Queueing models

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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 1900s by Erlang

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Queue
Delay box Multiplexer switch network
Message, packet, cell arrivals
Message, packet, cell departures
T seconds
Lost or blocked

• 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
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
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Littles Law

• Littles Law
• If the system does not block customers, then
• EN ? ET
• If the block rate is Pb, then
• EN (1-Pb) ? ET
• Proof

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Arrivals and Departures
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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 10/1000.1 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

• EN ? ET

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Basic Queueing Models
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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/EX

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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/Et
Service times X M exponential D
deterministic G general Service Rate m 1/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/?
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Queueing System Variables

• EN
• ENq
• ENs
• Traffic load
• Utilization

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The M/M/1/K Model

• Average packet transmission time
• EX EL/R
• Maximum service rate
• ?R/EL
• P1 arrival in ?t ? ?t o(?t)

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M/M/1 Steady State Probabilities

• Average number of customers in the system?
• Average delay?
• Average wait time?

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Example Effect of Scale

• 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.

• The average wait time

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Proof
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Erlang B Formula M/M/c/c
N(t)
Poisson arrivals
1
Limited number of trunks
Many lines
? (1Pb)
2
?
?
c
??Pb
EX1/?

• Blocked calls are cleared from the system no
  waiting allowed.
• Performance parameter Pb fraction of arrivals
  that are blocked
• Pb PN(t)c B(c,a) where a???
• The Erlang B formula, valid for any service time
  distribution

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State Transition Diagram