ELEN 602 Lecture 8

1. 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 ELEN 602 Lecture 8

2. Header Data payload Statistical Multiplexing Input lines A Output line B Buffer C

3. Dedicated versus Shared Lines (a) Dedicated Lines A1 A2 B1 B2 C1 C2 (b) Shared Line B2 C2 A2 A1 C1 B1

4. Number of Packets in System (a) Dedicated Lines A1 A2 B1 B2 C1 C2 (b) Shared Line B2 C2 A2 A1 C1 B1 (c) N(t)

5. 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 TDM/FDM/WDM Multiplexing

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

7. Arrival Rates and Interarrival Times n+1 A(t) n n-1 ••• 2 1 t 2 n 1 n+1 0 3 Time of nth arrival = 1 + 2 + . . . + n n arrivals 1 Arrival Rate 1 = = E[] 1 + 2 + . . . + nseconds (1+2 +...+n)/n Arrival Rate = 1 / mean interarrival time

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

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 Little’s Theorem

10. Arrivals and Departures in a FIFO System A(t) T7 Assumes first-in first-out T6 T5 T4 D(t) T3 T2 T1 Arrivals C1 C2 C3 C4 C5 C6 C7 C1 C2 C3 C4 C5 C6 C7 Departures

11. Exponentail interarrival Probability density e-t 0 t

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

13. Queuing System Variables N(t) = number in system N(t) = Nq(t) + Ns(t) Nq(t) = number in queue Ns(t) Nq(t) Ns(t) = number in service 1 Pb) 2   T = total delay c W X W = waiting time Pb T = W + X X = service time

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

15. A Markov State transition diagram 1 - (t 1 - (t 1 - (t 1 - (t 1 - t 1 - (t t t t t n 2 n-1 0 1 n+1 t t t t

16. Average Packet Delay vs. Load M/M/1/10 Finite buffer multiplexer Normalizedaveragedelay Load

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

18. Average Delay with infinite Buffers M/M/1 Normalizedaveragedelay M/D/1 Load