Teletra c theory I: Queuing theory · Teletra c theory I: Queuing theory D.Moltchanov, TUT, 2011 1. Place of the course TLT-2716 is a part of Teletra c theory ve courses set. 2011-2012

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


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?



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.



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



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.



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.



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.



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)?



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.



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.



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.



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.



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 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 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 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.



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.



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.



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!



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/


Documents

Teletra c theory I: Queuing theory · Teletra c theory I: Queuing theory D.Moltchanov, TUT, 2011 1. Place of the course TLT-2716 is a part of Teletra c theory ve courses set. 2011-2012