Introduction

Definition

M/M queues

M/M/1

M/M/S

M/M/infinity

M/M/S/K
1
Queuing system

A queuing system


is a place where customers arrive

According to an “arrival process”

To receive service from a service facility
Can be broken down into three major components

The input process

The system structure

The output process
Customer
Population
Waiting
queue
Service
facility
2
Characteristics of the system
structure
λ

μ
Queue


Infinite or finite
Service mechanism


λ: arrival rate
μ: service rate
1 server or S servers
Queuing discipline

FIFO, LIFO, priority-aware, or random
3
Queuing systems: examples

Multi queue/multi servers


Example:

Supermarket

Blade centers
 orchestrator
.
.
.
Multi-server/single queue

Bank

immigration
4
Kendall notation

David Kendall

A British statistician, developed a shorthand notation

To describe a queuing system

A/B/X/Y/Z
 A: Customer arriving pattern

B: Service pattern

X: Number of parallel servers

Y: System capacity

Z: Queuing discipline
M: Markovian
D: constant
G: general
Cx: coxian
5
Kendall notation: example

M/M/1/infinity


A queuing system having one server where

Customers arrive according to a Poisson process

Exponentially distributed service times
M/M/S/K
K

M/M/S/K=0

Erlang loss queue
6
Special queuing systems

Infinite server queue
μ
λ
.
.

Machine interference (finite population)
S repairmen
N
machines
7
M/M/1 queue
λ

μ
λ: arrival rate
μ: service rate
λn = λ, (n >=0); μn = μ (n>=1)
Pn 
 0 1 ...  n 1
 0  1 ...  n
P0  Pn 
 Pn   P0 ;  
n

n

n
P0


P0  P1  ...  Pn  ...  1
P0 (1      ...)  1  P0  1  
2
8
Traffic intensity

rho = λ/μ


It is a measure of the total arrival traffic to the system

Also known as offered load

Example: λ = 3/hour; 1/μ=15 min = 0.25 h
Represents the fraction of time a server is busy

In which case it is called the utilization factor

Example: rho = 0.75 = % busy
9
Queuing systems: stability

λ<μ


N(t)
busy
=> stable system

3
2
1
1
λ>μ
idle
2 3 4 5 6 7 8 9 10 11
Time
Steady build up of customers => unstable
N(t)
3
2
1
1
2 3 4 5 6 7 8 9 10 11
Time
10
Example#1

A communication channel operating at 9600 bps


Receives two type of packet streams from a gateway

Type A packets have a fixed length format of 48 bits

Type B packets have an exponentially distribution length
 With a mean of 480 bits
If on the average there are


20% type A packets and 80% type B packets
Calculate the utilization of this channel

Assuming the combined arrival rate is 15 packets/s
11
Performance measures

L


Lq


Mean queue length in the queue space
W


Mean # customers in the whole system
Mean waiting time in the system
Wq
 Mean waiting time in the queue
12
Mean queue length (M/M/1)

L  E[n] 


nP n 
n0

n  (1   )
n
n0

 (1   )   ( n 
n 1
n0

)  (1   )   (  )'
n
n0

  (1   )  (  )'
n
n0
  (1   )(
 L
1
1 
)' 

1 

 
13
Mean queue length (M/M/1)
(cont’d)

Lq 
 ( n  1) P
n
n 1


 nP
n 1

n

P
n
n 1
 L  (1  P0 )
 L  (1  (1   ))
 L
 L  Lq  
14