short messages, many long messages, and few in between
CSE-550-T062
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9. Queuing Models - Network of Queues -
Jackson's theorem states that:
In such a network of queues, each node is an independent queuing system, with a Poisson input determined by the principles of partitioning, merging, and tandem queuing
Each node may be analyzed separately from the others using the M/M/1 or M/M/N model
Results may be combined by ordinary statistical methods, e.g., mean delays at each node may be added to derive system delays
CSE-550-T062
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10. Application to a Packet-Switching Network
Consider a packet-switching network:
Consists of nodes interconnected by transmission links
Each node acts as the interface for zero or more attached systems, each of which functions as a source and destination of traffic
Each link is seen as a service station servicing packets
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11. Component Models Simplifications
Packets (requests) arrive according to a Poisson process (exponential interarrival times)
Infinite buffer size
Independent queues (just add delays induced in the different queues encountered on the path)
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12. Inside a Router
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13. Traffic Flow Analysis - Objective
Estimate:
Delay
Utilization of resources (links)
Traffic flow across a network depends on:
Topology
Routing
Traffic workload (from all traffic sources)
Desirable topology and routing are associated with:
Low delays
Reasonable link utilization (no bottlenecks)
CSE-550-T062
Lecture Notes - 3

14. Traffic Flow Analysis - Assumptions
Topology is fixed and stable
Links and routers are 100% reliable
Processing time at the routers is negligible
Capacity of all links is given (in bps)
Traffic workload is given Г = [γjk] (in pps)
Routing is given
Average packet size is given
CSE-550-T062
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15. Analyzing Throughput
The capacity of the network can also limit the number of connections/users it can handle for a particular type of service
This is determined by finding out the narrowest available bandwidth in the path
This is the network bottleneck
The narrowest bandwidth can be a router, switch, or link
CSE-550-T062
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16. External Workload The external workload offered to the network is:
Where:
γ = total workload in packets per second
γjk = workload between source j and destination k
N = total number of sources and destinations
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17. Internal Workload The internal workload on link i is:
λi = Σi Є jk γjk
Where:
γjk = workload between source j and destination k
jk = path followed by packets to go from source j and destination k
The total internal workload is:
λ = total load on all of the links in the network
λi = load on link i
L = total number of links
CSE-550-T062
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18. Link Utilization Utilization of link i is: ρi = λi * Tsi
Service time for link i is: Tsi = M / Bi
Where:
M = Average packet length (in bits)
Bi = Data rate on the link (in bps)
Average service rate: 1/Tsi = Bi / M
ρi = λi * M / Bi
ρb = max (ρi) – Link b is the primary bottleneck
Stability condition of a network is: ρb < 1
CSE-550-T062
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19. Path Length and Packets Waiting
Average length for all paths:
Average number of packets waiting and being served for link i is:
Number of packets waiting and being served in the network can be expressed as (using Little's formula):
γT =
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20. Link Delay
Because we are assuming that each queue can be treated as an independent M/M/1 model, we have:
The service time for link i is: Tsi = M / Bi ,Then:
CSE-550-T062
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21. Network Delay
Average delay experienced by a packet through the network:
Putting all of the elements together, we get:
CSE-550-T062
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22. Applying M/M/1 Results to a Single Network Link
Poisson packet arrivals with rate: λ = 2000 pps
Fixed link capacity: C = Mbps (T1 Carrier rate)
We approximate the packet length distribution by an exponential with
mean: L = 515 bits/packet
Thus, the service time is exponential with mean:
Ts = L/C = 0.33 ms/packet              
i.e., packets are served at a rate of: μ = 1/Ts = M / C = 3000 pps
Using our formulas for an M/M/1 queue:
                      ρ = λ/μ  = λ*Ts = 0.67
So,
r = ρ/(1- ρ) = 2 packets                
and:
Tr = r/ λ = 1 ms
CSE-550-T062
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23. Exercise 1
The problem consists of 3 Routers A, B, C, and 6 Switches, a, b, c, d, e, and f
Assume that the three Routers are connected according to a unidirectional ring topology (A-B-C-A) and that all links have the same capacity of 2 Mbps
Assume that the Switches are connected as follows: (a, C), (b, C), (c, A), (d, A), (e, B), (f, B)
The average packet size has been estimated equal to 2000 bits
It has also been observed that the traffic generated by the various switches is Poissonian with rates as indicated in the following table showing the Inter-switches traffic in pps:
Question: Find T, the average delay per packet a b c d e f - 20 50 10 30 40 60 80
100
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24. CSE-550-T062
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25. Animation of a Transmission Link
Play with animation of a transmission link at
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26. References William Stalling, “Queuing Analysis”, 2000
Dr. Khalid Salah (ICS, KFUPM), CSE 550 Lecture Slides, Term 032
CSE-550-T062
Lecture Notes - 3