More Queueing Theory Carey Williamson Department of Computer - - PowerPoint PPT Presentation

More Queueing Theory

Carey Williamson Department of Computer Science University of Calgary

Motivating Quote for Queueing Models

“Good things come to those who wait”

• poet/writer Violet Fane, 1892
• song lyrics by Nayobe, 1984
• motto for Heinz Ketchup, USA, 1980’s
• slogan for Guinness stout, UK, 1990’s

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▪ M/M/1 queue is the most commonly used type of queueing model ▪ Used to model single processor systems or to model individual devices in a computer system ▪ Need to know only the mean arrival rate λ and the mean service rate μ ▪ State = number of jobs in the system M/M/1 Queue

1 2 j-1 j j+1

…

l l l l l l m m m m m m

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▪ Mean number of jobs in the system: ▪ Mean number of jobs in the queue: Results for M/M/1 Queue (cont’d)

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▪ Probability of n or more jobs in the system: ▪ Mean response time (using Little’s Law):

—Mean number in the system

= Arrival rate × Mean response time

—That is:

Results for M/M/1 Queue (cont’d)

k k n n k n n

p k n P   

 

 =  =

= − = =  ) 1 ( ) (

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M/M/1/K – Single Server, Finite Queuing Space

l

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K K

▪ State-transition diagram: ▪ Solution Analytic Results

 

       = +  − − = = = =

+ − =



1 1 1 1 1 1 here w ,

1 1

     m l   K p p p

K K n n n n

1 K-1 K

…

l l l l m m m m

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M/M/m - Multiple Servers

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▪ State-transition diagram: ▪ Solution Analytic Results

       = =

− − = +



m n m m p m n n p p p

m n n n n j j j n

! 1 ! 1

1 1

  m l

1 m-1 m m+1

…

l l l l l l m 2m (m-1)m mm mm mm

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▪ Infinite number of servers - no queueing M/M/ - Infinite Servers

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▪ State-transition diagram: ▪ Solution ▪ Thus the number of customers in the system follows a Poisson distribution with rate 𝜍

Analytic Results



 

− −  =

=       = =



e n p n p p

n n n n 1

! 1 ! 1

1 j-1 j j+1

…

l l l l l l m 2m (j-1)m jm (j+1)m (j+2)m

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▪ Single-server queue with Poisson arrivals, general service time distribution, and unlimited capacity ▪ Suppose service times have mean

1 𝜈 and variance 𝜏2

▪ For 𝜍 < 1, the steady-state results for 𝑁/𝐻/1 are: M/G/1 Queue

) 1 ( 2 ) / 1 ( ] [ , ) 1 ( 2 ) / 1 ( 1 ] [ ) 1 ( 2 ) 1 ( ] [ , ) 1 ( 2 ) 1 ( ] [ 1 , /

2 2 2 2 2 2 2 2 2 2

  m l   m l m  m    m     m l  − + = − + + = − + = − + + = − = = w E r E n E n E p

q

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—No simple expression for the steady-state probabilities —Mean number of customers in service: 𝜍 = 𝐹 𝑜 − 𝐹 𝑜𝑟 —Mean number of customers in queue, 𝐹[𝑜𝑟], can be

rewritten as:

𝐹[𝑜𝑟] = 𝜍2 2 1 − 𝜍 + 𝜇2𝜏2 2 1 − 𝜍

▪ If 𝜇 and 𝜈 are held constant, 𝐹[𝑜𝑟] depends on the variability, 𝜏2, of the service times.

M/G/1 Queue

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▪ For almost all queues, if lines are too long, they can be reduced by decreasing server utilization (𝜍) or by decreasing the service time variability (𝜏2) ▪ Coefficient of Variation: a measure of the variability of a distribution 𝐷𝑊 = 𝑊𝑏𝑠 𝑌 𝐹[𝑌]

— The larger CV is, the more variable is the distribution relative to its

expected value.

▪ Pollaczek-Khinchin (PK) mean value formula:

Effect of Utilization and Service Variability

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) 1 ( 2 ) ) ( 1 ( ] [

2 2

   − + + = CV n E

▪ Consider 𝐹[𝑜𝑟] for M/G/1 queue: Effect of Utilization and Service Variability

        +         − = − + = 2 ) ( 1 1 ) 1 ( 2 ) 1 ( ] [

2 2 2 2 2

CV n E

q

   m  

Same as for M/M/1 queue Adjusts the M/M/1 formula to account for a non-exponential service time distribution

Mean no. of customers in queue Traffic Intensity (𝜍)

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