1 chapter 4 fundamental queueing system. 2 3 ref: mischa schwartz “telecommunication networks”...
DESCRIPTION
3 Ref: Mischa Schwartz “Telecommunication Networks” Addison-Wesley publishing company 1988TRANSCRIPT
1
Chapter 4
Fundamental Queueing System
2
3
Ref: Mischa Schwartz “Telecommunication Networks” Addison-Wesley publishing company 1988
4
5
mTTp
T
6
x
7rateserviceratearrived
::
8
9
10
1
11
211
nn
nnn
PPsurfacenPPPsurface
11
12
13
(input rate/output rate) (the probability that the system is nonempty)
The throughput (customer/see) = λ
14
net arrival rate= = r
The throughput r/μ= )1()1(
111)1( 110
N
N
NP
)1()/( BPseecustomerthroughput
15
16
17
i i
m
p
0
1n
np
18
Q
Q
NW
mP
WT Q11
19
20
21
Example 1 Statistical Multiplexing Compared with TDM and FDM
Assume m statistically iid Poisson packet streams each with an arrival rate of packets/sec.
The packet lengths for all streams are independent and exponentially distributed.
The average transmission time is . If the streams are merged into a single Poisson stream, with rate , the
average delay per packet is
If, the transmission capacity is divided into m equal portions, as in TDM and FDM, each portion behaves like an M/M/1 queue with arrival rate and average service rate . Therefore, the average delay per packet is
.
m
1
1T
mm
mT
22
Example 2 Using One vs. Using Multiple Channels Statistical MUX(1)
A communication link serving m independent Poisson traffic streams with overall rate .
Packet transmission times on each channel are exponentially distributed with
mean . The system can be modeled by the same Markov chain as the M/M/m queue. The average delay per packet is given by
An M/M/1 system with the same arrival rate and service rate (statistical multiplexing with one channel having m times larger capacity), the average delay per packet is
and denote the queueing probability
1
m
mP
T Q1
mP
mT Q
ˆ1ˆ
QP Qp̂
23
When << 1 (light load) , , and
At light load, statistical MUX with m channels produces a delay almost m timeslarger than the delay of statistical MUX with the m channels combined in one.
When , , , << , and
At heavy load, the ratio of the two delays is close to 1.
QP 0 0
QP 1 1 1 )(1 mQp̂
Qp̂
mTT
ˆ
1ˆ TT
1
24
4.6
existtdoesnxx
xf '1
11)( 22
25
26
27
28
WRW WRW
29
A.
30
second moment of service time and load
IF 2X W
31
2
32
33
34
35
Roll-call PollingStations are interrogated sequentially, one by one, by the central system, which
asks if they have any messages to transmit.
: walk time : frame transmission time
iW
it
time
36
The scan or cycle time is given by
The average scan time
, are the ave. walk time and the ave. time to transmit pkt at station .L is the total walk time of the complete poling system.
ct
ct
iw it i
37
For station , let : the ave. pkt arrival rate : the ave. packet length : the number of overhead bits C : the channel capacity in bps : the ave. frame length in time
The average number of packets waiting to be transmitted when station ispolled is , the time required to transmit is
With the traffic intensity, the average scan time is given
With representing the total traffic intensity on the common channel.
i
i
im Cmi /)(
ii ct
iii m ct
N
iii
N
ii m
11
ciicii tmtt
38
For small the average access delay should be .
Assume that each station has the same , same frame-length statistics,and the same .
The average access delay is
is the second moment of the frame length, .The access delay is the average time a packet must wait at a station from the time it first arrives until the time transmission begins. Access delay is thusthe average wait time in an M/G/1 queue.
Ref: Mischa Schwarty: “Telecommunication Networks, Protocols Modeling and Analysis”, Addison-Wesley Publishing Company, 1988, PP. 408-422
2/ct
w
mN 2m
39
Hub Polling� Control is passed sequentially from one station to another.
� Let the polling message be a fixed value, tp sec in length.
� The time required per station to synchronize to a polling message is ts sec.
� The total propagation delay for the entire N-station system is sec.
40
Hub Polling (Cont’)� The total walk time for roll-call polling is .
� Let the stations all be equally spaced, and the round-trip propagation delay between the controller and station N be τ sec.
� The overall propagation delay is just
� The analysis of the hub-polling strategy is identical to that of roll-call polling.
� The only difference is that the walk time L is reduced through the use of hub polling.
� For hub polling, .shub NtL
sp NtNtL
)1(2
N