Teletraffic theory I: Queuing theory

Transcription

1 Teletraffic theory I: Queuing theory Lecturer: Dmitri A. Moltchanov

2 1. Place of the course TLT-2716 is a part of Teletraffic theory five courses set 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 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

3 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

4 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

5 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 Output buffer 1 Input i.. p 11 p 1j p 1M p 11 p 1j p 1M.. Output buffer j.... Input N p 11 p 1j Output buffer M p 1M Internal Switching Lecture: Overview of the course 5

6 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

7 2. Aims of the course What 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 Lecture: Overview of the course 7

8 3. Queuing theory Queuing 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

9 3.1. Graphical representation Server(s) Arrivals Waiting positions Departures 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

10 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

11 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

12 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

13 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

14 4. Outline of the course Outline 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

15 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

16 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/E r /1 queuing system. Lecture: Overview of the course 16

17 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

18 5. Important information Pay attention: Lectures will be given once a week during periods 1 and 2: every Tuesday starting from ; 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 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

19 6. Expected knowledge and references Knowledge 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: hlynka/queue.html; everything starting from around 30 lecture sets to queuing software. Lecture: Overview of the course 19

20 7. Credit points Credit 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

21 8. Personal information: Lectures: Dmitri Moltchanov; course page: Exercises: Alexander Pyattaev, Tatiana Efimushkina; s: and course page: Lecture: Overview of the course 21