1 ELEN 602 Lecture 8 Review of Last lecture –HDLC, PPP –TDM, FDM Today’s lecture –Wavelength Division Multiplexing –Statistical Multiplexing –Preliminary Queuing theory Reading -- Chapter 4.3, 5.5.1, Appendix A.1 - A.3

2 HeaderData payload Buffer A B C Input lines Output line Statistical Multiplexing

3 A1A1 A2A2 B1B1 B2B2 C2C2 C1C1 A2A2 B1B1 B2B2 C2C2 C1C1 (a) (b) A1A1 Shared Line Dedicated Lines Dedicated versus Shared Lines

4 A1A1 A2A2 B1B1 B2B2 C2C2 C1C1 A2A2 B1B1 B2B2 C2C2 C1C1 (a) (b) A1A1 Shared Line Dedicated Lines (c) N(t) Number of Packets in System

5 TDM/FDM/WDM Multiplexing In TDM, FDM, and WDM link capacity is subdivided into m portions –A packet of length L takes L/(C/m) = Lm/C time Resources are allocated to individual streams –some streams may have empty queues while others may have long queues –Delay behavior dependent on individual stream arrival –Resources could be wasted Statistical multiplexing -- no resource wastage –smaller delays, but larger delay variance In TDM/FDM/WDM -- no need for packet headers –less overhead, simpler packet processing

6 Delay Box : Multiplexer Switch Network Message, Packet, Cell Arrivals Message, Packet, Cell Departures T seconds Lost or Blocked Network Delay Analysis

7 A(t) t n-1 n n+1 Time of nth arrival =  1 +   n Arrival Rate n arrivals  1 +   n seconds = 1 = 1 (  1 +   n )/n E[]E[] 11 22 33 nn  n+1 Arrival Rate = 1 / mean interarrival time Arrival Rates and Interarrival Times

8 A(t) D(t) Delay Box N(t) T Little’s Theorem

9 N = T N = Average Number of packets in the system = Packet Arrival rate T = Average Service Delay per packet Larger the service delay (queuing delay +service time), larger the number of waiting (or buffered) packets Higher the arrival rate, larger the number of buffered packets

10 A(t) D(t) T1T1 T2T2 T3T3 T4T4 T5T5 T6T6 T7T7 Assumes first-in first-out C1C1 C2C2 C3C3 C4C4 C5C5 C6C6 C7C7 C1C1 C2C2 C3C3 C4C4 C5C5 C6C6 C7C7 Arrivals Departures Arrivals and Departures in a FIFO System

11 0 Probability density e - t t Exponentail interarrival

12 Service times X M = exponential D = deterministic G = general Service Rate:  E[X] Arrival Process / Service Time / Servers / Max Occupancy Interarrival times  M = exponential D = deterministic G = general Arrival Rate:  E[  ] 1 server c servers infinite K customers unspecified if unlimited Multiplexer Models: M/M/1/K, M/M/1, M/G/1, M/D/1 Trunking Models: M/M/c/c, M/G/c/c User Activity: M/M/ , M/G/  Queuing Model Classification

c X N q (t) N s (t) N(t) = N q (t) + N s (t) T = W + X W  P b  P b ) N(t) = number in system N q (t) = number in queue N s (t) = number in service T = total delay W = waiting time X = service time  Queuing System Variables

14 Poisson arrivals rate K-1 buffer Exponential service time with rate  M/M/1K Queue

n-1 n n (  t 1 -  t  t  t A Markov State transition diagram

16 Finite buffer multiplexer Normalized average delay Load Average Packet Delay vs. Load M/M/1/10

17 Loss probability Load Packet loss probability vs. Load M/M/1/10

18 Normalized average delay M/M/1 M/D/1 Load Average Delay with infinite Buffers