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

λ

Characteristics of the system structure

μ λ: arrival rate μ: service rate  Queue  Infinite or finite  Service mechanism  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 λ .

.

μ  Machine interference (finite population)

N machines

S repairmen 7

λ

M/M/1 queue

μ λ: arrival rate μ: service rate  λ n = λ, (n >=0); μ n = μ (n>=1)

P n

   0 0  1  1 ...

 ...



n

 1

n P

0 

P n

 

n



n P

0 

P n

 

n P

0 ;    

P

0 

P

1  ...



P n P

0 ( 1     2  ...

 1  ...)  1 

P

0  1   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 N(t)

 λ<μ  => stable system

busy idle 3 2 1

 λ>μ 

1 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  Mean # customers in the whole system  L q  Mean queue length in the queue space  W  Mean waiting time in the system  W q  Mean waiting time in the queue 12

Mean queue length (M/M/1)

L



E

[

n

] 

n

   0

nP n



n

   0

n



n

( 1   )  ( 1   )

n

   0  (

n



n

 1 )  ( 1   )

n

   0  ( 

n

)'   ( 1   )

n

   0 ( 

n

)'    ( 1 

L

  )( 1  1  )'       1   13

Mean queue length (M/M/1) (cont’d)

L q



n

   1 (

n

 1 )

P n

   

n

   1

nP n L L



n

   1

P n

 ( 1   ( 1 

P

0 ) ( 1   ))

L

  

L



L q

  14