1. Introduction Lect01.ppt S-38.145 - Introduction to Teletraffic Theory – Spring 2004 1 1. Introduction Contents • Purpose of Teletraffic Theory • Teletraffic models • Classical model for telephone traffic • Classical model for data traffic 2 1. Introduction Traffic point of view • Telecommunication system from the traffic point of view: incoming outgoing users traffic system traffic • Ideas: – the system serves the incoming traffic – the traffic is generated by the users of the system 3 1. Introduction Interesting questions • Given the system and incoming traffic, what is the quality of service experienced by the user? • Given the incoming traffic and required quality of service, how should the system be dimensioned? • Given the system and required quality of service, what is the maximum traffic load? incoming outgoing users traffic system traffic 4 1. Introduction General purpose (1) • Determine relationships between the following three factors: – quality of service – system capacity – traffic load quality of service system capacity traffic load 5 1. Introduction General purpose (2) • System can be – a single device (e.g. a link between two telephone exchanges, a processor routing packets in a data network, a statistical multiplexer in ATM network) or – a whole communications network (e.g. telephone or data network) or a part of it • Typically a system consists of – a device (hardware) and – principles controlling it (software) • Traffic, on the other hand, consists of (depending on the case) – calls, packets, bursts, cells, etc. 6 1. Introduction General purpose (3) • Quality of service can be described – from the users point of view • e.g. call blocking probability, distribution of delay experienced by a packet stream – from the system point of view (= performance of the system) • e.g. utilization • It is also possible to think quality of service – from the offered traffic point of view • e.g. blocking experienced by ATM-connection requests, or other connection level property; grade of service – from the carried traffic point of view • e.g. cell loss during an accepted ATM connection, or other criteria during an accepted connection; quality of service 7 1. Introduction Example • Telephone traffic – traffic = telephone calls – system = telephone network – quality of service = probability that “the line is not busy” 1234567 PRRRR!!! 8 1. Introduction Relationships between the three factors • Qualitatively, the relationships are as follows: system capacity quality of service quality of service traffic load traffic load system capacity with given with given with given quality of service system capacity traffic load • To describe the relationships quantitatively, mathematical models are needed 9 1. Introduction Teletraffic models • Teletraffic models are usually stochastic (= probabilistic) – systems themselves are usually deterministic but traffic is typically stochastic • The underlying reason for it is thus the stochastic nature of traffic: – “you never know, who calls you and when” • It follows that the variables, which are needed to describe the quality of service, are also stochastic by nature, i.e. they are random variables: – number of ongoing calls – number of packets in a buffer • Random variable is described by its distribution, e.g. – probability that there are n ongoing calls – probability that there are n packets in the buffer •Astochastic process describes the temporal development of a random variable 10 1. Introduction Related fields • Probability Theory • Stochastic Processes • Queueing Theory • Statistical Analysis (of traffic measurements) • Operations Research • Optimization Theory • Decision Theory (Markov decision processes) • Simulation Techniques (object oriented programming) 11 1. Introduction Difference between the real system and the model • It is good to keep in mind the difference between the real system and the model – (Ideally), a model is a description of only a certain part or property of the real system – Often, due to various reasons, the model is not very accurate but rather approximative caution is needed when conclusions are drawn 12 1. Introduction Practical goals • Network planning – dimensioning – optimization – performance analysis • Network management and control – efficient operation – fault recovery – traffic management – routing – accounting 13 1. Introduction Literature • Teletraffic Theory – V. B. Iversen, Chapter 1 of “Teletraffic Engineering Handbook”, available from http://www.tele.dtu.dk/teletraffic/ – Teletronikk (1995) Vol. 91, Nr. 2/3, Special Issue on “Teletraffic” – COST 242, Final report (1996) “Broadband Network Teletraffic”, Eds. J. Roberts, U. Mocci, J. Virtamo, Springer – J.M. Pitts and J.A. Schormans (1996) “Introduction to ATM Design and Performance”, Wiley – J. Roberts, “Traffic Theory and the Internet”, available from http://www.comsoc.org/ci/public/preview/roberts.html • Queueing Theory – L. Kleinrock (1975) “Queueing Systems, Volume I: Theory”, Wiley – L. Kleinrock (1976) “Queueing Systems, Volume II: Computer Applications”, Wiley – D. Bertsekas and R. Gallager (1992) “Data Networks”, 2nd ed., Prentice-Hall – P.G. Harrison and N.M. Patel (1993) “Performance Modelling of Communication Networks and Computer Architectures”, Addison-Wesley 14 1. Introduction Contents • Purpose of Teletraffic Theory • Teletraffic models • Classical model for telephone traffic • Classical model for data traffic 15 1. Introduction Teletraffic models • Two phases of teletraffic modelling: – modelling of the incoming traffic traffic model – modelling of the system itself system model • Roughly, teletraffic models can be divided into two categories by the system model – loss systems (loss models) – waiting/queueing systems (queuing models) • During this course, we present simple teletraffic models describing a single resource • These models can be combined to create models for whole telecommunication networks – loss networks – queueing networks 16 1. Introduction Simple teletraffic model • Customers arrive at rate λ (customers per time unit on average) –1/λ = average inter-arrival time • Customers are served by n parallel servers • When busy, a server serves at rate µ (customers per time unit) –1/µ = average service time of a customer • There are m waiting places in the system • It is assumed that blocked customers (arriving in a full system) are lost µ 1 λ m n 17 1. Introduction Exercise • Consider the simple teletraffic model presented on the previous slide – What is the traffic model? – What is the system model? µ 1 λ m n 18 1. Introduction Pure loss system • No waiting places (m = 0) – If the system is full (with all n servers occupied) when a customer arrives, she is not served at all but lost. System is thus lossy. • From the customer’s point of view, it is interesting to know e.g. – What is the probability that the system is full when she arrives? • From the system’s point of view, it is interesting to know e.g. – What is the utilization factor of the servers? µ 1 λ n 19 1. Introduction Pure waiting system • Infinite number of waiting places (m = ∞) – If all n servers are occupied when a customer arrives, she occupies one of the waiting places. No customers are lost but some of them have to wait before getting served; system is thus lossless. • From the customer’s point of view, it is interesting to know e.g. – what is the probability that she has to wait “too long”? • From the system’s point of view, it is interesting to know e.g. – what is the utilization factor of the servers? µ 1 λ ∞ n 20 1. Introduction Mixed system • Finite number of waiting places (0 < m < ∞) – If all n servers are occupied but there are free waiting places when a customer arrives, she occupies one of the waiting places – If all n servers and all m waiting places are occupied when a customer arrives, she is not served at all but lost – Some customers are lost and some customers have to wait before getting served. Also this system is thus lossy. µ 1 λ m n 21 1. Introduction Infinite system • Infinite number of servers (n = ∞) – No customers are lost or even have to wait before getting served; it is a lossless system. – This (hypothetical) system is often much more simple to analyze than the corresponding real system with finite capacity. – Sometimes, this can be the only way to get even some approximate results for the real system µ 1 λ • • • ∞ 22 1. Introduction Little’s formula • Consider a system where λλ – new customers arrive at rate λ • Stability assumption: – Customers do not accumulate in the system, occasionally system is empty • Consequence: – Customers depart from the system at rate λ • Denote N = average number of customers in the system T = average time a customer spends in the system • Little’s formula: N = λT 23 1. Introduction Contents • Purpose of the Teletraffic Theory • Teletraffic models • Classical model for telephone traffic • Classical model for data traffic 24 1. Introduction Classical model for telephone traffic (1) • Loss models have traditionally been used to describe (circuit-switched) telephone networks – Pioneering work made by Danish mathematician A.K. Erlang (1878-1929) • Consider a link between two telephone exchanges (classical teletraffic problem) – traffic consists of the ongoing telephone calls on the link 25 1. Introduction Classical model for telephone traffic (2) • Erlang modelled this as a pure loss system (m = 0) –customer = call • λ = call arrival rate – service time = (call) holding time • h = 1/µ = average holding time – server = channel on the link