Transcript Appendix A Figures - Northwestern University

Chapter 8
Queueing models
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Delay and Queueing

Main source of delay






Transmission (e.g., n/R)
Propagation (e.g., d/c)
Retransmission (e.g., in ARQ)
Processing (e.g., running time of protocols)
Queueing
Queueing Theory


Study of mathematical queueing models
Since early 1900’s by Erlang
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Queue
Message,
packet,
cell
arrivals
Lost or
blocked



Delay box:
Multiplexer
switch
network
Message,
packet,
cell
departures
T seconds
Customer
System
Service
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Queueing Discipline



One customer at a time
 First-in first-out (FIFO)
 LIFO
 Round robin
 Priorities
Multiple customers at a time
 FIFO
 Separate queues/separate servers
Blocking rule
 Discard when full
 Drop randomly
 Block a certain class
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Definitions









T: Time spent in the system
A(t): # of arrivals in [0,t]
B(t): # of blocked customers in [0,t]
D(t): # of departures in [0,t]
N(t): # of customers in the system at t
N(t)=A(t)-D(t)-B(t)
Long term arrival rate
Throughput
Average number in the system
Fraction of blocked customers
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Arrivals
n+1
A(t)
n
n-1
•••
2
1
0 1
2
t
n
3
n+1
Time of nth arrival = 1 + 2 + . . . + n
Arrival
Rate =
n arrivals
1 + 2 + . . . + n seconds
=
1
(1+2 +...+n)/n
1
E[]
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Little’s Law

Little’s Law
If the system does not block customers, then
E{N} = λ E{T}
If the block rate is Pb, then
E{N} = (1-Pb) λ E{T}

Proof
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Arrivals and Departures
A(t)
Assumes
first-in
first-out
T6
T1
Arrivals
Departures
T2
T3
T4
T7
T5
D(t)
C1 C2 C3 C4 C5
C1 C2 C3
C6
C4 C5
C7
C6
C7
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Examples

Let the arrival rate be 100 packets/sec. If 10
packets are found in the queue in average,
then the average delay is 10/100=0.1 sec.

Traffic is bad in a rainy day.

For the same volume of customers, a fast
food restaurant requires smaller dining area.
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Example
E{N} = λ E{T}
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Basic Queueing Models
Servers
Arrival process
Queue
A(t)
2
i+1
t
B(t)
D(t)
t

i
1
c
X
Service time
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Arrival Processes



Interarrival times 1, 2,…
Arrival rate  =1/E{}
Statistics



Deterministic
Exponential interarrival times
Poisson arrival process
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Service Processes

Service times X1, X2,…

Processing capacity =1/E{X}
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Queueing System Classification
Arrival Process / Service Time / # of Servers / Max Occupancy
Interarrival times 
M = exponential
D = deterministic
G = general
Arrival Rate:
 = 1/E[]
Service times X
M = exponential
D = deterministic
G = general
Service Rate:
 = 1/E[X]
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/
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Queueing System Variables
N(t) = Nq(t) + Ns(t)
Ns(t)
Nq(t)
1
 Pb




Ns(t) = number in service
2
W

Nq(t) = number in queue


(1 – Pb)
N(t) = number in system
T=W+X
c
T = total delay
X
W = waiting time
X = service time
E{N}
E{Nq}
E{Ns}
Traffic load
Utilization
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The M/M/1/K Model
Poisson arrivals
rate 

K – 1 buffer
Exponential service
time with rate 
Average packet transmission time
E{X} = E{L}/R

Maximum service rate
=R/E{L}

P{1 arrival in Δt} =  Δt + o(Δt)
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M/M/1 Steady State Probabilities
1 - t
1 - (t
t
0



1 - (t
t
t
1 - (t
t
2
1
t
1 - (t
t
n
n-1
t
1 - (t
n+1
t
Average number of customers in the system?
Average delay?
Average wait time?
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Example: Effect of Scale

m
separate
systems





One
versus consolidated
system
m
m

What is the average delay of the combined
system?
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The M/G/1 Model

Similar to M/M/1 except that the service time X
may not be exponentially distributed.
Infinite buffer
 = 1/E[X]
Poisson arrivals
rate 

The average wait time:
2}

E
{
X
E{W }
2(1  )
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Proof
Wi  Ri 
i 1

j iN
i
X
j
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Erlang B Formula: M/M/c/c

N(t)
1
 = (1–Pb)
Pb




2

Many
lines
Limited
number of
trunks
Poisson
arrivals
c
E[X]=1/
Blocked calls are cleared from the system; no waiting allowed.
Performance parameter: Pb = fraction of arrivals that are blocked
Pb = P[N(t)=c] = B(c,a) where a=
The Erlang B formula, valid for any service time distribution
B(c,a) 
ac
c!
c

j=0
aj
j!
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State Transition Diagram
1 - t
1 - (t
t
2
1
t
1 - ((c-1) t 1 - ct
t
t
0
1 - (2t
2t
c
c-1
c  t
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