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Teletraffic theory I: Queuing theory
Lecturer: Dmitri A. Moltchanov
E-mail: [email protected]
http://www.cs.tut.fi/kurssit/TLT-2716/
Teletraffic theory I: Queuing theory D.Moltchanov, TUT, 2011
1. Place of the courseTLT-2716 is a part of Teletraffic theory five courses set.
2011-2012 academic year:
• Fall: TLT-2716 ”Teletraffic theory part I: Queuing theory”;
• Spring: TLT-2727 ”Teletraffic theory part II: Performance evaluation”;
• Spring: TLT-2786 ”Advanced topics in teletraffic theory: traffic modeling”.
2012-2013 academic year:
• Fall: TLT-2707 ”Network simulation techniques”
• Spring: TLT-2786 ”Advanced topics in teletraffic theory: advanced queues”
The ultimate goal: dimensioning of communications networks:
• queuing theory: solving models of servicing systems.
Lecture: Overview of the course 2
Teletraffic theory I: Queuing theory D.Moltchanov, TUT, 2011
1.1. What entities are interested in teletraffic?
Teletraffic theory is attractive for:
• service providers:
– how to best distribute service access points to facilitate the users’ requests?
– how many servers are needed to satisfy users’ request?
• networks operators:
– how to best distribute network load?
– how much buffer space should be assigned to traffic load?
– what are the optimal link rates?
• vendors:
– how to best utilize resources of the switching/routing equipment?
– what kind of improvements should be made to switching equipment?
• end users:
– what is actual quality of service obtained from the network?
Lecture: Overview of the course 3
Teletraffic theory I: Queuing theory D.Moltchanov, TUT, 2011
1.2. What it is complicated discipline?
Multidisciplinary in nature:
• General disciplines:
– probability theory;
– theory of stochastic processes
– statistics.
• Specific disciplines: parts of operations research:
– queuing theory:
– simulations;
– traffic modeling;
– reliability;
– optimization;
Note: all these allow to create models and analyze them.
Lecture: Overview of the course 4
Teletraffic theory I: Queuing theory D.Moltchanov, TUT, 2011
1.3. Why all these disciplines?
Classic problem: dimension the buffer of the hypothetical router:
• determine the buffer space and the link rate;
• arriving traffic and routing are known.
Input 1
Input i
Input N
Output buffer 1..
Internal Switching
Output buffer M
Output buffer j
.. ....
p11
p1j
p1M
p11
p1j
p1M
p11
p1j
p1M
Lecture: Overview of the course 5
Teletraffic theory I: Queuing theory D.Moltchanov, TUT, 2011
The step-by-step procedure:
Represent arrival traffic on each input link:
• we have to know: probability, stochastic process, statistics, traffic modeling;
Define superposition of processes entering the queue at the output port:
• we have to know: probability, stochastic process, statistics, traffic modeling.
Analyze the queue under defined load:
• we have to know: queuing theory, simulations, reliability theory;
Determine required buffer space and link rate share:
• we have to know: queuing theory, optimization methods.
Lecture: Overview of the course 6
Teletraffic theory I: Queuing theory D.Moltchanov, TUT, 2011
2. Aims of the courseWhat we study in the whole course ’Teletraffic Theory’:
• teletraffic theory part I: queuing theory:
– analytical tool to study the network.
• teletraffic theory part II: performance analysis of computer networks:
– application of queuing theory to dimensioning of real networks;
Aims of the whole course are:
• to give knowledge necessary to traffic management and network dimensioning.
This course is also tightly connected with:
• ’Network simulation techniques’ is up in fall 2012:
– complements queuing theory;
• ’Traffic modeling’ is up for spring 2012.
Lecture: Overview of the course 7
Teletraffic theory I: Queuing theory D.Moltchanov, TUT, 2011
3. Queuing theoryQueuing system is a complex system where:
• jobs/customers/users/calls/packets arrive to the some point;
• get service;
• depart once the service is provided.
Some examples:
• telephone systems:
– customers call gaining access to one of the finite set of lines going out from an exchange.
• computer networks:
– packets are forwarded from sources to destination through a number of intermediate nodes;
– queuing systems arise at each node where the buffering occurs.
• computer systems:
– computing jobs and operating system’s routines require service from central processor.
Lecture: Overview of the course 8
Teletraffic theory I: Queuing theory D.Moltchanov, TUT, 2011
3.1. Graphical representation
Arrivals Departures
Waiting positions
Server(s)
Figure 1: General model of the queuing system.
Questions to define:
• how does one describe the arrival and service processes?
• how many servers does the system have?
• are there waiting positions in the queue?
• are there any special local rules (order of service, priorities, vacations)?
Lecture: Overview of the course 9
Teletraffic theory I: Queuing theory D.Moltchanov, TUT, 2011
3.2. Specification of the queuing system
The queue is specified using the following:
• description of arrival process (interarrival time distribution);
• description of service process (service time distribution);
• number of severs (how many);
• number of waiting positions (how many);
• special queuing rules:
– service discipline (FCFS, LCFS, RANDOM);
– vacations (vacation time distribution, when the vacation starts/end);
– priorities (how many priorities);
– batch arrivals (batch distribution).
– other special rules...
Important note: some parameters are sometimes silently assumed.
Lecture: Overview of the course 10
Teletraffic theory I: Queuing theory D.Moltchanov, TUT, 2011
3.3. Network of queues
To specify network of queues additional information is required:
• interconnection between queues;
• routing strategy:
– deterministic;
– probabilistic;
– class-based probabilitic/deterministic.
• handling of blocking (if the buffer at destination is full):
– loss of customer;
– blocking of original queue (just waiting).
– re-routing (if the routing is probabilistic).
• number of customers classes.
Note: we consider some simple examples of queuing networks.
Lecture: Overview of the course 11
Teletraffic theory I: Queuing theory D.Moltchanov, TUT, 2011
3.4. Method of analysis
Analysis of queueing system or queuing network can be accomplished by:
• analytical analysis;
• simulation study;
• both means.
Analytical results are usually preferred:
• usually require less time to compute;
• usually require less effort to compute;
• usually require more time to analyze:
– depends on the complexity of the system.
• give exact results:
– no statistical errors are produced.
Lecture: Overview of the course 12
Teletraffic theory I: Queuing theory D.Moltchanov, TUT, 2011
3.5. Obtained results
Obtained results may be classified to two large groups:
• important for user:
– what is the performance level?
– application: how well the application perform.
• important for network operators:
– how much resources should be provided?
– application: link rates and buffers dimensioning.
• important for vendors:
– how to expand the capability of a given equipment?
– application: link rates and buffers dimensioning.
• important for service providers:
– how much resources should be provided?
– application: processors, links dimensioning.
Lecture: Overview of the course 13
Teletraffic theory I: Queuing theory D.Moltchanov, TUT, 2011
4. Outline of the courseOutline of the ’Teletraffic theory I: queuing theory’:
• Lecture 1: Introduction to the course
– objectives of queuing theory;
– motivation to study queuing theory;
– basic notations;
– parameters of interest;
– example of analysis of simple queuing system.
• Lecture 2: Reminder of probability theory
– definitions of probability through Kolmogorov’s axioms;
– combinatorial analysis, conditional probabilities;
– PDF, pdf, PF, moments, functions of RV;
– useful continuous-time distributions (uniform, exponential etc.);
– useful discrete-time distributions (geometric, phase-type etc.).
Lecture: Overview of the course 14
Teletraffic theory I: Queuing theory D.Moltchanov, TUT, 2011
• Lecture 3: Reminder of stochastic processes
– definition, overall description;
– classification (strict and second order stationary, ergodicity);
– moments and autocorrelation function;
– Markov property;
– continuous and discrete-time Markov chains, properties;
– birth-death processes.
• Lecture 4: Reminder of transforms
– Z-transform;
– Laplace transform.
• Lecture 5: Overview of arrival and service processes
– description of arrival and service processes;
– Poisson process;
– Markov modulated processes;
– basic notes on traffic modeling in real networks.
Lecture: Overview of the course 15
Teletraffic theory I: Queuing theory D.Moltchanov, TUT, 2011
• Lecture 6: Basic definitions of queuing theory
– Kendall’s notation of queuing systems;
– service disciplines (FCFS, RANDOM, LIFO);
– transient and equilibrium solutions;
– Little’s result with prove.
• Lecture 7: M/M/-/-/- queuing system, part I
– PASTA property with prove;
– M/M/1 queuing system;
– Delay performance.
• Lecture 8: M/M/-/-/- queuing system, part II
– M/M/1 queuing system with dependent arrivals and service;
– M/M/C queuing system;
– M/M/C/K (C=K) loss queuing system;
– M/M/1/K queuing system;
– M/Er/1 queuing system.
Lecture: Overview of the course 16
Teletraffic theory I: Queuing theory D.Moltchanov, TUT, 2011
• Lecture 9: M/G/-/-/- queuing system, part I
– description of M/G/1;
– methods of analysis;
– residual lifetime approach;
– transform approach based on imbedded Markov chain.
• Lecture 10: M/G/-/-/- queuing system, part II
– method of supplementary variables;
– direct approach based on imbedded Markov chain;
– delay performance of M/G/1 queuing system;
– M/G/1/K queuing system.
• Lecture 11: G/M/-/-/- queuing system
– direct approach based on imbedded Markov chain;
– G/M/m queuing system;
– G/M/m/m queuing system.
Lecture: Overview of the course 17
Teletraffic theory I: Queuing theory D.Moltchanov, TUT, 2011
5. Important informationPay attention:
• Lectures will be given once a week during periods 1 and 2:
– every Tuesday starting from 13.09.2011;
– Room TB219, time 16:15 – 17:45.
• Exercises will be given once a week during periods 1 and 2:
– on Thursdays, room TB222, time 16:15 – 17:45;
– starting from 22.09.2011.
• Two assignments:
– contains interesting practical examples;
– will be available at the course page soon.
• exam: date will be announced later:
– check POP system;
– you have to sign for exam at POP at least one week before.
Lecture: Overview of the course 18
Teletraffic theory I: Queuing theory D.Moltchanov, TUT, 2011
6. Expected knowledge and referencesKnowledge necessary to attend the course:
• all information necessary to understand the content of the course will be given;
• basic knowledge of probability theory and stochastic processes is appreciated.
References:
• lecture notes will be available at the course page;
• no general references: any book on queuing theory can be used:
– L. Kleinrock, ”Queuing systems”;
– H. Akimaru, K. Kawashima, ”Teletraffic: theory and applications”;
Ultimate source:
• http://www2.uwindsor.ca/˜hlynka/queue.html;
• everything starting from around 30 lecture sets to queuing software.
Lecture: Overview of the course 19
Teletraffic theory I: Queuing theory D.Moltchanov, TUT, 2011
7. Credit pointsCredit points:
• one can earn up to 6 CPs:
– minimum: 3 CPs;
– maximum: 6 CPs.
How you get it:
• 3 CPs: pass of exam only;
– this is base;
– you may not attend lectures, exercises, assignments!
• 1 CP: 70% of lecture and exercise attendance;
• 1 CP per correctly completed assignment.
Important note: if you fail to pass exam you get nothing!
Lecture: Overview of the course 20
Teletraffic theory I: Queuing theory D.Moltchanov, TUT, 2011
8. Personal information:Lectures:
• Dmitri Moltchanov;
• e-mail: [email protected];
• course page:
– http://www.cs.tut.fi/kurssit/TLT-2716/
Exercises:
• Alexander Pyattaev, Tatiana Efimushkina;
• e-mails: [email protected] and [email protected];
• course page:
– http://www.cs.tut.fi/kurssit/TLT-2716/
Lecture: Overview of the course 21