Application: Transmission System ueue---transmitter transmission line Average number of packets in the transmitter(or transmitter utilization: p=nX=n/u X: average transmission time u=1/X: average transmission rate Average number of packets in the buffer: N,=nw W: average time that a packet waits in the buffer Average number of packets in the system: N=n(W+X) p+N
Communication Networks Application: Transmission System queue transmitter transmission line 11 • Average number of packets in the transmitter (or transmitter utilization): 𝜌 = 𝜆𝑋ത = 𝜆/𝜇 – 𝑋ത: average transmission time – 𝜇 = 1/𝑋ഥ: average transmission rate • Average number of packets in the buffer: 𝑁𝑞 = 𝜆𝑊 – 𝑊: average time that a packet waits in the buffer • Average number of packets in the system: 𝑁 = 𝜆 𝑊 + 𝑋ത = 𝜌 + 𝑁𝑞 𝜆
Application: A K-server System Consider the queuing system N rooms for the customers (either in queue or in service) Starts with n customers Average service time is X a departing customer is immediately replaced by an arrival(saturated) What is the average customer time T in the system? Challenge: input rate is unknown
Communication Networks Application: A 𝐾-server System 12 Consider the queuing system: • 𝑁 rooms for the customers (either in queue or in service) • Starts with 𝑁 customers • Average service time is 𝑋ത • A departing customer is immediately replaced by an arrival (saturated) What is the average customer time 𝑇 in the system? • Challenge: input rate is unknown S1 SK S1
Solution (S The queuing system is equivalent to feedback system a departing customer re-enters the queue immediately Output rate= input rate Consider K servers as a closed system where there are K customers constantly and the average service time is X. Thus, we have n=KX. Again, it follows from Little's law that N NX T
Communication Networks S1 SK S1 l Solution 13 The queuing system is equivalent to feedback system: • A departing customer re-enters the queue immediately • Output rate = input rate Consider 𝐾 servers as a closed system where there are 𝐾 customers constantly and the average service time is 𝑋ത. Thus, we have 𝜆 = 𝐾/𝑋ത. Again, it follows from Little’s law that 𝑇 = 𝑁 𝜆 = 𝑁𝑋ത 𝐾
Application to a Complex system For each flow i, there is Ni =niTi arrival rate of stream i Ti: average delay of packets of stream i Ni: average number of packets of stream i in the network For the whole network there is N= nt N=∑Mand=∑2
Communication Networks Application to a Complex System R R R R R R R λ1 λ2 λ3 λ3 λ2 λ1 14 For each flow 𝑖, there is 𝑁𝑖 = 𝜆𝑖𝑇𝑖 • 𝜆𝑖 : arrival rate of stream 𝑖 • 𝑇𝑖 : average delay of packets of stream 𝑖 • 𝑁𝑖 : average number of packets of stream 𝑖 in the network For the whole network, there is 𝑁 = 𝜆𝑇 • 𝑁 = σ𝑖 𝑁𝑖 and 𝜆 = σ𝑖 𝜆𝑖
票 hanghai Jiao Tong University ARRIVAL MODEL
Communication Networks ARRIVAL MODEL Shanghai Jiao Tong University 15