Arrival process The arrival process can normally be described by the number of arrivals in a unit time or can be described by inter-arrival time Poisson process Most commonly used arrival model in telecom network Named after the french mathematician simeon - Denis Poisson(1781-1840)
Communication Networks Arrival Process • The arrival process can normally be described – by the number of arrivals in a unit time – or can be described by inter-arrival time • Poisson process – Most commonly used arrival model in telecom network – Named after the French Mathematician Simeon-Denis Poisson (1781 – 1840) 16
Examples of poisson process The number of page requests arriving at a web server(no attack, please) The number of telephone calls arrives at an switch The number of photons hitting a photon detector, when lit by a laser The execution of trades on a stock exchange
Communication Networks Examples of Poisson Process • The number of page requests arriving at a web server (no attack, please) • The number of telephone calls arrives at an switch • The number of photons hitting a photon detector, when lit by a laser • The execution of trades on a stock exchange • … 17
Definition 1 The number of arrivals N(t)in a finite interval of length t Obeys Poisson distribution with parameter nt The numbers of arrivals in non-overlapped intervals are independent PrN(t=n=e-t (λt) n! 72 Poisson(al1) Poisson(nt2)
Communication Networks Definition 1 The number of arrivals 𝑁 𝑡 in a finite interval of length 𝑡 – Obeys Poisson distribution with parameter 𝜆𝑡 – The numbers of arrivals in non-overlapped intervals are independent 𝑇1 Poisson (𝜆𝑇1) 𝑇2 Poisson (𝜆𝑇2) 18 Pr 𝑁 𝑡 = 𝑛 = 𝑒 −𝜆𝑡 𝜆𝑡 𝑛 𝑛!
Definition 2 The interval times are independent and obey negative exponential distribution with rate n Pr{τ≤t=1 ↓↓↓↓↓↓↓ · Proof of1→2 P{τ≤t}=1-P{>t} 1-PO arrival within t) -e·t
Communication Networks Definition 2 The interval times are independent and obey negative exponential distribution with rate 𝜆 𝜏 • Proof of 1→2 19 Pr 𝜏 ≤ 𝑡 = 1 − 𝑒 −𝜆𝑡 𝑃 𝜏 ≤ 𝑡 = 1 − 𝑃 𝜏 > 𝑡 = 1 − 𝑃 0 arrival within 𝑡 = 1 − 𝑒 −𝜆𝑡
Definition 3 In an infinitesimal time interval 8, there may occur only one arrival, and this happens with probability no ↓↓↓ PN(6)=0}=e16=1-6+0(6) 16 PN(6)=1}=e0=(1-6+0(6)6=6+0(6 PN(6)=2}=e-68(46)2
Communication Networks Definition 3 In an infinitesimal time interval 𝛿, there may occur only one arrival, and this happens with probability 𝜆𝛿 𝛿 20 𝑃 𝑁 𝛿 = 0 = 𝑒 −𝜆𝛿 = 1 − 𝜆𝛿 + 𝑜 𝛿 𝑃 𝑁 𝛿 = 1 = 𝑒 −𝜆𝛿 𝜆𝛿 1 = 1 − 𝜆𝛿 + 𝑜 𝛿 𝜆𝛿 = 𝜆𝛿 + 𝑜 𝛿 𝑃 𝑁 𝛿 = 2 = 𝑒 −𝜆𝛿 𝜆𝛿 2 2 = 𝑜 𝛿