ELEN 602 Lecture 8

1
ELEN 602 Lecture 8

• Review of Last lecture
• HDLC, PPP
• TDM, FDM
• Todays lecture
• Wavelength Division Multiplexing
• Statistical Multiplexing
• Preliminary Queuing theory
• Reading -- Chapter 4.3, 5.5.1, Appendix A.1 - A.3

2
Statistical Multiplexing
Input lines
A
Output line
B
Buffer
C
3
Dedicated versus Shared Lines
(a)
A1
Dedicated Lines
A2
B1
B2
C2
C1
(b)
Shared Line
B2
C2
A2
B1
C1
A1
4
Number of Packets in System
(a)
A1
A2
Dedicated Lines
B1
B2
C2
C1
(b)
Shared Line
A2
B1
B2
C2
C1
A1
(c)
N(t)
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
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
n1
A(t)
n
n-1

2
1
t
?2
?n
?1
?n1
0
?3
Time of nth arrival ?1 ?2 . . . ?n
n arrivals
1
Arrival Rate

1


E?
?1 ?2 . . . ?n seconds
(?1?2 ...?n)/n
Arrival Rate 1 / mean interarrival time
8
Littles Theorem
T
A(t)
D(t)
Delay Box
N(t)
9
Littles Theorem

• 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
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 ???????EX
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
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
n1
???t
???t
???t
???t
16
Average Packet Delay vs. Load M/M/1/10
Finite buffer multiplexer
Normalized average delay
Load
17
Packet loss probability vs. Load M/M/1/10
Loss probability
Load
18
Average Delay with infinite Buffers
M/M/1
Normalized average delay
M/D/1
Load