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Chapter 8. Queueing models. 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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chapter 8

Chapter 8

Queueing models

delay and queueing
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
queue
Queue
  • Customer
  • System
  • Service

Delay box:

Multiplexer

switch

network

Message,

packet,

cell

arrivals

Message,

packet,

cell

departures

T seconds

Lost or

blocked

queueing discipline
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
definitions
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
arrivals
Arrivals

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

little s law
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
arrivals and departures

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

Arrivals and Departures
examples
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.
example
Example

E{N} = λ E{T}

basic queueing models

Servers

Arrival process

1

Queue

A(t)

D(t)

2

t

t

i

i+1

c

B(t)

X

Service time

Basic Queueing Models
arrival processes
Arrival Processes
  • Interarrival times 1, 2,…
  • Arrival rate λ=1/E{}
  • Statistics
    • Deterministic
    • Exponential interarrival times
  • Poisson arrival process
service processes
Service Processes
  • Service times X1, X2,…
  • Processing capacity =1/E{X}
queueing system classification
Queueing System Classification

Arrival Process / Service Time / # of Servers / Max Occupancy

Interarrival times 

M = exponential

D = deterministic

G = general

Arrival Rate:

 = 1/E[t]

Service times X

M = exponential

D = deterministic G = general

Service Rate:

m = 1/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/

queueing 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

(1 – Pb)

2

T = total delay

c

W

X

W = waiting time

Pb

T = W + X

X = service time

Queueing System Variables
  • E{N}
  • E{Nq}
  • E{Ns}
  • Traffic load
  • Utilization
the m m 1 k model

Exponential service

time with rate

K – 1 buffer

Poisson arrivals

rate

The M/M/1/K Model
  • Average packet transmission time

E{X} = E{L}/R

  • Maximum service rate

=R/E{L}

  • P{1 arrival in Δt} =  Δt + o(Δt)
m m 1 steady state probabilities

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

M/M/1 Steady State Probabilities
  • Average number of customers in the system?
  • Average delay?
  • Average wait time?
example effect of scale

m

One

consolidated

system

m

separate

systems

versus

m

Example: Effect of Scale
  • What is the average delay of the combined system?
the m g 1 model

Infinite buffer

 = 1/E[X]

Poisson arrivals

rate

The M/G/1 Model
  • The average wait time:
  • Similar to M/M/1 except that the service time X may not be exponentially distributed.
erlang b formula m m c c

c

a

c

!

=

B

(

c

,

a

)

c

j

a

å

j

!

j=0

Erlang B Formula: M/M/c/c

N(t)

Poisson

arrivals

  • 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

1

Limited

number of

trunks

Many

lines

 = (1–Pb)

2

c

Pb

E[X]=1/

state transition diagram

1 - (t

1 - (2t

1 - t

1 - ((c-1) t

1 - ct

t

t

t

c

2

c-1

0

1

t

2t

c t

State Transition Diagram