CSE 550 Computer Network Design

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

Outline • Queuing Models • Application to Networks • Traffic Flow Analysis Lecture Notes - 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 whenρ = 1 • Practical considerationslimitthe input rate for a single server to 70-90% of the theoretical maximum • Little's formula (general relationship): r = λTr and w = λTw Lecture Notes - 3

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 Lecture Notes - 3

Queuing Models- Multiple Single-server queues - • Example of a Network of Queues • Traffic Partitioning • Traffic Merging • Queues in Tandem Lecture Notes - 3

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 Lecture Notes - 3

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 Lecture Notes - 3

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 Lecture Notes - 3

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 Lecture Notes - 3

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 Lecture Notes - 3

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) Lecture Notes - 3

Inside a Router Lecture Notes - 3

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) Lecture Notes - 3

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 Lecture Notes - 3

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 Lecture Notes - 3

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 Lecture Notes - 3

Internal Workload • The internal workload on link i is: λi =Σi Є jkγjk • 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: • Where: • λ = total load on all of the links in the network • λi = load on link i • L = total number of links Lecture Notes - 3

Link Utilization • Utilization of link i is:ρi = λi * Tsi • Service time for link iis: 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 primarybottleneck • Stability condition of a network is: ρb< 1 Lecture Notes - 3

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 = Lecture Notes - 3

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 iis: Tsi= M / Bi ,Then: Lecture Notes - 3

Network Delay • Average delay experienced by a packet through the network: • Putting all of the elements together, we get: Lecture Notes - 3

Applying M/M/1 Results to a Single Network Link • Poisson packet arrivals with rate: λ = 2000 pps • Fixed link capacity: C = 1.544 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 Lecture Notes - 3

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 Lecture Notes - 3

Lecture Notes - 3

Animation of a Transmission Link • Play with animation of a transmission link at http://poisson.ecse.rpi.edu/~vastola/pslinks/perf/hing/mm1animate.html Lecture Notes - 3

References • William Stalling, “Queuing Analysis”, 2000 • Dr. Khalid Salah (ICS, KFUPM), CSE 550 Lecture Slides, Term 032 Lecture Notes - 3

CSE 550 Computer Network Design