Networking Theory and Fundamentals - Lecture 2 potx

TCOM 501: Networking Theory & Fundamentals Lecture January 22, 2003 Prof Yannis A Korilis 2-2 Topics Delay in Packet Networks Introduction to Queueing Theory Review of Probability Theory The Poisson Process Little’s Theorem Proof and Intuitive Explanation Applications 2-3 Sources of Network Delay Processing Delay Assume processing power is not a constraint Queueing Delay Time buffered waiting for transmission Transmission Delay Propagation Delay Time spend on the link – transmission of electrical signal Independent of traffic carried by the link Focus: Queueing & Transmission Delay 2-4 Basic Queueing Model Buffer Server(s) Departures Arrivals Queued In Service A queue models any service station with: One or multiple servers A waiting area or buffer Customers arrive to receive service A customer that upon arrival does not find a free server is waits in the buffer 2-5 Characteristics of a Queue b m Number of servers m: one, multiple, infinite Buffer size b Service discipline (scheduling): FCFS, LCFS, Processor Sharing (PS), etc Arrival process Service statistics 2-6 Arrival Process n +1 τn n tn n −1 t τ n : interarrival time between customers n and n+1 τ n is a random variable {τ n , n ≥ 1} is a stochastic process Interarrival times are identically distributed and have a common mean E[τ n ] = E [τ ] = 1/ λ λ is called the arrival rate 2-7 Service-Time Process n +1 n −1 n sn t sn : service time of customer n at the server { s n , n ≥ 1} is a stochastic process Service times are identically distributed with common mean E [ sn ] = E [ s ] = µ µ is called the service rate For packets, are the service times really random? 2-8 Queue Descriptors Generic descriptor: A/S/m/k A denotes the arrival process For Poisson arrivals we use M (for Markovian) B denotes the service-time distribution M: exponential distribution D: deterministic service times G: general distribution m is the number of servers k is the max number of customers allowed in the system – either in the buffer or in service k is omitted when the buffer size is infinite 2-9 Queue Descriptors: Examples M/M/1: Poisson arrivals, exponentially distributed service times, one server, infinite buffer M/M/m: same as previous with m servers M/M/m/m: Poisson arrivals, exponentially distributed service times, m server, no buffering M/G/1: Poisson arrivals, identically distributed service times follows a general distribution, one server, infinite buffer */D/∞ : A constant delay system 2-10 Probability Fundamentals Exponential Distribution Memoryless Property Poisson Distribution Poisson Process Definition and Properties Interarrival Time Distribution Modeling Arrival and Service Statistics ... = k1; X2 = k2g = = P fX1 = k1; X2 = k2 jX = k1 + k2g P fX = k1 + k2g ³k + k ´ ¸k1+k2 k1 k2 ¡¸ = p p2 ¢ e k1 (k1 + k2)! (¸p1)k1 (¸p2)k2 ¢ e¡¸(p1+p2) = k1 !k2! k1 k2 ¡¸p1 (¸p1 ) ¡¸p2 (áp2 ) Âe... (áp2 ) Âe = e k1 ! k2! X1 and X2 are independent k1 k2 ² P fX1 = k1g = e¡¸p1 (¸pk 1!) , P fX2 = k2g = e¡¸p2 (¸pk 2! ) Xi follows Poisson distribution with parameter ¸pi  2- 2 0 Poisson Approximation... 2- 2 Topics Delay in Packet Networks Introduction to Queueing Theory Review of Probability Theory The Poisson Process Little’s Theorem Proof and Intuitive Explanation Applications 2- 3 Sources